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+{"text":"\n\\section{Introduction}\n\nIn mathematical imaging the problem of segmentation refers to the process of automatically detecting regions, objects or patterns of interest. This is of particular importance in biomedical imaging for cell or organ quantification as well as in materials science and engineering. The main goal of this work is to present a new multiscale segmentation method combining variational inverse scale-space methods with nonlinear spectral analysis.\n\n\n\\textit{Image segmentation.} The task of image segmentation can sometimes be addressed by simple methods which are solely based on intensity or histogram thresholds. However those methods quickly fail in more complex experimental scenarios, where challenges in terms of different sizes, intensities, contrasts or uncertainties like noise occur. Thus, a class of commonly used mathematical techniques to overcome some of the difficulties are nonlinear variational methods. In general there are two different ways of describing a region, either by its edge or by the interior of the region. Therefore two main concepts of addressing segmentation by an energy minimization problem evolved in the previous decades: edge-based and region-based segmentation. While edge-based segmentation separates regions based on discontinuity information \\cite{Kass1988,Kichenassamy1995,Caselles1997}, region-based segmentation separates by similarity measures within the regions. In this paper we focus on a region-based approach.\n\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Sizes]{\\includegraphics[height=0.15\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_1.png}}\\qquad\n  \\subfigure[Intensities]{\\includegraphics[height=0.15\\textwidth]{results_new\/balls_int_nonoise\/bregman_cv_1.png}}\\qquad\n  \\subfigure[Shapes]{\\includegraphics[height=0.15\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_1.png}}\\\\\n  \\caption{Minimal examples where multiscale image segmentation could offer added value}\n\\label{fig:overview_goal_segmentation}\n\\end{figure}\n\nThe proposed method in this work is mainly built upon the Chan-Vese (CV) model \\cite{Chan2001}. For a given image function $f:\\Omega \\rightarrow \\mathbb{R}$ the domain $\\Omega \\subset \\mathbb{R}^d$ should be separated into two regions $\\Omega_1$ and $\\Omega_2 := \\Omega \\setminus \\Omega_1$ via the following variational energy minimization method\n\\vspace{-2pt}\n\\begin{equation}\\label{eq:CVorig}\n\\int_{\\Omega_1} (f(x) - c_1)^2 \\mathrm{d}x + \\int_{\\Omega_2} (f(x) - c_2)^2 \\mathrm{d}x + \\alpha \\cdot Per(C)\\longrightarrow \\min_{C,c_1,c_2}\n\\end{equation}\n\\vspace{-2pt}\nwhere $C$ is the desired contour separating $\\Omega_1$ and $\\Omega_2$ and $c_i$ stands for a desired average intensity value within region $\\Omega_i$. $Per(C)$ denotes the length of the contour $C$ separating $\\Omega_1$ and $\\Omega_2$ and can be controlled via a regularization parameter $\\alpha > 0$. An easy example is the image $f$ given in Figure \\ref{fig:overview_goal_segmentation}(a) where the domain $\\Omega$ can be segmented with respect to foreground and background. In contrast to simple histogram thresholding methods this holds true even in the case of strong noise if the parameter $\\alpha$ is chosen adequately. A generalization of this model was introduced in \\cite{Vese2002} and in \\cite{Chan2000} the model has also been extended to vector-valued images like color images. Recently in \\cite{zosso2015} an extension of the CV model that can handle artifacts and illumination bias in images has been proposed.\n\n\n\\textit{Challenges\/Questions.} The CV model is very useful in segmenting regions of interest which have very similar intensity values, e.g. Figure \\ref{fig:overview_goal_segmentation}(a). However, automatically detecting single objects based on their size is more challenging. Even with a varying parameter $\\alpha$ controlling the contour length (forward scale-space), it is for example not possible to detect the smallest object as a singleton. A similar challenge occurs when segmenting separate objects due to their intensity values, e.g. Figure \\ref{fig:overview_goal_segmentation}(b). Increasing the number of constants $c_i$ to four is suboptimal because we usually a-priori don't know the number of objects. Varying a threshold or parameter $\\alpha$ could lead to a correct segmentation but the estimated intensity constants $c_1$, $c_2$ will likely be incorrect. \\textit{Hence, is it possible to automatically detect multiple scales in a nonlinear variational image segmentation model, for instance with respect to different object sizes or object intensities? Can the segmentation of an image automatically be decomposed with respect to those scales?}\n\nMany region-based segmentation methods only use constraints on the contour length or curvature as regularization. However, in view of shape optimization and dictionary learning an approach that could also automatically separate objects with respect to their shape (cf. Figure \\ref{fig:overview_goal_segmentation}(c)) would be very interesting. \\textit{Hence, what is the role of geometric shapes in a multiscale segmentation approach?}\n\n\n\\textit{Scale-space methods.}\nIn the previous decades there has been a continuous interest in the analysis of different scales and the construction of scale spaces in imaging. In general it is desired to automatically detect all scales present in an image and simultaneously determine which scales are informative and contribute most to the image. For segmentation this problem is addressed in the fundamental works by Witkin and Koenderink \\cite{Witkin1983,Koenderink1984}. In relation to those, several methods to detect and analyze interesting scales have been proposed, see for example \\cite{Lifshitz1990,Vincken1997,Tabb1997,Lindeberg1998,Florack2000,Letteboer2004}. The underlying scale-space that is examined is defined by a linear diffusion process. A drawback of those approaches is that linear diffusion smoothes edge information and is therefore in general not suitable for applications where one is interested in retaining sharp edge information. Especially in biomedical image applications this is often the case. Therefore those theories were extended to non-linear diffusion processes, see \\cite{Niessen1997,Niessen1999,Dam2000}. A drawback of these approaches is that their analysis of scales is not fully automatic and can only be used in a forward approach, thus going from fine to coarse scales and then trying to find a backward relation. In this work we concentrate on nonlinear diffusion processes for segmentation where scale automatically relates to intensity and size.\n\nA prominent example of a variational method for nonlinear diffusion is the ROF model \\cite{Osher2005}. With increasing regularization parameter $\\alpha$ a sequence of functionals generates a nonlinear forward scale space flow that filters signals from fine to coarse. However in this process the total variation regularization functional is known to lead to a systematic contrast loss in the filtered image $u$ \\cite{Meyer2001}, whereas the main discontinuities in the signal remain at their position in the domain. To tackle the problem of intensity loss Osher et al. proposed in \\cite{Osher2005} an iterative contrast enhancement procedure based on Bregman distances. This approach is known to generate a nonlinear inverse scale space flow generating filtered signals from coarse to fine and with improved quality. This idea was successfully applied to more general inverse problems. \\cite{Burger2007,Brune2011,Benning2013}\n\n\\textit{Spectral methods.} Recently Gilboa \\cite{Gilboa2013,Gilboa2014} developed a framework to detect scales based on the nonlinear total variation diffusion process. The total variation is known to retain edge information while smoothing the signal apart from the edges. In this framework scales are detected based on a spectral decomposition of the given image into TV eigenfunctions \\cite{Meyer2001}. This concept does not only hold true for higher-order regularization functionals \\cite{Papafitsoros2015511,Poeschl2015,Benning2013} but more generally for convex, one-homogenous functionals $J$ with corresponding nonlinear eigenvalue equations \\cite{Burger2015,Burger2016} of the following form\n\\vspace{-5pt}\n\\begin{equation*}\n\t\\lambda u \\in \\partial J(u) \\ \n\\end{equation*}\nwhere $\\partial J(\\cdot)$ denotes the subdifferential of $J$. When minimizing for instance the total variation $J$, those eigenfunctions $u$ simply loose contrast whereas the overall structure of the function remains the same. The magnitude of contrast loss is related to the eigenvalue $\\lambda$. The eigenfunctions shape is determined by the chosen norm in the TV functional which can be adapted to the application of interest. In this way signals are not linearly smoothed to overcome scales but are step-by-step transformed to a composition of nonlinear eigenfunctions at coarser scales. A spectral response function can be used to examine which scales have a strong contribution to the original signal and to design filters of certain scales. Moreover, such spectral method can be combined with forward and inverse scale space approaches \\cite{Burger2016}.\\newpage\n\n\n\\textit{Main contribution.} In this paper we will extend the idea of (inverse) scale space methods known for nonlinear diffusion processes to segmentation and shape detection problems.\n\\begin{itemize}\n\\setlength{\\itemsep}{3pt}\n\t\\setlength{\\itemindent}{-.2in}\n\t\\item First goal: A novel inverse multiscale segmentation method based on Bregman iterations\n\t\\item Second goal: An adaptive regularization parameter strategy for $\\alpha$ (independent of $c_1$, $c_2$)\n\t\\item Third goal: A spectral analysis for segmentation regarding shapes of eigenfunctions\n\\end{itemize}\nA TV-based forward scale space for segmentation can easily be derived from the CV model via an increasing regularization parameter. We extend this framework by an inverse scale space for segmentation, still based on the CV model and therefore on a nonlinear diffusion process. For this purpose, we make use of Bregman iterations, among others well-known for improving total variation denoising results. The relation between the forward scale space and the inverse scale space is examined. Both iterative strategies are accomplished by a spectral transform and response function, which are used to easily examine scales and to filter certain scales. Since our method uses the total variation as a nonlinear diffusion process, we can make use of relatively easy and fast numerical and parallel implementation schemes developed in recents years.\n\n\\textit{Organization.} This work is organized as follows.\nIn Section \\ref{sec:CVmodel} we start with a revision of the segmentation model by Chan and Vese including its convexification. Together with a revision of the iterative denoising strategy using Bregman iterations in Section \\ref{ref:bregmandnoising}, the combination of those two concepts forms the first ingredient of our new inverse scale space method for multiscale segmentation introduced in Section \\ref{sec:bregman-cv}. An interesting interpretation of the method as an adaptive regularization method is presented in Section \\ref{sec:reg-param}. \nSection \\ref{sec:spec-analysis} deals with nonlinear spectral methods and contains the second main ingredient of our new approach. In \\ref{sec:genTV} we start with a brief summary of generalized nonlinear total variation functionals addressing different eigenshapes and continue in \\ref{sec:spec-analysisdenoising} with recent works by Gilboa et al. solving related nonlinear eigenvalue problems in imaging. In Section \\ref{sec:spec-analysissegm} we extend those ideas from nonlinear image denoising to image segmentation.\nIn Section \\ref{sec:numrealization} we describe the numerical realization of our approach using primal-dual convex optimization methods. \nIn Section \\ref{sec:results} we first illustrate the strengths and limitations of our multiscale segmentation method by studies on synthetic datasets with a certain focus on eigenfunctions and shapes. Moreover, we underline the potential and wide applicability by three different biomedical imaging applications. Its reliable performance is demonstrated on real fluorescence microscopy images that contain Circulation Tumor Cells, with various shapes and sizes, among white blood cells and debris in \\ref{sec:resultsCTC}. Besides, we present results on electron microscopy images suffering from inhomogeneous backgrounds in \\ref{sec:resultsEM} and interesting results on network-like shapes representing vascular systems in \\ref{sec:resultsnetwork}.\nWe end with a conclusion and an outlook to future possible perspectives in Section \\ref{sec:conl}.\n\\section{Modeling Segmentation with Inverse Scale Spaces}%\nIn the following section we will shortly describe the model by Chan and Vese \\cite{Chan2001} for segmentation and the adaption of the ROF model for denoising using Bregman distances \\cite{Osher2005}. Afterwards we will introduce our novel Bregman-CV model for segmentation and show some advantages of our model. \n\\subsection{Globally Convex Segmentation}\\label{sec:CVmodel}\nThe idea of the CV model has originally been derived from the more general model for image segmentation introduced by Mumford and Shah (\\cite{Mumford1989}). Here, one seeks for a solution of the variational energy\n\\begin{equation*}\nJ^{\\text{MS}}(u,C) = \\int_{\\Omega} |f(x) - u(x)|^2 \\mathrm{d}x +  \\alpha \\cdot Per(C) + \\beta \\cdot \\int_{\\Omega\\setminus C} |\\nabla u(x)| ^2 \\mathrm{d}x \\longrightarrow \\min_{u,C}\n\\end{equation*}\nwhere $u$ is a differentiable function that is allowed to be discontinuous on $C$. $C$ describes the union of the boundaries and thereby represents the contour defining the segmentation. Thus, $u$ is a smooth approximation of the original image $f$ and is composed of several regions $\\Omega_i$. Within each region $\\Omega_i$, $u$ is smooth. If we restrict this model so that $u$ is composed of only two regions $\\Omega_1$ and $\\Omega_2$ we derive with $\\beta \\rightarrow \\infty$ the CV model for segmentation\n\\begin{equation*}\nJ^{\\text{CV}}(c_1,c_2,C) = \\int_{\\Omega_1} (f(x) - c_1)^2 \\mathrm{d}x + \\int_{\\Omega_2} (f(x) - c_2)^2 \\mathrm{d}x + \\alpha \\cdot Per(C)\\longrightarrow \\min_{C,c_1,c_2}.\n\\end{equation*}\nHere, $c_i$ is the intensity value of $u$ within the corresponding region $\\Omega_i$. These values need to be determined together with contour the $C$. Note that in comparison to the original model we omit the area regularization term (cf. \\eqref{eq:CVorig}).\n\nThe contour $C$ can be indirectly represented via a level-set function $\\Phi$ (\\cite{Osher1988}) with\n\\begin{equation*} \nC = \\{x \\in \\Omega: \\Phi(x) = 0\\} \n\\end{equation*} \nand $\\Phi(x)$ being positive if and only if $x \\in \\Omega_1$. Together with the Heaviside function\n\\begin{equation*}\n\tH(\\Phi(x)) = \\begin{cases}1 &\\text{ if }\\Phi(x)\\geq 0 \\\\ 0 &\\text{ if }\\Phi(x)< 0\\end{cases}\n\\end{equation*}\nand its regularized version $H_{\\epsilon}$ this results in\n\\begin{align*}\nJ^{\\text{CV2}}(c_1,c_2,\\Phi) = \\int_{\\Omega} (f(x) - c_1)^2 H_{\\epsilon}(\\Phi(x)) \\mathrm{d}x & + \\int_{\\Omega} (f(x) - c_2)^2 (1 - H_{\\epsilon}(\\Phi(x)))\\mathrm{d}x \\\\&+ \\alpha \\cdot \\int_{\\Omega}|\\nabla H_{\\epsilon}(\\Phi(x))|\\mathrm{d}x~\\longrightarrow~ \\min_{\\Phi,c_1,c_2}.\n\\end{align*}\nThe contour $C$ evolves during minimization until it reaches a minimum which, in the ideal case, describes the object boundaries. Besides the original minimization strategy by gradient descent, several minimization methods to solve the CV model have been developed, see for example \\cite{He2007,Zehiry2007,Badshah2008,Bae2009}. One disadvantage of the model is its non-convexity which makes the solution depending on the used initialization. With a badly chosen initialization the minimization might get stuck in a local minimum that corresponds to a bad or meaningless segmentation.\n\nFor a better understanding of the relation between the nonlinear denoising model by Rudin, Osher and Fatemi (\\cite{Rudin1992}) and the CV segmentation model we use the total variation defined as\n\\begin{equation}\\label{eq:TV}\nTV(u) := \\sup\\limits_{\\substack{\\varphi \\in C_0^{\\infty}(\\Omega; \\mathbb{R}^2)\\\\ ||\\varphi||_{\\infty} <1}} \\int_{\\Omega} u \\nabla\\cdot \\varphi \\;\\mathrm{d}\\mu \\text {\\ \\  with \\ \\ } BV(\\Omega) := \\{ u \\in L^{1}(\\Omega)|TV(u) < \\infty\\}.\n\\end{equation}\nThe total variation of a characteristic $u(x) = \\begin{cases} 1 &\\text{\\  if  \\ } x \\in \\Omega_1 \\cup C\\\\ 0  &\\text{ if } x \\in \\Omega_2\\end{cases}$ corresponds to the contour length $|C|$ which can be shown by the co-area-formula.\nTherefore we can formulate the segmentation problem as\n\\begin{equation} \\label{eq:CV3} J^{CV3}(c_1,c_2,u) = \\int_{\\Omega} u\\left((f(x)-c_1)^2 - (f(x) - c_2)^2\\right)\\mathrm{d}x + \\alpha \\; TV(u)\n\\longrightarrow \\min_{\\substack{u\\in BV(\\Omega),c_1,c_2\\\\u(x)\\in \\{0,1\\}}}.\n\\end{equation}\nFor fixed $c_1, c_2$ the solution of \\eqref{eq:CV3} corresponds to the solution of an ROF problem with binary constraint (\\cite{Burger2012}):\n\\begin{equation}\\label{eq:CV-ROF}\n \\min_{\\substack{u\\in BV(\\Omega)\\\\u(x)\\in \\{0,1\\}}}\\frac{1}{2}|| u(x) - r(x)||_2^2 + \\alpha  TV(u)\n \\end{equation}\nwith $r(x) = (f(x)-c_2)^2 \\!-\\! (f(x) - c_1)^2 \\!-\\!\\frac{1}{2}$.\n\nThe regularization parameter $\\alpha$ in the segmentation model \\eqref{eq:CV3} has the role of a scale parameter, meaning that $\\alpha$ determines the scale of the objects that are segmented. The CV model describes a forward scale approach, thus a small parameter $\\alpha$ corresponds to small scales that are segmented. An increased regularization parameter results in a solution where the smaller scales are not segmented but only larger ones. The meaning of scale is determined by the regularization functional, in this case the total variation. The total variation encodes a measure of the contour length as well as the height of piecewise constant areas. One disadvantage is that due to the 1-homogeneity of $TV$ our method cannot distinguish between height and contour length. Thus, a small object with a bright intensity can have the same scale as a large object with a less bright intensity. For more details see section \\ref{sec:results}.\n\\\n\\paragraph{Convexification}\nThe CV segmentation model \\eqref{eq:CV3} as well as the binary ROF model \\eqref{eq:CV-ROF} are both not convex. Even for fixed values of $c_1$ and $c_2$ both models are non-convex due to the binary constraint on $u$. As mentioned before this might result in local instead of global minimum solutions. Approaches to overcome this difficulty and find global minima of the CV model are presented for example in\\cite{Chan2006,Bresson2007,Goldstein2010,Brown2012}. In \\cite{Chan2006} the authors showed that global minimizers of \\eqref{eq:CV3} for any given fixed $c_1,c_2 \\in \\mathbb{R}$ can be found by solving\n \\begin{equation}\\label{eq:convCV}\nJ^{CV3}(c_1,c_2,v) = \\int_\\Omega v ((f(x)-c_1)^2 - (f(x)-c_2)^2) dx + \\alpha~TV(v) \\longrightarrow \\min_{v \\in BV(\\Omega),~v(x) \\in [ 0,1 ]}\n\\end{equation}\nand defining $ u(x) := v^{\\ast}(x) \\geq \\mu$ for a.e. $\\mu \\in [0,1]$.\n\nThus, the binary constraint can be relaxed and combined with a thresholding. Here, the variational model to solve is convex, though not strictly convex. One should bear in mind that the found solution is therefore not unique. Yet solutions of \\eqref{eq:convCV} are close to binary even if the constraint is relaxed. Thus for most choices of $\\mu$ we derive the same solution which means that the choice of $\\mu$ has only a very limited impact on our method. Therefore we don't see any disadvantages when choosing the global but not unique minimum $u(x) := v^{\\ast}(x) \\geq 0.5$. A fully convex formulation (including the constants $c_1$ and $c_2$) of problem \\eqref{eq:CV3} can be found in \\cite{Brown2012}. This method is computationally less efficient and currently we don't think that, in our method, the advantages of the full convexity outweigh the increased computational time. \n\\subsection{Inverse Scale Space for TV-Denoising}\\label{ref:bregmandnoising}\nBefore introducing our new segmentation model in the following subsection we will first recall some properties of the well-known ROF model \\cite{Rudin1992} and its extension by Bregman distances introduced in \\cite{Osher2005}. To denoise an image corrupted by additive Gaussian noise, \\cite{Rudin1992} proposed to solve the nonlinear variational problem  \n\\begin{equation}\\label{eq:ROF}\n\\frac{1}{2}||u - f||_2^2 + \\alpha \\ TV(u) \\longrightarrow \\min_{u \\in BV(\\Omega)}\n\\end{equation}\nreferred to as the ROF model. \nSimilar to the CV model this generates a forward scale space flow regarding the scale parameter $\\alpha$. An increased parameter $\\alpha$ leads again to a solution $u$ where fine scales are removed and vice versa. The total variation regularization functional is known to lead to a systematic contrast loss in the denoised image $u$ \\cite{Meyer2001}. To tackle this problem Osher et al. proposed in \\cite{Osher2005} an iterative contrast enhancement procedure based on Bregman distances. Instead of using the total variation regularization functional as before, information about the solution $u$ that we gained from a prior solution of problem \\eqref{eq:ROF} is included. Therefore, problem \\eqref{eq:ROF} is replaced by a sequence of variational problems \n\\begin{equation}\\label{eq:Bregman-ROF}\nu_{k+1} = \\argmin_{u\\in BV(\\Omega)} \\frac{1}{2}||u - f||_2^2 + \\alpha\\ D^{p_k}_{TV}(u,u_k).\n \\end{equation}\nThe regularization $D^{p_k}_{TV}(u,u_k):=TV(u)-TV(u_k)-\\langle p_k,u-u_k\\rangle$ is the Bregman distance of $u$ to the previous iterate $u_k$ with respect to the total variation. $p_k \\in \\partial TV(u_{k})$ is an element in the subdifferential of the total variation of the prior solution $u_{k}$. Although this subdifferential might be multivalued, the iterative regularization algorithm automatically selects a unique subgradient based on the optimality condition. For $k = 0$ we set $u_0 = p_0 = 0$.\nThe iterative strategy of this model is as follows: We start with a large parameter $\\alpha$ that results in an oversmoothed solution $u_1$ that consists of only large scales. In every iteration step finer scales are added back to the solution. Thus, the scale parameter that determines the range of the scales present in $u$ is the iteration parameter $k$. In contrast to the forward approaches presented before, a small $k$ corresponds to coarse scales and a large $k$ to very fine scales. Therefore, the Bregman-ROF denoising approach is an inverse scale space approach. The authors showed that this strategy leads to enhanced contrast of the final solution $u_{k_{\\text{max}}}$ compared to the solution of \\eqref{eq:ROF}. Hence, solving \\eqref{eq:Bregman-ROF} instead of \\eqref{eq:ROF} with increasing $\\alpha$ is not only an inverted way of detecting scales. This method rather allows for a detection of solutions which cannot be obtained by an adequate choice of $\\alpha$ in the original ROF model.\\\\\n\n\\subsection{Bregman-CV Segmentation Model}\\label{sec:bregman-cv}\nIn the following section we will introduce our new inverse scale space approach for segmentation. It is based on the similarity of the ROF functional and the CV functional shown in \\eqref{eq:CV-ROF}. Similar to the Bregman-ROF denoising problem we replace the total variation regularization by an iterative regularization based on Bregman distances. Thus, the resulting novel segmentation model is given by\n\\begin{equation*}\nu_{k+1} = \\argmin_{\\substack{u\\in BV(\\Omega)\\\\u(x)\\in [0,1]}} \\int_{\\Omega} u\\left((f-c_1)^2 - (f-c_2)^2\\right)\n+ \\alpha \\ D^{p_k}_{TV}(u,u_k)\n\\end{equation*}\nBy inserting the definition of the Bregman distance $D^{p_k}_{TV}(u,u_k):=TV(u)-TV(u_k)-\\langle p_k,u-u_k\\rangle$ and ignoring the parts independent of $u$ we derive the following model.\n\\begin{equation}\\label{eq:Bregman-CV}\nu_{k+1}  = \\argmin_{u\\in BV(\\Omega)} \\int_{\\Omega} u\\left((f-c_1)^2 - (f-c_2)^2\\right) + \\chi_{[0,1]}(u)\n+ \\alpha \\ \\left(TV(u)-<u,p_k>\\right)\n\\end{equation}\nwith $p_k \\in \\partial TV(u_k)$, $p_0 = 0$ and $ \\chi_{[0,1]}(u) = 0$ if $u(x) \\in [0,1]$ and equal to infinity elsewhere. The range of the scales present in $u_{k+1}$ is again determined by the iteration index $k$. This model is an inverse scale space approach, thus a small $k$  corresponds to a large scale segmentation and vice versa. \n\nBy definition of the Bregman distance it is $p_k \\in \\partial TV(u_k)$ where the subdifferential is multivalued. Therefore we need to determine a rule to choose $p_k$. One way is to derive an update strategy based on the optimality condition of \\eqref{eq:Bregman-CV} (cf. \\cite{Osher2005}):\n\\begin{equation*}\n0 = \\left((f-c_1)^2 - (f-c_2)^2\\right) + q_{k+1} + \\alpha p_{k+1} - \\alpha p_k\\text{\\ \\ \\ (opt.cond.)}.\n\\end{equation*}\nHere, $q_{k+1}$ is an element in the subdifferential of the characteristic function $\\chi_{[0,1]}(u_{k+1})$. The subdifferential of a characteristic function is a normal cone and is in our case given by \n\\begin{equation*}\nq_k(x) \\in \\begin{cases} (-\\infty,0] &\\mbox{if} \\quad u_k(x) = 0 \\\\ \\{ 0 \\} &\\mbox{if} \\quad 0 < u_k(x) < 1 \\\\ [0,\\infty) &\\mbox{if}\\quad u_k(x) = 1\\end{cases}.\n\\end{equation*}\nThus, we can choose $q_k = 0$ and neglect it from hereon. The update strategy for $p_{k+1}$ is then given by\n\\begin{equation}\np_{k+1} = p_k - \\frac{1}{\\alpha}\\left((f-c_1)^2 - (f-c_2)^2\\right) = -\\frac{k+1}{\\alpha}\\left((f-c_1)^2 - (f-c_2)^2\\right)\\label{eq:update_p}.\n\\end{equation}\nThis update is independent of $u_k$, thus it is a pointwise constant update in every iteration.\n\n\\subsection{Interpretation as an Adaptive Regularization Approach}\\label{sec:reg-param}\nOne important question is whether solutions of this model are in some sense improved compared to the solutions of the original CV model. We mentioned before that Bregman iterations lead to a contrast enhancement when applied to the ROF functional and that solving the CV model corresponds to solving a binary ROF. Yet, one should bear in mind that a contrast enhancement is meaningless in the case of a binary image since the contrast is already determined by the binary constraint. This is supported by the following observation: when inserting \\eqref{eq:update_p} into \\eqref{eq:Bregman-CV} we get\n\\begin{align*}\nu_{k+1} & = \\argmin_{\\substack{u\\in BV(\\Omega)\\\\u(x)\\in [0,1]}}\\int_{\\Omega} u\\left((f-c_1)^2 - (f-c_2)^2\\right) + \\alpha \\left(TV(u)-<u,p_k>\\right)\\\\\n& =  \\argmin_{\\substack{u\\in BV(\\Omega)\\\\u(x)\\in [0,1]}} \\int_{\\Omega} u\\left((f-c_1)^2 - (f-c_2)^2\\right) + \\frac{\\alpha}{(k+1)} \\ TV(u)\n\\end{align*}\nWith this, it is straightforward to see that all solutions $u \\in BV(\\Omega)$ derived by the Bregman-CV model can also be found by the original CV model. Nevertheless, there are advantages of using the iterative update strategy.\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Linearly spaced $\\alpha$'s from 50 to 1.]{\\includegraphics[width=0.35\\textwidth]{img\/normal_alphas.png}}\\label{fig:reg_params_linear }\\quad \n  \\subfigure[\"$\\alpha$'s\" resulting from 50 Bregman steps with $\\alpha = 50$.]{\\includegraphics[width=0.35\\textwidth]{img\/bregman_alphas.png}}\\label{fig:reg_params_bregman}\\quad \n  \\caption{Comparison between linearly spaced regularization parameters and the automatically chosen parameters in the Bregman-CV model.}\n\\label{fig:reg_params} \n\\end{figure}\nIn Figure \\ref{fig:reg_params} b) the resulting $\\tilde\\alpha = \\frac{\\alpha}{k+1}$ for $\\alpha = 50$ and 50 Bregman iterations are shown. It is obvious that in the first iterations the decrease of the regularization parameter is much larger compared to later iterations. This is reasonable since first the large scales are reconstructed and in later Bregman iterations the finer scales are incorporated in the result. Making large steps with $\\alpha$ in large scales and becoming finer is therefore a reasonable strategy. A large decrease in the later iterations would probably miss some scales in between while in the first iterations too small steps are not reasonable. With this strategy the problem of automatically choosing the regularization parameter is less severe. By choosing a large $\\alpha$ and performing multiple Bregman iterations, a broad spectrum of scales is detected. Yet one should bear in mind that a strategy to automatically detect important scales is needed for a fully automated framework. \n\\section{A Spectral Method for Multiscale Segmentation}\\label{sec:spec-analysis} \nThe analysis of eigenvalues and spectral decomposition is a well-known theory in the field of linear signal and image processing (see e.g. \\cite{MarpleJr1987} or \\cite{Stoica2005} for a more recent overview). Since nonlinear regularization became popular in the last years, there is a growing interest in generalizing this theory to nonlinear operators. In \\cite{Benning2012} Benning et al. examined singular values for nonlinear, convex regularization functionals and in  \\cite{Gilboa2013,Gilboa2014} Gilboa transferred the idea of spectral decompositions to the nonlinear total variation functional and related operators. The general idea is to examine solutions of the nonlinear eigenvalue problem \n\\begin{equation}\\label{eq:eigenval}\n\\lambda u \\in \\partial J(u)\n\\end{equation}\nwhere $\\partial J$ denotes the subdifferential of a (one-homogeneous) convex functional usually representing regularization in inverse imaging problems. \nBy transferring solutions of \\eqref{eq:eigenval} to sparse peaks in a spectral domain, advanced filters enhancing or suppressing certain image components can easily be designed. This concept was first introduced for the total variation functional and later generalized to one-homogenous functions and analyzed in \\cite{Burger2015}, \\cite{Gilboa2015} and \\cite{Burger2016}. \n\n\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Solution of the ROF model for different $\\alpha$.]{\\includegraphics[height=0.22\\textheight]{img\/tv_block_1d.png}}\\label{fig:TVblock }\\quad \n  \\subfigure[Solution of the inverse Bregman-ROF model for different Bregman iterations.]{\\includegraphics[height=0.22\\textheight]{img\/bregmantv_block_1d.png}}\\label{fig:BregmanTVblock}\\quad \n  \\caption{Solutions of the ROF and Bregman-ROF model in case the signal is a TV eigenfunction. Different regularization parameters $\\alpha$ in the forward scale (left) and Bregman steps in the inverse scale space (right).}\n\\label{fig:1dEF} \n\\end{figure}\nIn the case of a one dimensional signal the eigenfunction of the total variation corresponds to a single block with constant height, see Figure \\ref{fig:1dEF}. When increasing the regularization parameter in the ROF model \\eqref{eq:ROF}, i.e. when minimizing the total variation of this block signal, the block looses it height but the edges remain at the same position (cf. Figure \\ref{fig:1dEF} (a)). Therefore all signals can be seen as the original signal multiplied by a scalar and are eigenfunction of the total variation. The same holds true for solutions of the Bregman-ROF model \\eqref{eq:Bregman-ROF}. The only difference is that now eigenfunctions of the TV functional (blocks) are reshaped with increasing number of Bregman iterations instead of removed (see Figure \\ref{fig:1dEF}(b)). This reflects the inverse scale approach of the Bregman-ROF model. Different to our novel approach, the Bregman updates in the Bregman-ROF model cannot be reformulated as an adaptive regularization parameter choice. As far as we know it is not entirely clear how the forward and inverse TV flow for denoising relate to each other.\n\n\n\\subsection{Generalized Definition of the Total Variation}\\label{sec:genTV}\nIn the previous paragraph we mentioned that in the one dimensional case the eigenfunction of the total variation functional is a block with constant height. If we want to proceed to the two dimensional case  the eigenfunctions are not as clearly defined as in the 1D case. Different than before we now have the freedom to choose the norm body that is used within the infinity norm in the TV definition \\eqref{eq:TV}. This choice determines the shape of the eigenfunctions. To reflect this dependence on the chosen norm in the TV definition we introduce a generalized version of the TV definition:\n\\begin{equation}\\label{eq:TVgen}\nTV_\\gamma(u) :=  \\sup\\limits_{\\substack{\\varphi \\in C_C^{1}(\\Omega; \\mathbb{R}^d)\\\\ \\varphi(x) \\cdot n <\\gamma(n) \\forall n\\in\\mathbb{R}^{d}}}\\ -\\int_{\\Omega} u \\nabla\\cdot \\varphi  \\mathrm{d}x.\n\\end{equation}\nHere, $\\gamma : \\mathbb{R}^d \\rightarrow \\mathbb{R} $ is a convex, positively 1-homogeneous function such that  $\\gamma (x) > 0$ for $x \\neq 0$.\nIf $u$ is a function in $\\mathcal{W}^{1,1}(\\Omega)$ the primal definition of equation \\eqref{eq:TVgen} is given by\n\\begin{equation*}\nTV_\\gamma(u) :=  \\int_{\\Omega} \\gamma(\\nabla u) \\mathrm{d}x.\n\\end{equation*}\nIn both definitions the choice of $\\gamma$ determines the shape of the eigenfunctions. We refer to \n \\begin{equation*}\n \\mathcal{F}_{\\gamma} := \\large\\{ z \\in \\mathbb{R}^{d} : \\gamma(z) \\leq 1\\large\\}\n \\end{equation*}\nas the Frank diagram and the corresponding Wulff shape is defined as\n\\begin{align*}\\mathcal{W}_{\\gamma} :=& \\large\\{ z \\in \\mathbb{R}^{d} : z \\cdot x \\leq \\gamma(x) \\text{ for all } x\\in \\mathbb{R}^{d} \\large\\}\\\\\n= &\\large\\{ z \\in \\mathbb{R}^{d} : \\gamma^{\\ast}(x) := \\sup\\limits_{x\\in \\mathbb{R}^{d}} \\ \\frac{z \\cdot x}{ \\gamma(x)} \\leq 1 \\large\\}.\\end{align*}\nNote that from definition \\eqref{eq:TVgen} together with the definition of the Wulff shape we can conclude that $\\varphi(x) \\in \\mathcal{W}_{\\gamma}$. Therefore, for every choice of $\\gamma$, functions with the same shape as the Wulff shape are eigenfunctions of $TV_\\gamma$ (see \\cite{esedoglu2004} Theorem 4.1). The Frank diagram $\\mathcal{F}_{\\gamma}$ and the Wulff shape $\\mathcal{W}_{\\gamma}$ for the three most common choices of $\\gamma$ are presented in Table \\ref{tab:shapes}. For simplicity we will omit the subscript $\\gamma$ in $TV_\\gamma$ from hereon, but we will come back to it in the numerical results in section \\ref{sec:results}.\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n$\\gamma(x)$ & $\\mathcal{F}_{\\gamma}$ & $\\mathcal{W}_{\\gamma} $\\\\[7pt]\n\\hline\n$\\gamma = \\| \\cdot \\|_{1}$ & \\includegraphics[height = 0.05\\textheight]{img\/square_rot.png} & \\includegraphics[height = 0.05\\textheight]{img\/square.png}\\\\[6pt]\n\\hline\n$\\gamma = \\| \\cdot \\|_{2}$& \\includegraphics[height = 0.05\\textheight]{img\/circle.png} & \\includegraphics[height = 0.05\\textheight]{img\/circle.png}\\\\[6pt]\n\\hline\n $\\gamma= \\| \\cdot \\|_{\\infty}$ &\\includegraphics[height = 0.05\\textheight]{img\/square.png} & \\includegraphics[height = 0.05\\textheight]{img\/square_rot.png}\\\\\n\\hline\n\\end{tabular}\n\\caption{Examples for $\\gamma$ and the corresponding Frank diagrams $\\mathcal{F}_{\\gamma}$ and Wulff shapes $\\mathcal{W}_{\\gamma}$. The Wulff shape corresponds to the shape of the eigenfunctions of the $TV_{\\gamma}$ functional.}\n\\label{tab:shapes}\n\\end{table}\n\n\\subsection{Spectral Analysis for Nonlinear Functionals}\\label{sec:spec-analysisdenoising}\nIn the following section we will first review the basic ideas of spectral TV analysis before transferring those ideas to segmentation in the next section. The idea to decompose an image based on the basic TV elements was presented by Gilboa in \\cite{Gilboa2013,Gilboa2014}. These basic TV elements, called eigenfunctions of the total variation functional, are all functions $u \\in BV(\\Omega)$ that solve the nonlinear eigenvalue problem\n\\begin{equation}\\label{eq:eigenvalProb}\n\\lambda u \\in \\partial TV(u).\n\\end{equation}\nIn the previous section we already presented some examples of TV eigenfunctions (see e.g. Figure \\ref{fig:1dEF} and Table \\ref{tab:shapes}). A  more general description of these eigenfunctions in case $\\gamma$ is isotropic is given in \\cite{Bellettini2002}. Bellettini et al. showed that all indicator functions $ \\mathbbm{1}_C(x)$ of a convex and connected set $C$ with finite perimeter $Per(C)$ which admit\n\\begin{equation}\\label{eq:l2eig}\n\\esssup_{p \\in \\partial C} \\kappa (p) \\leq \\frac{Per(C)}{Area(C)}\n\\end{equation}\nwhere $Area(C)$ denotes the area of $C$ and $\\kappa$ the curvature of $\\partial C \\in C^{1,1}$ are solutions of \\eqref{eq:eigenvalProb} and therefore eigenfunctions. Obtaining an analog condition for anisotropic, smooth and strictly convex choices of $\\gamma$ is straightforward but more challenging in the case of non-smooth or non-strictly convex choices of $\\gamma$. Candidates of these shapes are presented in \\cite{bellettini2001}. In \\cite{esedoglu2004} Esedoglu and Osher gave an example of a TV eigenfunction for $\\gamma = \\| \\cdot \\|_{1}$ that is not a Wulff shape (see section \\ref{sec:resultsshapes} for more details).\n\nIn order to detect eigenshapes at different scales in a given signal $f$ a scale space approach is needed. One way to define a scale space based on the total variation is given by the total variation flow \\cite{Andreu2001,Andreu2002,Bellettini2002,Burger2007,Steidl2004}. The TV flow arises when minimizing the total variation with steepest descent method and is defined as \n\\begin{equation}\\label{eq:tvflow}\n\\begin{aligned}\nu_t(t,x) &= -p(t,x) \\hspace{15pt} \\text{for } p(t,x) \\in \\partial TV(u(t,x))\\\\\nu(0,x) &= f(x)\n\\end{aligned}\n\\end{equation}\nwith Neumann boundary conditions. For $f(x) = \\mathbbm{1}_C(x)$, with $C$ defined as above, the unique solution of \\eqref{eq:tvflow} is $u(t,x) = (1 - \\lambda_C t) ^{+}  \\mathbbm{1}_C(x)$ with $\\lambda_C = \\frac{Per(C)}{Area(C)}$ (cf. \\cite{Bellettini2002}). Hence, the time derivative $u_t(t,x)$ is given by the original signal multiplied with a scalar and $u$ is an eigenfunction. To obtain a suitable framework to decompose or filter images based on these eigenfunction Gilboa proposed to define a spectral framework that transforms eigenfunctions to peaks in the spectral domain. If $u(t,x)$ is a solution of \\eqref{eq:tvflow} the TV spectral transform and spectral response can be defined as\n\\begin{equation}\\label{eq:tvtransform}\n\\phi (t,x) = u_{tt}(t,x)\\cdot t \\quad\\text{and}\\quad S(t) = || \\phi(t,x) ||_{L^{1}(\\Omega)}.\n\\end{equation}\nNote, that there are alternative definitions of the spectral response presented in \\cite{Burger2015}. \n\nAnother approach to construct an forward scale space is based on the variational ROF problem. Instead of solving \\eqref{eq:tvflow} the ROF model\n\\begin{equation}\\label{eq:ROFscalespace}\n \\min_{u\\in BV(\\Omega)}\\frac{1}{2}|| u(x) - f(x)||_2^2 + t \\cdot  TV(u).\n\\end{equation}\nis solved for different regularization parameters. Hence, $t$ is the (artificial) time variable that determines the scale comparable to $t$ in the TV-flow approach \\eqref{eq:tvflow}. One drawback of this approach is that there is no clear rule for the choice of different $t$'s. The spectral transform and response function can be equivalently defined as in \\eqref{eq:tvtransform}. \n\nA third, but significantly different, scale space approach is an inverse scale approach. The inverse scale space flow is defined as\n\\begin{equation}\\label{eq:inverseflow}\n\\begin{aligned}\np_s(s,x) &= f(x) - u(s,x) \\hspace{15pt} \\text{for } p(s,x) \\in \\partial TV(u(s,x))\\\\\nu(0,x) &= p(0,x) = 0.\n\\end{aligned}\n\\end{equation}\nThus, the flow is now defined on the dual variable $p \\in \\partial TV(u)$ and $s$ is the time variable determining the scale. Note that in \\cite{Burger2005} it was shown that the iterative Bregman-L2-TV model \\eqref{eq:Bregman-ROF} can be associated with a discretization of \\eqref{eq:inverseflow} for $\\frac{1}{\\alpha} \\rightarrow 0$.\nAs \\eqref{eq:inverseflow} is an inverse approach the time variable $t$ can be associated with $\\frac{1}{s}$. In this case the spectral transform and response functions are defined as\n\\begin{equation}\\label{eq:tvtransform2}\n\\phi (s,x) = u_{s}(s,x) \\quad\\text{and}\\quad S(s) = || \\phi(s,x) ||_{L^{1}(\\Omega)},\n\\end{equation}\nwhere $u(s,x)$ is the solution of \\eqref{eq:inverseflow}. Note, that for small $s$ $\\phi(s,x)$ now measures changes in the coarse scales. See \\cite{Burger2015} for more details.\n\nWith all three approaches we are able to transform a signal to the spectral domain and detect different scales based on TV eigenfunctions. If we assume that $\\phi(t,x)$ is integrable over time, the original signal $f$ can be reconstruct via\n\\begin{equation*}\nf(x) = \\int_0^{\\infty} \\phi(t,x) dt + \\bar{f}\n\\end{equation*}\nwhere $\\bar{f}$ is the average of $f$. Filters can be defined with $\\phi_{H}(t,x) = H(t)\\phi(t,x)$ via \n\\begin{equation*}\nf_{H}(x) = \\int_0^{\\infty} \\phi_{H}(t,x) dt + H(\\infty)\\bar{f}.\n\\end{equation*}\n\n\n\\subsection{Spectral Response of Multiscale Segmentation}\\label{sec:spec-analysissegm} \nIn section \\ref{sec:bregman-cv} we presented two variational models to detect segmentations of a given image $f$ at different scales. To decompose the segmentation into different scales and detect important scales or clusters of scales in the segmentation we want to transfer the idea of spectral analysis based on the total variation (cf. sec. \\ref{sec:spec-analysisdenoising}) to segmentation. Therefore we need to find a suitable transformation of the segmentation $u$ to the spectral domain and vice versa. Note that our goal is not to reconstruct the original signal $f$ or filtered versions of $f$ but the reconstructed function should be a segmentation itself. To do so we make use of the idea that eigenfunctions of the TV functional should be transformed to peaks in the spectral domain. In the following we will derive this spectral transform function for a forward scale space and an inverse scale space approach. To represent the forward scale space we associate the regularization parameter $\\alpha$ in the convex version of the original model by Chan and Vese \\eqref{eq:convCV} with the artificial time variable $t$. That means, we solve\n \\begin{equation}\\label{eq:convCV-scale}\n\\int_\\Omega u ((f(x)-c_1)^2 - (f(x)-c_2)^2) dx + t \\cdot TV(u) \\longrightarrow \\min_{u \\in BV(\\Omega),~u(x) \\in [ 0,1 ]}\n\\end{equation}\nfor different $t$.This is comparable to the variational approach in \\eqref{eq:ROFscalespace}. So far, we could not find a forward scale space representation that can be associated with a flow on $u$ comparable to the TV-flow. One difficulty is that the optimality condition of this model, i.e. $0 = (f-c_1)^2 - (f-c_2)^2 + \\alpha p$ with $p \\in TV(u)$, has no direct dependence on $u$. The inverse scale space representation is based on the Bregman-CV model we introduced in \\eqref{eq:Bregman-CV}. The optimality condition in each step of the iterative Bregman-CV strategy is given as\n \\begin{equation*}\n 0 = \\left((f-c_1)^2 - (f-c_2)^2\\right) + \\alpha \\left( p_k - p_{k-1}\\right) \\text{ with } p_k \\in TV(u_k) \\ \\forall \\ k \\\\.\n \\end{equation*}\nThe resulting equation\n\\begin{equation*}\n\\frac{p_k - p_{k-1}}{\\frac{1}{\\alpha}} = (f-c_2)^2 - (f-c_1)^2\n \\end{equation*}\ncan be interpreted as a discretization with stepsize $\\frac{1}{\\alpha}$ of\n\\begin{equation}\\label{eq:invsegslow}\n\\begin{aligned}\np_s(s,x) &=  (f(x)-c_2)^2 - (f(x)-c_1)^2\\hspace{15pt} \\text{with } p(s,x) \\in \\partial TV(u(s,x))\\\\\np(0,x) &= 0.\n\\end{aligned}\n\\end{equation}\nAgain, $s$ is the time variable that is inverse to the time variable $t$ in \\eqref{eq:convCV-scale}. We refer to this flow as the inverse scale space segmentation flow. Note that within this flow description there is no direct dependence on $u$ but $u$ is only indirectly given by $p \\in \\partial TV(u)$. Yet, when looking for solutions of this flow at different times $s$, we solve the Bregman-CV model for multiple Bregman iterations and there the corresponding $u$ is available. \n\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Evolution of $u$ in \\eqref{eq:convCV-scale}.]{\\includegraphics[width=0.4\\textwidth]{img\/peak_segm.png}}\\quad \\quad\n  \\subfigure[Evolution of $u$ in \\eqref{eq:invsegslow}.]{\\includegraphics[width=0.4\\textwidth]{img\/peak_segm_bregman.png}}\n  \\caption{Illustration of the evolution of $u\\in\\{0,1\\}$ over time at a fixed point when $f$ is a TV eigenfunction shown for the forward (a) and inverse scale space approach (b). Note, due to the binary constraint on u the evolution of $u$ is already a step function, i.e. $u(x)$ is either part of the segmentation (u(x)=1) or not (u(x)=0).}\n\\label{fig:peaks} \n\\end{figure}\n\nTo transform the eigenfunctions to peaks in the spectral domain we define the spectral transform function as follows:\n\\begin{equation}\\label{eq:spectransseg}\n\\phi (t,x) =\\begin{cases} -u_{t}(t,x)& \\text{ (forward case)}\\\\\n\\ \\ \\ u_t(t,x)& \\text{ (inverse case)}\\end{cases}\\\\[3pt]\n\\end{equation}\nThis definition is motivated by Figure \\ref{fig:peaks} where the evolution of TV eigenfunctions over time in \\eqref{eq:convCV-scale} and \\eqref{eq:invsegslow} is illustrated. We can see, that the first derivative (in a distributional sense) is a peak at that time point where the eigenfunction vanishes or appears respectively. For the spectral response function we use the definition\n\\begin{equation}\n\tS(t) = || \\phi(t,x) ||_{L^{1}(\\Omega)}\n\t\\label{eq:spectralResponse}\n\\end{equation}\nintroduced by Gilboa (\\cite{Gilboa2013,Gilboa2014}). Here, the influence of certain scales to the segmentation is encoded. Using this function $\\phi$ and assuming integrability over time we can get the following backtransformation:\n\\begin{equation*}\nf^{\\text{seg}}(x) = \\int_0^{\\infty} \\phi(t,x) dt,\n\\end{equation*}\nwhere $f^{\\text{seg}}(x):= (f - c_2)^2 - (f-c_1)^2 + \\frac{1}{2} < \\frac{1}{2}$ is the results of the simple clustering problem\n$\\min_{u(x)\\in \\{0,1\\}} \\int_{\\Omega} u\\left((f(x)-c_1)^2 - (f(x) - c_2)^2\\right)\\mathrm{d}x$. By choosing a function $H$ with $H(t) \\in \\{ 0,1 \\}$ certain scales of interest can be filtered via\n\\begin{equation}\\label{eq:filterseg}\nf^{\\text{seg}}_{H}(x) = \\int_0^{\\infty} \\phi_{H}(t,x) dt \\ \\ \\text{ with } \\ \\ \\phi_{H}(t,x) = H(t)\\phi(t,x).\n\\end{equation}\nWith this framework we can easily segment an image and simultaneously detect important scales in this segmentation instead of using the spectral TV analysis as in Section \\ref{sec:spec-analysisdenoising} and segmenting separately. Moreover, using the filtering approach we can easily construct segmentations of only certain scales by filtering (cf. \\eqref{eq:filterseg}). Examples are shown in section \\ref{sec:results}.%\n\\section{Numerical Methods}\\label{sec:numrealization}\nOur novel approach introduced in the two previous sections consists of two parts. First we have to solve the Bregman-CV model  \\eqref{eq:Bregman-CV} introduced in section \\ref{sec:bregman-cv}. Afterwards we can analyze the detected scales using the spectral response function \\eqref{eq:spectralResponse} introduced in section \\ref{sec:spec-analysissegm}. Therefore an efficient solution of the Bregman-CV model is required. In the following section we will give a very brief introduction into primal-dual optimization schemes and then show how this can be used to solve \\eqref{eq:Bregman-CV}. We close the section by a speed comparison between a Matlab and a parallelized C\/Mex implementation of our code. \n\n\\subsection{Primal-Dual Optimization Methods}\nTo solve nonlinear problems of the form\n\\begin{equation}\\label{eq:primalprob}\n \\min_{x \\in X} \\ F(Kx) + G(x)\n \\end{equation}\nwith $F$ and $G$ being proper, convex, lower-semicontinuous functions, primal-dual optimization methods became very popular in the last years. Instead of minimizing the primal problem  \\eqref{eq:primalprob} they make use of the primal-dual formulation of this problem given by\n\\begin{equation}\\label{eq:primaldualprob}\n\\min_{x\\in X} \\max_{y\\in Y}\\  \\langle Kx,y\\rangle + G(x) - F^{\\ast}(y)\n\\end{equation}\nwith $G: X \\rightarrow [0, +\\infty]$ and $F^{\\ast}:Y \\rightarrow [0, +\\infty]$ being the convex-conjugate of $F$. By updating in every iteration step a primal and a dual variable these methods are able to avoid some difficulties that arise when working on the purely primal or dual problem. One example is the minimization of variational methods with TV regularization. If the gradient is zero, the TV functional is not differentiable which leads to problems in purely primal minimization schemes like gradient descent. Some examples for primal-dual minimization algorithms are the PDHG algorithm \\cite{Zhu2008}, a generalization of PDHG by Esser et al. \\cite{Esser2010}, the Split Bregman algorithm by Goldstein and Osher \\cite{Goldstein2009}, Bregman iterative algorithms \\cite{Yin2008}, the second-order CGM algorithm \\cite{Chan1999} and inexact Uzawa methods \\cite{Zhang2011}.\n\nIn \\cite{Chambolle2011} Chambolle and Pock proposed an algorithm which can be seen as a generalization of PDHG as well. This algorithm was originally proposed by Pock et al. in \\cite{Pock2009} to minimize a convex relaxation of the Mumford-Shah functional. The efficient first-order primal-dual algorithm to minimize general problems of the form \\eqref{eq:primaldualprob} is presented in Algorithm \\ref{alg:cp}. \n$ (I + \\sigma \\partial F^{\\ast})^{-1}$ and $(I + \\tau \\partial G)^{-1}$ are the resolvent operators of $F^{\\ast}$ and $G$ respectively which are defined through \n\\eq{y = (I + \\tau \\partial G)^{-1}(x) = \\argmin_{y} \\left\\{ \\frac{\\| y-x\\|^2}{2\\tau} + G(y) \\right\\}.}\n\\begin{algorithm}\n\t\\caption{First-order primal-dual algorithm by Chambolle and Pock (\\cite{Chambolle2011})}\n\t\\label{alg:cp}\n\t{\\fontsize{10}{10}\\selectfont\n\t\\begin{algorithmic}\n\t\t\\State \\textbf{Parameters:} $\\tau, \\sigma > 0$, $\\theta \\in [0,1]$\n\t\t\\State \\textbf{Initialization:} $n=0,\\ x^0 \\in X,\\ y^0 \\in Y, \\bar{x}^0 = x^0$\\\\\n\t\t\\State \\textbf{Iteration:}\\\\\n                \\State  \\textbf{for } $(n\\geq 0)$\\ \\textbf{ do }\n\t\t\\begin{enumerate}\\itemsep5pt\n\t\t\t\\item $y^{n+1} = (I + \\sigma \\partial F^{\\ast})^{-1}(y^n + \\sigma K \\bar{x}^n)$.\n\t\t\t\\item $x^{n+1} = (I + \\tau \\partial G)^{-1}(x^n - \\tau K^{\\ast} y^{n+1})$.\n\t\t\t\\item $\\bar{x}^{n+1} = x^{n+1} + \\theta (x^{n+1} - x^{n})$.\n\t\t\t\\item Set $n=n+1$.\n\t\t\\end{enumerate}\\\\\n               \n\t\t\\State \\textbf{end for}\\\\\n\t\t\\State \\Return $x^n$\n\t\\end{algorithmic}\n\t}\n\\end{algorithm}\n\n\n\\subsection{Numerical Realization Bregman-CV}\nIn order to use Algorithm \\ref{alg:cp} to solve the constraint problem \\eqref{eq:Bregman-CV} we reformulate the problem into\n\\begin{equation}\\label{eq:primBregCV}\n \\int_{\\Omega} u\\left((f-c_1)^2 - (f-c_2)^2\\right)\n+ \\text{id}_{[0,1]}(u) + \\alpha \\ \\left(TV(u)-<u,p_k>\\right) \\longrightarrow \\min_{u\\in BV(\\Omega)}\n\\end{equation}\nwith $p_k \\in \\partial TV(u_k)$ and $p_0 = 0$. $\\text{id}_{[0,1]}(u)$ is the indicator function of the interval $[0,1]$ defined as 0 if $u \\in [0,1]$ and $\\infty$ otherwise. To derive an minimization strategy based on Algorithm \\ref{alg:cp} we set $x = u$, $K(u) = \\grad u$ and\n\\begin{equation*}\nF(u) = \\| u\\|_{1} \\text{ and } G(u) = \\text{id}_{[0,1]}(u) +  \\int_{\\Omega} u\\left( (f - c_1)^2 - (f - c_2)^2 - \\alpha p_k\\right).\n\\end{equation*}\nThe convex-conjugate of $F(u) = \\| u\\|_{1}$ is given by $F^{\\ast}(p) = \\delta_{P}(p)$ with $P = \\left\\{p: \\| p\\|_{\\infty} \\leq 1\\right\\}$ and\n\\begin{equation}\n\t\\delta_{P}(p) = \\begin{cases} 0 &\\mbox{if } p \\in P\\\\ \\infty &\\mbox{if } p\\notin P\\end{cases} .\n\\end{equation}\nTogether with \\eqref{eq:primaldualprob} we derive the primal-dual variant of \\eqref{eq:primBregCV}:\n\\eqn{\\label{eq:dualBregCV}\\langle \\grad u,p\\rangle + \\text{id}_{[0,1]}(u) +  \\int_{\\Omega} u\\left[ (f - c_1)^2 - (f - c_2)^2 - \\alpha p_k\\right] - \\alpha \\delta_{P}(p) \\longrightarrow \\min_{u}\\max_{p}.}\nThe resolvent operators for $G$ and $F^{\\ast}$ are defined through\n\\begin{align}\nu = (I + \\tau \\partial G)^{-1}(\\tilde{u}) &= \\text{Proj}_{[0,1]}\\left[\\tilde{u} - \\tau\\left((f - c_1)^2 - (f - c_2)^2 - \\alpha p_k\\right)\\right]\\notag\\\\\n& = \\max\\left(0,\\min\\left(1, \\tilde{u} - \\tau\\left((f - c_1)^2 - (f - c_2)^2 - \\alpha p_k\\right)\\right)\\right)\\label{eq:proxprimal}\n\\end{align}\nand\n\\begin{equation}\np = (I + \\sigma \\partial F^{\\ast})^{-1}(\\tilde{p}) = \\text{Proj}_{\\left\\{\\left\\{p: \\| p\\|_{\\infty} \\leq 1\\right\\}\\right\\}}\\left(\\tilde{p_{ij}}\\right).\\label{eq:proxdual}\n\\end{equation}\nNote that the $L^{\\infty}$ norm $\\| p\\|_{\\infty}$ is in the discrete setting defined as $\\| p\\|_{\\infty} = \\max_{i,j} \\large\\{\\gamma^{\\ast}(p_{i,j})\\large\\}$ where the choice of $\\gamma$ determines the shape of the eigenfunctions of the TV functional. For $\\gamma = \\| \\cdot \\|_{\\ell^{p}}$ with $p = 1$ the unit ball defined by $$\\left\\{ (x,y) \\in \\Omega | \\max\\{|x|,|y|\\} \\leq 1\\right\\}$$ is an TV eigenfunction, for $p = 2$ the unit ball defined by $$\\left\\{ (x,y) \\in \\Omega | \\sqrt{|x|^2+|y|^2} \\leq 1\\right\\}$$ and for $p=\\infty$ the unit ball defined by $$\\left\\{ (x,y) \\in \\Omega | (|x|+|y|) \\leq 1\\right\\}.$$ However these are not the only eigenfunctions.\nWith \\eqref{eq:proxprimal} and \\eqref{eq:proxdual}, we derive the primal-dual algorithm presented in Algorithm \\ref{alg:cpforcv} to minimize  \\eqref{eq:Bregman-CV}. Note that we are not updating the constants $c_1$ and $c_2$, but start with a good estimate and leave it fixed. To a certain extend the varying regularization parameter can compensate for an error in those constants. Some examples are presented in Section \\ref{sec:results}.\n\\begin{algorithm}\n\t\\caption{First-order primal-dual algorithm to solve \\eqref{eq:Bregman-CV}.}\n\t\\label{alg:cpforcv}\n\t{\\fontsize{10}{10}\\selectfont\n\t\\begin{algorithmic}%\n\t\t\\State \\textbf{Parameters:} data $f$, reg. param. $\\alpha \\geq 0$, $\\tau, \\sigma > 0$, $\\theta \\in [0,1]$, \t\t\t   $maxIts \\in \\mathbb{N}, \\ maxBregIts \\in \\mathbb{N}$\n\t\t\\State \\textbf{Initialization:} $l=1,\\ u^0_1=0, \\ p_0:=0, \\bar{u}^0 = u^0$\\\\\n\t\t\\State \\textbf{Iteration:}\\\\\n\t\t\\State \\textbf{while } \\big($k<maxBregIts$\\big) \\textbf{ do }\n                \\begin{enumerate}\\itemsep5pt\n\t\t\t\\item Set n=0.\n\t\t\\end{enumerate}\n\t\t\\begin{quote}\n\t\t \\State  \\textbf{while } \\big($n<maxIts$\\big) \\textbf{ do }\\\\\n\t\t\\begin{enumerate}\\renewcommand{\\labelenumi}{\\alph{enumi})}\\itemsep5pt\n\t\t\t\\item $p^{n+1} = \\text{Proj}_{\\left\\{\\left\\{p: \\| p\\|_{\\infty} \\leq 1\\right\\}\\right\\}}\\left( p^n + \\sigma \\grad \\bar{u}^n \\right)$.\n\t\t\t\\item $u^{n+1} = \\max\\left(0,\\min\\left(1, u^n + \\tau \\div p^{n+1} - \\frac{\\tau}{\\alpha}\\left((f - c_1)^2 - (f - c_2)^2 - \\alpha p_k\\right)\\right)\\right)$.\n\t\t\t\\item $\\bar{u}^{n+1} = u^{n+1} + \\theta (u^{n+1} - u^{n})$.\n\t\t\t\\item Set $n=n+1$.\n\t\t\\end{enumerate}\\\\\n\t\t\\State \\textbf{end while}\\\\\n\t\t\\end{quote}\n\t\t\\begin{enumerate}\\itemsep5pt\n\t\t\\setcounter{enumi}{1}\n\t\t\\item Update $p_{k+1} = p_k + \\frac{1}{\\alpha}\\left((f - c_2)^2 - (f - c_1)^2\\right)$.\n\t\t\\item Set $u^0_{k+1}=u^n_{k}$.\n\t\t\\item Set $k=k+1$.\n\t\t\\end{enumerate}\\\\\n\t\t\\State \\textbf{end while}\\\\\n\t\t\\State \\Return $u^0_{k}$\n\t\\end{algorithmic}\n\t}\n\\end{algorithm}\n\\begin{figure}[t]\n  \\centering \n  \\includegraphics[width=0.6\\textwidth]{img\/speed_test.png}\n  \\caption{\\textit{Comparison of runtimes for Matlab and C\/Mex code.} The code was tested on 2 different computers: a Mac with an Intel Core i5 processor with two cores of 2.6 GHz and 8GB of memory (solid graphs) and a Linux computer with an Intel Xeon processor with 16 cores of  2.6GHz and 32 GB of memory (dashed graphs). On both computers we have tested a full Matlab implementation (blue), a C\/MEX implementation without parallelization (yellow) and a C\/MEX implementation with parallelization using openMP (green).}\n\\label{fig:speed} \n\\end{figure}\n\nWe implemented two versions of the code, the first one is written in MATLAB and the second one is written in C and called via MATLAB using the MEX interface. For speed comparisons we've tested both code versions (always with 1 Bregman iteration) on two different machines. First, a MacBook Pro with an Intel Core i5 processor with two cores of 2.6 GHz and 8GB of memory (solid graphs in Figure \\ref{fig:speed}) and the second machine was a Dell Linux workstation with in Intel Xeon processor with 16 cores of 2.6GHz and 32GB of memory (dashed graphs in Figure \\ref{fig:speed}). Moreover, we ran the C\/Mex code with and without parallelization using openMP. The results of our speed test can be seen in Figure \\ref{fig:speed}. We see that especially for large images the computational time is significantly decreased when using the C implementation (yellow and green graphs) compared to the pure MATLAB code (blue graphs). All C versions behave very similar for small images but deviate for larger images. Without parallelization the C code is slightly faster on the Mac (solid yellow graph). When activating openMP and parallelizing parts of the code the computational time could be further improved for both computers. The difference is much more significant when using the computer with more cores. For the Mac we see a speed-up of about 30 percent (solid green graph) for large images while on the Linux computer (dashed green graph) we could achieve a speed-up of about 65 percent compared to the C implementation without parallelization.\n\\section{Experimental Results}\\label{sec:results}\nIn this section we illustrate the main properties, advantages and limitations of our novel multiscale segmentation framework via several synthetic experiments as well as real experiments from biological cell imaging and retina imaging. The main framework, we apply and study throughout this section, consists of Algorithm \\ref{alg:cpforcv}, performing the inverse scale space method Bregman-CV in \\eqref{eq:Bregman-CV}, together with an evaluation of the spectral response $S$ in \\eqref{eq:spectralResponse}.\n\nIn the first set of segmentation experiments we focus on simple objects (e.g. discs) and investigate how underlying object intensities and sizes can automatically be detected as multiple scales. In the second part we concentrate on the stability of our method under uncertainty like noise. In the third part of this section we generalize our study to other simple shapes (e.g. blocks or diamonds), explore the connection to underlying eigenfunctions of the nonlinear $TV_{\\gamma}$ functional and the potential for shape detection respectively shape optimization. Finally, we apply our framework to real images obtained from the CellSearch system and other automatic scanning microscopes used for the enumeration of Circulating Tumor Cells \\cite{allard2004} offering new automatic procedures for tumor cell identification, quantification and classification. Moreover, we show potential applications of our algorithm in the field of electron microscopy of biological samples and retina imaging.\n\\subsection{Multiscale Segmentation: Size versus Intensity}\\label{sec:resultsscale}\nIn this first subsection we start with the investigation of basic segmentation tasks. We try to find round objects (discs) with varying scale (e.g. differences in size or intensity) in front of homogeneous backgrounds to verify the basic properties of our model. Since discs are the unit balls of the $\\ell^2$ norm we chose $\\gamma = \\gamma^{\\ast} = \\| \\cdot \\|_2$ in the TV norm \\eqref{eq:TVgen}.\n\\begin{figure}[htb]\n  \\centering \n    \\subfigure[1st Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_1.png}}\\quad \n  \\subfigure[9th Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_9.png}}\\quad \n  \\subfigure[12th Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_12.png}}\\quad \n    \\subfigure[15th Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_16.png}}\\quad \n      \\subfigure[22nd Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_24.png}}\\\\\n        \\subfigure[Spectral response $S$]{\\includegraphics[height=0.16\\textheight]{results_new\/balls_size_nonoise\/S_gesamt.png}}\\quad\n         \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[height=0.16\\textheight]{results_new\/balls_size_nonoise_new\/back_gray.png}}\n  \\caption{\\textit{Detection of size scales}. Automatic segmentation of discs of different sizes under fixed intensity (height). Segmentation is visualized via red contour. (a)-(e) Segmentation after different number of Bregman Iterations. (f) Associated spectral response function $S$. The number of Bregman iterations is plotted on the x-axis. Every peak in $S$ corresponds to one object that appears in the segmentation (compare to (a) - (e)). The decreasing size of the peaks reflects the decreasing disc size. }\n\\label{fig:ballssize} \n\\end{figure}\n\nIn the first example in Figure \\ref{fig:ballssize}(a) we can automatically identify four discs with different diameters (sizes) via the Bregman-CV inverse scale space algorithm \\ref{alg:cpforcv}. As visualized in Figure \\ref{fig:ballssize}(b)-(e) the method automatically picks out the discs one by one while iterating (w.r.t $t \\approx \\alpha_k$) through the scale space. At the end, the full foreground-background segmentation is available. In addition to this, the spectral response \\eqref{eq:spectralResponse} of our iterates regarding $t$, nicely reflects the sparse occurrence of new contours, see Figure \\ref{fig:ballssize}(f). Note that in this example the intensity of peaks decreases due to the different size of each disc. The height of $S$ at each time point reflects the number of pixels that occurred (or vanished) in the segmentation in this step. In Figure \\ref{fig:ballssize}(g) we plotted the function $\\int_{t}\\Phi(x,t) \\cdot t \\mathrm{d} t$ in color coding. To obtain an easily interpretable visualization we use a gray background. Since the function $\\Phi(x,t)$ for given $x$ is equal to 1 exactly at the time point $t_x$ where $x$ occurs in the segmentation and 0 elsewhere, $\\int_{t}\\Phi(x,t) \\cdot t \\mathrm{d} t$ corresponds for every point $x$ to $t_x$. We can see that apart from the four time steps where the four discs appear nearly nothing is appearing in between, reflecting the sparse behavior of our methods. Only very few pixels at the object boundaries occur in later iterations which is caused by discretization inaccuracies. In this way the multiscale approach of our method can be seen since each color reflects one segmented scale.\\\\\n\\begin{figure}[htb]\n  \\centering \n\\subfigure[1st Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_int_nonoise\/bregman_cv_1.png}}\\quad \n  \\subfigure[3rd Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_int_nonoise\/bregman_cv_3.png}}\\quad \n  \\subfigure[5th Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_int_nonoise\/bregman_cv_5.png}}\\quad \n    \\subfigure[10th Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_int_nonoise\/bregman_cv_10.png}}\\quad \n      \\subfigure[29th Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_int_nonoise\/bregman_cv_29.png}}\\\\\n        \\subfigure[Spectral response $S$.]{\\includegraphics[height=0.16\\textheight]{results_new\/balls_int_nonoise\/S_gesamt.png}}\\quad\n         \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[height=0.16\\textheight]{results_new\/balls_int_nonoise_new\/back_gray.png}}\n\n  \\caption{\\textit{Detection of intensity scales}. Automatic segmentation of discs of different intensities (heights) under fixed size. Segmentation is visualized via red contour. (a)-(e) Segmentation after different number of Bregman Iterations. (f) Associated spectral response function $S$. The number of Bregman iterations is plotted on the x-axis. Every peak in $S$ corresponds to one object that appears in the segmentation (compare to (a) - (e)).}\n\\label{fig:ballsint} \n\\end{figure}\n\nThe results of the same experiment but now with intensity scales are shown in the next Figure.  The four discs in Figure \\ref{fig:ballsint}(a) all have\nthe same size but vary in their intensity from $0.55$ to $1$. Note that in some sense this contradicts the assumption of the CV model that the data roughly consists of two different intensity levels $c_1$ and $c_2$. Nevertheless our multiscale approach is able to detect all scales reliably. Like in all our examples we estimated the constants beforehand as $c_1 = \\text{mean}{\\{f(x)| f(x) < \\frac{1}{2}\\cdot \\max{f(x)}\\}}$ and $c_2 = \\text{mean}{\\{f(x)| f(x) \\geq \\frac{1}{2}\\cdot \\max{f(x)}\\}} $ and leave them fixed throughout the whole iteration process. Figure \\ref{fig:ballsint}(b)-(e) visualizes the result of the four Bregman iterations in which the different discs appear. The color-coded result of multiples scales ($\\int_{t}\\Phi(x,t) \\cdot t \\mathrm{d} t$) is shown in Figure \\ref{fig:ballsint}(g). Again, only a few pixels occur at time steps apart from the four significant ones. It is remarkable that the first three scales appear shortly after each other while the last scale appears strikingly later. This can be explained by the way the regularization parameter is adapted automatically through the Bregman distance (cf. Figure \\ref{fig:reg_params} ). In later iterations the change in $\\alpha_k$ becomes smaller so that the time until a certain object appears is larger compared to the previous iterate. Although this might lead to late appearance of smaller scales we can thereby guarantee to not miss scale differences in the finer scales. The late appearance of the last disc is also reflected in the spectral response function $S$ in Figure \\ref{fig:ballsint}(f). Again, the response function has a sparse representation and the last peak occurs delayed. Apart from small variations caused by discretization artifacts the peaks now have all the same height reflecting their similar size.\n\n\\paragraph{Limitations:} In the previous two examples we showed how we can reliably detect different size scales and different intensity scales with our method. One limitation of our approach is the simultaneous detection of size and intensity scales in an image. \n\\begin{figure}[htb]\n  \\centering \n  \\subfigure[Data $f$ with intensity of 0.68 (large disc).]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_varying_size_and_int\/f_068_bw.png}}\\quad \\quad\n  \\subfigure[Data $f$ with intensity of 0.69 (large disc).]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_varying_size_and_int\/f_069_bw.png}}\\quad \\quad\n  \\subfigure[Data $f$ with intensity of 0.70 (large disc).]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_varying_size_and_int\/f_070_bw.png}}\\\\\n   \\subfigure[Spectral response $S$ for (a).]{\\includegraphics[width=0.2\\textwidth]{results_new\/balls_varying_size_and_int\/S_gesamt_068.png}}\\quad \n      \\subfigure[Spectral response $S$ for (b).]{\\includegraphics[width=0.2\\textwidth]{results_new\/balls_varying_size_and_int\/S_gesamt_069.png}}\\quad \n        \\subfigure[Spectral response $S$ for (c).]{\\includegraphics[width=0.2\\textwidth]{results_new\/balls_varying_size_and_int\/S_gesamt_070.png}}\n  \\caption{\\textit{Scale ambiguity w.r.t. size and intensity.} (a) - (c) Three different datasets that differ only in the intensity of the larger circle (0.68\/0.69\/0.7). The associated spectral responses $S$ are shown below. (a) together with (d) shows that out method is not able to reliably distinguish size and intensity scales but only a slight decrease of the intensity leads to two separate peaks in the spectral response function ((b) together with (e) and (c) together with (f)). }\n\\label{fig:criticalcase} \n\\end{figure}\nFigure \\ref{fig:criticalcase}(a)-(c) shows three different data sets, each with a small disc with intensity 1 and a larger discs with intensity $0.68-0.7$ respectively. For all three examples we computed the spectral response with fixed parameters $\\alpha = 200$ and 40 Bregman iterations shown in Figure \\ref{fig:criticalcase}(d)-(f). We can see that in (d) the spectral method was not able to detect the difference between a large object with a lower intensity and a smaller object with a high intensity. Both objects appear at exactly the same point in time. Figure \\ref{fig:criticalcase}(e) and (f) shows that already a small increase of the lower intensity circumvents the problem and both objects are represented as individual peaks in the spectral response. The height of each peaks reveals which object occurs. Yet we hope to solve this issue in future versions; solution ideas are presented in the summary and conclusion section \\ref{sec:conl}.\n\n\\subsection{Robustness against Noise}\nIn this second subsection we investigate the robustness of our methods with respect to uncertainties like noise. \n\\begin{figure}[htb]\n  \\centering \n\n  \\subfigure[Noise level $\\sigma = 0.25$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_littlenoise\/bregman_cv_1.png}}\\quad \n  \\subfigure[Noise level $\\sigma = 0.5$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_mediumnoise\/bregman_cv_1.png}}\\quad \n  \\subfigure[Noise level $\\sigma = 0.75$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_strongnoise\/bregman_cv_1.png}}\\quad \n  \\subfigure[Noise level $\\sigma = 1$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_verystrongnoise\/bregman_cv_1.png}}\\\\\n   \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_littlenoise\/S_gesamt.png}}\\quad \n  \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_mediumnoise\/S_gesamt.png}}\\quad \n  \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_strongnoise\/S_gesamt.png}}\\quad \n  \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_verystrongnoise\/S_gesamt.png}}\\\\\n   \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_littlenoise_new\/back_gray.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_mediumnoise_new\/back_gray.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_highnoise_new\/back_gray.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_veryhighnoise_new\/back_gray.png}}\n      \n\\caption{\\textit{Robustness against noise}. (a) - (d) shows the same underlying binary signal but with different levels of noise added. Exactly the same model parameters are used to process all 4 datasets. The associated spectral response functions are shown in (e) - (h) and the color-coded segmentation results are shown in (i) - (l). We can see that our method is very robust against noise, we only see a slight shift to the right in the spectral response function for higher noise levels. For high noise levels some artifacts at the objects boundary occur but we reliably find all 4 objects independent of the noise level. }\n\\label{fig:robustness} \n\\end{figure}\nFigure \\ref{fig:robustness}(a)-(d) shows the same underlying image but with different noise levels. The underlying signal is a binary signal showing four squares with different size. We added Gaussian noise with mean $0$ and standard deviation $\\sigma = 0.25, 0.5, 0.75$ and $1$ respectively. Since squares are the unit balls of the $\\ell^{\\infty}$ norm we chose $\\gamma = \\| \\cdot \\|_{1}$ and therefore $\\gamma^{\\ast} = \\| \\cdot \\|_{\\infty}$. The results of our inverse Bregman-CV model are plotted in terms of the spectral response function $S$ and the color-coded segmentation of all scales $\\int_{t}\\Phi(x,t) \\cdot t \\mathrm{d} t$. Note that we chose the same parameter setting for all four examples ($\\alpha = 100$ and 30 Bregman iterations). In Figure \\ref{fig:robustness}(e)-(h) we can see that the spectral response function has for all noise levels a sparse representation indicating very clearly the four scales in the underlying signal. It is also remarkable that the peaks are roughly at the same points in time, shifting only slightly to the right for higher noise levels. This is even more obvious in the color-coded representation Figure \\ref{fig:robustness}(i)-(l). The color bar next to each image shows that the color coding is consistent over all four examples. Thus, the same colors (see for example Figure \\ref{fig:robustness}(i) and (j)) reflect that the objects occur at the exact same iteration unaffected of the changed noise level. Even in case of high noise ($\\sigma$ = 0.75) only the second and the fourth largest object appear one iteration later although the smaller objects are now affected by an incomplete segmentation. In case of very high noise ($\\sigma = 1$) we cannot perfectly segment the objects, especially the smaller ones \"get lost\" in the noise. Nevertheless the spectral response function is still close to sparse and for all four objects we reliably segment most of the object. These results show that our method is very robust to noise and circumvents to some extend the problem of choosing a good regularization parameter $\\alpha$. The parameter setting that we used yielded a reliable extraction of all scales in the underlying signal independent of the noise present in our data. \n\\subsection{Segmentation regarding Shapes of Eigenfunctions}\\label{sec:resultsshapes}\nThis subsection is about the choice of the $\\gamma$-function and how this choice is reflected in our spectral analysis framework. As mentioned in Section \\ref{sec:genTV} by varying $\\gamma$ in the generalized definition of the total variation \\eqref{eq:TVgen} the shape of the eigenfunctions is changed. In the following two examples we will first show different examples of eigenfunctions that lead to a sparse representation of the spectral response function. Here we differentiate between eigenfunctions that are Wulff shapes of the chosen $\\gamma$ function and those that are not. In addition, we investigate what happens if the choice of $\\gamma$ does not match the shapes present in the image.\n\n\\begin{figure}[htb]\n  \\centering \n  \\subfigure[Data $f$ with segmentation shown in red.]{\\includegraphics[width=0.22\\textwidth]{results_new\/l1_size_nonoise\/bregman_cv_10.png}}\\quad \n  \\subfigure[Data $f$ with segmentation shown in red.]{\\includegraphics[width=0.22\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_24.png}}\\quad \n  \\subfigure[Data $f$ with segmentation shown in red.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_nonoise\/bregman_cv_24.png}}\\\\\n    \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/l1_size_nonoise\/S_gesamt.png}}\\quad \n      \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/balls_size_nonoise\/S_gesamt.png}}\\quad \n        \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_nonoise\/S_gesamt.png}}\\\\\n    \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/l1_size_nonoise_new\/back_gray.png}}\\quad \n      \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/balls_size_nonoise_new\/back_gray.png}}\\quad \n        \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_nonoise_new\/back_gray.png}}\n  \\caption{\\textit{Varying norm bodies and eigenshapes.} Automatic detection of different signals composed of eigenshapes for different norm bodies. Column one shows the results for $\\gamma^{\\ast} = \\| \\cdot \\|_1$, column two for $\\gamma^{\\ast} = \\| \\cdot \\|_2$ and column three for $\\gamma^{\\ast} = \\| \\cdot \\|_{\\infty}$. (a) - (c) Segmentation results visualized via red contour after 30 Bregman Iterations. Associated spectral response function $S$ is shown in (d) - (f) and the color-coded segmentation in (g) - (i). With the correct choice of $\\gamma$ all eigenshapes can be detected in a single step. }\n\\label{fig:differentnorms} \n\\end{figure}\nIn Figure \\ref{fig:differentnorms}(a)-(c) each images shows a composite of eigenfunction for three different $\\ell^p$-norm choices for $\\gamma$ (namely $p = \\infty$, $2$ and $1$). In each of the images, the input data is shown together with the final segmentation result in red. The associated spectral responses are shown below (Figure \\ref{fig:differentnorms}(d)-(f)). We can see that as long as the choice of $\\gamma$ and the given dataset match (in the sense that the components of $f$ have the same shape as the unit-ball of $\\gamma^{\\ast}$ (cf. Table \\ref{tab:shapes})) we get a sparse response function. In Figure \\ref{fig:differentnorms}(g)-(i) the color-coded segmentation of all scales is shown. We can see that for rectangular shapes ((g) and (i)) no discretization artifacts at the boundary occur. Moreover, we can again use the same parameter setting for all three data sets. The similar color coding in (h) and (i) indicates that the objects occur roughly at the same time point independent of their shape and $\\gamma$. The data set in the first column is larger than the other ones thus the objects occur earlier and we need less iterations. \n\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[TV eigenfunction for $\\gamma = \\| \\cdot \\|_2$.]{\\includegraphics[height=0.14\\textheight]{results_new\/nonWulf_l2_new\/nonWulff.png}}\\quad \n   \\subfigure[Spectral response $S$.]{\\includegraphics[height=0.14\\textheight]{results_new\/nonWulf_l2_new\/S_gesamt.png}}\\quad \n \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[height=0.14\\textheight]{results_new\/nonWulf_l2_new\/back_gray.png}}\\\\\n    \\subfigure[TV eigenfunction for $\\gamma = \\| \\cdot \\|_1$.]{\\includegraphics[width=0.235\\textwidth]{results_new\/nonWulf_linf_new\/nonWulff2.png}}\\quad \n \\subfigure[Spectral response $S$.]{\\includegraphics[height=0.14\\textheight]{results_new\/nonWulf_linf_new\/S_gesamt.png}} \\quad \n \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[height=0.091\\textheight]{results_new\/nonWulf_linf_new\/back_gray.png}}\n  \\caption{\\textit{Reconstruction of eigenfunctions that are not Wulff shapes.} (a) and (d) show two TV eigenfunctions that are not Wulff shapes for $\\gamma = \\| \\cdot \\|_2$ and $\\gamma = \\| \\cdot \\|_1$ respectively. The associated spectral responses are shown in (b) and (e) and the color-coded segmentation results in (c) and (f). It is obvious that eigenfunctions that are not Wulff shapes do have a sparse spectral response representation and appear in one step in the corresponding segmentation.}\n\\label{fig:nonWulff} \n\\end{figure}\n In Figure \\ref{fig:nonWulff} (a) and (d) we present two eigenfunctions which are not Wulff shapes. The round object in (a) fulfills condition \\eqref{eq:l2eig} and is therefore a TV eigenfunction for $\\gamma = \\| \\cdot \\|_2$. The corresponding spectral response function in (b) and the color-coded segmentation in (c) show that not only eigenfunctions that are Wulff shapes appear in one step but also more general eigenfunctions. Another example (now for $\\gamma = \\| \\cdot \\|_1$) is shown in Figure \\ref{fig:nonWulff} (d). Esedoglu and Osher showed in \\cite{esedoglu2004} that a rectangle whose contours are parallel to the x- and y-axis respectively is an eigenfunction of the anisotropic TV functional. This is reflected in the spectral response function Figure \\ref{fig:nonWulff} (e) and the color-coded segmentation in \\ref{fig:nonWulff} (f) which show that the rectangle appears in one step although it is not a Wulff shape of the 1-norm. For natural images, where objects are often not Wulff shapes, this allows objects with various shapes to appear in one step. Some examples can be seen in the subsections \\ref{sec:resultsCTC} - \\ref{sec:resultsnetwork}.\n\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Given Data.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_1.png}}\\quad \n  \\subfigure[Bregman iteration 10.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_10.png}}\\quad \n  \\subfigure[Bregman iteration 16.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_16.png}}\\\\\n  \\subfigure[Bregman iteration 18.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_18.png}}\\quad \n  \\subfigure[Bregman iteration 31.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_31.png}}\\quad \n  \\subfigure[Bregman iteration 41.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_41.png}}\\\\\n  \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/S_gesamt.png}}\\quad\n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.31\\textwidth]{results_new\/varying_shapes_nonoise_new\/back_gray.png}}  \n  \\caption{\\textit{Reconstruction of mixed shapes.} Automatic detection of a signal consisting of different size scales and shapes. The binary signal is shown in (a). In (b) - (f) the segmentation results of the Bregman iterations where a significant peak in the spectral response $S$ in (g) appeared are shown. The color-coded segmentation is shown in (h). Here the behavior of non-eigenfunction over time can be seen. They do not appear in one step but reshape and evolve into the shape of the original signal.  }\n\\label{fig:scalesmix} \n\\end{figure}\nIn Figure \\ref{fig:scalesmix} we investigated how our multiscale approach behaves in case of shape mixtures and shapes that do not match the prior shape assumption made in $\\gamma$. The given data (Figure \\ref{fig:scalesmix}) consists of different size scales and different sizes with only two objects matching the prior assumption of round objects ($\\gamma = \\| \\cdot \\|_2$). Figure \\ref{fig:scalesmix}(b)-(f) shows all time steps where a new scale occurs in the segmentation (shown in red). We can see that the round objects appear in one step while all other ones appear with rounded edges and reshape over time. This can be seen more clearly in the color-coded segmentation of all steps shown in Figure \\ref{fig:scalesmix}(h). Especially for the large triangle in the middle we can nicely see how non-eigenfunctions are reshaped over time and how the segmentation is very round at the beginning and then propagates into the edges. Nevertheless we cannot get clear edges, neither for the triangle nor for the squares. The spectral transform function is nevertheless close to sparse but due to this reshaping behavior there is a small signal also between the clear peaks. It might be suitable to use the $\\ell_0$-norm of $S$ as a measure in order to evaluate if a certain signal matches the prior shape assumption made. However for this idea the influence of discretization artifacts as shown in some examples before, would have to be clarified in further detail.\n\\subsection{Fluorescence microscopy images containing Circulating Tumor Cells}\\label{sec:resultsCTC}\nIn this subsection we demonstrate how our multiscale segmentation approach can be applied to the analysis of Circulating Tumor Cells (CTCs). Before presenting some experimental data sets we will first give a short introduction on CTCs, associated research questions and challenges for diagnosis and treatment of cancer patients.\n\nCirculating Tumor Cells are cells that dissociate from a primary tumor and invade the blood stream. In recent literature it was shown that the number of CTCs present in the blood is associated with the survival chance of a patient and can be used to guide therapy of cancer patients. The elimination of CTCs indicates effective therapy whereas increase or failure to eliminate indicates a futile therapy. A challenge in the identification of CTCs is that they are very rare and therefore difficult to detect among other cells in the pool of blood cells. The gold standard for CTC enumeration is the CellSearch system and in prospective multicenter studies a threshold of 5 CTCs \/ 7.5 ml of blood was used to separate patients with metastatic disease into those with favorable and unfavorable prognosis \\cite{cristofanilli2004,debono2008}. Currently all known CTC analysis tools are based on subjective morphological criteria and objects can be classified differently by different operators which results in different interpretation of the status of the patients. The low cell counts and subsequent statistical analyses further increases the chances of mistakes. The problem of subjectivity in the CTC analysis is currently addressed by the development of an open-source toolbox to automatically detect and classify CTCs in datasets from various machines and institutes. This project is part of a European consortium called CANCER-ID that aims at validating blood-based biomarkers and is funded by the European Union. One key component of this toolbox is the development of a (nearly) parameter-free segmentation algorithm which performs reliably and efficiently on multiple tumor cell datasets (see Figure \\ref{fig:cells}).\n\\begin{figure}[t]\n\\centering\n \\includegraphics[width=0.4\\textwidth]{img\/CTC_classes.png}\n   \\caption{\\textit{Schematic overview of different CTC classes.} For each subclass a cartoon image of a typical cell is shown. Next to it on the right different examples for every class are shown. All images consist of two overlaid color channels where green is the tumor cell marker and magenta is the nucleus marker. The relative frequencies of each subclass are shown on the left. By looking only at the green signal it is obvious that the size scale varies between different subclasses.}\n\\label{fig:cellclasses} \n\\end{figure}\nMoreover, in recent studies \\cite{coumans2010} it was hypothesized that not only the definition used to classify objects as CTC predicted clinical outcome but also those objects that did not match the criteria of an intact cell. A schematic overview of different cell classes and their relative frequencies is shown in Figure \\ref{fig:cellclasses}. Here, green is a fluorescence marker indicating the presence of cytokeratin expressed in cancer cells but not on white blood cells and magenta is a fluorescence marker indicating the presence of DNA thus showing the nucleus of the cell. When comparing the green marker among the different CTC classes we observe that the signal clusters strongly vary in size (area) and therefore motivate the usage of our multiscale segmentation approach.\n\n\\begin{figure}[ht]\n  \\centering \n  \\subfigure[Fluorescence image from the CellSearch system.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp1\/cell_comp1.png}}\\quad \n  \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp1\/S_gesamt_jet.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp1\/back_gray.png}}\\\\\n  \n    \\subfigure[Fluorescence image from a filter system.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp2\/cell_comp2.png}}\\quad \n  \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp2\/S_gesamt_jet.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp2\/back_gray.png}}\\\\\n  \n    \\subfigure[Fluorescence image from the CellSearch system with more clustered cells.]{\\includegraphics[width=0.27\\textwidth]{results_new\/cell_comp3\/cell_comp3.png}}\\quad \n  \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp3\/S_gesamt_jet.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp3\/back_gray.png}}\\\\\n  \n    \\subfigure[Fluorescence image from the ARIOL system.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp4\/cell_comp4.png}}\\quad \n  \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp4\/S_gesamt_jet.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp4\/back_gray.png}}\\\\\n\n   \\caption{\\textit{Experimental cell data.} Automatic detection of (a) different sparse cell sizes (scales) with nearly constant background, (d) cells in front of an inhomogeneous background, (g) clustered cells with nearly constant background and (j) different intensity and size scales. All results are obtained with the exact same parameters. In column one single channel images obtained from different fluorescence microscopes are shown. The corresponding spectral responses are shown in the second column and the color-coded segmentation in column three. It can be seen that more complex images (inhomogeneous background or mixtures of scales) lead to more complexity in the spectral response function (response becomes less sparse).}\n\\label{fig:cells} \n\\end{figure}\nIn Figure \\ref{fig:cells} (a), (d), (g) and (j) experimental data sets are shown. From the original image sets, consisting of three different fluorescent color channels, we extracted the tumor cell marker (green) and used those images as input for our multiscale segmentation approach. Although the difficulties vary between all images (inhomogeneous background, noise, cell clusters, mixture of size and intensity scales), we can process all images with our multiscale segmentation approach with the \\textit{exact same parameters}. This is essential  for the development of a user-friendly (parameter-free) toolbox for CTC analysis. Note that the dim spots in image (d) are not cells but only pores of a filter used to collect cells (bright spots) and therefore it is not desired to segment them. The resulting spectral response functions for all four images are shown in Figure \\ref{fig:cells} (b), (e), (h) and (k) with a color coding corresponding to the coding used in the segmentation results in (c), (f), (i) and (l). The color coding of the response function shows that all objects which appear later in our segmentation and therefore belong to finer scales have a yellow to reddish color in the color-coded segmentation. The very large and intact cells are blue (with some small artifacts at the boundary) and smaller cells (or large fragments) are shown in light blue to green. We can nicely observe that the object colors cover the whole color scale range. For images that are more complex (e.g. (d) and (j)) also the spectral response function is more complex but the color-coded segmentation shows that nearly every object appears in one step and thereby has a clearly defined scale that we use as a feature in our classification approach. Here, we profit from the fact that not only Wulff shapes (perfectly circular objects) but also other eigenshapes appear in one step in our segmentation.\nHence, this segmentation approach not only provides all contours automatically without any parameter adaptations but simultaneously also a simple classification of cells based on their size (scale resp. color\/appearance time). This analysis can be applied to all color channels separately and be used together with more features in a subsequent automatic classification approach. The constants $c_1$ and $c_2$ can again simply be estimated a-priori from the data by a simple thresholding and averaging approach and are fixed throughout the iterative process.\n\n\\subsection{Electron microscopy cell images with nonuniform background}\\label{sec:resultsEM}\nIn this subsection we show that our multiscale segmentation approach can also be used for real-world images where the background is less homogenous as in the cell images before. In Figure \\ref{fig:emcells} (a) we show an electron microcopy dataset of acinar cells provided by \\cite{riedel}. For a more obvious visual interpretation we reverted the contrast from the original EM dataset (bright background with dark cells) to a dark, but not uniform, background with brighter cells. The intensity of the cells is also inhomogeneous and varies from very bright cells (left) to cells with an intensity closer to the background (right). Figure \\ref{fig:emcells} (b) shows the color-coded spectral response function and (c) the color-coded segmentation of the cells. We see that the inhomogeneous intensity of the cells is reflected in the scales found by our method. The inhomogeneity of the background is not influencing the segmentation result and the constants $c_1$ and $c_2$ can still be estimated beforehand from the data without further adaption. One problem that occurs is the ambiguity of size and intensity scales which can be observed when comparing small but bright cells on the right (visualized in light green) with larger cell clusters on the right (also visualized in light green). Currently our method is not able to differentiate their scales. A useful extension of our method would be a combination of our multiscale approach with a watershed type algorithm that is able to split cell clusters into individual cells of different scales.\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Electron microscopy data of acinar cells in the parotid gland provided by \\cite{riedel}.]{\\includegraphics[height=0.15\\textheight]{results_new\/em1\/em1.png}}\\quad \n   \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[height=0.15\\textheight]{results_new\/em1\/S_gesamt_jet.png}}\\quad \n \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[height=0.15\\textheight]{results_new\/em1\/back_gray.png}}\n  \\caption{\\textit{Segmentation of cells in front of an inhomogeneous background.} (a) shows the input image obtained by electron microscopy, (b) the corresponding spectral response function and (c) the color-coded segmentation. This result shows that our method can detect inhomogeneities in the object we want to segment while inhomogeneities in the background (up to a certain degree) do not affect the result of our segmentation method. }\n\\label{fig:emcells} \n\\end{figure}\n\n\\subsection{Segmentation and clustering of biological network structures}\\label{sec:resultsnetwork}\nIn many biological and medical applications network-like structures occur as part of vascular systems. Examples are blood-vessels in retinal images, anisotropic structures in brain or myocardium imaging or at the microscopic level even cells consisting of a round core with several meandering arms of different diameters. Although these networks have a much more complex shape than the examples before, our multiscale method is very useful for a segmentation or clustering of these networks. An example of a round object with \"arms\" of different diameters is shown in Figure \\ref{fig:cellarms} (a). In (b) the corresponding spectral response function is shown and the color-coded segmentation in (c). We can see that the spectral response function has sparse clusters where each cluster corresponds to branches of a certain diameter. Although these \"arms\" are not eigenfunctions, they do appear in one step and therefore have a distinct scale if they have constant diameter. A similar behavior can be seen in Figure \\ref{fig:retina}. As input for our multiscale method we used a manual blood vessel segmentation (b) of a retinal image (a) from the STARE dataset \\cite{hoover2000}. The resulting spectral response is shown in Figure \\ref{fig:retina} (c) and the color-coded segmentation in (d). We see that the different times of appearance of the vessels (indicated by the different colors) lead to a clustering of the underlying network based on the diameter of each branch. With this approach we can get segmentations of the network that contain only branches of certain diameter scales or directly get an estimate of how many different vessel sizes are present and how often they occur. In order to apply our segmentation approach directly on raw retinal images, instead of using it as a postprocessing step, a decomposition method for retinal images similar to Zosso \\cite{zosso2015} could be combined with our multiscale approach.\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Given Data.]{\\includegraphics[height=0.2\\textheight]{results_new\/network\/network.png}}\\quad \n   \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[height=0.2\\textheight]{results_new\/network\/S_gesamt_jet.png}}\\quad \n \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[height=0.2\\textheight]{results_new\/network\/back_gray.png}}\n  \\caption{\\textit{Reconstruction of different scales in network-like structures.} The input data is shown in (a) with the corresponding spectral response function (b) and the color-coded segmentation (c). In this example the usability of our method in case of more complex, network-like structures is demonstrated. Interestingly, the spectral response function shows distinct sparse clusters where every cluster corresponds to \"arms\" or \"branches\" of a specific diameter. Although they are not shaped as an eigenfunction they appear (nearly) in one step and are represented by one specific scale.}\n\\label{fig:cellarms} \n\\end{figure}\n\n\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Retinal image of the STARE dataset \\cite{hoover2000}.]{\\includegraphics[height=0.2\\textheight]{results_new\/network_retina2\/network_retina_orig.png}}\\quad \n     \\subfigure[Manually segmented blood vessel network \\cite{hoover2000}.]{\\includegraphics[height=0.2\\textheight]{results_new\/network_retina2-2\/network_retina2-2.png}}\\quad \n   \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[height=0.2\\textheight]{results_new\/network_retina2-2\/S_gesamt_jet.png}}\\quad \n \\subfigure[Color-coded clustering of network in (b).]{\\includegraphics[height=0.2\\textheight]{results_new\/network_retina2-2\/back_gray.png}}\n  \\caption{\\textit{Clustering of networks.}  (a) A retinal image of a human eye taken from the STARE dataset \\cite{hoover2000} with a manual segmentation of the blood vessel network in (b). We applied our multiscale segmentation method to the binary image in (b) and thereby obtain the spectral response function (c) and a clustering of the network based on the diameter of the network branches (d). We see that large blood vessels are represented by blue colors, medium vessels by light blue to green colors and the very fine branches by yellow to red colors. Although the representation of (c) is not as sparse as in the previous example we can get a meaningful clustering of the graph based on the resulting colors in (d).}\n\\label{fig:retina} \n\\end{figure}\n\n\n\\section{Summary and Conclusion}\\label{sec:conl}\n\nIn this paper, we have studied a novel multiscale segmentation method which is a combination of an inverse scale space variational approach and a nonlinear spectral analysis. Using our approach we can automatically detect objects of different scales where scale can refer to intensity or size. These scales can be evaluated using a spectral response function and we can easily decompose the segmentation with respect to those scales using the spectral transform function. Thus we have extended the useful spectral analysis that was introduced for $TV$ denoising applications before to a segmentation approach. This novel approach can to some extent circumvent the problem of choosing a suitable regularization parameter. The iterative procedure using Bregman distances can be interpreted as an automatic regularization parameter choice method and is very robust with respect to noise or errors in the estimated $c_1$ and $c_2$ values in the CV model. We have shown that our method works very reliable even for applications with more than two intensity scales although this actually contradicts the models binary assumption. The framework can easily be adapted to different prior shape assumptions (Wulff shapes). With regard to real experimental datasets such an assumption is not limiting the usefulness of the method, due to its ability to even detect compositions of eigenfunctions not belonging to the restricted class of Wulff shapes. For the numerical implementation we used a primal-dual optimization scheme that, especially with a C\/MEX version, lead to a very efficient framework, even for large datasets. We illustrated the strengths and limitations of our method on synthetic datasets with a certain focus on eigenfunctions, shapes as well as robustness against noise and background artifacts. Three different biomedical imaging applications with different shapes and challenges emphasize the potential and wide applicability of our approach. The code is also part of an open-source toolbox called ACCEPT. With this software users are able to automatically detect and classify CTCs from fluorescence images, but can also extract important information related to the level of expression of certain therapy targets on the CTC (see the application and results described in section \\ref{sec:resultsCTC}).\n\nOne limitation that we currently have in our model is an ambiguity in scale. It is known for variational methods with an $l_2$ type data fidelity term combined with total variation regularization that these methods are not able to distinguish between intensity and size scales. In practice it seems not very likely that two objects with different size and different intensity fall onto exactly the same peak in the spectral response. Nevertheless to reliably interpret the spectral response function it is advantageous to have only one dominating scale in the data that is analyzed. Another limitation is the flexibility with regard to different shapes. Currently only one shape assumption at the moment is possible. \n\n\n\\textit{Outlook.} There are four main questions that we want to address in our future research. To allow a clear interpretation of the spectral response function even in the case of size and intensity scales we want to adapt the dataterm in the way that it can differentiate both scales. For denoising this can be done using an $l_1$ type data fidelity and we want to generalize this to our segmentation model (which is not obvious in terms of gradient flows). For the experimental cell datasets we currently analyze every fluorescent channel separately. In a future work we would like to investigate if a combined segmentation in all channels can lead to an improved segmentation and a richer response function. Moreover, we are interested in how far we can use the spectral response function to investigate if a shape assumption matches the data. One idea is to use the $l_0$ norm of the spectral response function as a measure for shape optimization. A further development of this idea could lead to an approach where we could learn an optimal $\\gamma$ function and thereby an optimal shape of eigenfunctions for a specific dataset. Another very interesting consideration is to transfer our method from the currently local setup to a nonlocal setup. A first foundation for this was recently laid with a work on non-local TV spectral theory by Aujol et al. in \\cite{Aujol2015}. More mathematical spectral theory for nonlocal TV, its usefulness for solving and improving complex segmentation tasks and a direct comparison with spectral clustering and graph cuts will be of high interest for the community and our future research.\n\\section*{Acknowledgements}%\nLZ, LT, CB acknowledge support by the EUFP7  program \\#305341 CTCTrap and the IMI EU program \\#115749 CANCER-ID. CB acknowledges support by the NWO via Veni grant 613.009.032 within the NDNS+ cluster.\n\n\\bibliographystyle{alpha}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Supranuclear density matter}\n\n\\subsection{Introduction}  \n\nNeutron stars are the densest observable objects in the Universe, attaining physical conditions of matter that cannot be replicated on Earth. Inside neutron stars, the state of matter ranges from ions (nuclei) embedded in a sea of electrons at low densities in the outer crust, through increasingly neutron-rich ions in the inner crust and outer core, to the supranuclear densities reached in the center, where particles are squeezed together more tightly than in atomic nuclei, and theory predicts a host of possible exotic states of matter (Figure \\ref{nscut}). The nature of matter at such densities is one of the great unsolved problems in modern science, and this makes neutron stars unparalleled laboratories for nuclear physics and quantum chromodynamics (QCD) under extreme conditions.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{nsV3less.jpeg}\n\\caption{Schematic structure of a neutron star. The outer layer is a solid ionic crust supported by electron degeneracy pressure. Neutrons begin to leak out of ions (nuclei) at densities $\\sim 4 \\times 10^{11} $ g\/cm$^3$ (the neutron drip density, which separates inner from outer crust), where neutron degeneracy also starts to play a role.  At densities $\\sim 2\\times 10^{14}$ g\/cm$^3$, the nuclei dissolve completely.  This marks the crust-core boundary. In the core, densities reach several times the nuclear saturation density $\\rho_\\mathrm{sat} = 2.8\\times 10^{14}$ g\/cm$^3$ (see text).}\n\\label{nscut}\n\\end{figure}\n\nThe most fundamental macroscopic diagnostic of dense matter is the pressure-density-temperature relation of bulk matter, the equation of state (EOS). The EOS can be used to infer key aspects of the microphysics, such as the role of many-body interactions at nuclear densities or the presence of deconfined quarks at high densities (Section \\ref{nmat}).  Measuring the EOS of supranuclear density matter is therefore of major importance to nuclear physics.  However it is also critical to astrophysics.  The dense matter EOS is clearly central to understanding the powerful, violent, and enigmatic objects that are neutron stars. However, neutron star\/neutron star and neutron star\/black hole binary inspiral and merger, prime sources of gravitational waves and the likely engines of short gamma-ray bursts \\citep{Nakar07}, also depend sensitively on the EOS \\citep{Shibata11,Faber12,Bauswein12,Lackey12,Takami14}.  The EOS affects merger dynamics, black hole formation timescales, the precise gravitational wave and neutrino signals, any associated mass loss and r-process nucleosynthesis, and the attendant gamma-ray bursts and optical flashes \\citep{Metzger10,Hotokezaka11,Rosswog15,Kumar15}.  The EOS of dense matter is also vital to understanding core collapse supernova explosions and their associated gravitational wave and neutrino emission \\citep{Janka07}\\footnote{Note that whilst most neutron stars, even during the binary inspiral phase, can be described by the cold EOS that is the focus of this Colloquium (see Section \\ref{mrtoeos}), temperature corrections must be applied when describing either newborn neutron stars in the immediate aftermath of a supernova, or the hot differentially rotating remnants that may survive for a short period of time following a compact object merger.  The cold and hot EOS must of course connect and be consistent with one another.}.\n\n\\subsection{The nature of matter:  major open questions}\n\\label{nmat}\n\nThe properties of neutron stars, like those of atomic nuclei, depend\ncrucially on the interactions between protons and neutrons (nucleons)\ngoverned by the strong force. This is evident from the seminal work of\nOppenheimer and Volkoff \\citep{Oppenheimer39}, which showed\nthat the maximal mass of neutron stars consisting of non-interacting\nneutrons is 0.7 M$_\\odot$. To stabilize heavier neutron stars, as\nrealized in nature, requires repulsive interactions between nucleons,\nwhich set in with increasing density. At low energies, and thus low\ndensities, the interactions between nucleons are attractive, as they\nhave to be to bind neutrons and protons into nuclei. However, to\nprevent nuclei from collapsing, repulsive two-nucleon and\nthree-nucleon interactions set in at higher momenta and\ndensities.   Because neutron stars reach densities exceeding those in atomic nuclei, this makes them particularly sensitive to many-body forces \\citep[see, for example,][]{Akmal98}, and recently it was shown that the dominant uncertainty at nuclear densities is due to three-nucleon forces \\citep{Hebeler10b,Gandolfi12}\n\nAt low energies, effective field theories based on QCD provide a systematic basis for nuclear forces \\citep{Epelbaum09}, which make unique predictions for many-body forces \\citep{Hammer13} and\nneutron-rich matter \\citep{Tolos08,Hebeler14,Hebeler15}. While two-nucleon interactions are well constrained,\nthree-nucleon forces are a frontier in nuclear physics, especially for\nneutron-rich nuclei \\citep[see, for example,][]{Wienholtz13}. Such exotic nuclei are the focus of present and upcoming\nlaboratory experiments. Neutron star observations probe the same\nnuclear forces at extremes of density and neutron richness. In\naddition, to effective field theories, there are nuclear potential\nmodels, such as the Argonne two-nucleon and Urbana\/Illinois\nthree-nucleon potentials, which are fit to two-body scattering data and\nlight nuclei \\citep{Gandolfi14,Carlson14}.\n\nAt high densities, neutron stars may be affected by exotic states of\nmatter. This regime is not accessible to first principle QCD\ncalculations due to the fermion sign problem \\citep[see, for example, the discussions in][]{Hands07,Miller13a}. Therefore, at present, one has to resort to models, and experiment and observation are vital\nto test theories and drive progress. In addition, recently,\nperturbative QCD calculations have been performed at very high\ndensities (above 10 GeV\/fm$^3$,  $\\sim 64 \\rho_\\mathrm{sat}$), and used to interpolate to the EOS at low densities\n\\citep{Kurkela14}.\n\nFor symmetric matter (with an equal number of neutrons and protons) at the\nnuclear saturation density $\\rho_\\mathrm{sat} = 2.8 \\times 10^{14}$ g\/cm$^3$ (the central density in very large nuclei when the Coulomb interaction is neglected)\nthere is a range of experimental constraints. This includes nuclear\nmasses and charge\nradii~\\citep[see, for example,][]{Klupfel09,Kortelainen10,Kortelainen14,Niksic15} as well\nas giant dipole resonances and dipole\npolarizabilities~\\citep{Trippa08,Tamii11,Piekarewicz12}. Neutron-rich\nmatter can be probed by measuring the neutron skin thickness of heavy\nnuclei \\citep{Horowitz01,RocaMaza11}. However, all of these laboratory\nexperiments probe only matter at nuclear densities and\nbelow. Low-energy heavy-ion collisions probe hot and dense matter, but\nhave uncontrolled extrapolations to zero temperature and to extreme\nneutron-richness \\citep{Tsang09}.  Neutron stars therefore provide a\nunique environment for testing our understanding of the physics of the\nstrong interaction and dense matter.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{Trhom_RMP2.pdf}\n\\caption{Hypothetical states of matter accessed by neutron stars and \ncurrent or planned laboratory experiments (Large Hadron Collider and other heavy ion collision experiments, shown by black arrows), in the parameter space of\ntemperature against baryon chemical potential (1-2 GeV corresponds to\n$\\sim$ 1-6 times the density of normal atomic nuclei). Quarkyonic\nmatter: a hypothesised phase where cold dense quarks experience\nconfining forces \\citep{McLerran07,Fukushima11}. The stabilizing\neffect of gravitational confinement in neutron stars permits\nlong-timescale weak interactions (such as electron captures) to reach\nequilibrium, generating matter that is neutron rich (see Figure 2 of\n\\citealt{Watts14}) and may involve matter with strange quarks. This\nmeans that neutron stars access unique states of matter that can only\nbe created with extreme difficulty in the laboratory: nuclear\nsuperfluids, strange matter states with hyperons, deconfined quarks,\nand color superconducting phases.}\n\\label{paramspace}\n\\end{figure}\n\nAt very high densities, possibly reached in neutron star cores,\ntransitions to non-nucleonic states of matter may occur. Some of\nthe possibilities involve strange quarks: unlike heavy-ion collision\nexperiments, which always produce very short-lived and hot dense\nstates, the stable gravitationally confined environment of a neutron\nstar permits slow-acting weak interactions that can form states of\nmatter with a high net strangeness. Strange matter possibilities\ninclude the formation of hyperons \\citep[strange baryons,][]{Ambartsumyan60,Glendenning82,Balberg99,Vidana15}, deconfined quarks \\citep[forming a hybrid star,][]{Collins75}, or color superconducting\nphases \\citep{Alford08}. It is even possible that the entire star\nmight convert into a lower energy self-bound state consisting of up,\ndown and strange quarks, known as a strange quark star\n\\citep{Bodmer71,Witten84,Haensel86}. Other states that have been hypothesized\ninclude Bose-Einstein condensates of mesons \\citep[pions or kaons, the\nlatter containing a strange quark, see for example][]\n{Kaplan86,Kunihiro93}. The densities at which such phases may\nappear are highly uncertain.\n\nFigure \\ref{paramspace} compares the parameter space that can be\naccessed within the laboratory to that which can be explored with\nneutron stars. The physical ground state of dense matter is neutron\nrich, which develops via weak interactions, and it is unbound so \ngravitational confinement is necessary to realize the ground state\nof dense matter in nature. Only neutron stars sample this low\ntemperature regime of the dense matter EOS. The exotic non-nucleonic\nstates of matter described in the previous paragraph can be reached\nonly with extreme difficulty in the laboratory.\n\n\\subsection{Methodology: how neutron star mass and radius specify the EOS}\n\\label{mrtoeos}\n\nThe relativistic stellar structure equations relate the EOS to macroscopic observables including the mass $M$ and radius $R$ of the neutron star.  The dependence of the EOS on temperature can be neglected in computing bulk structure for neutron stars older than $\\sim 100$ s: by this point the neutron star has cooled far below the Fermi temperature of the particles involved, the matter is degenerate, and hence temperature effects are negligible \\citep[see for example][]{Haensel07}.  For non-rotating and non-magnetic stars, the classic Tolman-Oppenheimer Volkoff stellar structure equations would apply \\citep{Tolman39,Oppenheimer39}.  However rotation is important, and the equations must be modified accordingly. For neutron stars spinning at a few hundred Hz the slow rotation (to second order) Hartle-Thorne metric is appropriate for most applications \\citep{Hartle68}.  One can also compute full GR models for stars spinning at up to break-up speeds using a variety of methods implemented in well-tested codes \\citep[for a review see][]{Stergioulas03}.  Codes such as \\texttt{rotstar} \\citep{Bonazzola98} and \\texttt{rns} \\citep{Stergioulas95} generate masses and radii for rapidly rotating neutron stars that are accurate to better than one part in $10^{-4}-10^{-5}$. \n\nThere is a one to one map from the EOS to the $M$-$R$ relation \\citep[see, for example,][]{Lindblom92}. Some examples are shown in Figure \\ref{eos2mr}.  A few general features are worthy of note.  For each EOS there is a maximum mass that is a direct consequence of General Relativity \\citep[for a review of this topic see][]{Chamel13}, and there are plausible astrophysical mechanisms (formation or accretion) that might lead to this being reached in real neutron stars.  The minimum observable mass, by contrast, is more likely to be set by evolution than by stability.   Radius tends to reduce as mass increases (although for some EOS models, radius increases slightly with increasing mass in the mid-range of masses), and current models suggest radii in the range 8-15 km for masses above 1 M$_\\odot$.   \n\nIn terms of dependence on the nuclear physics, the maximum mass is determined primarily by the behavior of the cold EOS at the very highest densities \\citep[$\\sim 5-8 \\rho_\\mathrm{sat}$,][]{Lattimer05,Ozel09,Read09,Hebeler13}. The presence of non-nucleonic phases (such as hyperons or condensates) softens the EOS, reducing pressure support and leading to a smaller maximum mass. Radius on the other hand, depends more strongly on the behavior of the EOS at $\\sim (1-2) \\rho_\\mathrm{sat}$ \\citep{Lattimer01}. The nucleonic EOS at these densities is highly sensitive to three-nucleon forces \\citep{Hebeler10,Gandolfi12}, whilst the presence of non-nucleonic phases tends to reduce $R$. The slope of the $M$-$R$ relation (i.e. whether $R$ increases or decreases with $M$), for masses $\\gtrsim 1.2$ M$_\\odot$ (the observed minimum, consistent with expectations from formation models), depends on the pressure at $\\sim 4 \\rho_\\mathrm{sat}$ \\citep{Ozel09}.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{eossmall.png}\n\\caption{The pressure density relation (EOS, left) and the corresponding $M$-$R$ relation (right) based on models\nwith different microphysics. Red: nucleonic EOS from \\citet{Lattimer01}. Black solid: Hybrid models (strange quark core; \\citealt{Zdunik13}). Black dashed: Hyperon core models \\citep{Bednarek12}. Magenta: A self-bound strange quark star model \\citep{Lattimer01}. Grey band: range of a parameterized family of nucleonic EOS based on chiral effective field theory at low densities, which provides\na systematic expansion for nuclear forces that allows one to estimate the theoretical uncertainties involved, combined with using a general extrapolations to high densities \\citep[see Figure 12 of][for examples of specific representative EOS lying within this band]{Hebeler13}.}\n\\label{eos2mr}\n\\end{figure*}\n\nIn testing EOS models, there are two potential approaches.  One is simply to compute, for a given EOS model, the resulting $M$-$R$ relation, and then determine the likelihood of obtaining the measured values of $M$, $R$ (with uncertainties) if this model is correct.   The other option is to perform the inverse process, and to map from the measured values of $M$-$R$ (with their uncertainties) to the EOS.  The first attempt to address this problem, which made no assumptions about the form of the EOS, was made by \\citet{Lindblom92}.  Since then the approach has been refined by several authors.  Newer analyses rely on parameterized representations of the EOS that are a good characterization of many specific EOS models, but contain the lowest possible number of adjustable parameters (e.g. piecewise polytropic fits as employed by \\citet{Ozel09,Read09,Steiner10}, or spectral representations as employed by \\citet{Lindblom12,Lindblom14}).  \\citet{Ozel09} showed that given three measurements of $M$, $R$ with accuracies of $\\sim 5$\\%, current EOS models could be distinguished at the 3$\\sigma$ level using a piecewise-polytropic representation.    Using spectral representations, \\citet{Lindblom12} showed that the inversion process itself (reliance on discrete measurements, and the use of a generalized model) introduces errors that are typically less than $\\sim 1$ \\%\\footnote{This was the case for all models tested apart from those with a very strong phase transition, which are not well described by spectral representations with only a few parameters, and are better tested using the alternative method.}.  At this level the anticipated measurement errors on $M$, $R$ (which are at the few percent level) would be the primary determinant of the uncertainty on the inferred EOS.  \n\n\n\\subsection{Current observational constraints on the cold dense EOS}\n\\label{currentobs}\n\nThe cleanest constraints on the EOS to date have come from radio pulsar timing, where the mass of neutron stars in compact binaries can be measured very precisely using relativistic effects \\citep{Lorimer08}. Since any given EOS has a maximum stable mass (Figure \\ref{eos2mr}), high mass stars can rule out particular EOS.   The most massive pulsars have masses $\\approx 2$ M$_\\odot$ \\citep{Demorest10,Antoniadis13}, and that these results would have an impact on EOS studies was immediately clear \\citep{Demorest10,Ozel10}.    The requirement to generate neutron stars with masses of at least  2 M$_\\odot$ is now an integral part of EOS model development.  \\citet{Hebeler13} and \\citet{Lattimer14}, for example, have combined insights from  nuclear physics with the new maximum mass requirement in their models. Meanwhile \\citet{Kurkela14} have generated models that are constrained to approach the EOS of quark matter at high density computed from state of the art perturbative QCD calculations (which are robust in the very highest density regime).   \n\nOne of the largest impacts of the maximum mass measurements has been on studies of hyperons. The presence of hyperons in neutron stars is energetically probable, but should induce a strong softening of the EOS that would lead to maximum masses below 2 M$_\\odot$. The solution of this {\\it hyperon puzzle} \\citep[see for example][]{Chen11,Weissenborn12} requires some additional repulsion to make the EOS stiffer. Possible mechanisms include stiffer hyperon-nucleon and\/or hyperon-hyperon interactions, the inclusion of three-body forces with one or more hyperons \\citep{Takatsuka08,Logoteta13}, or the appearance of a phase transition to quark matter (see discussion of hybrid stars in Section \\ref{nmat}). \n\nWhilst measuring masses using radio pulsars yields very precise results, measuring radius (or mass and radius simultaneously) is more challenging and, with the methods currently in use, more model-dependent. \nCurrent efforts to constrain the full $M$-$R$ relation focus primarily on spectroscopic measurements of the surface emission from accreting neutron stars in quiescence \\citep{Rutledge99,Heinke03,Heinke06,Webb07,Servillat12,Catuneanu13,Guillot13,Guillot14,Heinke14}, and when they exhibit thermonuclear (Type I) X-ray bursts due to unstable nuclear burning in accreted surface layers \\citep{vanParadijs79,vanParadijs87,Ozel06,Ozel09a,Guver10a,Guver10b,Suleimanov11,Ozel12,Guver13}. \n\nThe essence of the method is the fact that the surface spectrum is\nclose to a diluted blackbody with a color temperature that is larger\nthan the effective temperature of the star $T_{\\rm c}=f_c T_{\\rm eff}$\nby the color-correction factor $f_{\\rm c}\\approx 1.3-2$ \\citep{London86,Zavlin96,Madej04,Heinke06,Suleimanov11b,Suleimanov12}.  This factor depends on the\natmospheric composition and on the effective surface gravity. For a slowly\nspinning neutron star at a known distance $D$, measuring the flux $F$\nof surface emission and the color temperature $T_{\\rm c}$ gives a relation between $M$ and $R$:\n\\begin{equation}\nR^2\\left(1-\\frac{2GM}{Rc^2}\\right)^{-1}=\\frac{F D^2 f_{\\rm c}^4}\n{\\sigma T_{\\rm c}^4}\\;.\n\\end{equation}\nwhere $\\sigma$ is the Stefan-Boltzmann constant.  Spectroscopic measurements using quiescent neutron stars, for which no additional information is available, result only in broad\nconstraints between the neutron star masses and radii \\citep[see][]{Heinke14,Guillot14}, although ensemble constraints may be tighter \\citep{Ozel15}.   \n\n\nStudies of the spectra during thermonuclear X-ray bursts allow \nadditional constraints to be obtained that can break the degeneracy\nbetween $M$ and $R$ \\citep[see reviews by][]{Lewin93,Ozel06}.  \nThe radius expansion bursts (during which radiation forces lift\nthe neutron star photospheres) serve as a good laboratory\nbecause, from the burst flux $F_{\\rm Edd}$ at the touchdown point,\none can get a measurement of the Eddington luminosity, which is a\ndifferent function of $M$ and $R$\n\\begin{equation}\nL_{\\rm Edd} = 4\\pi D^2 F_{\\rm Edd} = \\frac{4\\pi GMc}{\\kappa_e} \\left(\n1-\\frac{2GM}{Rc^2}\\right)^{1\/2}\\;.\n\\end{equation}\nHere $\\kappa_e=0.2(1+X)$~cm$^2$~g$^{-1}$ is the electron scattering\nopacity and $X$ is the hydrogen mass function.  Combining this with\nthe measurement of the surface area based on the thermal flux leads to\na weakly correlated inference of both $M$ and $R$ \\citep{Ozel09,Guver10a,Guver10b,Suleimanov11,Ozel12,Guver13,Poutanen14}.\n\nDifferent spectroscopic methods and burst selection criteria have been used to obtain constraints on the EOS. \\citet{Ozel10,Steiner10,Steiner13} find that large ($\\ge 13$ km) radii are ruled out, while others \\citep{Suleimanov11,Poutanen14} get radii in the range 11-15 km. Clearly, the $\\approx 5\\%$ accuracy in radius required to pinpoint the value of the pressure beyond the nuclear saturation density has not yet been achieved. \n\nBoth source types are affected by uncertainties in the composition of the atmosphere and (in many cases) lack of prior knowledge of the distance to each source  \\citep[see for example][]{Heinke14}\\footnote{Note that in principle, spectroscopic measurements rely on the absolute flux calibration of X-ray telescopes.  However, a limited number of studies currently seem to indicate that there is no significant flux calibration bias \\citep{Suleimanov11,Guver15}.}.  In the case of quiescent neutron stars, there are additional sources of uncertainty related to the composition of the intervening interstellar medium \nas well as the effects of residual accretion \\citep{Heinke14,Guillot14}, but see also \\citet{Bahramian15}. In the case of the X-ray\nbursters, there are several uncertainties: a systematic spread in the angular size of the source and the Eddington flux \\citep{Guver12a,Guver12b} which may reflect some level of non-uniform emission over the stellar surface;  identification of the touchdown point \\citep{Galloway08b,Guver12a,Miller13a}; and the role of accretion \\citep{Kajava14,Poutanen14}.  \n\n \nPerhaps more importantly, both types of spectroscopic measurements of\nneutron star masses and radii rely on the particular astrophysical\ninterpretations of two types of observations: that the quiescent\nemission is powered by deep crustal heating and that the Eddington critical flux can be estimated accurately from the data.\nThe timing techniques discussed later in this article, on the other\nhand, do not rely on these interpretations and, therefore, provide an\nindependent measurement of masses and radii, with orthogonal\nsystematic uncertainties and biases. Obtaining multiple measurements\nof masses and radii with different techniques, and in many cases for\nthe same sources, will allow us to address and correct for the\nsystematic uncertainties of each method.\n\nTiming-based techniques for constraining $M$ and $R$ rely on the presence of surface inhomogeneities, leading to emission that varies periodically as the star rotates. Three types of neutron star systems are suitable for a timing analysis, and several attempts have already been made:  for the accretion-powered pulsars \\citep{Poutanen03,Leahy04,Leahy09,Leahy11,Morsink11}; for accreting neutron stars that show oscillations during their thermonuclear X-ray bursts \\citep{Bhattacharyya05c}; and for rotation-powered X-ray pulsars \\citep{Bogdanov07,Bogdanov08,Bogdanov09,Bogdanov13}.  All of the $M$-$R$ constraints coming from these observations to date have very large error bars. However the method has great promise (see Section \\ref{pulse}). \n\nLaboratory experiments have also provided some constraints on the EOS.  Neutron star radii, for example, have been shown to depend strongly on the density dependence of the nuclear symmetry energy close to $\\rho_\\mathrm{sat}$ \\citep[see, for example,][]{Lattimer14}, and the behaviour of this quantity for densities up to $\\rho_\\mathrm{sat}$ is now being probed by laboratory experiments \\citep{Tsang12}.  Other experimental observables, such as K$^+$ meson production in nuclear collisions at subthreshold energies \\citep{Sturm01} and the nuclear elliptic flow in heavy ion collisions \\citep{Danielewicz02}, have been used as a sensitive probe for the stiffness of nuclear matter for high densities.  K$^+$ meson production seems to suggest a soft EOS for two to three times saturation density.  Indeed, these results on heavy-ion collisions have been used\nto constrain the features of neutron stars \\citep{Sagert12}.  Meanwhile results on elliptic flow rule out strongly repulsive nuclear EOS from relativistic mean field theory and weakly repulsive EOS with phase transitions at densities less than three times that of stable nuclei, but not EOS softened at higher densities due to a transformation to quark matter.  However, heavy-ion observables are obtained in highly-symmetric systems at high temperatures, thus the analysis of the EOS for asymmetric zero-temperature systems must be treated with caution. A reliable analysis of the EOS for zero temperature in asymmetric matter can be only achieved by neutron star measurements.\n\n\n\\subsection{Future observational constraints on the cold dense EOS}\n\nThe dense matter EOS will be a target science area for a number of different telescopes over the next decade, operating in very different wavebands.  To set the techniques that can be exploited in the hard X-ray band in context, we first review the advances that are expected in other wavebands. \n     \nThe next decade will see a major expansion in our ability to detect galactic radio pulsars, with the advent of the Square Kilometer Array \\citep[SKA,][]{Bourke15} and its pathfinders LOFAR \\citep{vanHaarlem13}, ASKAP \\citep{Schinckel12} and MeerKAT \\citep{Booth12}. The SKA is both more sensitive than current telescopes and will have increased timing precision.  By finding more radio pulsars in binary systems, and being able to determine post-Keplerian orbital parameters more precisely, the SKA should increase the number of measured neutron star masses by a factor of at least $\\sim 10$ \\citep{Watts14}.  New EOS constraints will result if the maximum mass record is broken, as discussed in Section \\ref{currentobs}.   Radio observations can also deliver radius via measurements of the neutron star moment of inertia (determined from spin-orbit coupling).  The moment of inertia of the only known double pulsar system PSR J0737-3039 \\citep{Burgay03}, will be determined to within 10\\% within the next 20 years \\citep{Lattimer05b,Kramer09}, resulting in a constraint on $R$ $\\sim 5$\\% \\citep[see also the discussion in Section 4.2 of][]{Watts14}.  The SKA may discover more systems for which a measurement of moment of inertia is possible, although it will be challenging since the requirements on system geometry are quite restrictive \\citep{Watts14}.  \n\nThe upgraded gravitational wave telescopes Advanced LIGO \\citep{LIGO15} and Advanced VIRGO \\citep{Acernese15} begin to enter service from 2015 and will operate well into the next decade. Gravitational waves from the late inspirals of binary neutron stars are sensitive to the EOS, with departures from the point particle waveform constraining $M$ and $R$. Global seismic oscillations excited by coalescence also depend on the EOS \\citep{Bauswein12}.  Estimates are that Advanced LIGO and VIRGO could achieve uncertainties of ~10\\% (1$\\sigma$) in $R$, for the closest detected binaries \\citep{Read13}, although event rates are highly uncertain. \n\nNICER-SEXTANT is a NASA Explorer Mission of Opportunity experiment that is due to be mounted on the International Space Station in late 2016 \\citep{Arzoumanian14}.  NICER has an effective area that is 0.2~m$^2$ at 2~keV dropping to 0.06~m$^2$ at 6~keV.  The primary focus of NICER will therefore be on soft X-ray sources, in particular rotation-powered pulsars that emit in both the X-ray and radio bands.   The X-ray emission from rotation-powered pulsars is expected to be steady, which means that long accumulation of data is possible, even though they are dim. Radii will then be inferred using soft X-ray waveform modeling \\citep[][and see Section \\ref{pulse}]{Bogdanov08}.  If the mass of a neutron star and the pattern of radiation from its surface are known accurately a priori, NICER observations will\nachieve an accuracy of $\\simeq 2$\\% in the measurement of  radius \\citep{Gendreau12,Bogdanov13}. In practice, the measurement will limited by uncertainties in these two\nrequirements.   The uncertainty in the mass measurement of\nNICER's primary target, the bright pulsar PSR~J0437$-$4715, is $\\sim$5\\% \\citep{Reardon15}. Other main targets of NICER will be the nearby isolated radio millisecond pulsars PSR J0030+0451 and J2124-3358, however these will produce less stringent constraints since there is no prospect for measuring their masses. The temperature profile on the\nneutron star surface, which is believed to be determined by the flux\nof return currents circulating in the magnetosphere \\cite[see e.g.][]{Bai10,Philippov14} is also highly uncertain, and development of ab initio numerical models that might fully address these questions is still a work in progress.  Waveform modelling for accretion-powered millisecond pulsars may also be possible \\citep[see the discussion in][and Section \\ref{pulse}]{Gendreau12}.\n\nAthena, the soft X-ray observatory selected for the European Space Agency's L2 launch slot in the late 2020s, has an effective area requirement of 2 m$^2$ at 1 keV, dropping to 0.25 m$^2$ at 6 keV \\citep{Nandra13}.  Dense matter is not a primary science goal for Athena.  However it could in principle be used for waveform modelling for the same isolated X-ray pulsars as NICER, and spectral modeling of neutron stars in quiescence or the cooling tails of X-ray bursts \\citep{Motch13}.  These techniques have already been discussed in detail both above and in Section \\ref{currentobs}.    \n\nASTROSAT \\citep{Singh14}, a multiwavelength Indian astronomy\nsatellite was launched in September 2015. LAXPC, a 3--80 keV X-ray\ntiming instrument, has an effective area of 0.8 m$^2$ in the 5--20 keV\nrange.  SXT, a co-aligned 0.3-8 keV X-ray imager, has an effective\narea of 0.01 m$^2$ in the 1--2 keV range.  This combination is\nwell-suited for spectral modeling of the cooling tails in X-ray\nbursts.  LAXPC can also in principle be used for waveform modeling of\naccretion-powered millisecond pulsars or X-ray burst oscillations.\n\n\\section{Hard X-ray timing techniques that deliver $M$ and $R$}\n\nHard X-ray timing enables three primary techniques that have the capability of delivering $M$ and $R$:  waveform (or {\\it pulse profile}) modeling, spin measurements, and asteroseismology.  These techniques involve different classes of neutron star: accreting neutron stars with thermonuclear bursts, accretion-powered X-ray pulsars, and isolated highly magnetic neutron stars known as magnetars.  The use of multiple techniques and different source types allows cross-calibration of techniques, and independent cross-checks on the EOS.  In this Section we will explore each technique in turn, and discuss the contraints on the EOS that would be achievable with a large area  ($\\sim 10$ m$^2$) hard X-ray (2-30 keV) timing telescope.  Much of the work presented here was developed as part of the science case for the proposed Large Observatory for X-ray Timing \\citep[LOFT:][and the LOFT ESA M3 Yellow Book, \\texttt{http:\/\/sci.esa.int\/loft\/53447-loft-yellow-book}]{Feroci12,Feroci14}.  \n\n\\subsection{Waveform modelling}\n\nMillisecond X-ray oscillations are observed from accretion-powered pulsars \\citep{Patruno12}, from some thermally-emitting rotation-powered (non-accreting) pulsars \\citep{Becker01} and during some thermonuclear bursts on accreting neutron stars \\citep[burst oscillations, see][]{Watts12}.  These oscillations are thought to be produced by X-ray emission from a region on the surface of the star that is hotter than the rest of the stellar surface and is offset from the rotational pole of the star.  As the hotter region rotates around the star, the X-ray flux seen by a distant observer is modulated at or near the rotation frequency of the star (Figure \\ref{pulse}).   Accretion-powered pulsations are formed as material is channeled onto the magnetic poles of accreting X-ray pulsars.  X-ray emission comes from both a hotspot that forms at the magnetic poles, and from the shock that forms just above the star's surface as the channeled material decelerates abruptly.  Burst oscillations, by contrast, are due to hotspots that form during thermonuclear X-ray bursts on accreting neutron stars \\citep{Lewin93,Galloway08}.  Ultimately one would hope to be able to use both types of pulsation when modelling the resulting pulse profile, or waveform, to recover $M$ and $R$: several sources show both phenomena, providing an important cross-check on the results.   \n\n\\subsubsection{Factors affecting the waveform}\n\\label{factorswaveform}\n\nAs the photons propagate through the curved space-time of the star, information about $M$ and $R$ is encoded into the shape and energy-dependence of the waveform.  General relativistic light-bending, which depends on compactness $M$\/$R$, affects the amplitude of the pulsations.  Special relativistic Doppler boosting, aberration, and the magnitude of time delays depend on the relative orientation of the hot spot and\nthe line of sight \\citep{Braje00}. This introduces a number of effects that lead to a direct measurement of the neutron star\nradius. \n\nIn the absence of special relativity, the peak flux occurs when the hot spot is facing the observer and the time for the\nflux to rise from minimum to maximum is the same as the time to fall from maximum to minimum. The Doppler boosting effect makes the\nblueshifted side of the star appear brighter than the redshifted side, so that the flux maximum occurs earlier in phase than if the special\nrelativistic effects did not occur. This asymmetry between the rise and fall time is approximately proportional to the projected line-of-sight velocity. Since the angular velocity is known from the pulse frequency, this provides a constraint on $R$ if the inclinations of the\nspot and observer are known. The great advantage of the very rapidly rotating neutron stars (with spin frequencies of 400 Hz and larger) is that the Doppler boosting effect is more pronounced in the data which reduces degeneracies. The asymmetry in the rise\/fall times is independent of the normalization, so errors in flux calibration are not important.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{newfig.pdf}\n\\caption{Left: As the neutron star rotates, emission from a surface hotspot generates a pulsation. The figure shows observer inclination $i$ and hotspot inclination $\\alpha$.  The invisible surface is smaller than a hemisphere due to relativistic light-bending.  Right: A schematic from \\citet{Psaltis14b} to illustrate the effects on the waveform arising from relativistic effects: the waveform is modified from a pure sinusoid, and the temperature (indicated by the color, which is the ratio of the number of photons with energies above to those below the blackbody temperature) also varies even though the underlying hotspot has a uniform temperature. The shape of the waveform and its energy dependence can be used to recover $M$ and $R$. See also \\citet{Viironen04} for examples of waveforms that include the expected variation in X-ray polarization.}\n\\label{pulse}\n\\end{figure*}\n\n\nThere are, of course, other factors that affect the waveforms and these must be taken into account when fitting for $M$ and $R$. The\ngeometric parameters $\\alpha$, the angle between the spin axis and the spot center, and $i$, the observer's inclination angle (see Figure \\ref{pulse}) introduce\ndegeneracies with $M$ and $R$. As the compactness ratio $M\/R$ increases, the pulse fraction decreases. However a decrease in the quantity\n$\\sin(\\alpha)\\sin(i)$ also decreases the pulse fraction. Similarly an increase in $\\sin(\\alpha)\\sin(i)$ increases the projected line-of-sight\nvelocity, leading to a degeneracy with the star's equatorial radius. Other geometric parameters such as the hotspot's shape and\nsize as well as emission from the rest of the star and the disk affect the shape of the light curve. Fortunately, the resulting parameter\ndependencies can be resolved when the detailed structure of the waveforms and their dependence on photon energy is taken into account (see below), allowing us to recover $M$ and $R$.\n\nA very important contribution to the parameter degeneracy is the beaming pattern of the radiation. A highly beamed emission pattern could, in some ways mimic the effects of decreased gravitational light bending and special relativistic Doppler boosting.  In the case of the thermal emission from X-ray bursts, this pattern is very well understood from theoretical modelling: it is very close to the limb-darkened pattern of a scattering dominated atmosphere \\citep{Madej91,Suleimanov12}, and is not a significant source of uncertainty \\citep{Miller13b}.  In contrast, in the case of the accretion-powered pulsations, the theoretical beaming pattern due to Compton scattering introduces a free parameter that is poorly constrained by the observations, leading\nto the weak constraints on $M$ and $R$ that result for these stars \\citep{Poutanen03, Leahy09, Leahy11, Morsink11}. The beaming pattern for the hydrogen atmosphere models used to compute the light curves of the rotation-powered pulsars \\citep{Bogdanov08}  do not formally have free parameters, although there are still some open questions regarding the beaming from such atmospheres that are thought to be heated by relativistic particles from the magnetosphere.  \n\nAdditional complications to the waveform modeling from accreting sources come from the fact that the neutron star is surrounded by the accretion disk,  which may block radiation coming from the `southern hemisphere'.  For accreting pulsars, the eclipses by the disk may lead to appearance of strong harmonic structure to the waveform that completely dominates all other sources producing harmonics \\citep{Poutanen08}. The evolution of the waveform from the best-studied pulsar SAX J1808.4$-$3658 during the outburst can be explained by varying just one parameter - the disk inner radius \\citep{Poutanen09,Ibragimov09,Kajava11}. The way out is to either use data at high accretion rate, when the southern pole  is completely blocked, or to model the eclipse by the disk, but this requires knowledge of the inclination.  In addition the accretion steam may block hotspot emission at some phases,  introducing a dip \\citep{Ibragimov09}. It is therefore important to use constraints from different techniques where possible to cross-check.\n\n\n\\subsubsection{Space-time of spinning neutron stars}\n\nAs mentioned in Section \\ref{factorswaveform}, rapid rotation is desirable for waveform modelling since it can break degeneracies\\footnote{The known accretion-powered pulsars and rapidly-rotating burst oscillation sources, the prime targets for this technique, have spin rates in the range 180-620 Hz \\citep{Patruno12,Watts12}.  More rapid rotation rates are seen in the radio pulsar population, and rotation rates exceeding 1000 Hz are theoretically possible (see Section \\ref{spinm})}.  Extensive work on gravitational lensing in spinning neutron star spacetimes has fully quantified the various levels of approximation  and their effects on the generation of pulse profiles.  The qualitative character of the general-relativistic effects depends on the ratio\n\n \\begin{equation}\n\\frac{f_{\\rm s}}{f_0}=0.24\n\\left(\\frac{f_{\\rm s}}{600~{\\rm Hz}}\\right)\n\\left(\\frac{M}{1.8 ~\\mathrm{M}_\\odot}\\right)^{-1\/2}\n\\left(\\frac{R}{10~{\\rm km}}\\right)^{3\/2}\\;,\n\\end{equation}\nof the spin frequency of the neutron star, $f_{\\rm s}$, and the characteristic frequency $f_0=\\sqrt{GM\/R^3}\/(2\\pi)$. To zeroth order in $f_{\\rm s}\/f_0$, the\nneutron star is spherically symmetric and its external spacetime is described by the Schwarzschild metric, which depends only on the mass\nof the star. Waveform calculations in this limit were performed in the pioneering work of \\citet{Pechenick83}. A simple approximation for\nthe light-bending integral, introduced by \\citet{Beloborodov02}, is also useful in many cases.\n\nTo first order in $f_{\\rm s}\/f_0$, the external spacetime of a slowly rotating neutron star and the exterior of the Kerr metric are the same \\citep{Hartle67}.  In this case, the amount of gravitational lensing depends also on the spin angular momentum of the star (because of the effects of\nframe dragging), which in turn depends on the density profile inside the star and hence on the equation of state. The effects of frame\ndragging on gravitational lensing are negligible \\citep{Braje00} and, therefore, the external spacetime (for the purpose of calculating\nwaveforms) is still well-described by the Schwarzschild metric. However, at the same order, special relativistic effects (Doppler boosts and aberration) as well as time delays become significant and can be calculated in the Schwarzschild + Doppler approximation \\citep{Miller98,Poutanen03,Poutanen06}.  This Schwarzschild + Doppler approach has been shown to be an excellent approximation for spin frequencies less than about 300 Hz \\citep{Cadeau07} through comparisons with waveforms calculated in exact, numerically generated neutron star spacetimes.\n\nTo second order in $f_{\\rm s}\/f_0$, the neutron star becomes oblate and its external spacetime is described by the Hartle--Thorne metric \\citep{Hartle68},\nwhich depends on the mass of the neutron star, on its spin angular momentum, and on its quadrupole mass moment. For spin frequencies\n$\\lesssim 600$~Hz, the effect of the stellar oblateness on the waveforms can be as large as 10-30\\% \\citep{Morsink07,Psaltis14a,Miller15} whereas the effect of the spacetime quadrupole is of the order of $1-5$\\% \\citep{Psaltis14a}. The\neffects of the stellar oblateness alone can be incorporated into an approximation that makes use of the light-bending formula arising from\nthe Schwarzschild metric, and a shape function that only depends on $M\/R$ and the spin frequency \\citep{Morsink07,Baubock13,AlGendy14}.\n\nFinally, to even higher orders in $f_{\\rm s}\/f_0$, the external spacetime of the neutron star depends on multipole moments that are of\nincreasing order. In principle the external spacetime of a rapidly spinning neutron star can be accurately described by the analytic\nsolution of \\citet{Manko00a,Manko00b}, see \\citep{Berti04,Berti05}.  However, the form of this metric is impractical for use in ray-tracing applications. On the other hand, spacetimes of rapidly spinning neutron stars can be calculated numerically \\citep[see for example][]{Cook94a,Stergioulas95,Bonazzola98}.  Simulations of waveforms for such rapidly rotating neutron stars with particular choices of the equation of state have been performed by \\citet{Cadeau07}.\n\n\\subsubsection{Inversion: from waveform to $M$ and $R$}\n\n The key question, when considering how to apply the waveform modelling technique in practice, is how many photons must be accumulated for a given source.  Some insight is given by the study of \\citet{Psaltis14b}, which generated simulated waveforms under various assumptions (including that of a small hotspot - angular radius less than 20$^\\circ$ - and isotropic emission from the hotspot), and studied the dependence of some key waveform properties on $M$, $R$ and other relevant parameters.  By using information about the shape and energy-dependence of the waveform, the different dependence on $M$ and $R$ of the various observables can in principle be used to break the degeneracies with the geometric parameters to recover $M$ and $R$.   This study also resulted in an order of magnitude estimate for the number of photons from the hotspot that would need to be accumulated in order to reach precisions of a few \\% in $R$, $\\sim 10^6$ counts.  \n\nThe inversion problem has also been examined by \\citet{Lo13} and \\citet{Miller15}.  These studies employed a Bayesian approach and Markov Chain Monte Carlo sampling methods in an extensive parameter estimation study that estimated $M$ and $R$ by fitting waveform models to synthetic waveform data, and determined confidence regions in the $M$-$R$ plane. Some example results are shown in Figure \\ref{contours}.  The uncertainties in $M$ and $R$ estimates are most sensitive to the stellar rotation rate (rapid rotation leading to smaller uncertainties), the spot inclination, and the observer inclination. They also depend on the background (be that from the accretion disk, astrophysical background, or instrumental background), but much more weakly \\citep[see also][]{Psaltis14b}. This technique does not require knowledge of the distance because one fits the fractional amplitude of the pulsations, not the absolute value. \\citet{Lo13} showed that for fixed values of the other system properties that affect the waveform, uncertainties in $M$ and $R$ estimates scale as ${\\cal R}^{-1}$, where ${\\cal R} \\equiv N_{\\rm osc}\/\\sqrt{N_{\\rm tot}} = 1.4f_{\\rm rms}\\sqrt{N_{\\rm tot}}$. Here $N_{\\rm osc}$ is the total number of counts in the oscillating component of the waveform, $N_{\\rm tot}$ is the total number of counts collected, and $f_{\\rm rms}$ is the fractional rms amplitude of the oscillation. If the stellar rotation rate is $\\gtrsim\\,$300~Hz and the spot center and the observer's sightline are both within 30$^\\circ$ of the star's rotational equator, burst oscillation waveform data with ${\\cal R} \\gtrsim\\,$400 allow $M$ and $R$ to be determined with uncertainties $\\lesssim\\,$10\\%.\n\n\\begin{figure*}\n\\centering\n\\includegraphics{LOFT3panel.pdf}\n\\caption{Constraints on $M$ and $R$ obtained by fitting a waveform model to synthetic observed waveform data \\citep[from][]{Miller15}. The black square indicates the mass and radius used to construct the synthetic data, the solid red lines show the 1$\\sigma$ constraints, and the blue dashed lines show the 2$\\sigma$ constraints. Here the spin rate $f_s$=600 Hz, hotspot inclination $\\alpha$ and observer inclination $i$ are 90$^\\circ$, the fractional rms amplitude of the oscillations is 10.8\\%, and the total number of counts is $10^7$. $M$ and $R$ are tightly constrained: $\\Delta M\/M$=2.8 \\% and $\\Delta R\/R$=2.9\\%. The parameter values used to generate the waveform data used in Panel (b) are the same as in Panel (a), except the rotation rate, which is much lower (300 Hz), causing the constraints on $M$ and $R$ to be weaker: $\\Delta M\/M$=5.6\\% and $\\Delta R\/R$=7.6 \\%. Panel (c) shows that when the spot is at an intermediate colatitude (here 60$^\\circ$), the constraints on $M$ and $R$ are weaker ($\\Delta M\/M$=6.5\\% and $\\Delta R\/R$=6.7 \\%), even if the star has a large radius (here 15 km) and is rapidly rotating (here, at 600 Hz).}\n\\label{contours}\n\\end{figure*}\n\nA key result from the \\citet{Lo13} and \\citet{Miller15} studies is their conclusions on the effects of systematic errors. They considered the consequences of differences in the actual spot shape, beaming pattern and energy spectrum from what was assumed in the model.  None of these cases yielded simultaneously (1) a statistically good fit, (2) apparently tight constraints (at the desired few percent level) on $M$ and $R$, and (3) significantly biased masses and radii.  Thus if an analysis yields a good fit with tight constraints, the inferred mass and radius are reliable.  This statement is currently unique among proposed methods to measure neutron star radii.\n\n\\subsubsection{Instrument requirements and observing strategy}\n\nIt is the necessity of obtaining ${\\cal R} \\approx 400$ for systems with favorable geometry that drives the requirement for large-area instruments to properly exploit this technique.  The observing times necessary to achieve this have been studied in detail as part of the LOFT assessment process (see LOFT ESA M3 Yellow Book, \\texttt{http:\/\/sci.esa.int\/loft\/53447-loft-yellow-book}), with observing strategy focusing on the burst oscillation sources given their well understood spectrum \\citep{Suleimanov12} and very low ($\\sim$ 1\\%) atmospheric model uncertainties \\citep{Miller13b}.  Using burst and burst oscillation properties observed with RXTE \\citep[burst brightness, burst oscillation fractional amplitude, as summarized in][]{Galloway08} for the known burst oscillation sources (such as 4U 1636-536), it was shown that with a $\\sim 10$ m$^2$ instrument one would need to combine data from $\\sim 10-30$ bursts with oscillations to meet the required target\\footnote{For a more detailed discussion of how one would combine data from different bursts, and how one can account for potential changes in the size and position of the hotspot, see \\citet{Lo13}.}.  Given the percentage of bursts that show oscillations (which is not 100\\%), and the mean burst rate \\citep[again using properties observed with RXTE,][]{Galloway08} this would require observing sources for a few hundred ks, easily feasible within anticipated mission lifetimes.  Burst oscillations do seem to occur preferentially in certain accretion states \\citep{Muno04}, and targeting these states with a suitable all-sky monitor would reduce the necessary observing time.  There are at present 27 known sources with burst oscillations and\/or accretion-powered pulsations spinning at 100 Hz or faster.  Given that some are transient sources with long periods of quiescence, new discoveries are to be expected.   Having such a large number of sources to choose from will help to select a sample with optimal observational characteristics (such as flux, pulse amplitude and harmonic content), the latter being easily achievable. \n\nAs indicated above, for favorable geometry that will be reflected in higher harmonic content of the pulsations, ${\\cal R} \\approx 400$ is sufficient to measure $M$ and $R$ to a few percent precision (Figure \\ref{ellipses}).  Less favorable geometries would require more observing time, since errors on $M$ and $R$ scale roughly as the inverse square root of the total number of counts  \\citep{Lo13,Psaltis14b}. Since a mix of geometries among sources is a reasonable expectation, observing strategy must be flexible enough to ensure that the goals can be met no matter what system geometries we encounter. Flexibility can also allow responsiveness to preliminary findings:  ideally one would schedule longer observations of the sources that are most constraining in the $M$-$R$ plane, in order to further reduce the size of their error ellipses. By thus tailoring observations one can confirm key findings at a much higher confidence. Figure \\ref{ellipses} illustrates the type of constraints on the $M$-$R$ relation that could be delivered by such a strategy.  \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{m4ellipses.png}\n\\caption{$1\\sigma$ confidence regions illustrating the constraints on the EOS that would be expected from a $\\sim 10$ m$^2$ hard X-ray timing telescope. For the sake of illustration, the regions are assumed to trace the $M$--$R$ curve of a proposed EOS that produces a neutron star with a strange-matter core. The larger regions assume observing time has been dedicated to reach ${\\cal R}\\approx 400$ for the 50\\% of known burst oscillation systems (here 10) that have optimal spot and observer inclinations ($i=60^\\circ$--$90^\\circ$) if their orientations are randomly distributed. The size and shape of these regions are from the Bayesian analyses of waveform data by \\citet{Lo13,Miller15} that assume the hot spots and observers are in the equatorial plane and that there is no independent knowledge of any of the model waveform parameters. The smaller $1\\sigma$ confidence regions show the constraints that could be achieved by a deep follow-up to halve the fractional uncertainties in $M$ and $R$ for the 3--4 stars that are most constraining, in terms of their locations in the $M$--$R$ plane, the potential for reducing their uncertainties, and the availability of complementary constraints from other observations. Constraints like those shown will exclude many of the EOS models shown in this figure. In this illustration, the models shown as grey lines are excluded. }\n\\label{ellipses}\n\\end{figure}\n\nIndependent knowledge of any of the relevant parameters improves the uncertainties, with the biggest improvement coming from knowledge of the observer inclination.  There are very good prospects in the next $\\sim 10$ years for determining the angle of our line of sight to the axis of the binary orbit using Fe line modeling \\citep{Cackett10,Egron11}, Doppler shifting of burst oscillation frequencies \\citep{Strohmayer02,Casares06}, and burst echo mapping \\citep[using the time delay between X-ray burst emission from the neutron star and the optical echo of that emission as it is reprocessed by the surface of the companion star,][note that this requires simultaneous optical observations]{Casares10}.  Since mass transfer is expected to cause the stellar rotation axis to align with the orbital axis in our target systems \\citep{Hills83,Bhattacharya91,Guillemot14}, this will yield the observer inclination. \n\nUsing accreting sources has the advantage that it enables independent cross-checks. Several potential targets show both accretion-powered pulsations and burst oscillations, allowing checks using two independent waveform models.   One can also use the continuum spectral modelling constraints outlined in Section \\ref{currentobs}, and there are potential new continuum fitting techniques that would be enabled by high quality spectra enabled from a large area detector \\citep{Lo13}.  Observation of an identifiable surface atomic line in the hot-spot emission would also provide a tight contraint \\citep[see for example][]{Rauch08}.   The rotational broadening of an atomic line depends on $R$, so combining this with its centroid (which depends on $M\/R$) yields a measurement of $M$ and $R$ independently, modulo the unknown inclination \\citep{Ozel03}.  To avoid the degeneracy with the inclination, one can also use the equivalent width of the line, which depends on the effective gravitational acceleration $g_\\mathrm{eff} = GM (1-2GM\/Rc^2)^{-1\/2}\/R^2$ \\citep{Chang05}.  Combining these two pieces of information yields separate measurements of $M$ and $R$, that are complementary to and independent of the constraints obtained from waveform fitting.   Note that whilst the effects of frame dragging can be neglected \\citep{Bhattacharyya05b}, the effect of stellar oblateness and the space-time quadrupole on the line profile must be taken into account for rotation rates $\\gtrsim 300$ Hz \\citep{Baubock13}.  \\citet{Lo13} explore how the constraints that would result from the detection of an atomic line could be combined with the results of waveform fitting.  \n\n\\subsection{Spin measurements}\n\\label{spinm}\n\nThe spin distribution of neutron stars offers another way of constraining the EOS, and a large-area instrument, with a correspondingly high sensitivity to pulsations, offers a unique opportunity to fully characterise this function.  \n\n\\subsubsection{Rapid rotation}\n\nAt the very simplest level, one can obtain constraints from the most rapidly rotating neutron stars.  The limiting spin rate $f_\\mathrm{max}$, at which the equatorial surface velocity is comparable to the local orbital velocity and mass-shedding occurs, is a function of $M$ and $R$ and hence fast spins constrain the EOS.  The mass-shedding frequency is given to good approximation \\citep{Haensel09}  by the empirical formula \n\n\\begin{equation}\nf_\\mathrm{max} \\approx {\\cal C} \\left[\\frac{M}{M_\\odot}\\right]^\\frac{1}{2} \\left[\\frac{R}{10 ~\\mathrm{km}} \\right]^{-\\frac{3}{2}} ~\\mathrm{kHz}\n\\end{equation}\nwhere $R$ is the radius of the non-rotating star of mass $M$.  Softer EOS have smaller $R$ for a given $M$ and hence have higher limiting spin rates.  More rapidly spinning neutron stars place increasingly stringent constraints on the EOS.  The deviation of ${\\cal C}$ from its Newtonian value of 1.838 depends, in GR, \\citep[as computed by][]{Haensel09} on the neutron star interior mass distribution.  For a hadronic EOS (one that consists of baryons or mesons), ${\\cal C}$ = 1.08, whilst for a strange star with a crust, ${\\cal C} \\approx$ 1.15.   This can be recast as a limit on $R$\n\n\\begin{equation}\nR<10 {\\cal C}^\\frac{2}{3} \\left[\\frac{M}{M_\\odot}\\right]^\\frac{1}{3} \\left[\\frac{f_s}{1 ~\\mathrm{kHz}}\\right]^{-\\frac{2}{3}} ~\\mathrm{km} \n\\end{equation}\nwhich places a constraint on the EOS (Figure \\ref{spinlim}). \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{spind.png}\n\\caption{Spin limits on the EOS, reproduced from \\citet{Watts14}.  Neutron stars of a given spin rate must lie to the left of the relevant limiting line in the $M$-$R$ plane (shown in blue for various spins). The current record holder, which spins at 716 Hz \\citep{Hessels06} is not constraining.  However given a high enough spin, individual EOS can be ruled out.  Between 1 kHz and 1.25 kHz, for example, some individual EOS in the grey band \\citep[which represents a parameterized family of EOS models from][]{Hebeler13} would be excluded.}\n\\label{spinlim}\n\\end{figure}\n\nThe most rapidly spinning neutron star known, a 716 Hz radio pulsar \\citep{Hessels06}, is not spinning rapidly enough to be constraining.  However the discovery of a neutron star with sub-millisecond spin would pose a strong and clean constraint on the EOS.  What then are the prospects therefore for finding more rapidly spinning stars?  Since the standard formation route for the millisecond radio pulsars (MSPs) is via spin-up due to accretion \\citep{Alpar82,Radhakrishnan82,Bhattacharya91}, it is clear that we should look in the X-ray as well as the radio, and theory has long suggested that accretion could spin stars up close to the break-up limit \\citep{Cook94b}.  \n\nFigure \\ref{spindist} shows the current spin distributions for the MSPs and their proposed progenitors, the rapidly spinning accreting neutron stars. For the far more numerous radio pulsars, there is a clear fall off in the distribution at high spin.  Until a few years ago, the sometimes prohibitive computational costs of large surveys sensitive to very rapidly rotating objects limited radio pulsar searches, and this is likely still reflected in the measured distributions \\citep[for a more in-depth discussion, see][]{Watts14}.  The improvement in computational capabilities in recent years, and targeted searches for sources from the Fermi catalogues have led to the discovery of several new binary millisecond radio pulsars with spin frequencies above 500 Hz.  Interestingly the most rapidly rotating stars, including the 716 Hz spin record holder, appear to be in eclipsing systems, where matter from the companion obscures the radio pulsations for large fractions of the orbit.  This suggests that there may be an observational bias against finding these systems in the radio \\citep[see][for further discussion]{Watts14}.  \n\nFor the accreting systems, it is clear that we are still in the regime of small number statistics: however the drop-off at high spin rates seen in the radio is not apparent.  There are moreover physical reasons why it may have been difficult to find the most rapidly-spinning accreting sources.  Rapid spin is most likely in sources that accrete at high rates, for example, and when accretion rate is high episodes of channeled accretion are expected to be intermittent \\citep{Romanova08}.   Strong accretion may also drive a star towards alignment, making pulsations weak \\citep{Ruderman91,Lamb09b}.   For bursters, the most rapid spins may suppress flame spread \\citep{Spitkovsky02,Cavecchi13}, weakening and shortening bursts.  This means that the existence of rapidly spinning stars, that have been out of reach of current detectors, is certainly plausible both on observational and theoretical grounds.  \n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{mergedcorr.pdf}\n\\caption{Spin distributions of neutron stars with rotation rates above 100 Hz.  Top panel:  radio pulsars.  Lower panel:  accreting neutron stars (accretion powered millisecond pulsars and burst oscillation sources).}\n\\label{spindist}\n\\end{figure}\n\nAccretion-powered pulsations and burst oscillations are strongest in the hard X-ray (2-30 keV) band, and sensitivity scales directly with signal to noise.  As such, a large area hard X-ray timing instrument is necessary to discover more neutron star spins.  Intermittent accretion-powered pulsations have already been detected from a number of rapidly spinning sources \\citep{Galloway07,Casella08,Altamirano08}:  to detect more of these events, sensitivity to brief pulsation trains is key.  A 10 m$^2$ instrument with response similar to that of RXTE would be able to detect 100-s duration pulse trains down to an amplitude of $\\sim$ 0.4\\% rms for a 100 mCrab source (5$\\sigma$), $\\sim 100$ s being the duration of intermittent pulsations observed from Aql X-1 by \\citet{Casella08}.   RXTE would have needed 15 times as long to reach the same sensitivity, so that 100 s pulse trains were severely diluted; longer pulse trains suffered from Doppler smearing.   Searches for weak (rather than intermittent) accretion-powered pulsations using the sophisticated techniques being used for the Fermi pulsar surveys \\citep{Atwood06,Abdo09,Messenger11,Pletsch12}, which compensate for orbital Doppler smearing, would yield 5 $\\sigma$ multi-trial pulsation sensitivities of $\\sim 0.03-0.003$ \\% (rms) in bright neutron stars ($>$ 100 mCrab) and $\\sim 0.2-0.04$ \\% in faint neutron stars (10-100 mCrab), taking into account expected observing times and prior knowledge of the orbits.   A 10 m$^2$ instrument would also be able to detect oscillations in individual Type I X-ray bursts down to amplitudes of 0.6\\% rms (for a 1s exposure and a typical burst of brightness of 4 Crab); by stacking bursts sensitivity would improve.  \n\n\\subsubsection{Spin distribution and evolution}\n\nMapping the spin distribution more fully, so that the accreting neutron star sample is no longer limited by small number statistics, is also extremely valuable.  One of the big open questions in stellar evolution is how precisely the recycling scenario progresses - and whether it does indeed account for the formation of the entire MSP population.  The discovery of the first accreting millisecond X-ray pulsar by \\citet{Wijnands98}, and the recent detection of transitional pulsars, that switch from radio pulsars to accreting X-ray sources \\citep{Archibald09,Papitto13,Bassa14,Patruno14,Stappers14,Bogdanov14,Bogdanov15}, seems to confirm the basic picture.  However key details of the evolutionary process, in particular the specifics of mass transfer and magnetic field decay, remain to be resolved \\citep[see for example the discussion in][]{Tauris12}. Comparison of the spin distributions of the MSPs and the accreting neutron stars is a vital part of that effort.  \n\nThe torques that operate on rapidly spinning accreting neutron stars also remain an important topic of investigation.  Accretion torques, mediated by the interaction between the star's magnetic field and the accretion flow \\citep[first explored in detail by][]{Ghosh78,Ghosh79a,Ghosh79b}, clearly play a very large role \\cite[see][for a review of more recent work]{Patruno12}.  There are also several mechanisms, such as core r-modes \\citep[a global oscillation of the fluid, restored by the Coriolis force, see][for a recent review]{Haskell15} and crustal mountains \\citep[see][for a recent review]{Chamel13}, that may generate gravitational waves and hence a spin-down torque.  These mechanisms are expected to depend in part on the EOS \\citep[see, for example,][]{Ho11,Moustakidis15}.  In addition there are potential interactions between internal magnetic fields and an unstable r-mode that may be important \\citep[see for example][]{Mendell01}, and the physics of the weak interaction at high densities also becomes relevant, since weak interactions control the viscous processes that are an integral part of the gravitational wave torque mechanisms \\citep{Alford12}.  \n\nTorque mechanisms can be probed in two ways:  firstly, by examining the maximum spin reached, which may be below theoretical break-up rates, since both magnetic torques and gravitational wave torques may act to halt spin-up \\citep{Bildsten98,Lamb05,Andersson05}; and secondly by high precision tracking of spin evolution, enabled by increased sensitivity to pulsations.  Whilst extracting EOS information from the spin distribution and spin evolution will clearly be more challenging than the clean constraint that would come from the detection of a single rapid spin, it is nonetheless an important part of the models and one that can be tested.  Ultimately, more and better quality timing data are needed to confirm if it is, indeed, the magnetic field that regulates the spin of the fastest observed accreting neutron stars or if additional torques are needed. On the one hand, it has been argued that the spin-evolution during and following an accretion outburst of IGR~J00291+5934 is consistent with the `standard' magnetic accretion model \\cite{Falanga05,Patruno10,Hartman11}. On the other hand, the results are not quite consistent and there is still room for refinements and\/or additional torques \\cite{Andersson14,Ho14}. Whether this means that there is scope for a gravitational-wave element or not remains unclear \\cite{Ho11,Haskell12}, but a large area X-ray instrument should take us much closer to the answer.\n\nPrecision ephemerides from X-ray timing are very important enablers for simultaneous gravitational wave searches, since one has to fold long periods of data to detect the weak signals, and the gravitational wave frequency depends on spin rate in both mountain and r-mode mechanisms. Without such ephemerides, the number of templates that must be searched makes detection of continuous wave emission from sources like Sco X-1 very difficult \\citep{Watts08b}. This is very clear when one compares the limits currently obtained for continuous wave gravitational wave searches where ephemerides are known \\citep[the radio pulsars,][]{Abbott07,Abbott08} to those obtained for systems where the spin is not known (non-pulsing systems like Cas A, \\citealt{Abadie10} and Sco X-1, \\citealt{Aasi14}). A direct detection of gravitational waves from such a system would of course have immediate consequences for potential gravitational wave emission mechanisms, and any EOS dependence.  \n\n\n\\subsection{Asteroseismology}\n\nAsteroseismology is now firmly established as a precision technique for the study of the interiors of normal stars.  As such the detection of seismic vibrations in neutron stars was one of RXTE's most exciting discoveries.  They were found in magnetars, young, highly magnetized neutron stars that emit bursts of hard X-ray\/gamma-rays powered by decay of the strong magnetic field \\citep[see][for a review]{Woods06}.  What triggers the flares remains unknown, but most likely involves either starquakes or magnetospheric instabilities.  Rapid reconfiguration\/reconnection powers the electromagnetic burst:  however the events are so powerful that it had already been suggested that they might set the star vibrating \\citep{Duncan98}.   These vibrations, which manifest as Quasi-Periodic Oscillations (QPOs) in hard X-ray emission, were first detected in the several hundred second long tails of the most energetic giant flares from two magnetars \\citep{Israel05,Strohmayer05,Strohmayer06a,Watts06}.  Similar QPOs have since been discovered during storms of short, low fluence bursts from several magnetars \\citep{Huppenkothen13,Huppenkothen14a,Huppenkothen14b}.   The QPOs have frequencies that range from 18 to 1800 Hz.  \n\nSeismic vibrations offer us a unique way to explore the interiors of neutron stars.  The QPOs were initially tentatively identified with torsional shear modes of the neutron star crust, and torsional Alfv\\'en modes of the highly magnetized fluid core. These identifications were based on the expected mode frequencies, which are set by both the size of the resonant volume (determined by the star's radius) and the relevant wave speed. The fact that the oscillations must be computed in a relativistic framework introduces additional dependences, and for this reason they can be used to diagnose $M$ and $R$ (see for example \\citealt{Samuelsson07} for relativistic crust modes, and \\citealt{Sotani08} for relativistic core Alfv\\'en modes). Seismic vibrations also take us beyond the simple $M$-$R$ relation, constraining the non-isotropic components of the stress tensor of supranuclear density material.\n\nIn fact, for a star with a magnetar strength magnetic field, crustal vibrations and core vibrations should couple together on very short timescales \\citep{Levin07}. The current viewpoint is that the QPOs are associated with global magneto-elastic axial (torsional) oscillations of the star \\citep{Glampedakis06,Lee08,Andersson09,Steiner09,vanHoven11,vanHoven12,Colaiuda11,Colaiuda12,Gabler12,Gabler13a, Passamonti13,Passamonti14,Asai14,Glampedakis14}. Since coupled oscillations depend on the same physics, they have frequencies in the same range as the natural frequencies of the isolated elements. \n\nCurrent magneto-torsional oscillation models can in principle easily explain the presence of oscillations at 155 Hz and below. Until recently there was a significant problem with the higher frequency QPOs, which appeared to persist much longer than the models predicted, but this has now been resolved \\citep{Huppenkothen14c}. Issues currently being addressed include questions of emission \\citep{Timokhin08,DAngelo12,Gabler14}, excitation \\citep{Link14}, coupling to polar Alfv\\'en modes \\citep{Lander10,Lander11,Colaiuda12}, and resonances between the crust and core that might develop as a result of superfluid effects \\citep{Gabler13b,Passamonti14}. The latter in particular can have a large effect on the characteristics of the mode spectrum, and since superfluidity is certainly present in neutron stars, mode models must start to take this into account properly before we can make firm mode identifications. What is now clear is that mode frequencies depend not only on $M$ and $R$, but also on magnetic field strength\/geometry, superfluidity, and crust composition.\n\nSeveral papers have specifically explored EOS dependencies in neutron star asteroseismology \\citep{Strohmayer05,Strohmayer06a,Watts07,Samuelsson07,Sotani08,Steiner09,Gabler12}.  Figure \\ref{seis} illustrates the constraints that result when one models the QPOs detected in the SGR 1806--20 hyperflare as torsional shear oscillations of the neutron star crust, \\citep{Samuelsson07}.   This model is simple, in that it does not include magnetic coupling between crust and core.  However it gives some idea of the types of constraints on $M$ and $R$ that can result from the detection of several frequencies in a single event, where having multiple simultaneous frequencies assists mode identification (the burst storm identifications discussed above involve combining data from multiple bursts, so are less useful in this regard).  \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{seis.png}\n\\caption{$M$-$R$ diagram showing the seismological constraints for the soft gamma-ray repeater SGR 1806--20 using the relativistic torsional crust oscillation model of \\citet{Samuelsson07}, in which the 29 Hz QPO is identified as the fundamental and the 625 Hz QPO as the first radial overtone. The neutron star lies in the box where the constraints from the two frequency bands overlap. Once QPOs are detected, frequency measurement errors are negligible for this purpose.  This model is very simple (it does not include) crust-core coupling, but it gives some idea of the type of constraints that might result from the detection of a harmonic sequence of seismic vibrations.  More sophisticated models that take into account coupling and the other relevant physical effects are under development.}\n\\label{seis}\n\\end{figure}\n\nSadly giant flares are rare, occurring only every $\\sim$ 10 years.   Ideally therefore we would like the ability to make similar detections in the more frequent but less bright events.    Intermediate flares, which are detected roughly once per year, have similar peak fluxes and spectra to the tails of the giant flares, but are too brief ($\\sim$ 1 s) to permit detection of similar QPOs with current instrumentation. A $\\sim 10$ m$^2$ hard X-ray timing instrument would be sensitive to QPOs in intermediate flares with similar fractional amplitudes as those observed in the tails of giant flares, provided that the collimator permitted the transmission of higher energy (above 30 keV) photons.  The latter is important since intermediate flares are unpredictable and likely to be observed off-axis, although one can increase the odds of capturing them by scheduling pointed observations during periods of high burst activity \\citep{Israel08}.  Theoretically the expectation of similar fractional amplitudes is justified: mode excitation at substantial amplitude even by events releasing energies typical of intermediate flares is feasible \\citep{Duncan98}.  Empirically, QPOs in giant flares tend to appear rather late in the tails, when luminosities are similar to those in intermediate flares, and given that they appear and disappear multiple time in these tails, may be triggered by magnetic starquakes at these 'low' fluxes \\citep{Strohmayer06a}.  The development of similar fractional amplitude QPOs in intermediate flares is thus considered plausible.  This idea has also been given a boost by the discovery of QPOs in short burst storms from two different magnetars \\citep{Huppenkothen13,Huppenkothen14a,Huppenkothen14b} including one that had also shown QPOs in a giant flare, since individually these bursts are much less energetic than the intermediate flares. The amplitudes at which the oscillations were detected in the burst storms are comparable to those of the detections in the giant flares. Upper limits on the presence of QPOs in the intermediate flares observed by current instruments, however, are above this level.  \n\n\n\\section{Summary}\n\nNeutron stars are unique testing grounds for fundamental nuclear physics, the only place where one can study the  equation of state of cold matter in equilibrium, at up to ten times normal nuclear densities.  The stable gravitational confinement permits the formation of matter which is extremely neutron rich, and which may involve matter with strange quarks.  The relativistic stellar structure equations show that there is a one to one mapping between the bulk properties of neutron stars, in particular their mass and radius, and the dense matter EOS.   Efforts to measure neutron star properties for this purpose are being made by both electromagnetic and gravitational wave astronomers.  In this Colloquium, we explored the techniques available using hard X-ray timing instruments:  waveform fitting, spin measurements, and asteroseismology. Hard X-ray timing offers unique advantages in terms of the numbers of known sources, and the potential for cross-checks using independent techniques and source classes.   \n\nThe previous generation of hard X-ray timing telescopes, in particular RXTE (a 0.6 m$^2$ telescope which operated from 1995 to 2012), uncovered many of the phenomena described in this Colloquium.  To exploit them to measure the EOS, however, requires larger area instruments, and various mission concepts are now being proposed.  These have included the 3 m$^2$ Advanced X-ray Timing Array \\citep[AXTAR:][]{Ray10}, and the 8.5-10 m$^2$ Large Observatory for X-ray Timing, LOFT \\citep[][see also the LOFT ESA M3 Yellow Book \\texttt{http:\/\/sci.esa.int\/loft\/53447-loft-yellow-book}]{Feroci12,Feroci14}.   None have yet been successful in securing a launch slot.   However the advantages that such a telescope would offer in terms of measuring the dense matter equation of state are sufficiently highly compelling that mission concept development continues apace.  \n\n\\begin{acknowledgments}\nThe authors would like to thank all of the members of the LOFT Consortium, in particular the members of the LOFT Dense Matter Working Group, for useful discussions.  ALW acknowledges support from NWO Vidi Grant 639.042.916, NWO Vrije Competitie Grant 614.001.201, and ERC Starting Grant 639217 CSINEUTRONSTAR.  The work of KH and AS is supported by ERC Grant No. 307986 STRONGINT and the DFG through Grant SFB 634.  MF, GI, and LS acknowledge support from the Italian Space Agency (ASI) under contract I\/021\/12\/0.  LT acknowledges support from the Ramon y Cajal Research Programme and from Contracts No. FPA2010-16963 and No. FPA2013-43425-P of Ministerio de Economia y Competitividad, from FP7-PEOPLE-2011-CIG under Contract No. PCIG09-GA-2011-291679 as well as NewCompStar (COST Action MP1304).  SMM acknowledges support from NSERC. JP acknowledges the Academy of Finland grant 268740. AP acknowledges support from NWO Vidi Grant 639.042.319.  AWS was supported by the U.S. Department of Energy Office of Nuclear Physics.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nThe field of gravitational lensing has seen exponential growth\nin its physical \\cite{SEF,KSW} and mathematical \\cite{PLW,PWreview,Petters} infrastructure,\nyielding diverse applications in astronomy and cosmology.\nIn this paper we address gravitational lensing in the setting of one of the\nmost important non-spherically symmetric and non-static solutions\nof the Einstein equations, namely, Kerr black holes.  This has already been the focus of many studies.  Indeed, several authors have explored the Kerr's caustic structure, as well as Kerr black hole lensing in the\nstrong deflection limit, focusing on leading order effects in light\npassing close to the region of photon capture\n(e.g., Rauch \\& Blandford \\cite{RB}, Bozza \\cite{Bozza,Bozza2,Bozza3}, V\\'asquez \\& Esteban \\cite{VqzE}, \nBozza, De Luca, Scarpetta, and Sereno \\cite{Betal}, Bozza, De Luca, and Scarpetta \\cite{Betal2}, and Bozza \\& Scarpetta \\cite{BS2}).\n\nStudies of Kerr lensing have also been undertaken in the weak deflection limit.  In particular, Sereno \\& De Luca \\cite{SerenoDeLuca,SerenoDeLuca2} gave an analytic treatment of caustics and two lensing observables for Kerr lensing in the weak deflection limit, while Iyer \\& Hansen \\cite{Iyer1,Iyer2} found an asymptotic expression for the equatorial bending angle.  Werner \\& Petters \\cite{WernerPetters} used magnification relations for weak-deflection Kerr lensing to address the issue of Cosmic Censorship (for lensing and Cosmic Censorship in the spherically symmetric case, see Virbhadra \\& Ellis \\cite{VE2}).  \n\nIn Papers I and II of our series, we are developing a comprehensive analytic framework for Kerr black\nhole lensing, with a focus on regimes\nbeyond the weak deflection limit (but not restricted to the strong\ndeflection limit).  In three earlier papers \\cite{KP1,KP2,KP3}, Keeton \\& Petters\nstudied lensing by static, spherically symmetric compact objects in\ngeneral geometric theories of gravity.  In \\cite{KP1,KP2}, the authors found\nuniversal relations among lensing observables for Post-Post-Newtonian (PPN) models that\nallowed them to probe the PPN spacetime geometry beyond the\nquasi-Newtonian limit.  In \\cite{KP3} they considered braneworld gravity,\nwhich is an example of a model outside the standard PPN family as it\nhas an additional independent parameter arising from an extra\ndimension of space.  They developed a wave optics theory (attolensing)\nto test braneworld gravity through its signature in interference\npatterns that are accessible with the Fermi Gamma-ray Space\nTelescope.  Papers I and II present a similar analysis of Kerr black hole lensing beyond the weak deflection limit.\n\nThe outline of this paper is as follows.  In Section~\\ref{sec:gen-lenseq} we present\na new, general lens equation and magnification formula governing lensing by a thin deflector.\nBoth equations are applicable for non-equatorial observers and \nassume that the source and observer are in the asymptotically\nflat region.\nIn addition, our lens equation incorporates \nthe displacement for a general setting that Bozza \\& Sereno \\cite{BS} \nintroduced for the case of a spherically symmetric deflector.\nThis occurs when the light ray's tangent lines\nat the source and observer do not intersect on the lens plane.\nSection~\\ref{sec:lenseq-Kerr} gives an explicit expression for this\ndisplacement when the observer is in the equatorial plane\nof a Kerr black hole as well as\nfor the case of spherical symmetry.   The lens equation itself is derived in Appendix~\\ref{app:Kerr-null-geodesics}.\n\nIn Paper II we solve our lens equation perturbatively to obtain analytic\nexpressions for five lensing observables (image positions, magnifications,\ntotal unsigned magnification, centroid, and time delay) for the regime of\nquasi-equatorial lensing.\n\n\n\n\n\\section{General Lens Equation with Displacement}\n\\label{sec:gen-lenseq}\n\n\n\\subsection{Angular Coordinates on the Observer's Sky}\n\\label{sec:obs-coords}\n\nLet us define Cartesian coordinates $(x,y,z)$ centered on the\ncompact object and oriented such that the observer lies on the\npositive $x$-axis.  We assume that gravitational lensing will take place outside the photon region, which is outside the ergosphere, so that $(x,y,z)$ are always spatial coordinates.  We also point out that we will not be considering distances more than a Hubble time, so that our formalism ignores the expansion of the universe.\n\nAssume that the observer in the asymptotically flat region is at\nrest relative to the $(x,y,z)$ coordinates.  {\\em All equations\nderived in this section are relative to the asymptotically flat\ngeometry of such an observer.}\nThe natural coordinates for the observer to use in gravitational\nlensing are angles on the sky.  To describe these angles, we\nintroduce ``spherical polar\" coordinates\ndefined with respect to the observer and the optical axis (from the observer to\nthe lens), and the $yz$-planes at the deflector and the light source.  The vector to the image position is then described by the\nangle $\\vth$ it makes with the optical axis, and an azimuthal\nangle $\\vphi$.  Similarly, the vector to the source position is\ndescribed by the angle $\\cb$ it makes with the optical axis and\nby an azimuthal angle $\\chi$.  These angles are shown in\n\\reffig{ObsCoords}.  Note that the optical axis is the $x$-axis.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.64]{ObsCoords.eps}\n\\end{center}\n\\caption{\nAngles on the observer's sky.\nAn image's position is determined by $(\\vth, \\vphi)$.  The source's position is given by $(\\cb,\\chi)$.}\n\\label{fig:ObsCoords}\n\\end{figure}\nWe adopt the usual convention for spherical polar coordinates:\nthe image position has $\\vth > 0$ and $0 \\le \\vphi < 2\\pi$,\nwhile the source position has $\\cb \\ge 0$ and $0 \\le \\chi < 2\\pi$.\nIn fact, since we only need to consider the ``forward'' hemisphere from the observer\nwe can limit $\\vth$ to the interval $(0,\\pi\/2)$ and $\\cb$ to the interval $[0,\\pi\/2)$.\n\nThe\n``lens plane'' is the plane perpendicular to the optical axis\ncontaining the lens, and the ``source plane'' is the plane\nperpendicular to the optical axis containing the source; these are also shown in \\reffig{ObsCoords}.  Define\nthe distances $d_L$ and $d_S$ to be the perpendicular distances\nfrom the observer to the lens plane and source plane, respectively,\nwhile $d_{LS}$ is the perpendicular distance from the lens plane\nto the source plane.  Some investigators define $d_S$ to be the\ndistance from the observer to the source itself, as opposed to\nthe shortest distance to the source plane.  We shall comment on\nthis distinction in Section~\\ref{sec:spherical}.\n\n\n\n\n\\subsection{General Lens Equation via Asymptotically Flat Geometry}\n\\label{sec:gen-lens-eqn}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.67]{LensGeom.eps}\n\\end{center}\n\\caption{\nA lensing scenario demonstrating that the tangent line to the segment of the ray arriving at the observer and the tangent line of the ray at the source need not intersect on the lens plane; i.e., $A' \\neq B'$ in general.  The angles $\\cb$ and $\\vartheta$ are as in \\reffig{ObsCoords} (or rather, they are their projections onto the $xz$-plane), $\\hat{\\alpha}$ is the ``bending angle,\" and $d_L$, $d_{S}$, and $d_{LS}$ are the perpendicular distances between the lens plane and observer, the source plane and observer, and the lens and source planes, respectively.}\n\\label{fig:LensGeom}\n\\end{figure}\n\nConsider the lensing geometry shown in \\reffig{LensGeom}.  With respect to the light ray being lensed, there are two tangent lines of interest: the tangent line to the segment of the ray arriving at the observer and the tangent line to the ray emanating from the source.  \nAs first emphasized in \\cite{BS}, {\\it it is important to realize that\nthese two tangent lines need not intersect.}  If they do intersect (as\nfor a spherical lens, since in that case the tangent lines are\ncoplanar), the intersection point need not lie in the lens plane.\nThis effect has often been neglected, and while it may be small in the\nweak deflection limit (see Section~\\ref{sec:spherical} below) we should include it for greater generality.\nA simple way to capture this displacement is to\nconsider the points where the two tangent lines cross the lens\nplane, namely, the points $A'$ and $B'$ in \\reffig{LensGeom}.\nIf the tangent lines do intersect in the lens\nplane, then $A' = B'$.  Otherwise, as can be seen in greater detail in \\reffig{Lens-Geom3}, there is a displacement on the lens plane that\nwe quantify by defining\n\\beq \\label{eq:disp-def}\n\\cd_y = B'_y - A'_y\\,, \\qquad \\cd_z = B'_z - A'_z\\,.\n\\eeq\nNote from \\reffig{Lens-Geom3} that the\ntangent line to the segment of the ray arriving at the observer \nis determined by $(\\vth,\\vphi)$.  The tangent line to the ray\nemanating from the source can likewise be described by the angles\n$(\\vth_S,\\vphi_S)$, where $-\\pi\/2 < \\vth_S < \\pi\/2$ and $0 \\leq \\vphi_S < 2\\pi$.  As shown in \\reffig{Lens-Geom3}, $\\vth_S$ has vertex $B'$ and is measured from the line joining the points $B'$ and $B''$, which runs parallel to the optical axis.  We adopt the following sign convention for $\\vth_S$: if $\\vth_S$ goes {\\it toward} the optical axis, then it will be positive; otherwise it is negative (e.g., the $\\vth_S$ shown in \\reffig{Lens-Geom3} is positive).\nWe will obtain the general lens equation by considering the coordinates\nof the points $A'$ and $B'$ in \\reffig{Lens-Geom3}.  Using the\nasymptotically flat geometry of the observer, we have\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.62]{LensGeom-without-nu.eps}\n\\end{center}\n\\caption{\nA detailed diagram of lensing with displacement.  The tangent line to the segment of the ray arriving at the observer is determined by $(\\vth,\\vphi)$ and intersects the lens plane at $A'$, while the tangent line to the ray emanating from the source is determined by $(\\vth_S,\\vphi_S)$ and intersects the lens plane at $B'$.  The distance between these two points is quantified by the displacement amplitude $\\cd$, whose horizontal and vertical components we denote by $\\cd_y$ and $\\cd_z$, respectively.  The deflector could be a Kerr black hole and the light ray may dip below the $xy$-plane.}\n\\label{fig:Lens-Geom3}\n\\end{figure}\n\\beqa\nA'_x &=& 0\\,, \\nonumber\\\\\nA'_y &=& d_L \\tan\\vth \\, \\cos\\vphi\\,, \\label{eq:Acoordsy}\\\\\nA'_z &=& d_L \\tan\\vth \\, \\sin\\vphi\\,,\n  \\label{eq:Acoords}\\\\\nB'_x &=& 0\\,, \\nonumber\\\\\nB'_y &=& d_S \\tan\\cb \\, \\cos\\chi + d_{LS} \\tan\\vth_S \\, \\cos(\\pi-\\vphi_S)\\,,\n  \\\\\nB'_z &=& d_S \\tan\\cb \\, \\sin\\chi - d_{LS} \\, \\tan \\vth_S \\, \\sin (\\pi-\\vphi_S)\\,.\n  \\label{eq:Bcoords}\n\\eeqa\nSubstituting eqns.~(\\ref{eq:Acoordsy})--(\\ref{eq:Bcoords}) into eqn.~(\\ref{eq:disp-def}) yields\n\\beqa\nd_S \\tan\\cb \\, \\cos\\chi & = & d_L \\tan\\vth \\, \\cos\\vphi\n  \\ + \\ d_{LS} \\tan \\vth_S \\, \\cos \\vphi_S\n  \\ + \\ \\cd_y\\,,\n  \\label{eq:le-h} \\\\\nd_S \\tan\\cb \\, \\sin\\chi & = & d_L  \\tan\\vth \\, \\sin\\vphi\n  \\ + \\ d_{LS} \\tan \\vth_S \\, \\sin \\vphi_S\n  \\ + \\ \\cd_z\\,.\n  \\label{eq:le-v}\n\\eeqa\nThe left-hand sides involve only the source position, while the\nright-hand sides involve only the image position.\nIn other words, {\\em this pair of equations\nis the general form of the gravitational lens equation for\nsource and observer in the asymptotically flat region, for a general isolated compact object.}  Note that apart from the asymptotic flatness assumption, these equations use no properties specific to Kerr black holes; and if the deflector was a Kerr black hole, then neither the observer nor the source has been assumed to be equatorial.\nWe shall refer to eqns.~(\\ref{eq:le-h}) and (\\ref{eq:le-v}), respectively, as the\n``horizonal'' and ``vertical'' components of the lens equation\ndue to the cosine\/sine dependence on $\\chi$.\n\nConsider now the case when the light ray and\nits tangent lines lie in a plane which contains the optical axis.  This forces $\\chi = \\vphi$ or $\\chi = \\vphi+\\pi$ depending on whether\nthe source is on the same or opposite side of the lens as the image.  To distinguish these two cases, it is useful to define\n$\\snq = \\cos(\\chi-\\vphi)$ to be a sign that indicates whether the\nsource is on the same side of the lens as the image ($\\snq=+1$)\nor on the opposite side ($\\snq=-1$).  The condition $A' \\neq B'$ will still hold in general, but the line in the lens plane from the origin to the point $B'$ will now make the same angle with respect to the $y$-axis as the point $A'$, namely, the angle $\\vphi$ (see Fig.~\\ref{fig:Lens-Geom3}).  As a result, the line in the source plane from the origin to the point $B''$ will also make the angle $\\vphi$ with respect to the $y$-axis.  Thus we will have $\\vphi_S = \\vphi+\\pi$.  Given these conditions,\neqns.~(\\ref{eq:le-h}) and (\\ref{eq:le-v}) reduce to the single lens\nequation\n\\beq \\label{eq:le-sph}\n  d_S\\,\\snq\\,\\tan\\cb = d_L \\tan\\vth - d_{LS} \\tan(\\hat{\\alpha}-\\vth) + \\cd \\,,\n\\eeq\nwhere the displacement amplitude is $\\cd = \\cd_y\/\\!\\cos\\vphi = \\cd_z\/\\!\\sin\\vphi$ (in the case of planar rays),\nand to connect with traditional descriptions of gravitational\nlensing we have introduced the light bending angle\n$\\hat{\\alpha} \\equiv \\vth + \\vth_S$.  (If desired, the sign $\\snq$ can be\nincorporated into the tangent so that the left-hand side is\nwritten as $\\tan(\\snq\\cb)$, where we think of $\\snq\\cb$ as the\nsigned source position.)  {\\it Eqn.~(\\ref{eq:le-sph}) is the general form of the lens equation in the case of planar rays.}  If the displacement $\\cd$ is ignored,\nthen eqn.~(\\ref{eq:le-sph}) matches the spherical lens equation given\nby \\cite{virbetal2}.  We consider the displacement term in\n\\refsec{spherical}.\n\n\n\n\\subsection{General Magnification Formula}\n\\label{sec:gen-mag}\n\nThe magnification of a small source is given by the ratio of the\nsolid angle subtended by the image to the solid angle subtended\nby the source\n(e.g.,\n\\cite[p.~82]{PLW}).  As measured by the observer, if $\\ell$ is the\ndistance to the image (as opposed to the perpendicular distance),\nthen the small solid angle subtended by the image is\n\\beqa\n  d\\Omega_I = \\frac{|(\\ell \\, d \\vth)\\, (\\ell \\sin\\vth \\, d\\vphi)|}{\\ell^2}\n=|\\sin\\vth\\ d\\vth\\ d\\vphi| = |d(\\cos\\vth)\\ d\\vphi| .\\nonumber\n\\eeqa\nSimilarly, the small solid angle subtended by the source is\n\\beqa\n  d\\Omega_S = |\\sin\\cb\\ d\\cb\\ d\\chi| = |d(\\cos\\cb)\\ d\\chi| .\\nonumber\n\\eeqa\nWe then have the absolute magnification\n\\beqa\n  |\\mu| = \\frac{d\\Omega_I}{d\\Omega_S} = |\\det J|^{-1} ,\\nonumber\n\\eeqa\nwhere $J$ is the Jacobian matrix\n\\beqa\n  J = \\frac{\\partial (\\cos \\cb, \\chi)}{\\partial(\\cos\\vth, \\vphi)}\n  = \\left[\\matrix{\n    \\frac{\\partial\\cos\\cb}{\\partial\\cos\\vth}\n  & \\frac{\\partial\\cos\\cb}{\\partial\\vphi   } \\cr\n    \\frac{\\partial\\chi   }{\\partial\\cos\\vth}\n  & \\frac{\\partial\\chi   }{\\partial\\vphi   }\n  }\\right].\\nonumber\n\\eeqa\nWriting out the determinant and dropping the absolute value in\norder to obtain the signed magnification, we get\n\\beq \\label{eq:mu-general}\n  \\mu = \\left[ \\frac{\\sin\\cb}{\\sin\\vth} \\left(\n      \\frac{\\partial\\cb}{\\partial\\vth }\\ \\frac{\\partial\\chi}{\\partial\\vphi}\n    - \\frac{\\partial\\cb}{\\partial\\vphi}\\ \\frac{\\partial\\chi}{\\partial\\vth }\n  \\right) \\right]^{-1} .\n\\eeq\nIn the case of spherical symmetry, the image and source lie in\nthe same plane, so $\\partial\\cb\/\\partial\\vphi=0$ and\n$\\partial\\chi\/\\partial\\vphi=1$, recovering  the familiar result\n\\beqa\n  \\mu = \\left( \\frac{\\sin\\cb}{\\sin\\vth}\\ \\frac{\\partial\\cb}{\\partial\\vth}\n    \\right)^{-1} .\\nonumber\n\\eeqa\n\n\n\n\n\n\\section{Lens Equation for Kerr Black Holes}\n\\label{sec:lenseq-Kerr}\n\n\n\\subsection{Kerr Metric}\n\\label{sec:Kerr-metric}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.62]{BHCoords2.eps}\n\\end{center}\n\\caption{\nCartesian $(X,Y,Z)$ and spherical polar $(r,\\kpolar,\\kazym)$\ncoordinates centered on the black hole, where $\\kpolar = \\pi\/2 - \\wp$ with $\\wp$ the polar angle; note that $-\\pi\/2 \\leq \\kpolar \\leq \\pi\/2$.  The black hole spins about the $Z$-axis, which corresponds to\n$\\kpolar=\\pi\/2$, in the direction of\nincreasing $\\kazym$.  The equatorial plane\nof the black hole corresponds to $\\kpolar = 0$ or the\n$(X,Y)$-plane.}\n\\label{fig:BHCoords}\n\\end{figure}\n\nNow let the deflector in Fig.~\\ref{fig:Lens-Geom3} be a Kerr black hole.  The Kerr metric is the unique axisymmetric,\nstationary, asymptotically flat, vacuum solution of the Einstein\nequations describing a stationary black hole with mass $\\bhpm$\nand spin angular momentum $\\bhpJ$ (see, e.g., \\cite[pp.~322-324]{wald}).\nConsider the Kerr metric in Boyer-Lindquist coordinates\n$(t,r,\\wp,\\kazym)$, where $\\wp$ is the polar angle\nand $\\kazym$ the azimuthal angle.  For our purposes, it is\nconvenient to transform $\\wp$ to $\\kpolar = \\pi\/2-\\wp$\nand work with the slightly  modified Boyer-Lindquist coordinates\n$(t,r,\\kpolar,\\kazym)$; note that $-\\pi\/2 \\leq \\kpolar \\leq \\pi\/2$.  The spatial coordinates are shown in\n\\reffig{BHCoords}.\n\nThe metric takes the form\n\\beqa\n\\label{eq:app:kerr-metric}\n  ds^2  = g_{tt}\\, dt^2  + g_{rr}\\,dr^2 \n    + g_{\\kpolar \\kpolar}\\,d\\kpolar^2\n    + g_{\\kazym \\kazym}\\,d\\kazym^2  \n    + 2\\,g_{t \\kazym}\\,dt\\,d\\kazym\\,,\\nonumber\n\\eeqa\nwhere $t = c {\\tt t}$ with ${\\tt t}$ being physical time.  The\nmetric coefficients are\n\\beqa \\label{eq:app:kerr-components}\n  g_{tt} &=& - \\frac{r(r-2\\gravr) + \\bha^2 \\sin^2\\kpolar}\n    {r^2 + \\bha^2 \\sin^2\\kpolar}\\ , \\label{gtt}\\\\\n  g_{rr} &=& \\frac{r^2 + \\bha^2 \\sin^2\\kpolar}{r(r-2\\gravr) + \\bha^2}\\ , \\label{grr}\\\\\n  g_{\\kpolar \\kpolar} &=& r^2 + \\bha^2 \\sin^2\\kpolar\\,, \\label{gss}\\\\\n  g_{\\kazym \\kazym} &=& \\frac{(r^2+\\bha^2)^2\n    - \\bha^2(\\bha^2+r(r-2\\gravr))\\cos^2\\kpolar}\n    {r^2 + \\bha^2 \\sin^2\\kpolar}\\ \\cos^2\\kpolar\\,, \\label{gphiphi}\\\\\n  g_{t \\kazym} &=& - \\frac{2 \\gravr \\bha \\, r \\cos^2\\kpolar}\n    {r^2 + \\bha^2 \\sin^2\\kpolar}\\ .\\label{gtphi}\n\\eeqa\nThe parameter $\\gravr$ is the gravitational radius, and $\\bha$ \nis the angular momentum per unit mass:\n\\beqa\n  \\gravr = \\frac{G \\bhpm}{c^2}\\ , \\qquad\n  \\bha = \\frac{\\bhpJ}{c \\bhpm}\\ .\\nonumber\n\\eeqa\nNote that both $\\gravr$ and $\\bha$ have dimensions of length.\nIt is convenient to define a dimensionless spin parameter:\n\\beqa\n\\label{eq:ahat}\n  \\ahat = \\frac{\\bha}{\\gravr}\\ .\\nonumber\n\\eeqa\nUnless stated to the contrary, the black hole's spin is\nsubcritical; i.e., $\\ahat^2 < 1$.\n\n\n\n\\subsection{Lens Equation for an Equatorial Observer}\n\\label{sec:Kerr-lens-eqn}\n\n{\\em We now specialize to the case when the observer lies\nin the equatorial plane of the Kerr black hole,} so the coordinates\n$(x,y,z)$ in \\reffig{Lens-Geom3} coincide with the coordinates\n$(X,Y,Z)$ in \\reffig{BHCoords}.  Note that we still consider\ngeneral source positions.\n\nIn \\refapp{Kerr-null-geodesics} we carefully analyze null\ngeodesics seen by an observer in the equatorial plane.  By\nconsidering constants of the motion, we derive the following\nlens equation:\n\\beqa\n  d_S \\tan\\cb \\cos\\chi &=& d_{LS} \\tan\\vth_S \\cos\\vphi_S\n    \\ + \\ d_L\\ \\frac{\\sin\\vth \\cos\\vphi}{\\cos\\vth_S}\\ ,\n    \\label{eq:le-h-Kerr} \\\\\n  d_S \\tan\\cb \\sin\\chi &=& d_{LS} \\tan\\vth_S \\sin\\vphi_S\n    \\ + \\ \\frac{d_L \\sin\\vth}{1 - \\sin^2\\vth_S \\sin^2\\vphi_S} \\times\n    \\label{eq:le-v-Kerr} \\\\\n  &&\\qquad \\left[ \\cos\\vphi \\sin\\vth_S \\tan\\vth_S \\sin\\vphi_S \\cos\\vphi_S\n    + \\left( \\sin^2\\vphi - \\sin^2\\vth_S \\sin^2\\vphi_S \\right)^{1\/2} \\right] .\n    \\nonumber\n\\eeqa\n{\\it This is the general form of the lens equation for an equatorial\nobserver in the Kerr metric for observer and source in the\nasymptotically flat region.}  \nIt is valid for all light rays, whether they loop\naround the black hole or not, as long as they lie outside the\nregion of photon capture.  No small-angle approximation is required.\n\n\n\nNote that eqns. (\\ref{eq:le-h}) and (\\ref{eq:le-v}) represent the general\nform of the lens equation, with the displacement terms explicitly\nwritten, while eqns. (\\ref{eq:le-h-Kerr}) and (\\ref{eq:le-v-Kerr}) give\nthe exact lens equation for an equatorial Kerr observer, with\nthe displacement terms implicitly included.  Demanding that these\ntwo pairs of equations be equivalent allows us to identify the\ndisplacement terms for an equatorial Kerr observer:\n\\beqa\n  \\cd_y &=& d_L \\sin\\vth \\cos\\vphi \\left( \\frac{1}{\\cos\\vth_S}\n    - \\frac{1}{\\cos\\vth} \\right) , \\label{eq:dispy}\\\\\n  \\cd_z &=& - d_L \\tan\\vth \\sin\\vphi \\ + \\ \n    \\frac{d_L \\sin\\vth}{1 - \\sin^2\\vth_S \\sin^2\\vphi_S} \\times \\label{eq:dispz}\\nonumber\\\\\n  &&\\qquad \\left[ \\cos\\vphi \\sin\\vth_S \\tan\\vth_S \\sin\\vphi_S \\cos\\vphi_S\n    + \\left( \\sin^2\\vphi - \\sin^2\\vth_S \\sin^2\\vphi_S \\right)^{1\/2} \\right].\\label{eq:dispz}\n\\eeqa\n\n\n\n\\subsection{Schwarzschild Case}\n\\label{sec:spherical}\n\nIn the case of a spherically symmetric lens we have $\\vphi_S = \\vphi + \\pi$, and either $\\chi = \\vphi$\nor $\\chi = \\vphi+\\pi$, depending on whether the source lies on the\nsame or opposite side of the lens as the image.  Once again, we define $\\snq = \\cos(\\chi-\\vphi)$ to be\na sign that distinguishes these two cases.  With these conditions\neqns.~(\\ref{eq:le-h-Kerr}) and (\\ref{eq:le-v-Kerr}) combine to form the\nsingle lens equation with displacement for a Schwarzschild black hole:\n\\beqa \\label{eq:le-sph-Kerr}\n  d_S\\,\\snq\\,\\tan\\cb = d_L\\ \\frac{\\sin\\vth}{\\cos\\vth_S} \\ - \\ \n    d_{LS}\\,\\tan\\vth_S\\,.\n\\eeqa\nTwo comments are in order.  First, our spherical lens equation\n(\\ref{eq:le-sph-Kerr}) is equivalent to the spherical lens\nequation recently derived by Bozza \\& Sereno \\cite{BS,Bozza2} (up to the sign $\\snq$,\nwhich was not discussed explicitly in \\cite{BS,Bozza2}).\nThe second comment refers to the amplitude of the displacement.\nBy comparing our general planar-ray lens equation (\\ref{eq:le-sph}) with eqn.~(\\ref{eq:le-sph-Kerr}),\nwe can identify the displacement\n\\beqa\n\\label{eq:disp-sph}\n  \\cd = d_L \\sin\\vth \\left[ \\frac{1}{\\cos(\\alpha-\\vth)} - \\frac{1}{\\cos\\vth}\n    \\right]\\,,\n\\eeqa\nwhere we have switched from $\\vth_S$ to the bending angle\n$\\alpha = \\vth+\\vth_S$.  Now let\n$\\delta\\alpha = \\alpha \\, {\\rm mod} \\, 2 \\pi$, and assume that\n$\\vth$ and $\\delta\\alpha$ are small.  (Note that we need not\nassume $\\alpha$ itself is small, only that $\\delta\\alpha$ is small.\nThis means that our analysis applies to all light rays, including those that loop\naround the lens.)  Taylor expanding the displacement in $\\vth$\nand $\\alpha$ yields\n\\beqa\n  \\cd = \\frac{d_L}{2} \\, (\\vth \\ \\delta \\alpha) \n         (\\delta \\alpha - 2 \\vth)  \\ + \\ {\\cal O}(4).\\nonumber\n\\eeqa\n\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nRecently a lens equation was developed for Schwarzschild lensing with displacement (see Section~\\ref{sec:spherical} above), when the light ray's tangent lines\nat the source and observer do not meet on the lens plane.  In this paper we found a new generalization of the lens equation with displacement for axisymmetric lenses that extends the previous work to a fully three-dimensional setting.  Our formalism assumes that the source and observer are in the asymptotically flat region, and does not require a small angle approximation.  Furthermore, we found a new magnification formula applicable to this more general context.  Our lens equation is thus applicable to non-spherically symmetric compact bodies, such as Kerr black holes.  We gave explicit formulas for the\ndisplacement when the observer is in the equatorial plane\nof a Kerr black hole and \nfor the situation of spherical symmetry.  \n\n\n\n\n\\begin{acknowledgments}\n\nABA and AOP would especially like to thank Marcus C. Werner for helpful discussions.  AOP acknowledges the support of NSF Grant DMS-0707003.\n\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLanguage and facial expression are strong indicators for behavior analysis. There is numerous research trying to solve the emotion recognition problem based on these data. However, the non-linguistic vocalizations are understudied even though they are very useful information. Analyzing and applying these signals are interesting topics and require more attention from researchers. The A-VB competition was conducted for that reason and is expected to explore advanced improvements in emotion science. The competition \\cite{b2} consists of 4 individual tasks as below:\n\n\\begin{itemize}\n\\item High-dimension task (A-VB-HIGH): a multi-output regression task generating 10 values in the range of [0,1] corresponding to levels of Awe, Excitement, Amusement, Awkwardness, Fear, Horror, Distress, Triumph, Sadness, and Surprise.\n\\item Two-dimension task (A-VB-TWO): a multi-output regression task generating 2 values in the range of [0,1] corresponding to levels of Valence and Arousal.\n\\item Culture task (A-VB-CULTURE): a multi-output regression task generating 40 values in the range of [0,1] corresponding to levels of culture including China, US, South Africa, and Venezuela combined with  levels of emotion including Awe, Excitement, Amusement, Awkwardness, Fear, Horror, Distress, Triumph, Sadness, Surprise.\n\\item Type task (A-VB-TYPE): a multi-class classification task predicting the type of expressive vocal including Gasp, Laugh, Cry, Scream, Grunt, Groan, Pant, Other.\n\\end{itemize}\n\nThe evaluation metric for three regression tasks is the Concordance Correlation Coefficient (CCC) and the metric for the categorical task is the Unweighted Average Recall (UAR). The detail is listed below:\n\\begin{itemize}\n\\item A-VB-HIGH: The metric is the average CCC score of 10 emotions.\n\\item A-VB-TWO: The metric is the average CCC score of valence-arousal.\n\\item A-VB-CULTURE: The metric is the average CCC score of 40 culture-emotion levels.\n\\item A-VB-TYPE: The metric is the UAR score of 8 classes of vocalizations.\n\\end{itemize}\n\nAll of the above metrics are in the range of [0,1]. The greater the score is, the better the model performs. We should also note that all the results in this paper are in percentage. \n\nIn this paper, we propose a straightforward approach using a pre-trained Wav2vec network to resolve the problem. The model accomplishes a noticeable improvement compared to the baseline provided by the organizers. Because of its simplicity, our model can be considered a new baseline for all tasks in the competition.\n\n\\section{Related work}\nIn the baseline paper \\cite{b2}, the authors introduce two different approaches, which are feature-based and end-to-end approaches. In the feature-based option, the OpenSMILE toolkit \\cite{b11} is leveraged to extract the COMPARE (COMputational PARalinguistics ChallengE \\cite{b12}) feature and EGEMAPS (The extended Geneva Minimalistic Acoustic Parameter Set \\cite{b13}) feature from an input sample. The features are then fed to a 3-layer fully-connected neural network. Mean Squared Error (MSE) loss is used for regression tasks while the classification task applies the Cross-entropy (CE) loss function.\n\nIn the end-to-end manner, the baseline model uses End2You \\cite{b14}, a multimodal profiling toolkit that is capable of training and evaluating models from raw input. Particularly, Emo-18 architecture \\cite{b15} is chosen for the competition. The model includes 1-D Convolutional Neural Network (CNN) layers to derive the features from audio frames and a Recurrent Neural Network to learn the temporal information.\n\nFor the speaker recognition task, Nik Vaessen and David A. van Leeuwen \\cite{b4} conducted fine-tuning the Wav2vec2 model by using a shared fully-connected layer. Their model and ours have one thing in common: exploiting the pre-trained Wav2vec. However, there are considerable differences between the two methods. Basically, speaker recognition is a classification problem so the authors optimize the model with CE or Binary Cross-entropy (BCE) loss. In our method, we consider two loss options for the regression tasks, which are MSE and CCC loss. Additionally, besides using a shared fully-connected layer, we also take advantage of the RNN to explore the temporal behavior.\n\\section{Method}\nThe sequence embeddings are obtained from the waveform signal by the audio extractor. They are then fed into an RNN to enrich the sequence information. Afterward, a fully connected layer changes the embeddings' dimension to the output sizes depending on the particular task. Finally, a pooling layer is used to reduce variable-length sequence embeddings to fix-size speaker embedding. The dimension of the final prediction would be 2, 10, 40, or 8 corresponding to A-VB-TWO, A-VB-HIGH, A-VB-CULTURE or A-VB-TYPE task, respectively. Figure~\\ref{fig:model} describes the architecture of our method.\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[height=8cm]{model.PNG}}\n\\caption{Block diagram of our proposed model.}\n\\label{fig:model}\n\\end{figure}\n\n\\subsection{Audio extractor}\n\nWe take advantage of the Pre-trained Wav2vec 2.0 models \\cite{b3} provided by Pytorch. They are trained with large unlabeled audio corpora so they can effectively capture the audio features. We conducted experiments with 4 versions of the Wav2vec2 model described below:\n\\begin{itemize}\n\\item BASE: use the base configuration of transformer trained with 960 hours of unlabeled audio from LibriSpeech \\cite{b5}.\n\\item LARGE: use the large configuration of transformer trained with 960 hours of unlabeled audio from LibriSpeech \\cite{b5}.\n\\item LARGE-LV60K: use the large configuration of transformer trained with 60,000 hours of unlabeled audio from Libri-Light \\cite{b6}.\n\\item XLSR53: use base configuration of transformer \\cite{b10} trained with 56,000 hours of unlabeled audio from multiple datasets (Multilingual LibriSpeech  \\cite{b7}, CommonVoice  \\cite{b8} and BABEL  \\cite{b9}).\n\\end{itemize}\n\n\\subsection{Pooling Method}\nInspired by the model of \\cite{b4}, we use 4 options of pooling to fix the length of embedding: first (take the first sequence embedding to be the speaker embedding), last (take the last sequence embedding to be the speaker embedding), max, and average pooling. The performance of models with various types of pooling layers is recorded to study their impact on the result. The operation of the pooling can be described as:\n\\begin{equation}\ns_{i} = Pooling(e_1, e_2, ..., e_{m_i})\n\\end{equation}\nwhere $s_{i}$ is the speaker embedding of $i^{th}$ sample; $e_1, e_2, ..., e_{m_i}$ are the temporal embeddings and $m_i$ is the sequence length of corresponding sample.\n\n\n\\subsection{Loss function}\nWe use the CE loss for the classification task. In the remaining tasks, we want to test the effect of the loss function on the performance of the model so we did the experiments and compared the result of the model using Mean Square Error (MSE) and Concordance Correlation Coefficient (CCC) loss. The CCC loss is formulated as below:\n\\begin{equation}\n\\mathcal{L} = 1 - CCC = 1 - \\frac{2s_{xy}}{s^{2}_x + s^{2}_y  + \\left(\\overline{x}-\\overline{y}\\right)^2}\n\\end{equation}\nwhere $\\overline{x}$ and $\\overline{y}$ are the mean values of ground truth and predicted values, respectively, $s_x$ and $s_x$ are corresponding variances and $s_{xy}$ is the covariance value.\n\n\\section{Dataset and Experiments}\n\\subsection{Dataset}\nThe database used for the competition is the HUME-VB dataset  \\cite{b1} which consists of 59201 audio files and is split into 3 sets (training, validation, and test) of similar size. Each file has 53 labels corresponding to 4 tasks, one label is a categorical label that is used in the classification task and the remains are values in the range [0,1] representing the emotional level. The organizers provide 2 versions of the HUME-VB dataset: the raw version sampled at 48kHz and the processed version where audio files are converted to 16kHz and normalized to -3 decibels. In our experiments, we take the processed version as the input of our model.\n\n\n\\subsection{Experiments}\nOur model was implemented using the Pytorch framework. The experiments were conducted on a machine with NVIDIA RTX 2080 Ti GPU. All scenarios were run in 20 epochs, and the model with the best performance on the validation set was recorded. The batch size is 16 and the initial learning rate is $1e-4$. We used Adam optimizer with the weight decay coefficient of $0.0625$.\n\nIn our setting, we take the output from the $12^{th}$ layer of the Wav2vec network to be the sequence embeddings. The input size of the RNN network depends on the configuration of the pre-trained audio extractor, which is 768 for base configuration and 1024 for large configuration. It includes 2 LSTM layers and the hidden size is fixed to 512.\n\n\\section{Discussion}\nWe tested the performance of the model with various audio extractors to explore their effect. Table~\\ref{tab:result} shows the performance on four tasks of the competition with 4 pre-trained Wav2vec extractors. As a result, the XLSR53 pre-trained model achieves the best performance in A-VB-TWO and A-VB-TYPE when LARGE-LV60K attains the highest scores in A-VB-HIGH and A-VB-CULTURE. In the meanwhile, the BASE model produces the lowest score in A-VB-TWO and A-VB-TYPE due to its simple architecture.\n\n\\begin{table}\n \\caption{Evaluation score on the HUME-VB validation set with different extractors. Experimented with CCC loss and Last Pooling}\n\\begin{center}\n  \\centering\n  \\begin{tabular}{|l|l|l|l|l|l|}\n    \\hline\n    Model & TWO & HIGH & CULTURE & TYPE \\\\\n    \\hline\n    \\textit{End2You Baseline} & \\textit{49.88}  & \\textit{56.38} & \\textit{43.59} & \\textit{41.66} \\\\\n    \\hline\n    BASE & 54.65  & 58.69 & 47.18 & 41.65     \\\\\n    \\hline\n    LARGE & 55.42 & 58.00 & 47.12 & 43.96      \\\\\n    \\hline\n    LARGE-LV60K & 61.59 & \\textbf{65.41} & \\textbf{53.39} & 47.22  \\\\\n    \\hline\n    XLSR53 & \\textbf{61.94} & 65.32 & 52.50 & \\textbf{49.89}  \\\\\n    \\hline\n  \\end{tabular}\n  \\label{tab:result}\n\\end{center}\n\\end{table}\n\nRegarding the pooling method, we examined their influence on the results in A-VB-HIGH. As shown in Table~\\ref{tab:pool}, in both LARGE-LV60K and XLSR53 model, the Last pooling outperforms the other options while the First pooling gets the lowest CCC score among the four methods. The result of Avg pooling is slightly better than Max pooling in both LARGE-LV60K and XLSR53 scenarios. It can be inferred that the last embedding of the sequence contains the most useful information for the prediction when using other embeddings or combining them may degrade the accuracy.\n\n\\begin{table}\n \\caption{Evaluation score on the HUME-VB validation set with different pooling methods. Experimented on A-VB-HIGH with CCC loss}\n\\begin{center}\n  \\centering\n  \\begin{tabular}{|l|l|l|}\n    \\hline\n    Pooling & LARGE-LV60K & XLSR53 \\\\\n    \\hline\n     First & 53.56 & 58.20 \\\\\n    \\hline\n     Max & 60.08 & 61.41 \\\\\n    \\hline\n     Avg & 61.49 & 62.40 \\\\\n    \\hline\n     Last & \\textbf{65.41} & \\textbf{65.32} \\\\\n    \\hline\n  \\end{tabular}\n  \\label{tab:pool}\n\\end{center}\n\\end{table}\nNext, we conducted the training processes with MSE and CCC to explore their advantage. As a consequence, the model trained with CCC loss gives better performance on the validation set compared to the one trained with MSE loss. The detail is shown in Table~\\ref{tab:loss}.\n\\begin{table}\n \\caption{Evaluation score on the HUME-VB validation set with different loss functions. Experimented on A-VB-HIGH with Last pooling}\n\\begin{center}\n  \\centering\n  \\begin{tabular}{|l|l|l|}\n  \\hline\n    Loss function & LARGE-LV60K & XLSR53 \\\\\n    \\hline\n    MSE & 63.77 & 63.94 \\\\\n    \\hline\n    CCC & \\textbf{65.41} & \\textbf{65.32} \\\\\n    \\hline\n  \\end{tabular}\n  \\label{tab:loss}\n\\end{center}\n\\end{table}\n\nIn addition, we carried out the ablation study to analyze the role of the RNN. According to Table~\\ref{tab:rnn}, using the RNN can significantly boost the accuracy of the model in all four tasks. It can be explained by the capability of learning temporal information of the LSTM, which can enhance the overall operation of the model.\n\n\\begin{table}\n \\caption{Evaluation score on the HUME-VB validation set with and without RNN. Experimented on A-VB-HIGH with LARGE-LV60K model, CCC loss, and Last pooling}\n\\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|}\n    \\hline\n    Model & TWO & HIGH & CULTURE & TYPE \\\\\n    \\hline\n    Without RNN & 47.38 & 50.70 & 40.20 & 39.10 \\\\\n    \\hline\n    With RNN & \\textbf{61.59} & \\textbf{65.41} & \\textbf{53.39} & \\textbf{47.22} \\\\\n    \\hline\n  \\end{tabular}\n  \\label{tab:rnn}\n\\end{center}\n\\end{table}\n\nAfter conducting the above experiments, we concluded that the best configuration of our model is combining either LARGE-LV60K or XLSR53 pre-trained model with last pooling method and utilizing CCC loss. This setting was used to train separated model for each task in order to obtain unbiased evaluation on test set. We decided to choose LARGE-LV60K for A-VB-HIGH and A-VB-CULTURE, XLSR53 for A-VB-TWO and A-VB-TYPE. This time we trained each model for 50 epochs and applied early stopping by monitoring the validation result. Our best models and their evaluations on test set and validation set are listed in  Table~\\ref{tab:test}.\n\n\\begin{table}\n\\caption{Evaluation score on the HUME-VB validation and test sets. Experimented with CCC loss and Last pooling for 50 epochs}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n Task & Pre-trained &\\multicolumn{2}{|c|}{Performance} \\\\\n\\cline{3-4} \n name & Audio Extractor & Val set & Test set \\\\\n\\hline\n TWO & XLSR53 & 61.94 & 62.02 \\\\\n\\hline\n HIGH & LARGE-LV60K & 66.76 & 66.77 \\\\\n\\hline\n CULTURE & LARGE-LV60K & 54.93 & 54.95 \\\\\n\\hline\n TYPE & XLSR53 & 49.89 & 49.70 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:test}\n\\end{center}\n\\end{table}\n\n\\section{Conclusion}\nThis paper presents our proposed method for all sub-challenges of the A-VB competition. Particularly, we fine-tuned the pre-trained Wav2vec and combined it with basic neural networks and a proper pooling method. The CCC loss and Last pooling show the best performance on four tasks among the other options. Our model outperforms the baseline of the organizer on the test set, with the CCC score of 62.02 for A-VB-TWO, 66.77 for A-VB-HIGH, 54.95 for A-VB-CULTURE and UAR metric of 49.70 for A-VB-TYPE.\n\n\\section*{Acknowledgment}\n\nThis work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT). (NRF-2020R1A4A1019191).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMassive multiple-input multiple-output (MIMO), also known as\nlarge-scale or very-large MIMO, is a promising technology to meet\nthe ever growing demands for higher throughput and better\nquality-of-service of next-generation wireless communication\nsystems \\cite{RusekPersson13,LarssonEdfors14,ChenSun16}. Massive\nMIMO systems are those that are equipped with a large number of\nantennas at the base station (BS) simultaneously serving a much\nsmaller number of single-antenna users sharing the same\ntime-frequency slot. By exploiting the asymptotic orthogonality\namong channel vectors associated with different users, massive\nMIMO systems can achieve almost perfect inter-user interference\ncancelation with a simple linear precoder and receive combiner\n\\cite{Marzetta10}, and thus have the potential to enhance the\nspectrum efficiency by orders of magnitude.\n\n\n\nDespite all these benefits, massive MIMO systems pose new\nchallenges for system design and hardware implementation. Due to\nthe large number of antennas at the BS, the hardware cost and\npower consumption could become prohibitively high if we still\nemploy expensive and power-hungry high-resolution\nanalog-to-digital convertors (ADCs) \\cite{ChenZhao14}. To address\nthis obstacle, recent studies (e.g.\n\\cite{RisiPersson14,FanJin15,ZhangDai16,JacobssonDurisi15,WangLi15,WenWang16,ChoiMo16})\nconsidered the use of low-resolution ADCs (e.g. 1-3 bits) for\nmassive MIMO systems. It is known that the hardware complexity and\npower consumption grow exponentially with the resolution (i.e. the\nnumber of bits per sample) of the ADC. Therefore lowering the\nresolution of the ADC can effectively reduce the hardware cost and\npower consumption. In particular, for the extreme one-bit case,\nthe ADC becomes a simple analog comparator. Also, automatic gain\ncontrol (AGC) is no longer needed when one-bit ADCs are used,\nwhich further simplifies the hardware complexity.\n\n\n\nMassive MIMO with low-resolution ADCs has attracted much attention\nover the past few years. Great efforts have been made to\nunderstand the effects of low-resolution ADCs on the performance\nof MIMO and massive MIMO systems. Specifically, by assuming full\nknowledge of channel state information (CSI), the capacity at both\nfinite and infinite signal-to-noise ratio (SNR) was derived in\n\\cite{MoHeath15} for one-bit MIMO systems. For massive MIMO\nsystems with low-resolution ADCs, the spectral efficiency and the\nuplink achievable rate were investigated in\n\\cite{RisiPersson14,FanJin15,ZhangDai16,LiangZhang16} under\ndifferent assumptions. The theoretical analyses suggest that the\nuse of the low cost and low-resolution ADCs can still provide\nsatisfactory achievable rates and spectral efficiency.\n\n\n\n\n\n\n\n\n\nIn this paper, we consider the problem of channel estimation for\nuplink multiuser massive MIMO systems, where one-bit ADCs are used\nat the BS in order to reduce the cost and power consumption.\nChannel estimation is crucial to support multi-user MIMO operation\nin massive MIMO systems\n\\cite{AdhikaryNam13,ChoiLove14,SunGao15,GaoDai15,FangLi17}. To\nreach the full potential of massive MIMO, accurate downlink CSI is\nrequired at the BS for precoding and other operations. Most\nliterature on massive MIMO systems, e.g.\n\\cite{Marzetta10,RusekPersson13,YinGesbert13,MullerCottatellucci14},\nassumes a time division duplex (TDD) mode in which the downlink\nCSI can be immediately obtained from the uplink CSI by exploiting\nchannel reciprocity. Nevertheless, channel estimation for massive\nMIMO systems with one-bit ADCs is challenging since the magnitude\nand phase information about the received signal are lost or\nseverely distorted due to the coarse quantization. It was shown in\n\\cite{RisiPersson14} that one-bit massive MIMO systems require an\nexcessively long training sequence (e.g. approximately 50 times\nthe number of users) to achieve an acceptable performance. The\nwork \\cite{JacobssonDurisi15} showed that for one-bit massive MIMO\nsystems, a least-squares channel estimation scheme and a\nmaximum-ratio combining scheme are sufficient to support both\nmultiuser operation and the use of high-order constellations.\nNevertheless, a long training sequence is still a requirement. To\nalleviate this issue, a Bayes-optimal joint channel and data\nestimation scheme was proposed in \\cite{WenWang16}, in which the\nestimated payload data are utilized to aid channel estimation. In\n\\cite{ChoiMo16}, a maximum likelihood channel estimator, along\nwith a near maximum likelihood detector, were proposed for uplink\nmassive MIMO systems with one-bit ADCs.\n\n\n\nDespite these efforts, channel estimation using one-bit quantized\ndata still incur much larger estimation errors as compared with\nusing the original unquantized data, and require considerably\nhigher training overhead to attain an acceptable estimation\naccuracy. To address this issue, in this paper, we study one-bit\nquantizer design and examine the impact of the choice of\nquantization thresholds on the estimation performance.\nSpecifically, the optimal design of quantization thresholds as\nwell as the training sequences is investigated. Note that one-bit\nquantization design is an interesting and important issue but\nlargely neglected by existing massive MIMO channel estimation\nstudies. In fact, most channel estimation schemes, e.g.\n\\cite{RisiPersson14,JacobssonDurisi15,WenWang16,ChoiMo16}, assume\na fixed, typically zero, quantization threshold. The optimal\nchoice of the quantization threshold was considered in\n\\cite{KochLapidoth13,Verdu02}, but addressed from an\ninformation-theoretic perspective. Our theoretical results reveal\nthat, given that the quantization thresholds are optimally\ndevised, using one-bit ADCs can achieve an estimation error close\nto (with an increase only by a factor of $\\pi\/2$) the minimum\nachievable estimation error attained by using infinite-precision\nADCs. The optimal quantization thresholds, however, are dependent\non the unknown channel parameters. To cope with this difficulty,\nwe propose an adaptive quantization (AQ) scheme by which the\nthresholds are dynamically adjusted in a way such that the\nthresholds converge to the optimal thresholds, and a random\nquantization (RQ) scheme which randomly generates a set of\nnon-identical thresholds based on some statistical prior knowledge\nof the channel. Simulation results show that our proposed schemes,\nbecause of their wisely devised quantization thresholds, present a\nsignificant performance improvement over the fixed quantization\nscheme that use a fixed (say, zero) quantization threshold. In\nparticular, the AQ scheme, even with a moderate number of pilot\nsymbols (about 5 times the number of users), can provide an\nachievable rate close to that of the perfect CSI case.\n\n\nThe rest of the paper is organized as follows. The system model\nand the problem of channel estimation using one-bit ADCs are\ndiscussed in Section \\ref{sec:system-model}. In Section\n\\ref{sec:MLE-CRB}, we develop a maximum likelihood estimator and\ncarry out a Cram\\'{e}r-Rao bound analysis of the one-bit channel\nestimation problem. The optimal design of quantization thresholds\nand the pilot sequences is studied in Section\n\\ref{sec:optimal-design}. In Section \\ref{sec:AQ-RQ}, we develop\nan adaptive quantization scheme and a random quantization scheme\nfor practical threshold design. Simulation results are provided in\nSection \\ref{sec:experiments}, followed by concluding remarks in\nSection \\ref{sec:conclusion}.\n\n\n\n\n\n\\section{System Model and Problem Formulation} \\label{sec:system-model}\nConsider a single-cell uplink multiuser massive MIMO system, where\nthe BS equipped with $M$ antennas serves $K$ ($M\\gg K$)\nsingle-antenna users simultaneously. The channel is assumed to be\nflat block fading, i.e. the channel remains constant over a\ncertain amount of coherence time. The received signal at the BS\ncan be expressed as\n\\begin{align}\n\\boldsymbol{Y}=\\boldsymbol{H}\\boldsymbol{X}+\\boldsymbol{W}\n\\label{data-model}\n\\end{align}\nwhere $\\boldsymbol{X}\\in\\mathbb{C}^{K\\times L}$ is a training\nmatrix and its row corresponds to each user's training sequence\nwith $L$ pilot symbols, $\\boldsymbol{H}\\in\\mathbb{C}^{M\\times K}$\ndenotes the channel matrix to be estimated, and\n$\\boldsymbol{W}\\in\\mathbb{C}^{M\\times L}$ represents the additive\nwhite Gaussian noise with its entries following a circularly\nsymmetric complex Gaussian distribution with zero mean and\nvariance $2\\sigma^2$.\n\nTo reduce the hardware cost and power consumption, we consider a\nmassive MIMO system which uses one-bit ADCs at the BS to quantize\nthe received signal. Specifically, at each antenna, the real and\nimaginary components of the received signal are quantized\nseparately using a pair of one-bit ADCs. Thus in total $2M$\none-bit ADCs are needed. The quantized output of the received\nsignal, $\\boldsymbol{B}\\triangleq [b_{m,l}]$, can be written as\n\\begin{align}\n\\boldsymbol{B}=\\mathcal{Q}(\\boldsymbol{Y})\n\\label{conventional-quantizer}\n\\end{align}\nwhere $\\mathcal{Q}(\\boldsymbol{Y})$ is an element-wise operation\nperformed on $\\boldsymbol{Y}$, and for each element of\n$\\boldsymbol{Y}$, $y_{m,l}$, we have\n\\begin{align}\n\\mathcal{Q}(y_{m,l})=\\text{sgn}(\\Re(y_{m,l}))+j\\text{sgn}(\\Im(y_{m,l}))\n\\end{align}\nin which $\\Re(y)$ and $\\Im(y)$ denote the real and imaginary\ncomponents of $y$, respectively, and the sign function\n$\\text{sgn}(\\cdot)$ is defined as\n\\begin{align}\n\\text{sgn}(y) \\triangleq \\left\\{ \\begin{array}{ll}\n1 & \\textrm{if $y\\ge 0$}\\\\\n-1 & \\textrm{otherwise}\n\\end{array} \\right.\n\\end{align}\nTherefore the quantized output belongs to the set\n\\begin{align}\nb_{m,l}\\in \\{1+j,-1+j,1-j,-1-j\\}\\quad \\forall m,l\n\\end{align}\nNote that in (\\ref{conventional-quantizer}), we implicitly assume\na zero threshold for one-bit quantization. Nevertheless, using\nidentically a zero threshold for all measurements is not\nnecessarily optimal, and it is interesting to analyze the impact\nof the quantization thresholds on the channel estimation\nperformance. Such an issue (i.e. choice of quantization\nthresholds), albeit important, was to some extent neglected by\nmost existing studies. To examine this problem, let\n$\\boldsymbol{T}\\triangleq [\\tau_{m,l}]$ denote the thresholds used\nfor one-bit quantization. The quantized output of the received\nsignal, $\\boldsymbol{B}$, is now given as\n\\begin{align}\n\\boldsymbol{B}=\\mathcal{Q}(\\boldsymbol{Y}-\\boldsymbol{T})\n\\label{quantizer}\n\\end{align}\n\nTo facilitate our analysis, we first convert (\\ref{data-model})\ninto a real-valued form as follows\n\\begin{align}\n\\boldsymbol{\\widetilde{Y}}=\\boldsymbol{\\widetilde{A}}\\boldsymbol{\\widetilde{H}}+\\boldsymbol{\\widetilde{W}}\n\\end{align}\nwhere\n\\begin{align}\n\\boldsymbol{\\widetilde{Y}}\\triangleq & [ \\Re(\\boldsymbol{Y}) \\\n\\Im(\\boldsymbol{Y})]^T \\nonumber\\\\\n\\boldsymbol{\\widetilde{H}} \\triangleq & [ \\Re(\\boldsymbol{H}) \\\n\\Im(\\boldsymbol{H})]^T \\nonumber\\\\\n\\boldsymbol{\\widetilde{W}} \\triangleq & [ \\Re(\\boldsymbol{W}) \\\n\\Im(\\boldsymbol{W})]^T \\nonumber\n\\end{align}\nand\n\\begin{align}\n\\boldsymbol{\\widetilde{A}} \\triangleq \\left[\\begin{array}{ccc}\n\\Re(\\boldsymbol{X}) & \\Im(\\boldsymbol{X}) \\\\\n-\\Im(\\boldsymbol{X}) & \\Re(\\boldsymbol{X})\n\\end{array}\\right]^T \\label{A-X-relationship}\n\\end{align}\nVectorizing the real-valued matrix $\\boldsymbol{\\widetilde{Y}}$,\nthe received signal can be expressed as a real-valued vector form\nas\n\\begin{align}\n\\boldsymbol{y}=\\boldsymbol{A}\\boldsymbol{h}+\\boldsymbol{w}\n\\label{data-model-vector}\n\\end{align}\nwhere\n$\\boldsymbol{y}\\triangleq\\text{vec}(\\boldsymbol{\\widetilde{Y}})$,\n$\\boldsymbol{A}\\triangleq\\boldsymbol{I}_{M}\\otimes\\boldsymbol{\\widetilde{A}}$,\n$\\boldsymbol{h}\\triangleq\\text{vec}(\\boldsymbol{\\widetilde{H}})$,\nand\n$\\boldsymbol{w}\\triangleq\\text{vec}(\\boldsymbol{\\widetilde{W}})$.\nIt can be easily verified $\\boldsymbol{y}\\in\\mathbb{R}^{2ML}$,\n$\\boldsymbol{A}\\in\\mathbb{R}^{2ML\\times 2MK}$, and\n$\\boldsymbol{h}\\in\\mathbb{R}^{2MK}$. Accordingly, the one-bit\nquantized data can be written as\n\\begin{align}\n\\boldsymbol{b}=\\text{sgn}(\\boldsymbol{y}-\\boldsymbol{\\tau})\n\\label{quantized-data-model-vector}\n\\end{align}\nwhere $\\boldsymbol{\\tau}\\triangleq \\text{vec}([\n\\Re(\\widetilde{\\boldsymbol{T}}) \\\n\\Im(\\widetilde{\\boldsymbol{T}})]^T)$ and\n$\\boldsymbol{\\tau}\\in\\mathbb{R}^{2ML}$. For simplicity, we define\n$N\\triangleq 2ML$. \n\nOur objective in this paper is to estimate the channel\n$\\boldsymbol{h}$ based on the one-bit quantized data\n$\\boldsymbol{b}$, examine the best achievable estimation\nperformance and investigate the optimal thresholds\n$\\boldsymbol{\\tau}$ as well as the optimal training sequences\n$\\boldsymbol{X}$. To this objective, in the following, we first\ndevelop a maximum likelihood (ML) estimator and carry out a\nCram\\'{e}r-Rao bound (CRB) analysis. The optimal choice of the\nquantization thresholds as well as the training sequences is then\nstudied based on the CRB matrix of the unknown channel parameter\nvector $\\boldsymbol{h}$.\n\n\n\n\n\\section{ML Estimator and CRB Analysis} \\label{sec:MLE-CRB}\n\\subsection{ML Estimator}\nBy combining (\\ref{data-model-vector}) and\n(\\ref{quantized-data-model-vector}), we have\n\\begin{align}\nb_n=\\text{sgn}(y_n-\\tau_n)\n=\\text{sgn}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}+w_n-\\tau_n), \\quad\n\\forall n\n\\end{align}\nwhere, by allowing a slight abuse of notation, we let $b_n$,\n$y_n$, $\\tau_n$, and $w_n$ denote the $n$th entry of\n$\\boldsymbol{b}$, $\\boldsymbol{y}$, $\\boldsymbol{\\tau}$, and\n$\\boldsymbol{w}$, respectively; and $\\boldsymbol{a}_n^T$ denotes\nthe $n$th row of $\\boldsymbol{A}$. It is easy to derive that\n\\begin{align}\nP(b_n=1;\\boldsymbol{h}) & =P(w_n\\geq-(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n);\\boldsymbol{h}) \\nonumber \\\\\n& =F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)\n\\end{align}\nand\n\\begin{align}\nP(b_n=-1;\\boldsymbol{h}) & =P(w_n < -(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n);\\boldsymbol{h}) \\nonumber \\\\\n& =1-F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)\n\\end{align}\nwhere $F_{w}(\\cdot)$ denotes the cumulative density function (CDF)\nof $w_n$, and $w_n$ is a real-valued Gaussian random variable with\nzero-mean and variance $\\sigma^2$. Therefore the probability mass\nfunction (PMF) of $b_n$ is given by\n\\begin{align}\np(b_n;\\boldsymbol{h})=& [1-F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)]^{(1-b_n)\/2} \\nonumber \\\\\n&\\cdot[F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)]^{(1+b_n)\/2}\n\\end{align}\nSince $\\{b_n\\}$ are independent, the log-PMF or log-likelihood\nfunction can be written as\n\\begin{align}\nL(\\boldsymbol{h}) & \\triangleq  \\log p(b_1,\\dots,b_N;\\boldsymbol{h}) \\nonumber \\\\\n& = \\sum_{n=1}^{N} \\bigg\\{ \\frac{1-b_n}{2}\\log [1-F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)]   \\nonumber \\\\\n& \\qquad \\quad + \\frac{1+b_n}{2} \\log\n[F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)] \\bigg\\}\n\\label{log-PMF}\n\\end{align}\nThe ML estimate of $\\boldsymbol{h}$, therefore, is given as\n\\begin{align}\n\\hat{\\boldsymbol{h}}=\\arg\\max_{\\boldsymbol{h}} \\ L(\\boldsymbol{h})\n\\label{MLE}\n\\end{align}\nIt can be proved that the log-likelihood function\n$L(\\boldsymbol{h})$ is a concave function. Hence computationally\nefficient search algorithms can be used to find the global\nmaximum. The proof of the concavity of $L(\\boldsymbol{h})$ is\ngiven in Appendix \\ref{appA}.\n\n\n\n\n\n\n\\subsection{CRB}\nWe now carry out a CRB analysis of the one-bit channel estimation\nproblem (\\ref{quantized-data-model-vector}). The CRB results help\nunderstand the effect of different system parameters, including\nquantization thresholds as well as training sequences, on the\nestimation performance. We first summarize our derived CRB results\nin the following theorem.\n\n\n\\newtheorem{theorem}{Theorem}\n\\begin{theorem} \\label{theorem1}\nThe Fisher information matrix (FIM) for the estimation problem\n(\\ref{quantized-data-model-vector}) is given as\n\\begin{align}\n\\boldsymbol{J}(\\boldsymbol{h})=\\sum_{n=1}^{N}\ng(\\tau_n,\\boldsymbol{a}_n)\\boldsymbol{a}_n\\boldsymbol{a}_n^T\n\\end{align}\nwhere $g(\\tau_n,\\boldsymbol{a}_n)$ is defined as\n\\begin{align}\ng(\\tau_n,\\boldsymbol{a}_n) \\triangleq \\frac {f_{w}^2\n(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)}\n{F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)(1-F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n))}\n\\label{g-function}\n\\end{align}\nin which $f_{w}(\\cdot)$ denotes the probability density function\n(PDF) of $w_n$. Accordingly, the CRB matrix for the estimation\nproblem (\\ref{quantized-data-model-vector}) is given by\n\\begin{align}\n\\text{CRB}(\\boldsymbol{h})=\\boldsymbol{J}^{-1}(\\boldsymbol{h}) =\n\\left( \\sum_{n=1}^{N} g(\\tau_n,\\boldsymbol{a}_n) \\boldsymbol{a}_n\n\\boldsymbol{a}_n^T \\right)^{-1} \\label{CRB}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nSee Appendix \\ref{appB}.\n\\end{proof}\n\nAs is well known, the CRB places a lower bound on the estimation\nerror of any unbiased estimator \\cite{Kay93} and is asymptotically\nattained by the ML estimator. Specifically, the covariance matrix\nof any unbiased estimate satisfies:\n$\\text{cov}(\\hat{\\boldsymbol{h}})-\\text{CRB}(\\boldsymbol{h})\n\\succeq \\boldsymbol{0}$. Also, the variance of each component is\nbounded by the corresponding diagonal element of\n$\\text{CRB}(\\boldsymbol{h})$, i.e., $\\text{var}(\\hat{h}_i) \\ge\n[\\text{CRB}(\\boldsymbol{h})]_{ii}$.\n\nWe observe from (\\ref{CRB}) that the CRB matrix of\n$\\boldsymbol{h}$ depends on the quantization thresholds\n$\\boldsymbol{\\tau}$ as well as the matrix $\\boldsymbol{A}$ which\nis constructed from training sequences $\\boldsymbol{X}$ (cf.\n(\\ref{A-X-relationship})). Naturally, we wish to optimize\n$\\boldsymbol{\\tau}$ and $\\boldsymbol{A}$ (i.e. $\\boldsymbol{X}$)\nby minimizing the trace of the CRB matrix, i.e. the overall\nestimation error asymptotically achieved by the ML estimator. The\noptimization therefore can be formulated as follows\n\\begin{align}\n\\min_{\\boldsymbol{X},\\boldsymbol{\\tau}}\\quad &\n\\text{tr}\\left\\{\\text{CRB}(\\boldsymbol{h})\\right\\} = \\mathrm{tr}\n\\left\\{ \\left( \\sum_{n=1}^{N} g(\\tau_n,\\boldsymbol{a}_n)\n\\boldsymbol{a}_n \\boldsymbol{a}_n^T \\right)^{-1} \\right\\}\n\\nonumber\\\\\n\\text{s.t.} \\quad &\n\\boldsymbol{A}=\\boldsymbol{I}_M\\otimes\\boldsymbol{\\widetilde{A}}\n\\nonumber\\\\\n& \\boldsymbol{\\widetilde{A}} \\triangleq \\left[\\begin{array}{ccc}\n\\Re(\\boldsymbol{X}) & \\Im(\\boldsymbol{X}) \\\\\n-\\Im(\\boldsymbol{X}) & \\Re(\\boldsymbol{X})\n\\end{array}\\right]^T \\nonumber\\\\\n& \\text{tr}(\\boldsymbol{X}\\boldsymbol{X}^H)\\leq P\n \\label{opt1}\n\\end{align}\nwhere $\\text{tr}(\\boldsymbol{X}\\boldsymbol{X}^H)\\leq P$ is a\ntransmit power constraint imposed on the pilot signals. Such an\noptimization is examined in the following section, where it is\nshown that the optimization of $\\boldsymbol{X}$ can be decoupled\nfrom the optimization of the threshold $\\boldsymbol{\\tau}$.\n\n\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{g-function.eps}\n\\caption{The function value of $g(\\tau_n,\\boldsymbol{a}_n)$ vs.\n$(\\boldsymbol{a}_n^T\\boldsymbol{h}-\\tau_n)$, where $\\sigma^2=1$.}\n\\label{fig1}\n\\end{figure}\n\n\n\n\n\\section{Optimal Design and Performance Analysis} \\label{sec:optimal-design}\n\\subsection{Optimal Quantization Thresholds and Pilot\nSequences} Before proceeding, we first introduce the following\nresult.\n\n\\newtheorem{proposition}{Proposition}\n\\begin{proposition} \\label{proposition1}\nFor the Gaussian random variable $w_n$,\n$g(\\tau_n,\\boldsymbol{a}_n)$ defined in (\\ref{g-function}) is a\npositive and symmetric function attaining its maximum when\n$\\tau_n=\\boldsymbol{a}_n^{T}\\boldsymbol{h}$ (see Fig. \\ref{fig1}).\n\\end{proposition}\n\\begin{proof}\nPlease see Appendix \\ref{appE}.\n\\end{proof}\n\nHence, given a fixed $\\boldsymbol{A}$ (i.e. $\\boldsymbol{X}$), the\noptimal quantization thresholds conditional on $\\boldsymbol{A}$\nare given by\n\\begin{align}\n\\tau_n^{\\star}=\\boldsymbol{a}_n^{T}\\boldsymbol{h}, \\quad \\forall\nn\\in\\{1,\\ldots,N\\} \\label{optimumthreshold}\n\\end{align}\nThe result (\\ref{optimumthreshold}) comes directly by noting that\n\\begin{align}\n\\sum_{n=1}^N\ng_n(\\tau_n^{\\star},\\boldsymbol{a}_n)\\boldsymbol{a}_n\\boldsymbol{a}_n^T-\\sum_{n=1}^N\ng_n(\\tau_n,\\boldsymbol{a}_n)\\boldsymbol{a}_n\\boldsymbol{a}_n^T\\succeq\\mathbf{0}\n\\end{align}\nand resorting to the convexity of $\\text{tr}(\\boldsymbol{P}^{-1})$\nover the set of positive definite matrix, i.e. for any\n$\\boldsymbol{P}\\succ\\mathbf{0}$, $\\boldsymbol{Q}\\succ\\mathbf{0}$,\nand $\\boldsymbol{P}-\\boldsymbol{Q}\\succeq \\mathbf{0}$, the\nfollowing inequality\n$\\text{tr}(\\boldsymbol{P}^{-1})\\leq\\text{tr}(\\boldsymbol{Q}^{-1})$\nholds (see \\cite{BoydVandenberghe03}).\n\nWe see that the optimal choice of the quantization threshold\n$\\tau_n$ is dependent on the unknown channel $\\boldsymbol{h}$. To\nfacilitate our analysis, we, for the time being, suppose\n$\\boldsymbol{h}$ is known. Substituting (\\ref{optimumthreshold})\ninto (\\ref{opt1}) and noting that\n\\begin{align}\ng(\\tau_n^{\\star},\\boldsymbol{a}_n)=\\frac {f_{w}^2 (0)}\n{F_{w}(0)(1-F_{w}(0))}  = \\frac{2}{\\pi\\sigma^2} \\quad \\forall n\n\\end{align}\nthe optimization (\\ref{opt1}) reduces to\n\\begin{align}\n\\min_{\\boldsymbol{X}} \\quad & \\frac{\\pi\\sigma^2}{2} \\text{tr}\n\\left\\{ \\left( \\boldsymbol{A}^T \\boldsymbol{A} \\right)^{-1}\n\\right\\}\n\\nonumber\\\\\n\\text{s.t.} \\quad &\n\\boldsymbol{A}=\\boldsymbol{I}_M\\otimes\\boldsymbol{\\widetilde{A}}\n\\nonumber\\\\\n& \\boldsymbol{\\widetilde{A}} \\triangleq \\left[\\begin{array}{ccc}\n\\Re(\\boldsymbol{X}) & \\Im(\\boldsymbol{X}) \\\\\n-\\Im(\\boldsymbol{X}) & \\Re(\\boldsymbol{X})\n\\end{array}\\right]^T \\nonumber\\\\\n& \\text{tr}(\\boldsymbol{X}\\boldsymbol{X}^H)\\leq P \\label{opt2}\n\\end{align}\nwhich is now independent of $\\boldsymbol{h}$. We have the\nfollowing theorem regarding the solution to the optimization\n(\\ref{opt2}).\n\n\n\n\\begin{theorem} \\label{theorem2}\nThe minimum achievable objective function value of (\\ref{opt2}) is\ngiven by $(\\pi\\sigma^2 MK^2)\/P$ and can be attained if the pilot\nmatrix $\\boldsymbol{X}$ satisfies\n\\begin{align}\n\\boldsymbol{X}\\boldsymbol{X}^H = (P\/K) \\boldsymbol{I}\n\\label{theorem2:eqn1}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nSee Appendix \\ref{appC}.\n\\end{proof}\n\nTheorem \\ref{theorem2} reveals that, for one-bit massive MIMO\nsystems, users should employ orthogonal pilot sequences in order\nto minimize channel estimation errors. Although it is a convention\nto use orthogonal pilots to facilitate channel estimation for\nconventional massive MIMO systems, to our best knowledge, its\noptimality in one-bit massive MIMO systems has not been\nestablished before.\n\n\n\\subsection{Performance Analysis}\nWe now investigate the estimation performance when the optimal\nthresholds are employed, and its comparison with the performance\nattained by a conventional massive MIMO system which assumes\ninfinite-precision ADCs. Substituting the optimal thresholds\n(\\ref{optimumthreshold}) into the CRB matrix (\\ref{CRB}), we have\n\\begin{align}\n\\text{CRB}_{\\text{OQ}}(\\boldsymbol{h})=\\frac{\\pi\\sigma^2}{2}\n\\left( \\boldsymbol{A}^T \\boldsymbol{A} \\right)^{-1} \\label{CRB-Q}\n\\end{align}\nwhere for clarity, we use the subscript OQ to represent the\nestimation scheme using optimal quantization thresholds. On the\nother hand, when the unquantized observations $\\boldsymbol{y}$ are\navailable, it can be readily verified that the CRB matrix is given\nas\n\\begin{align}\n\\text{CRB}_{\\text{NQ}}(\\boldsymbol{h})=\\sigma^2 \\left(\n\\boldsymbol{A}^T \\boldsymbol{A} \\right)^{-1} \\label{CRB-NQ}\n\\end{align}\nwhere we use the subscript NQ to represent the scheme which has\naccess to the unquantized observations. Comparing (\\ref{CRB-Q})\nwith (\\ref{CRB-NQ}), we can see that if optimal thresholds are\nemployed, then using one-bit ADCs for channel estimation incurs\nonly a mild performance loss relative to using infinite-precision\nADCs, with the CRB increasing by only a factor of $\\pi\/2$, i.e.\n\\begin{align}\n\\text{CRB}_{\\text{OQ}}(\\boldsymbol{h})=\\frac{\\pi}{2}\n\\text{CRB}_{\\text{NQ}}(\\boldsymbol{h})\n\\end{align}\nWe also take a glimpse of the estimation performance as the\nthresholds deviate from their optimal values. For simplicity, let\n$\\tau_n=\\tau_n^{\\star}+\\delta=\\boldsymbol{a}_n^{T}\\boldsymbol{h}+\\delta,\n\\forall n$, in which case the CRB matrix is given by\n\\begin{align}\n\\text{CRB}_{\\text{Q}}(\\boldsymbol{h})=\\frac\n{F_{w}(\\delta)(1-F_{w}(\\delta))}{f_{w}^2 (\\delta)}\\left(\n\\boldsymbol{A}^T \\boldsymbol{A} \\right)^{-1}\n\\end{align}\nSince $(F_{w}(\\delta)(1-F_{w}(\\delta)))\/f_{w}^2(\\delta)$ is the\nreciprocal of $g(\\tau_n,\\boldsymbol{a}_n)$, from Fig. \\ref{fig1},\nwe know that the function value\n$(F_{w}(\\delta)(1-F_{w}(\\delta)))\/f_{w}^2(\\delta)$ grows\nexponentially as $|\\delta|$ increases. This indicates that a\ndeviation of the thresholds from their optimal values results in a\nsubstantial performance loss.\n\n\nIn summary, the above results have important implications for the\ndesign of one-bit massive MIMO systems. It points out that a\ncareful choice of quantization thresholds can help improve the\nestimation performance significantly, and help achieve an\nestimation accuracy close to an ideal estimator which has access\nto the raw observations $\\boldsymbol{y}$.\n\n\n\nThe problem lies in that the optimal thresholds\n$\\boldsymbol{\\tau}$ are functions of $\\boldsymbol{h}$, as\ndescribed in (\\ref{optimumthreshold}). Since $\\boldsymbol{h}$ is\nunknown and to be estimated, the optimal thresholds\n$\\boldsymbol{\\tau}$ are also unknown. To address this difficulty,\nwe, in the following, propose an adaptive quantization (AQ) scheme\nby which the thresholds are dynamically adjusted from one\niteration to another, and a random quantization (RQ) schme which\nrandomly generates a set of nonidentical thresholds based on some\nstatistical prior knowledge of the channel.\n\n\n\n\n\n\n\n\n\\section{Practical Threshold Design Strategies} \\label{sec:AQ-RQ}\n\\subsection{Adaptive Quantization}\nOne strategy to overcome the above difficulty is to use an\niterative algorithm in which the thresholds are iteratively\nrefined based on the previous estimate of $\\boldsymbol{h}$.\nSpecifically, at iteration $i$, we use the current quantization\nthresholds $\\boldsymbol{\\tau}^{(i)}$ to generate the one-bit\nobservation data $\\boldsymbol{b}^{(i)}$. Then a new estimate\n$\\hat{\\boldsymbol{h}}^{(i)}$ is obtained from the ML estimator\n(\\ref{MLE}). This estimate is then plugged in\n(\\ref{optimumthreshold}) to obtain updated quantization\nthresholds, i.e. $\\boldsymbol{\\tau}^{(i+1)}=\\boldsymbol{A}\n\\hat{\\boldsymbol{h}}^{(i)}$, for subsequent iteration. When\ncomputing the ML estimate $\\hat{\\boldsymbol{h}}^{(i)}$, not only\nthe quantized data from the current iteration but also from all\nprevious iterations can be used. The ML estimator (\\ref{MLE}) can\nbe easily adapted to accommodate these quantized data since the\ndata are independent across different iterations. Due to the\nconsistency of the ML estimator for large data records, this\niterative process will asymptotically lead to optimal quantization\nthresholds, i.e. $\\boldsymbol{\\tau}^{(i)} \\stackrel{i \\to\n\\infty}{\\longrightarrow} \\boldsymbol{A} \\boldsymbol{h}$. In fact,\nour simulation results show that the adaptive quantization scheme\nyields quantization thresholds close to the optimal values within\nonly a few iterations.\n\nFor clarity, we summarize the adaptive quantization (AQ) scheme as\nfollows.\n\n\\begin{center}\n\\textbf{Adaptive Quantization Scheme}\n\\end{center}\n\\vspace{0cm} \\noindent\n\\begin{tabular}{lp{7.7cm}}\n\\hline 1.& Select an initial quantization threshold\n$\\boldsymbol{\\tau}^{(0)}$ and the maximum number of iterations $i_{\\text{max}}$. \\\\\n2.& At iteration $i=1,2,\\ldots$: Based on $\\boldsymbol{y}$ and\n$\\boldsymbol{\\tau}^{(i)}$, calculate the new binary data\n$\\boldsymbol{b}^{(i)}=\\text{sgn}(\\boldsymbol{y}-\\boldsymbol{\\tau}^{(i)})$. \\\\\n3.& Compute a new estimate of $\\boldsymbol{h}$,\n$\\hat{\\boldsymbol{h}}^{(i)}$,\nvia (\\ref{MLE}). \\\\\n4.& Calculate new thresholds according to $\\boldsymbol{\\tau}^{(i+1)}=\\boldsymbol{A}\\hat{\\boldsymbol{h}}^{(i)}$. \\\\\n5.& Go to Step 2 if $i < i_{\\text{max}}$. \\\\\n\\hline\n\\end{tabular}\n\n\nNote that during the iterative process, the channel\n$\\boldsymbol{h}$ is assumed constant over time. Thus the AQ scheme\ncan be used to estimate channels that are unchanged or slowly\ntime-varying across a number of consecutive frames. For example,\nfor the scenario where the relative speeds between the mobile\nterminals and the base station are slow, say, 2 meters per second,\nthe channel coherence time could be up to tens of milliseconds,\nmore precisely, about 60 milliseconds if the carrier frequency is\nset to 1GHz, according to the Clarke's model\n\\cite{TseViswanath05}. Suppose the time duration of each frame is\n10 milliseconds which is a typical value for practical LTE\nsystems. In this case, the channel remains unchanged across 6\nconsecutive frames. We can use the AQ scheme to update the\nquantization thresholds at each frame based on the channel\nestimate obtained from the previous frame. In this way, we can\nexpect that the quantization thresholds will come closer and\ncloser to the optimal values from one frame to the next, and as a\nresult, a more and more accurate channel estimate can be obtained.\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{AQ.eps}\n\\caption{An off-line implementation of the AQ scheme.}\n\\label{fig2}\n\\end{figure}\n\nThe above scheme assumes a static or slowly time-varying channel\nacross multiple frames. Another way of implementing the AQ scheme\nrequires no such an assumption, but at the expense of increased\nhardware complexity. The idea is to use a number of\nsample-and-hold (S\/H) circuits to sample the analog received\nsignals and to store their values for subsequent offline\nprocessing. Specifically, each antenna\/RF chain is followed by\n$2L$ S\/H circuits which are equally divided into two groups to\nsample and store the real and imaginary components, respectively\n(see Fig. \\ref{fig2}). Through a precise timing control, we ensure\nthat at each antenna, say, the $m$th antenna, the $l$th S\/H\ncircuit pair in the two groups are controlled to store the real\nand imaginary components of the $l$th received pilot symbol, i.e.\n$\\Re(y_{m,l})$ and $\\Im(y_{m,l})$, respectively. Also, to avoid\nusing a one-bit ADC for each S\/H circuit, a switch can be used to\nconnect a single one-bit ADC with multiple S\/H circuits. Once the\nanalog signals $\\boldsymbol{y}$ have been stored, the AQ scheme\ncan be implemented in an offline manner. Clearly, this offline\napproach can be implemented on a single frame basis, and thus no\nlonger requires a static channel assumption. Nevertheless, such an\nimplementation requires a number of S\/H circuits as well as\nprecise timing control for sampling and quantization. Also, this\noffline processing may cause a latency issue which should be taken\ncare of in practical systems.\n\n\n\n\n\n\n\n\n\\subsection{Random Quantization}\nThe AQ scheme requires the channel to be (approximately)\nstationary, or needs to be implemented with additional hardware\ncircuits. Here we propose a random quantization (RQ) scheme that\ndoes not involve any iterative procedure and is simple to\nimplement. The idea is to randomly generate a set of non-identical\nthresholds based on some statistical prior knowledge of\n$\\boldsymbol{h}$, with the hope that some of the thresholds are\nclose to the unknown optimal thresholds. For example, suppose each\nentry of $\\boldsymbol{h}$ follows a Gaussian distribution with\nzero mean and variance $\\sigma_h^2$. Note that different entries\nof $\\boldsymbol{h}$ may have different variances due to the reason\nthat they may correspond to different users. Nevertheless, we\nassume the same variance for all entries for simplicity. We\nrandomly generate $N$ different realizations of $\\boldsymbol{h}$,\ndenoted as $\\{\\boldsymbol{\\tilde{h}}_n\\}$, following this known\ndistribution. The $N$ quantization thresholds are then devised\naccording to\n\\begin{align}\n\\tau_n=\\boldsymbol{a}_n^{T}\\boldsymbol{\\tilde{h}}_n, \\quad \\forall\nn\\in\\{1,\\ldots,N\\} \\label{multi-thresholding}\n\\end{align}\nOur simulation results suggest that this RQ scheme can achieve a\nconsiderable performance improvement over the conventional fixed\nquantization scheme which uses a fixed (typically zero) threshold.\nThe reason is that the thresholds produced by\n(\\ref{multi-thresholding}) are more likely to be close to their\noptimal values.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Simulation Results} \\label{sec:experiments}\nWe now carry out experiments to corroborate our theoretical\nanalysis and to illustrate the performance of our proposed one-bit\nquantization schemes, i.e. the AQ and the RQ schemes. We compare\nour schemes with the conventional fixed quantization (FQ) scheme\nwhich employs a fixed zero threshold for one-bit quantization, and\na no-quantization scheme (referred to as NQ) which uses the\noriginal unquantized data for channel estimation. For the NQ\nscheme, it can be easily verified that its ML estimate is given by\n\\begin{align}\n\\boldsymbol{\\hat{h}}=(\\boldsymbol{A}^T\\boldsymbol{A})^{-1}\\boldsymbol{A}^T\\boldsymbol{y}\n\\end{align}\nand its associated CRB is given by (\\ref{CRB-NQ}). For other\nschemes such as the RQ and the FQ, although a close-form\nexpression is not available, the ML estimate can be obtained by\nsolving the convex optimization (\\ref{MLE}). In our simulations,\nwe assume independent and identically distributed (i.i.d.)\nrayleigh fading channels, i.e. all elements of $\\boldsymbol{H}$\nfollow a circularly symmetric complex Gaussian distribution with\nzero mean and unit variance. Training sequences $\\boldsymbol{X}$\nwhich satisfy (\\ref{theorem2:eqn1}) are randomly generated. The\nsignal-to-noise ratio (SNR) is defined as\n\\begin{align}\n\\text{SNR}=\\frac{P}{KL\\sigma^2}\n\\end{align}\n\n\n\\begin{figure}[!t]\n \\centering\n\\begin{tabular}{c}\n\\includegraphics[width=3.5in]{msevsitr1}\\\\\n(a). $K=8$, $L=32$ and $\\text{SNR}=15$ dB. \\\\\n\\includegraphics[width=3.5in]{msevsitr2}\\\\\n(b). $K=16$, $L=40$ and $\\text{SNR}=15$ dB.\n\\end{tabular}\n  \\caption{MSEs of the AQ scheme as a function of the number of iterations.}\n   \\label{fig3}\n\\end{figure}\n\n\nWe first examine the estimation performance of our proposed AQ\nscheme which adaptively adjusts the thresholds based on the\nprevious estimate of the channel. Fig. \\ref{fig3} plots the\nmean-squared errors (MSEs) vs. the number of iterations for the AQ\nscheme, where we set $K=8$, $L=32$ for Fig. (a) and $K=16$, $L=40$\nfor Fig. (b). The SNR is set to 15dB. The MSE is calculated as\n\\begin{align}\n\\text{MSE}=\\frac{1}{K M}\n\\|\\boldsymbol{H}-\\boldsymbol{\\hat{H}}\\|_F^2\n\\end{align}\nTo better illustrate the effectiveness of the AQ scheme, we also\ninclude the CRB results in Fig. \\ref{fig3}. in which the CRB-OQ,\ngiven by (\\ref{CRB-Q}), represents the theoretical lower bound on\nthe estimation errors of any unbiased estimator using optimal\nthresholds for one-bit quantization, and the CRB-NQ, given by\n(\\ref{CRB-NQ}), represents the lower bound on the estimation\nerrors of any unbiased estimator which has access to the original\nobservations. From Fig. \\ref{fig3}, we see that our proposed AQ\nscheme approaches the theoretical lower bound CRB-OQ within only a\nfew (say, 5) iterations, and achieves performance close to the CRB\nassociated with the NQ scheme. This result demonstrates the\neffectiveness of the AQ scheme in searching for the optimal\nthresholds. In the rest of our simulations, we set the maximum\nnumber of iterations, $i_{\\text{max}}$, equal to 5 for the AQ\nscheme.\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{msevsL}\n\\caption{MSEs vs. number of pilot symbols, where $K=8$ and\n$\\text{SNR}=15$ dB.} \\label{fig4}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{msevsSNR}\n\\caption{MSEs vs. SNR(dB), where $K=8$ and $L=96$.} \\label{fig5}\n\\end{figure}\n\n\n\nWe now compare the estimation performance of different schemes.\nFig. \\ref{fig4} plots the MSEs of respective schemes as a function\nof the number of pilot symbols, $L$, where we set $K=8$ and\n$\\text{SNR}=15\\text{dB}$. The corresponding CRBs of these schemes\nare also included. Note that the CRBs for the FQ and the RQ\nschemes can be obtained by substituting the thresholds into\n(\\ref{CRB}). Results are averaged over $10^3$ independent runs,\nwith the channel and the pilot sequences randomly generated for\neach run. From Fig. \\ref{fig4}, we can see that the proposed AQ\nscheme outperforms the FQ and RQ schemes by a big margin. This\nresult corroborates our analysis that an optimal choice of the\nquantization thresholds helps achieve a substantial performance\nimprovement. In particular, the AQ scheme needs less than 30 pilot\nsymbols to achieve a decent estimation accuracy with a MSE of\n0.01, while the FQ and RQ schemes require a much larger number of\npilot symbols to attain a same estimation accuracy. On the other\nhand, we should note that although the AQ scheme has the potential\nto achieve performance close to the NQ scheme, the implementation\nof the AQ is more complicated since it involves an iterative\nprocess to learn the optimal thresholds. In contrast, our proposed\nRQ scheme is as simple as the FQ scheme to implement, meanwhile it\npresents a clear performance advantage over the FQ scheme. We can\nsee from Fig. \\ref{fig4} that the RQ requires about 100 symbols to\nachieve a MSE of 0.1, whereas the FQ needs about 250 pilot symbols\nto reach a same estimation accuracy. The reason why the RQ\nperforms better than the FQ is that some of the thresholds\nproduced according to (\\ref{multi-thresholding}) are likely to be\nclose to the optimal thresholds. In Fig. \\ref{fig5}, we plot the\nMSEs of respective schemes under different SNRs, where we set\n$K=8$ and $L=96$. Similar conclusions can be made from Fig.\n\\ref{fig5}.\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{servsL}\n\\caption{SERs vs. number of pilot symbols, where $K=8$, $M=64$ and\n$\\text{SNR}=5\\text{dB}$.} \\label{fig6}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{ratevsL}\n\\caption{Achievable rates vs. number of pilot symbols, where\n$K=8$, $M=64$ and $\\text{SNR}=5\\text{dB}$.} \\label{fig7}\n\\end{figure}\n\n\n\n\nNext, we examine the effect of channel estimation accuracy on the\nsymbol error rate (SER) performance. For each scheme, after the\nchannel is estimated, a near maximum likelihood detector\n\\cite{ChoiMo16} developed for one-bit massive MIMO is adopted for\nsymbol detection. For a fair comparison, in the symbol detection\nstage, the quantization thresholds are all set equal to zero, as\nassumed in \\cite{ChoiMo16}. In our experiments, QPSK symbols are\ntransmitted by all users. Fig. \\ref{fig6} plots the SERs of\nrespective schemes vs. the number of pilot symbols, where we set\n$K=8$, $M=64$, and $\\text{SNR}=5\\text{dB}$. Results are averaged\nover all $K$ users. The SER performance obtained by assuming\nperfect channel knowledge is also included. It can be seen that\nthe SER performance improves as the number of pilot symbols\nincreases, which is expected since a more accurate channel\nestimate can be obtained when more pilot symbols are available for\nchannel estimation. We also observe that the AQ scheme, using a\nmoderate number (about 120 symbols that is only 15 times the\nnumber of users) of pilot symbols, can achieve SER performance\nclose to that attained by assuming perfect channel knowledge.\nMoreover, the SER results, again, demonstrate the superiority of\nthe RQ over the FQ scheme. In order to attain a same SER, say,\n$10^{-3}$, the RQ requires about 60 pilot symbols, whereas the FQ\nrequires about 100 pilot symbols.\n\nIn Fig. \\ref{fig7}, the achievable rates of respective schemes vs.\nthe number of pilot symbols are depicted, where we set $K=8$,\n$M=64$, and $\\text{SNR}=5\\text{dB}$. The achievable rate for the\n$k$th user is calculated as \\cite{MollenChoi17}\n\\begin{align}\nR_k \\triangleq \\log_2 \\left( 1+ \\frac{\n|\\mathbb{E}\\left[s_k^*(t)\\hat{s}_k(t)\\right]|^2 }\n{\\mathbb{E}\\left[|\\hat{s}_k(t)|^2\\right] -\n|\\mathbb{E}\\left[s_k^*(t)\\hat{s}_k(t)\\right]|^2} \\right)\n\\end{align}\nwhere $s_k(t)$ is the transmit symbol of the $k$th user at time\n$t$, $()^{*}$ denotes the conjugate, and $\\hat{s}_k(t)$ is the\nestimated symbol of $s_k(t)$, which is obtained via the near\nmaximum likelihood detector by using the channel estimated by\nrespective schemes. The achievable rate we plotted is averaged\nover all $K$ users. It can be seen that, even with a moderate\nnumber of pilot symbols (about 5 times the number of users), the\nAQ scheme can provide an achievable rate close to that of the\nperfect CSI case, whereas the achievable rates attained by the\nother two schemes are far below the level of the AQ scheme.\nCompared to the FQ, the RQ scheme achieves an increase of about 30\npercent in the achievable rate.\n\n\n\n\n\n\\section{Conclusions} \\label{sec:conclusion}\nAssuming one-bit ADCs at the BS, we studied the problem of one-bit\nquantization design and channel estimation for uplink multiuser\nmassive MIMO systems. Specifically, based on the derived CRB\nmatrix, we examined the impact of quantization thresholds on the\nchannel estimation performance. Our theoretical analysis revealed\nthat using one-bit ADCs can achieve an estimation error close to\nthat attained by using infinite-precision ADCs, given that the\nquantization thresholds are optimally set. Our analysis also\nsuggested that the optimal quantization thresholds are dependent\non the unknown channel parameters. We developed two practical\nquantization design schemes, namely, an adaptive quantization\nscheme which adaptively adjusts the thresholds such that the\nthresholds converge to the optimal thresholds, and a random\nquantization scheme which randomly generates a set of\nnon-identical thresholds based on some statistical prior knowledge\nof the channel. Simulation results showed that the proposed\nquantization schemes achieved a significant performance\nimprovement over the fixed quantization scheme that uses a fixed\n(typically zero) quantization threshold, and thus can help\nsubstantially reduce the training overhead in order to attain a\nsame estimation accuracy target.\n\n\n\n\n\n\n\n\n\\useRomanappendicesfalse\n\\appendices\n\n\\section{Proof of Concavity of The Log-Likelihood Function (\\ref{log-PMF})} \\label{appA}\nIt can be easily verified that\n$f_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)$ is log-concave\nin $\\boldsymbol{h}$ since the Hessian matrix of $\\log\nf_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)$, which is given\nby\n\\begin{align}\n\\frac {\\partial^2 \\log\nf_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)} {{\\partial\n\\boldsymbol{h} \\partial \\boldsymbol{h}^T}} = -\n\\frac{\\boldsymbol{a}_n \\boldsymbol{a}_n^{T}} {\\sigma^2}\n\\end{align}\nis negative semidefinite. Consequently the corresponding\ncumulative density function (CDF) and complementary CDF (CCDF),\nwhich are integrals of the log-concave function\n$f_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)$ over convex\nsets $(-\\infty,\\tau_n)$ and $(\\tau_n,\\infty)$ respectively, are\nalso log-concave, and their logarithms are concave. Since\nsummation preserves concavity, $L(\\boldsymbol{h})$ is a concave\nfunction of $\\boldsymbol{h}$.\n\n\n\\section{Proof of Theorem \\ref{theorem1}} \\label{appB}\nDefine a new variable $z_n\\triangleq\n\\boldsymbol{a}_n^{T}\\boldsymbol{h}$ and define\n\\begin{align}\nl(z_n) & \\triangleq  \\frac{1-b_n}{2}  \\mathrm{log} [1-F_{w}(z_n-\\tau_n)]   \\nonumber \\\\\n& \\quad + \\frac{1+b_n}{2} \\mathrm{log} [F_{w}(z_n-\\tau_n)].\n\\end{align}\nThe first and second-order derivative of $L(\\boldsymbol{h})$ are\ngiven by\n\\begin{align}\n\\frac {\\partial L(\\boldsymbol{h})} {\\partial \\boldsymbol{h}} =\n\\sum_{n=1}^{N} \\frac {\\partial l(z_n)} {\\partial z_n} \\frac\n{\\partial z_n} {\\partial \\boldsymbol{h}} = \\sum_{n=1}^{N} \\frac\n{\\partial l(z_n)} {\\partial z_n} \\boldsymbol{a}_n\n\\end{align}\nand\n\\begin{align}\n\\frac {\\partial^2 L(\\boldsymbol{h})} {\\partial \\boldsymbol{h}\n\\partial \\boldsymbol{h}^T}\n&= \\sum_{n=1}^{N} \\boldsymbol{a}_n \\frac {\\partial^2 l(z_n)}\n{\\partial z_n^2}\n\\frac {\\partial z_n} {\\partial \\boldsymbol{h}^T} \\nonumber \\\\\n&= \\sum_{n=1}^{N} \\frac {\\partial^2 l(z_n)} {\\partial z_n^2}\n\\boldsymbol{a}_n \\boldsymbol{a}_n^T .\n\\end{align}\nwhere\n\\begin{align}\n\\frac {\\partial l(z_n)} {\\partial z_n} &= \\frac{1-b_n}{2}  \\frac{f_{w}(z_n-\\tau_n)}{F_{w}(z_n-\\tau_n)-1}   \\nonumber \\\\\n& \\quad + \\frac{1+b_n}{2}\n\\frac{f_{w}(z_n-\\tau_n)}{F_{w}(z_n-\\tau_n)} \\label{deriv1}\n\\end{align}\nand\n\\begin{align}\n\\frac {\\partial^2 l(z_n)} {\\partial z_n^2} =& \\frac{1-b_n}{2}\n\\bigg[\n\\frac{f'_{w}(z_n-\\tau_n)}{F_{w}(z_n-\\tau_n)-1} \\nonumber \\\\\n& -\\frac{f_{w}^2 (z_n-\\tau_n)}{(F_{w}(z_n-\\tau_n)-1)^2} \\bigg] + \\frac{1+b_n}{2}   \\nonumber \\\\\n&  \\cdot \\bigg[ \\frac{f'_{w}(z_n-\\tau_n)}{F_{w}(z_n-\\tau_n)} -\n\\frac{f_{w}^2 (z_n-\\tau_n)}{F_{w}^2 (z_n-\\tau_n)} \\bigg]\n\\label{deriv2}\n\\end{align}\nwhere $f_{w}(x)$ denotes the probability density function (PDF) of\n$w_n$, and $f'_{w}(x)\\triangleq\\frac{\\partial f_{w}(x)}{\\partial\nx}$.\n\nTherefore, the Fisher information matrix (FIM) of the estimation\nproblem is given as\n\\begin{align}\nJ(\\boldsymbol{h}) & = -E \\left[\\frac {\\partial^2\nL(\\boldsymbol{h})} {\\partial \\boldsymbol{h} \\partial\n\\boldsymbol{h}^T} \\right]\n= - \\sum_{n=1}^{N} E_{b_n} \\left[ \\frac {\\partial^2 l(z_n)} {\\partial z_n^2} \\right]\n\\boldsymbol{a}_n \\boldsymbol{a}_n^T  \\nonumber \\\\\n& \\stackrel {(a)}{=} \\sum_{n=1}^{N} \\frac {f_{w}^2\n(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)}\n{F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)(1-F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n))}\n\\boldsymbol{a}_n \\boldsymbol{a}_n^T\n\\end{align}\nwhere $E_{b_n}[\\cdot]$ denotes the expectation with respect to the\ndistribution of $b_n$, and $(a)$ follows from the fact that\n$b_{n}$ is a binary random variable with\n$P(b_{n}=1|\\tau_n,z_n)=F_{w}(z_n-\\tau_n)$ and\n$P(b_{n}=-1|\\tau_n,z_n)=1-F_{w}(z_n-\\tau_n)$. This completes the\nproof.\n\n\n\n\n\n\n\n\\section{Proof of Proposition \\ref{proposition1}} \\label{appE}\nBefore proceeding, we first introduce the following lemma.\n\\newtheorem{lemma}{Lemma}\n\\begin{lemma} \\label{lemma2}\nFor $x\\ge 0$, define\n\\begin{align}\n\\bar{F}(x) \\triangleq \\int_{0}^{x} f(u)\\mathrm{d}u\n\\end{align}\nwhere $f(\\cdot)$ denotes the PDF of a real-valued Gaussian random\nvariable with zero-mean and unit variance. We have $\\bar{F}(x)$\nupper bounded by\n\\begin{align}\n\\bar{F}(x) \\le \\frac{1}{2} \\sqrt{ 1-e^{-\\frac{2x^2}{\\pi}} } .\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{appF}.\n\\end{proof}\n\n\nDefine the function\n\\begin{align}\n\\bar{g}(x) \\triangleq \\frac {f^2 (x)} {F(x)(1-F(x))}\n\\end{align}\nwhere $f(\\cdot)$ and $F(\\cdot)$ denotes the PDF and CDF of a\nreal-valued Gaussian random variable with zero-mean and unit\nvariance respectively. Invoking Lemma \\ref{lemma2}, we have\n\\begin{align}\n\\bar{g}(x) =\\frac {f^2 (x)} {\\frac{1}{4}-\\bar{F}^2(x)} \\le\n\\frac{2}{\\pi} e^{-(1-\\frac{2}{\\pi})x^2} \\le \\frac{2}{\\pi}.\n\\end{align}\nand $\\bar{g}(x) =\\frac{2}{\\pi}$ if and only if $x=0$. Noting that\n\\begin{align}\n\\frac{1}{\\sigma^2} \\bar{g} \\left(\n\\frac{\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n}{\\sigma} \\right) =\ng(\\tau_n,\\boldsymbol{a}_n) .\n\\end{align}\nTherefore $g(\\tau_n,\\boldsymbol{a}_n)$ attains its maximum when\n$\\tau_n=\\boldsymbol{a}_n^{T}\\boldsymbol{h}$. The proof is\ncompleted here.\n\n\n\\section{Proof of Lemma \\ref{lemma2}} \\label{appF}\nDefine two i.i.d. Gaussian random variables with zero-mean and\nunit variance, namely, $X$ and $Y$. The joint distribution\nfunction of $X$ and $Y$ is $f_{XY}(x,y)=f(x)f(y)$. Define two\nregions $D_1 \\triangleq \\{(u,v)\\mid 0 \\le u \\le x , 0 \\le v \\le x\n\\}$ and $D_2 \\triangleq \\{(u,v)\\mid u \\ge 0 , v \\ge 0, u^2+v^2\\le\n\\frac{4x^2}{\\pi} \\}$. Obviously, the areas of $D_1$ and $D_2$ are\nthe same, i.e., $\\mu(D_1)=\\mu(D_2)$, where $\\mu(\\cdot)$ denote the\narea of a region. The probabilities of $(X,Y)$ belonging in these\ntwo regions can be computed as\n\\begin{align}\nP((X,Y)\\in D_1) &= \\iint_{D_1} f_{XY}(u,v) \\mathrm{d}u\\mathrm{d}v \\nonumber \\\\\n&= \\bar{F}^2(x) \\\\\nP((X,Y)\\in D_2) &= \\iint_{D_2} f_{XY}(u,v) \\mathrm{d}u\\mathrm{d}v \\nonumber \\\\\n&= \\frac{1}{4} \\left(1-e^{-\\frac{2x^2}{\\pi}}\\right)\n\\end{align}\nLet $S_1\\setminus S_2$ denote the set obtained by excluding\n$S_2\\cap S_1$ from $S_1$. Clearly, we have\n\\begin{align}\n\\mu(D_1 \\setminus D_2)=\\mu(D_2 \\setminus D_1) \\label{appF:eqn1}\n\\end{align}\nAlso, according to the definition of $D_1$ and $D_2$, we have\n\\begin{align}\nf_{XY}(u,v)&\\le \\frac{1}{2\\pi} e^{-\\frac{2x^2}{\\pi}}, \\quad (u,v)\\in D_1 \\setminus D_2 \\\\\nf_{XY}(u,v)&\\ge \\frac{1}{2\\pi} e^{-\\frac{2x^2}{\\pi}}, \\quad\n(u,v)\\in D_2 \\setminus D_1 \\label{appF:eqn2}\n\\end{align}\nCombining (\\ref{appF:eqn1})--(\\ref{appF:eqn2}), we arrive at\n\\begin{align}\n\\iint_{D_1 \\setminus D_2} f_{XY}(u,v) \\mathrm{d}u\\mathrm{d}v \\le\n\\iint_{D_2 \\setminus D_1} f_{XY}(u,v) \\mathrm{d}u\\mathrm{d}v\n\\end{align}\nFrom the above inequality, we have $P((X,Y)\\in D_1)\\le P((X,Y)\\in\nD_2)$, i.e.\n\\begin{align}\n\\bar{F}^2(x)\\leq\\frac{1}{4}\n\\left(1-e^{-\\frac{2x^2}{\\pi}}\\right)\\Rightarrow \\bar{F}(x) \\le\n\\frac{1}{2} \\sqrt{ 1-e^{-\\frac{2x^2}{\\pi}} }\n\\end{align}\nThis completes the proof.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem \\ref{theorem2}} \\label{appC}\nNote that from the constraint\n$\\text{tr}(\\boldsymbol{X}\\boldsymbol{X}^H)\\leq P$, we can easily\nderive that\n\\begin{align}\n\\text{tr}(\\boldsymbol{A}^T \\boldsymbol{A})\\leq 2M P\n\\end{align}\nTo prove Theorem \\ref{theorem2}, let us first consider a new\noptimization that has the same objective function as (\\ref{opt2})\nwhile with a relaxed constraint:\n\\begin{align}\n\\min_{\\boldsymbol{A}} \\quad & \\frac{\\pi\\sigma^2}{2}\\text{tr}\n\\left\\{ \\left( \\boldsymbol{A}^T \\boldsymbol{A} \\right)^{-1}\n\\right\\}\n\\nonumber\\\\\n\\text{s.t.} \\quad & \\text{tr}(\\boldsymbol{A}^T\\boldsymbol{A})\\leq\n2M P  \\label{appC:opt1}\n\\end{align}\nClearly, the feasible region defined by the constraints in\n(\\ref{opt2}) is a subset of that defined by (\\ref{appC:opt1}).\nSince $\\text{tr}(\\boldsymbol{Z}^{-1})$ is convex over the set of\npositive definite matrix, the optimization (\\ref{appC:opt1}) is\nconvex. Its optimum solution is given as follows.\n\\begin{lemma} \\label{lemma1}\nConsider the following optimization problem\n\\begin{align}\n\\min_{\\boldsymbol{Z}}\\quad &\\text{tr}(\\boldsymbol{Z}^{-1})\n\\nonumber\\\\\n\\text{s.t.}\\quad & \\text{tr}(\\boldsymbol{Z})\\leq P_0\n\\label{appC:opt2}\n\\end{align}\nwhere $\\boldsymbol{Z}\\in\\mathbb{R}^{p\\times p}$ is positive\ndefinite. The optimum solution to (\\ref{appC:opt2}) is given by\n$\\boldsymbol{Z}=(P_0\/p)\\boldsymbol{I}$ and the minimum objective\nfunction value is $p^2\/P_0$.\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{appD}.\n\\end{proof}\n\nFrom Lemma \\ref{lemma1}, we know that any $\\boldsymbol{A}$\nsatisfying\n\\begin{align}\n\\boldsymbol{A}^T \\boldsymbol{A} = (P\/K) \\boldsymbol{I}\n\\label{appC:eqn1}\n\\end{align}\nis an optimal solution to (\\ref{appC:opt1}). Note that the set of\nfeasible solutions (\\ref{appC:opt1}) subsumes the feasible\nsolution set of (\\ref{opt2}). Hence, if the optimal solution to\n(\\ref{appC:opt1}) is meanwhile a feasible solution of\n(\\ref{opt2}), then this solution is also an optimal solution to\n(\\ref{opt2}). It is easy to verify that if (\\ref{theorem2:eqn1})\nholds valid, the resulting $\\boldsymbol{A}$ satisfies\n(\\ref{appC:eqn1}) and is thus an optimal solution to\n(\\ref{appC:opt1}). As a consequence, it is also an optimal\nsolution to (\\ref{opt2}). This completes the proof.\n\n\n\n\n\n\n\n\n\\section{Proof of Lemma \\ref{lemma1}} \\label{appD}\nLet $\\boldsymbol{Z}=\\boldsymbol{U}\\boldsymbol{D}\\boldsymbol{U}^T$\ndenote the eigenvalue decomposition of $\\boldsymbol{Z}$, where\n$\\boldsymbol{U}\\in\\mathbb{R}^{p\\times p}$ and\n$\\boldsymbol{D}\\in\\mathbb{R}^{p\\times p}$. By replacing\n$\\boldsymbol{Z}$ with\n$\\boldsymbol{U}\\boldsymbol{D}\\boldsymbol{U}^T$, the optimization\n(\\ref{appC:opt2}) is reduced to determining the diagonal matrix\n$\\boldsymbol{D}\\triangleq \\text{diag}(d_1,\\dots,d_{p})$\n\\begin{align}\n\\min_{\\{d_i\\}} \\ & \\sum_{i=1}^{p} \\frac{1}{d_i}\n\\nonumber\\\\\n\\text{s.t.} \\ \\,\n& \\sum_{i=1}^{p} {d_i} \\leq P_0 \\nonumber\\\\\n& d_i > 0, \\qquad \\forall i\\in\\{1,\\dots,p\\}\\label{opt5}\n\\end{align}\nThe Lagrangian function associated with (\\ref{opt5}) is given by\n\\begin{align}\nL(d_i;\\lambda;\\nu_i)=\\sum_{i=1}^{p} \\frac{1}{d_i} + \\lambda \\left(\n\\sum_{i=1}^{p} {d_i} - P_0 \\right) - \\sum_{i=1}^{p} {\\nu_i d_i}\n\\end{align}\nwith KKT conditions \\cite{BoydVandenberghe03} given as\n\\begin{align}\n-\\frac{1}{d_i^2}+\\lambda-\\nu_i=0 & , \\quad \\forall i \\nonumber\\\\\n\\lambda\\left(\\sum_{i=1}^{p} {d_i} - P_0\\right)  =0 & \\nonumber\\\\\n\\lambda\\geq 0 & \\nonumber\\\\\n\\nu_i d_i=0 & ,\\quad \\forall i \\nonumber\\\\\nd_i >0 & , \\quad \\forall i \\nonumber\\\\\n\\nu_i\\ge 0 & , \\quad \\forall i \\nonumber\n\\end{align}\nFrom the last three equations, we have $\\nu_i=0$, $\\forall i$.\nThen from the first equation we have\n\\begin{align}\n\\lambda=\\frac{1}{d_i^2}>0 \\label{lambda}\n\\end{align}\nand\n\\begin{align}\nd_1=d_2=\\dots=d_{p}.\n\\end{align}\nFrom (\\ref{lambda}) and the second equation, we have\n$\\sum_{i=1}^{p} {d_i} - P_0 =0$, from which $d_i$ can be readily\nsolved as $d_i=P_0\/p$, $\\forall i$, i.e., the optimal\n$\\boldsymbol{D}$ is given by $\\boldsymbol{D}^{\\star}= (P_0\/p)\n\\boldsymbol{I}$. Consequently we have $\\boldsymbol{Z}^{\\star}=\n(P_0\/p) \\boldsymbol{I}$. This completed the proof.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}