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import torch
from einops import einsum, rearrange, reduce
from jaxtyping import Float
from scipy.spatial.transform import Rotation as R
from torch import Tensor
def interpolate_intrinsics(
initial: Float[Tensor, "*#batch 3 3"],
final: Float[Tensor, "*#batch 3 3"],
t: Float[Tensor, " time_step"],
) -> Float[Tensor, "*batch time_step 3 3"]:
initial = rearrange(initial, "... i j -> ... () i j")
final = rearrange(final, "... i j -> ... () i j")
t = rearrange(t, "t -> t () ()")
return initial + (final - initial) * t
def intersect_rays(
a_origins: Float[Tensor, "*#batch dim"],
a_directions: Float[Tensor, "*#batch dim"],
b_origins: Float[Tensor, "*#batch dim"],
b_directions: Float[Tensor, "*#batch dim"],
) -> Float[Tensor, "*batch dim"]:
"""Compute the least-squares intersection of rays. Uses the math from here:
https://math.stackexchange.com/a/1762491/286022
"""
# Broadcast and stack the tensors.
a_origins, a_directions, b_origins, b_directions = torch.broadcast_tensors(
a_origins, a_directions, b_origins, b_directions
)
origins = torch.stack((a_origins, b_origins), dim=-2)
directions = torch.stack((a_directions, b_directions), dim=-2)
# Compute n_i * n_i^T - eye(3) from the equation.
n = einsum(directions, directions, "... n i, ... n j -> ... n i j")
n = n - torch.eye(3, dtype=origins.dtype, device=origins.device)
# Compute the left-hand side of the equation.
lhs = reduce(n, "... n i j -> ... i j", "sum")
# Compute the right-hand side of the equation.
rhs = einsum(n, origins, "... n i j, ... n j -> ... n i")
rhs = reduce(rhs, "... n i -> ... i", "sum")
# Left-matrix-multiply both sides by the inverse of lhs to find p.
return torch.linalg.lstsq(lhs, rhs).solution
def normalize(a: Float[Tensor, "*#batch dim"]) -> Float[Tensor, "*#batch dim"]:
return a / a.norm(dim=-1, keepdim=True)
def generate_coordinate_frame(
y: Float[Tensor, "*#batch 3"],
z: Float[Tensor, "*#batch 3"],
) -> Float[Tensor, "*batch 3 3"]:
"""Generate a coordinate frame given perpendicular, unit-length Y and Z vectors."""
y, z = torch.broadcast_tensors(y, z)
return torch.stack([y.cross(z), y, z], dim=-1)
def generate_rotation_coordinate_frame(
a: Float[Tensor, "*#batch 3"],
b: Float[Tensor, "*#batch 3"],
eps: float = 1e-4,
) -> Float[Tensor, "*batch 3 3"]:
"""Generate a coordinate frame where the Y direction is normal to the plane defined
by unit vectors a and b. The other axes are arbitrary."""
device = a.device
# Replace every entry in b that's parallel to the corresponding entry in a with an
# arbitrary vector.
b = b.detach().clone()
parallel = (einsum(a, b, "... i, ... i -> ...").abs() - 1).abs() < eps
b[parallel] = torch.tensor([0, 0, 1], dtype=b.dtype, device=device)
parallel = (einsum(a, b, "... i, ... i -> ...").abs() - 1).abs() < eps
b[parallel] = torch.tensor([0, 1, 0], dtype=b.dtype, device=device)
# Generate the coordinate frame. The initial cross product defines the plane.
return generate_coordinate_frame(normalize(a.cross(b)), a)
def matrix_to_euler(
rotations: Float[Tensor, "*batch 3 3"],
pattern: str,
) -> Float[Tensor, "*batch 3"]:
*batch, _, _ = rotations.shape
rotations = rotations.reshape(-1, 3, 3)
angles_np = R.from_matrix(rotations.detach().cpu().numpy()).as_euler(pattern)
rotations = torch.tensor(angles_np, dtype=rotations.dtype, device=rotations.device)
return rotations.reshape(*batch, 3)
def euler_to_matrix(
rotations: Float[Tensor, "*batch 3"],
pattern: str,
) -> Float[Tensor, "*batch 3 3"]:
*batch, _ = rotations.shape
rotations = rotations.reshape(-1, 3)
matrix_np = R.from_euler(pattern, rotations.detach().cpu().numpy()).as_matrix()
rotations = torch.tensor(matrix_np, dtype=rotations.dtype, device=rotations.device)
return rotations.reshape(*batch, 3, 3)
def extrinsics_to_pivot_parameters(
extrinsics: Float[Tensor, "*#batch 4 4"],
pivot_coordinate_frame: Float[Tensor, "*#batch 3 3"],
pivot_point: Float[Tensor, "*#batch 3"],
) -> Float[Tensor, "*batch 5"]:
"""Convert the extrinsics to a representation with 5 degrees of freedom:
1. Distance from pivot point in the "X" (look cross pivot axis) direction.
2. Distance from pivot point in the "Y" (pivot axis) direction.
3. Distance from pivot point in the Z (look) direction
4. Angle in plane
5. Twist (rotation not in plane)
"""
# The pivot coordinate frame's Z axis is normal to the plane.
pivot_axis = pivot_coordinate_frame[..., :, 1]
# Compute the translation elements of the pivot parametrization.
translation_frame = generate_coordinate_frame(pivot_axis, extrinsics[..., :3, 2])
origin = extrinsics[..., :3, 3]
delta = pivot_point - origin
translation = einsum(translation_frame, delta, "... i j, ... i -> ... j")
# Add the rotation elements of the pivot parametrization.
inverted = pivot_coordinate_frame.inverse() @ extrinsics[..., :3, :3]
y, _, z = matrix_to_euler(inverted, "YXZ").unbind(dim=-1)
return torch.cat([translation, y[..., None], z[..., None]], dim=-1)
def pivot_parameters_to_extrinsics(
parameters: Float[Tensor, "*#batch 5"],
pivot_coordinate_frame: Float[Tensor, "*#batch 3 3"],
pivot_point: Float[Tensor, "*#batch 3"],
) -> Float[Tensor, "*batch 4 4"]:
translation, y, z = parameters.split((3, 1, 1), dim=-1)
euler = torch.cat((y, torch.zeros_like(y), z), dim=-1)
rotation = pivot_coordinate_frame @ euler_to_matrix(euler, "YXZ")
# The pivot coordinate frame's Z axis is normal to the plane.
pivot_axis = pivot_coordinate_frame[..., :, 1]
translation_frame = generate_coordinate_frame(pivot_axis, rotation[..., :3, 2])
delta = einsum(translation_frame, translation, "... i j, ... j -> ... i")
origin = pivot_point - delta
*batch, _ = origin.shape
extrinsics = torch.eye(4, dtype=parameters.dtype, device=parameters.device)
extrinsics = extrinsics.broadcast_to((*batch, 4, 4)).clone()
extrinsics[..., 3, 3] = 1
extrinsics[..., :3, :3] = rotation
extrinsics[..., :3, 3] = origin
return extrinsics
def interpolate_circular(
a: Float[Tensor, "*#batch"],
b: Float[Tensor, "*#batch"],
t: Float[Tensor, "*#batch"],
) -> Float[Tensor, " *batch"]:
a, b, t = torch.broadcast_tensors(a, b, t)
tau = 2 * torch.pi
a = a % tau
b = b % tau
# Consider piecewise edge cases.
d = (b - a).abs()
a_left = a - tau
d_left = (b - a_left).abs()
a_right = a + tau
d_right = (b - a_right).abs()
use_d = (d < d_left) & (d < d_right)
use_d_left = (d_left < d_right) & (~use_d)
use_d_right = (~use_d) & (~use_d_left)
result = a + (b - a) * t
result[use_d_left] = (a_left + (b - a_left) * t)[use_d_left]
result[use_d_right] = (a_right + (b - a_right) * t)[use_d_right]
return result
def interpolate_pivot_parameters(
initial: Float[Tensor, "*#batch 5"],
final: Float[Tensor, "*#batch 5"],
t: Float[Tensor, " time_step"],
) -> Float[Tensor, "*batch time_step 5"]:
initial = rearrange(initial, "... d -> ... () d")
final = rearrange(final, "... d -> ... () d")
t = rearrange(t, "t -> t ()")
ti, ri = initial.split((3, 2), dim=-1)
tf, rf = final.split((3, 2), dim=-1)
t_lerp = ti + (tf - ti) * t
r_lerp = interpolate_circular(ri, rf, t)
return torch.cat((t_lerp, r_lerp), dim=-1)
@torch.no_grad()
def interpolate_extrinsics(
initial: Float[Tensor, "*#batch 4 4"],
final: Float[Tensor, "*#batch 4 4"],
t: Float[Tensor, " time_step"],
eps: float = 1e-4,
) -> Float[Tensor, "*batch time_step 4 4"]:
"""Interpolate extrinsics by rotating around their "focus point," which is the
least-squares intersection between the look vectors of the initial and final
extrinsics.
"""
initial = initial.type(torch.float64)
final = final.type(torch.float64)
t = t.type(torch.float64)
# Based on the dot product between the look vectors, pick from one of two cases:
# 1. Look vectors are parallel: interpolate about their origins' midpoint.
# 3. Look vectors aren't parallel: interpolate about their focus point.
initial_look = initial[..., :3, 2]
final_look = final[..., :3, 2]
dot_products = einsum(initial_look, final_look, "... i, ... i -> ...")
parallel_mask = (dot_products.abs() - 1).abs() < eps
# Pick focus points.
initial_origin = initial[..., :3, 3]
final_origin = final[..., :3, 3]
pivot_point = 0.5 * (initial_origin + final_origin)
pivot_point[~parallel_mask] = intersect_rays(
initial_origin[~parallel_mask],
initial_look[~parallel_mask],
final_origin[~parallel_mask],
final_look[~parallel_mask],
)
# Convert to pivot parameters.
pivot_frame = generate_rotation_coordinate_frame(initial_look, final_look, eps=eps)
initial_params = extrinsics_to_pivot_parameters(initial, pivot_frame, pivot_point)
final_params = extrinsics_to_pivot_parameters(final, pivot_frame, pivot_point)
# Interpolate the pivot parameters.
interpolated_params = interpolate_pivot_parameters(initial_params, final_params, t)
# Convert back.
return pivot_parameters_to_extrinsics(
interpolated_params.type(torch.float32),
rearrange(pivot_frame, "... i j -> ... () i j").type(torch.float32),
rearrange(pivot_point, "... xyz -> ... () xyz").type(torch.float32),
)
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