Update index.html

index.html

@@ -264,7 +264,101 @@
             </div>
         </div>
     </section>
+    <section class="bg-gray-100 py-16" id="research">
+        <div class="container mx-auto px-4">
+            <h2 class="text-3xl font-bold text-center mb-12">Topological Analysis Mathematics</h2>
+
+            <div class="bg-white rounded-xl shadow-lg p-8 mb-8">
+                <h3 class="text-2xl font-bold mb-4">1. Persistent Homology</h3>
+                <p class="mb-4">Our system uses algebraic topology to quantify shape characteristics through persistent homology:</p>
+                
+                <div class="grid grid-cols-1 md:grid-cols-2 gap-8">
+                    <div class="bg-blue-50 p-4 rounded-lg">
+                        <h4 class="font-bold mb-2">Filtration Process</h4>
+                        <p>For image I: Ω → ℝ, we construct filtration:</p>
+                        <p class="text-center my-2 font-mono">
+                            F(α) = {p ∈ Ω | I(p) ≥ α}, α ∈ [0, 255]
+                        </p>
+                        <p>Tracking topological features across scales α</p>
+                    </div>
+                    
+                    <div class="bg-green-50 p-4 rounded-lg">
+                        <h4 class="font-bold mb-2">Persistence Diagram</h4>
+                        <p>For each topological feature (component, hole):</p>
+                        <p class="text-center my-2 font-mono">
+                            D(I) = {(b<sub>i</sub>, d<sub>i</sub>) ∈ ℝ² | i ∈ features}
+                        </p>
+                        <p>Where b=birth time, d=death time</p>
+                    </div>
+                </div>
+            </div>
+
+            <div class="bg-white rounded-xl shadow-lg p-8 mb-8">
+                <h3 class="text-2xl font-bold mb-4">2. Betti Numbers Analysis</h3>
+                <p>We compute Betti numbers for tumor characterization:</p>
+                
+                <div class="overflow-x-auto mt-4">
+                    <table class="min-w-full bg-white">
+                        <thead class="bg-gray-800 text-white">
+                            <tr>
+                                <th class="py-3 px-4">Betti Number</th>
+                                <th class="py-3 px-4">Description</th>
+                                <th class="py-3 px-4">Tumor Significance</th>
+                            </tr>
+                        </thead>
+                        <tbody>
+                            <tr class="border-b">
+                                <td class="py-3 px-4 font-bold">β₀</td>
+                                <td class="py-3 px-4">Connected components</td>
+                                <td class="py-3 px-4">Tumor count/multiplicity</td>
+                            </tr>
+                            <tr class="border-b">
+                                <td class="py-3 px-4 font-bold">β₁</td>
+                                <td class="py-3 px-4">1-dimensional holes</td>
+                                <td class="py-3 px-4">Tumor morphology complexity</td>
+                            </tr>
+                            <tr>
+                                <td class="py-3 px-4 font-bold">β₂</td>
+                                <td class="py-3 px-4">Void spaces</td>
+                                <td class="py-3 px-4">3D tumor structure analysis</td>
+                            </tr>
+                        </tbody>
+                    </table>
+                </div>
+            </div>
 
+            <div class="bg-white rounded-xl shadow-lg p-8">
+                <h3 class="text-2xl font-bold mb-4">3. Topological Loss Function</h3>
+                <p>Our model optimizes using persistent homology-based loss:</p>
+                
+                <div class="mt-4 space-y-4">
+                    <div class="bg-purple-50 p-4 rounded-lg">
+                        <p class="font-mono text-center">
+                            L<sub>topo</sub> = ∑|D<sub>pred</sub> - D<sub>true</sub>|² + λW(D<sub>pred</sub>, D<sub>true</sub>)
+                        </p>
+                    </div>
+                    
+                    <div class="grid grid-cols-1 md:grid-cols-2 gap-4">
+                        <div class="p-4 border rounded-lg">
+                            <h4 class="font-bold mb-2">Wasserstein Distance</h4>
+                            <p>Measures similarity between persistence diagrams:</p>
+                            <p class="mt-2 font-mono">
+                                W(D₁,D₂) = inf<sub>η</sub> ∑||x - η(x)||<sub>p</sub>
+                            </p>
+                        </div>
+                        
+                        <div class="p-4 border rounded-lg">
+                            <h4 class="font-bold mb-2">Euler Characteristic</h4>
+                            <p>Alternating sum of Betti numbers:</p>
+                            <p class="mt-2 font-mono">
+                                χ = β₀ - β₁ + β₂
+                            </p>
+                        </div>
+                    </div>
+                </div>
+            </div>
+        </div>
+    </section>
     <!-- Footer -->
     <footer class="bg-gray-900 text-white py-12">
         <div class="container mx-auto px-4">
@@ -312,7 +406,7 @@
                 </div>
             </div>
             <div class="border-t border-gray-800 mt-8 pt-8 text-center text-gray-500">
-                <p>&copy; 2023 NeuroScan. All rights reserved.</p>
+                <p>&copy; 2025 NeuroScan. All rights reserved.</p>
             </div>
         </div>
     </footer>