NeuroScan

Advanced Brain Tumor Detection

Using topological image segmentation to identify tumors with 94% accuracy

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How Our Topological Analysis Works

Topological Feature Extraction

We compute persistence diagrams and Betti numbers to capture the essential shape characteristics of brain structures.

Neural Network Analysis

Our deep learning model processes topological features to identify abnormal patterns indicative of tumors.

Confidence Scoring

The system provides a confidence score based on the persistence of topological features in abnormal regions.

Topological Analysis Mathematics

1. Persistent Homology

Our system uses algebraic topology to quantify shape characteristics through persistent homology:

Filtration Process

For image I: Ω → ℝ, we construct filtration:

F(α) = {p ∈ Ω | I(p) ≥ α}, α ∈ [0, 255]

Tracking topological features across scales α

Persistence Diagram

For each topological feature (component, hole):

D(I) = {(bi, di) ∈ ℝ² | i ∈ features}

Where b=birth time, d=death time

2. Betti Numbers Analysis

We compute Betti numbers for tumor characterization:

Betti Number Description Tumor Significance
β₀ Connected components Tumor count/multiplicity
β₁ 1-dimensional holes Tumor morphology complexity
β₂ Void spaces 3D tumor structure analysis

3. Topological Loss Function

Our model optimizes using persistent homology-based loss:

Ltopo = ∑|Dpred - Dtrue|² + λW(Dpred, Dtrue)

Wasserstein Distance

Measures similarity between persistence diagrams:

W(D₁,D₂) = infη ∑||x - η(x)||p

Euler Characteristic

Alternating sum of Betti numbers:

χ = β₀ - β₁ + β₂