TDA Theory

This section provides background on Topological Data Analysis (TDA) concepts used in MedTDA.

Note

This section provides conceptual overviews and intuitive explanations. For rigorous mathematical treatment, see the References page.

Overview

Topological Data Analysis (TDA) is a field of mathematics that studies the shape and structure of data using techniques from algebraic topology. TDA provides tools to:

  • Quantify global structure and connectivity

  • Identify holes, voids, and higher-dimensional features

  • Capture multi-scale geometric and topological properties

  • Provide robust descriptors invariant to small perturbations

Why TDA for Medical Images?

Medical images contain rich structural information that traditional methods may miss. TDA excels at:

Capturing Complex Anatomy

Medical structures often have intricate topological features:

  • Blood vessels: Networks with loops and branching

  • Organs: Cavities and internal voids

  • Tumors: Irregular boundaries and internal structure

  • Tissue: Porous structures and connectivity patterns

Traditional image analysis focuses on intensity values or local features, while TDA captures global topological properties.

Multi-Scale Analysis

Diseases often manifest at different scales:

  • Microscopic: Cell-level changes

  • Mesoscopic: Tissue structure alterations

  • Macroscopic: Organ-level deformations

Persistent homology naturally analyzes structures across scales through filtration.

Robustness to Noise

TDA methods are inherently robust:

  • Persistence distinguishes signal from noise

  • Short-lived features (low persistence) are likely noise

  • Long-lived features (high persistence) represent stable structure

Clinical Applications

TDA has been successfully applied to:

  • Cancer detection: Tumor microenvironment characterization

  • Neuroscience: Brain connectivity and structure

  • Cardiology: Vessel network analysis

  • Radiology: Texture and structure quantification

  • Pathology: Tissue organization patterns

Core Concepts

The TDA pipeline in MedTDA follows these steps:

  1. Filtration Construction

    Build a sequence of topological spaces from image intensities.

    See: Filtrations

  2. Persistent Homology Computation

    Track topological features (connected components, loops, voids) across the filtration.

    See: Persistent Homology

  3. Barcode/Diagram Generation

    Represent persistent features as barcodes or diagrams.

    See: Barcodes and Diagrams

  4. Vectorization

    Convert variable-length barcodes to fixed-length feature vectors for machine learning.

    See: Vectorization Methods

Topological Features in Medical Images

H0: Connected Components

Dimension 0 homology counts connected regions:

  • Number of separate structures

  • Component sizes and distributions

  • Merging patterns across scales

Medical examples:

  • Individual cells or nuclei

  • Separate lesions or tumors

  • Disconnected tissue regions

H1: Loops and Holes

Dimension 1 homology detects loops and tunnels:

  • Circular structures

  • Network connectivity

  • Enclosed regions

Medical examples:

  • Blood vessel loops

  • Glandular structures

  • Circular tissue arrangements

  • Airway branching patterns

H2: Voids and Cavities

Dimension 2 homology identifies enclosed voids (3D only):

  • Internal cavities

  • Hollow structures

  • Enclosed volumes

Medical examples:

  • Cysts and fluid-filled regions

  • Hollow organs (stomach, bladder)

  • Air-filled structures (lungs)

  • Necrotic tumor cores

Mathematical Foundation

MedTDA is built on well-established results from algebraic topology:

  • Simplicial and cubical homology — the algebraic machinery for counting topological features

  • Persistent homology — tracking how features are born and die across a filtration

  • Persistence modules and stability — guaranteeing that small data perturbations produce small changes in barcodes (bottleneck stability theorem)

This documentation focuses on:

  • Intuitive understanding of concepts

  • Practical usage in medical imaging

  • Implementation details in MedTDA

For rigorous mathematical treatment, see the References page.

Key Concepts Summary

Filtration

A nested sequence of topological spaces built from the image.

Persistent Homology

Tracking topological features across filtration scales.

Birth Time

Filtration value when a feature first appears.

Death Time

Filtration value when a feature disappears.

Persistence

Lifespan of a feature: death - birth.

Barcode

Visual representation of features as horizontal bars.

Persistence Diagram

Scatter plot of (birth, death) pairs.

Betti Numbers

Count of features at each dimension (β₀, β₁, β₂).

Theory Sections

Further Reading

Next Steps

After understanding the theory, see:

See Also