GeoTop: Advancing Image Classification with Geometric-Topological Analysis

GeoTop is a mathematically principled framework that unifies Topological Data Analysis and Lipschitz-Killing Curvatures to resolve the diagnostic ambiguity of topologically equivalent structures by integrating robust topological signatures with precise geometric features, thereby achieving superior accuracy and interpretability in image classification tasks such as skin lesion diagnosis.

The Gist

Imagine you are a detective trying to solve a mystery: Is this skin spot a harmless mole (benign) or a dangerous cancer (malignant)?

For a long time, doctors and computer programs have tried to solve this by looking at two things:

1. The Shape: Is it round? Is it jagged?
2. The Structure: Does it have holes? Is it one big piece or many small pieces?

The problem is that some dangerous cancers look exactly like harmless moles when you only look at their basic structure. It's like trying to tell the difference between a perfectly smooth, round marble and a rough, jagged rock just by counting how many pieces they are made of. If both are "one single piece," a simple count can't tell them apart. This is called the "Topological Equivalence" problem.

Enter GeoTop, a new mathematical detective tool created by the authors. Here is how it works, explained simply:

1. The Two Detectives: "The Counting Guy" and "The Measuring Guy"

The GeoTop framework combines two different ways of looking at an image:

• Detective A (Topological Data Analysis): This detective is like a counting game. They look at the image and ask: "How many separate islands are there? How many holes are in the middle?" They are great at seeing the big picture and ignoring small scratches or noise. However, they are "blind" to details like how bumpy the edge is. To them, a smooth circle and a jagged star might look the same if they both have one piece and no holes.
• Detective B (Lipschitz-Killing Curvatures): This detective is a precise measurer. They don't just count; they measure the perimeter, the area, and the bounciness (curvature) of the edges. They can tell you if a boundary is smooth like a river or jagged like a broken glass.

The Problem: Detective A misses the jagged edges. Detective B gets confused by too much noise.
The Solution: GeoTop forces them to work together. It says, "Count the holes, but also measure how rough the edges of those holes are."

2. The Creative Analogy: The "Cookie Cutter" vs. The "Dough"

Imagine you have two cookies:

• Cookie A (Benign): A perfect, smooth circle cut out with a cookie cutter.
• Cookie B (Malignant): A circle that has been crumbled and reformed with jagged, uneven edges, but it's still roughly the same size and shape.

If you just look at the outline (Topology), they might look similar enough to confuse a computer. But if you run your finger along the edge (Geometry), Cookie B feels rough and irregular, while Cookie A feels smooth.

GeoTop is the tool that does both at once. It counts the cookie (1 cookie) and feels the texture of the edge simultaneously.

3. What Did They Find?

The researchers tested this on thousands of real skin images (moles and cancers) and even on plant proteins. Here is what happened:

• Better Accuracy: By combining the two detectives, the system got 3.6% more accurate than using just one method. In the world of medicine, that's a huge jump.
• Fewer Mistakes: It made 15–18% fewer mistakes.
  • It stopped missing dangerous cancers (False Negatives) because the "Measuring Guy" saw the jagged, scary edges.
  • It stopped scaring people with harmless moles (False Positives) because the "Counting Guy" saw that the structure was actually stable and simple.
• The "Synthetic" Test: To prove it worked, they created fake images where a perfect square was hidden inside a smooth cloud. A normal topological tool saw them as identical. GeoTop saw the square immediately because the square has sharp, straight edges, while the cloud is soft and round.

4. Why Does This Matter?

• It's Not a "Black Box": Many modern AI tools are like magic boxes; they give an answer but you don't know why. GeoTop is different. It can show you a diagram saying, "I called this cancer because the edge was too jagged (Geometry) even though the shape looked simple (Topology)." This helps doctors trust the computer.
• It's Fast: It can analyze a high-quality medical image in less than half a second.
• It Works Everywhere: It didn't just work on skin; it also worked on identifying specific plant proteins, showing that this "Shape + Texture" logic is a universal truth for biology.

The Bottom Line

GeoTop is like upgrading a security camera. Old cameras just took a picture and counted faces. GeoTop takes a picture, counts the faces, and analyzes the texture of their clothes and the shape of their shoes to tell if they are a friend or a threat.

By mathematically fusing structure (what things look like) with geometry (how things feel), GeoTop solves a problem that has stumped doctors and computers for years: telling the difference between things that look the same but are actually very different.

Technical Summary

Here is a detailed technical summary of the paper "GeoTop: Advancing Image Classification with Geometric-Topological Analysis."

1. Problem Statement

The paper addresses a fundamental limitation in diagnostic imaging: Topological Equivalence.

• The Challenge: Conventional methods, including deep learning and standard Topological Data Analysis (TDA), often fail to distinguish between benign and malignant structures that share identical global topology (e.g., the same number of connected components or holes) but differ critically in local geometric details (e.g., boundary irregularity, surface curvature, or surface regularity).
• The Gap:
  • Pure TDA: Robust to noise and invariant under continuous deformation (homeomorphism), but discards metric information (curvature, scale, perimeter). It cannot distinguish a smooth circle from a jagged square if they share the same Betti numbers.
  • Pure Geometric Descriptors: Sensitive to curvature and scale but lack the hierarchical stability and noise robustness provided by TDA.
  • Deep Learning: Often operates as a "black box," lacking the mathematical interpretability required for clinical trust and failing to explicitly model the interaction between global topology and local geometry.

2. Methodology: The GeoTop Framework

GeoTop is a mathematically principled framework that unifies Persistent Homology (from TDA) and Lipschitz-Killing Curvatures (LKCs) to create a hybrid geometric-topological representation.

A. Topological Component: Cubical Persistent Homology

• Process: The framework applies a filtration process to grayscale images (treating them as unions of cubes). It iterates through pixel intensity thresholds to generate "superlevel sets" (or sublevel sets).
• Feature Extraction: It tracks the "birth" and "death" of topological features (connected components $H_0$, loops $H_1$, voids $H_2$) across these thresholds.
• Representation: Features are condensed into Persistence Diagrams and Barcodes.
• Vectorization: The diagrams are converted into feature vectors using amplitude metrics (bottleneck norm, Wasserstein norm, persistence entropy) and 7 additional amplitude metrics per dimension, resulting in 64 topological features per image (across RGB and grayscale channels).

B. Geometric Component: Lipschitz-Killing Curvatures (LKCs)

• Concept: LKCs quantify intrinsic volumes of excursion sets (thresholded images) at 200 equidistant thresholds.
• Three Key Curvatures:
  1. $L_2$ (Area): Linked to occupation density.
  2. $L_1$ (Perimeter): Reflects boundary regularity and complexity.
  3. $L_0$ (Euler-Poincaré Characteristic): A topological invariant representing connectivity (components minus holes).
• Feature Extraction: For each LKC curve, the framework computes statistical summaries including $L_2$ norms, trapezoidal integrals, sums, entropy, and derivative summaries. This yields 120 geometric features per image.

C. The Fusion Strategy

• GeoTop Vector: The final feature vector is the concatenation of the topological (TDA) and geometric (LKC) vectors.
• Theoretical Basis: The authors provide Lemma 1, formally proving that while two shapes may have a bottleneck distance of zero (identical topology), their LKC vectors will be distinct if their intrinsic volumes (area, perimeter, Euler characteristic) differ. This guarantees geometric separability where topology fails.

D. Classification

• Models: The primary classifier used is a Random Forest (chosen for interpretability and stability), supplemented by MLPs and Logistic Regression for robustness checks.
• Datasets:
  1. Skin Lesions: 3,297 images (1,800 benign, 1,497 malignant) from the ISIC/Kaggle dataset.
  2. Synthetic Benchmark: 4,000 pairs of topology-equivalent but geometric-distinct shapes (e.g., Gaussian peak vs. Square).
  3. Plant Signaling Peptides (SSPs): 5,596 sequences converted to image models to test cross-domain generalizability.

3. Key Results

A. Skin Lesion Classification (Medical Imaging)

• Accuracy: GeoTop achieved a mean accuracy of 0.87, outperforming standalone TDA (0.84) and standalone LKC (0.84) by a consistent 3.6%.
• Error Reduction: The framework reduced both False Positives (unnecessary biopsies) and False Negatives (missed cancers) by 15–18%.
  • Mechanism: TDA features primarily reduced False Positives by identifying globally coherent benign structures. LKC features reduced False Negatives by detecting the irregular, infiltrating boundaries characteristic of malignancy.
• Interpretability: The model successfully decomposed discriminative signals, showing that malignant lesions exhibit higher geometric variance (instability in perimeter profiles) and more abrupt transitions in Euler characteristics compared to benign lesions.

B. Synthetic Equivalence Benchmark

• Goal: To test if GeoTop can distinguish shapes that are topologically identical (bottleneck distance < 0.01) but geometrically different.
• Performance: GeoTop achieved 92% accuracy, significantly outperforming TDA-only (76%) and LKC-only (79%).
• Significance: This empirically validates Lemma 1, proving that GeoTop resolves the "topological equivalence" roadblock where pure topology fails.

C. Cross-Domain Generalization (Plant Signaling Peptides)

• Performance: GeoTop achieved 99% accuracy in classifying signaling peptides, outperforming sequence-based methods by 17% and TDA-only by 13%.
• Insight: In this domain, topological features were more dominant (62% importance) than in skin lesions, likely because peptide function relies heavily on 3D folding patterns captured well by persistent homology.

4. Key Contributions

1. Novel Framework: Introduction of GeoTop, the first unified framework to systematically fuse Persistent Homology with Lipschitz-Killing Curvatures for image classification.
2. Theoretical Guarantee: Formalization of Lemma 1, proving that geometric descriptors (LKCs) can separate classes that are topologically indistinguishable, providing a mathematical solution to the topological equivalence problem.
3. Clinical Utility: Demonstrated a significant reduction in diagnostic errors (15–18%) in skin cancer detection, offering a path toward more reliable AI-assisted diagnosis.
4. Interpretability: Unlike deep learning black boxes, GeoTop provides intrinsic interpretability via persistence diagrams (global structure) and LKC profiles (local geometry), allowing clinicians to understand why a classification was made.
5. Efficiency: The framework is computationally efficient, processing 224×224 pixel images in ≤0.5 seconds, making it suitable for real-time clinical workflows.

5. Significance and Future Impact

• Paradigm Shift: GeoTop moves beyond simple feature concatenation, offering a cohesive framework where topology and geometry inform and strengthen each other. It bridges the gap between discrete connectivity (topology) and continuous curvature (geometry).
• Broad Applicability: The framework is domain-agnostic. Its success in both macroscopic tumor analysis and molecular-level peptide classification suggests it can be applied to diverse fields, including 3D volumetric imaging (MRI), spatial transcriptomics, and structural biology.
• Future Directions: The authors identify scaling to 3D volumetric data and integrating GeoTop features into deep learning architectures (hybrid models) as key next steps to further enhance scalability and representation power.

In summary, GeoTop provides a mathematically rigorous, interpretable, and highly effective solution for distinguishing complex biological structures that traditional methods cannot separate, fundamentally advancing the state of the art in diagnostic image analysis.