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Hippocampal surface analysis using discrete Ricci flow and Riemannian metrics for the diagnosis of Alzheimer's disease

This paper proposes a geometry-aware framework for diagnosing Alzheimer's disease by extracting hippocampal surface features via discrete Ricci flow, representing them as Symmetric Positive Definite matrices, and achieving 96% classification accuracy on the ADNI dataset using KNN with the Affine-Invariant Riemannian Metric.

Original authors: Fatemeh Ahmadi, Behroz Bidabad

Published 2026-09-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Fatemeh Ahmadi, Behroz Bidabad

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

The human brain is a landscape of intricate folds and valleys, and in the early stages of Alzheimer's disease, this landscape begins to shift long before memory loss becomes obvious. One of the first places to show these subtle changes is the hippocampus, a small, seahorse-shaped structure deep inside the brain that is vital for forming new memories. For decades, doctors and researchers have tried to spot the disease by measuring the volume of this structure, essentially counting how much space it takes up. However, volume is a blunt instrument; it can miss the delicate, complex warping of the surface that often signals the onset of neurodegeneration. To see these hidden deformations, scientists are turning to a field that treats the brain not just as a collection of tissues, but as a geometric shape that can be mapped, stretched, and analyzed with mathematical precision.

A team of researchers at Amirkabir University of Technology in Iran has developed a new way to examine these shapes, moving beyond simple measurements to study the very geometry of the hippocampus. Their approach relies on a powerful mathematical concept known as discrete Ricci flow. Imagine a crumpled piece of paper that needs to be flattened out without tearing it; Ricci flow is a process that does something similar for complex 3D surfaces. It gradually smooths out the curves and bumps of a shape, spreading the curvature evenly across the surface until it settles into a standard, flat form. By applying this process to the 3D models of hippocampi derived from MRI scans, the researchers can watch how the surface changes as it is smoothed out. This transformation reveals specific geometric fingerprints that are unique to the disease.

In their study, the team analyzed MRI data from 200 individuals, half of whom had Alzheimer's disease and half of whom were cognitively normal. They used a standard medical software tool to extract the 3D surface of the left hippocampus from each scan. Then, they applied the Ricci flow process, which acts like a slow-motion animation of the brain surface flattening itself out. As this smoothing happened, the researchers tracked three specific features at every step of the journey. First, they looked at how heat would theoretically spread across the surface, a method that highlights the shape's natural structure. Second, they measured how much the surface stretched or shrank in specific areas as it was flattened. Third, they recorded the "conformal factor," a value that describes how the local geometry of the surface was being altered by the smoothing process.

These three features were not treated as simple numbers. Instead, the researchers grouped them together to create a statistical map for each person, known as a covariance matrix. This map captures how the different geometric features relate to one another, creating a unique signature for every brain. Because these signatures exist on a complex, curved mathematical space rather than a flat grid, the researchers had to be careful about how they compared them. They tested several different ways of measuring the distance between these signatures, including methods that treat the data as if it were on a flat plane and methods that respect the curved nature of the data.

The results of their experiments were clear and compelling. When they used a classification tool to sort the brains into "Alzheimer's" or "healthy" categories, the method that respected the curved geometry of the data performed far better than the others. Specifically, a technique called the Affine-Invariant Riemannian Metric, which calculates distance while accounting for the unique shape of the data space, achieved an accuracy of 96 percent. This was the highest performance among all the methods tested. Other approaches, such as those that ignored the curved nature of the data and treated it as a simple flat grid, performed significantly worse, with accuracy dropping to around 82 percent. This gap in performance suggests that the subtle differences between a healthy brain and one with Alzheimer's are hidden in the complex, non-linear relationships between the geometric features, relationships that only a geometry-aware approach can uncover.

The study also visualized these findings by projecting the complex data into a simpler, three-dimensional space to see if the two groups of people naturally separated. The images showed that when the data was treated with the correct geometric tools, the healthy brains and the diseased brains formed distinct clusters, whereas simpler methods failed to separate them clearly. This confirms that the disease leaves a specific geometric mark on the hippocampus that is distinct from normal aging. The researchers found that their method was effective for the diagnosis of Alzheimer's disease, offering a sensitive tool for detection based on structural changes.

While the study focused on a specific dataset of 200 people, the implications point toward a future where brain imaging is analyzed with a deeper understanding of shape and form. The researchers did not claim to have found a cure, nor did they suggest this method is ready for immediate use in every clinic. Instead, they demonstrated that by combining the smoothing power of Ricci flow with advanced geometric learning, it is possible to extract highly accurate diagnostic information from brain scans. Their work suggests that the key to understanding Alzheimer's may lie not just in how much brain tissue is lost, but in how the remaining tissue is shaped and how that shape evolves under mathematical scrutiny. By treating the brain as a dynamic geometric object, this approach opens a new window into the earliest stages of one of the world's most challenging diseases.

Technical Summary: Hippocampal Surface Analysis Using Discrete Ricci Flow and Riemannian Metrics for Alzheimer's Diagnosis

Problem Statement

Alzheimer's Disease (AD) is a leading cause of dementia characterized by progressive neurodegeneration, with the hippocampus being one of the first anatomical structures to exhibit morphological degeneration. While Magnetic Resonance Imaging (MRI) is widely used for AD diagnosis, traditional volumetric analysis often fails to capture subtle geometric deformations that contain discriminative pathological information. Consequently, surface-based analysis methods have emerged to study local geometric and topological variations. However, existing methods face challenges: some rely on deep learning frameworks requiring large datasets and lacking interpretability, while others may not adequately preserve the intrinsic non-Euclidean structure of geometric data when performing classification.

Methodology

The authors propose a geometry-aware framework that integrates discrete Ricci flow for surface parameterization with Symmetric Positive Definite (SPD) manifold learning for classification. The pipeline consists of four primary stages:

1. Data Preprocessing and Surface Reconstruction

Using 3D T1-weighted MRI data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), the authors employ the FreeSurfer pipeline to perform skull stripping, tissue classification, and surface extraction. The hippocampal region of the left hemisphere is isolated, and the resulting surface meshes are normalized (via down-sampling or up-sampling) to ensure a consistent number of vertices.

2. Discrete Ricci Flow Parameterization

The core geometric processing utilizes discrete Ricci flow to conformally parameterize the hippocampal surface meshes. This process involves:

Initial Circle Packing: An inversive distance circle packing metric is constructed on the triangular mesh to serve as the initial state.
Ricci Energy Optimization: The conformal metric is optimized via Newton's method to minimize Ricci energy, effectively smoothing the curvature distribution across the surface.
Planar Embedding: The optimized metric is embedded into a 2D Euclidean plane, creating a conformal mapping that preserves intrinsic geometric features.

3. Feature Extraction and Covariance Representation

During the Ricci energy optimization process (specifically over 30 iterative steps), three dynamic geometric descriptors are extracted for each vertex:

Heat Kernel Signature (HKS): Derived from the Laplace–Beltrami operator, capturing multi-scale local and global shape information.
Area Distortion: Measuring the local stretching or shrinking of the surface during conformal parameterization.
Conformal Factor: Representing the local scaling changes ( $u_i = \log \gamma_i$ ) during the flow.

These features are aggregated into feature vectors at each optimization step. To capture the statistical relationships between these features, the authors compute covariance matrices. Since these matrices are Symmetric Positive Definite (SPD), they inherently reside on a Riemannian manifold rather than a Euclidean vector space. Each subject is thus represented by a sequence of $T$ SPD covariance matrices.

4. Manifold-Based Classification

To respect the intrinsic geometry of the SPD descriptors, the authors employ a K-Nearest Neighbor (KNN) classifier using various distance metrics defined on the SPD manifold:

Affine-Invariant Riemannian Metric (AIRM): Calculates geodesic distance preserving affine invariance.
Log-Euclidean Metric (LEM): Maps SPD matrices to a tangent Euclidean space via matrix logarithm.
Wasserstein Metric: Based on optimal transport theory.
Euclidean Metric: Standard Frobenius norm (used as a baseline).
Tangent Space Representation: Projecting matrices to tangent space for Euclidean processing.

Additionally, the authors evaluated Support Vector Machine (SVM), Logistic Regression (LR), and Extreme Gradient Boosting (XGBoost) classifiers using these manifold-aware embeddings to compare against the KNN approach.

Key Contributions

The paper outlines four main contributions:

Novel Framework: A method combining discrete Ricci flow with covariance-based SPD representation specifically for hippocampal surface analysis.
Dynamic Feature Extraction: The extraction of dynamic geometric features (HKS, Area Distortion, Conformal Factor) during the Ricci flow optimization process to track surface evolution.
Metric Comparison: A comprehensive evaluation of multiple manifold-aware metrics (AIRM, LEM, Wasserstein, etc.) for classifying SPD-based descriptors.
Validation of Riemannian Learning: Demonstration that Riemannian manifold learning effectively captures subtle structural changes in the hippocampus associated with AD.

Experimental Results

The framework was tested on a dataset of 200 subjects (100 AD, 100 Cognitively Normal) from the ADNI database. Experiments were conducted using an 80/20 train-test split repeated 10 times.

Performance: The AIRM metric combined with the KNN classifier achieved the highest performance, reaching 96% accuracy, with corresponding precision, recall, and F1-scores of 0.96.
Comparative Performance:
- The Log-Euclidean metric with KNN achieved 94% accuracy.
- The Euclidean metric (baseline) yielded the lowest performance (82% accuracy with KNN), highlighting the limitations of ignoring the non-Euclidean structure of SPD matrices.
- Among other classifiers, XGBoost with Wasserstein embedding achieved 88% accuracy, while SVM and Logistic Regression generally underperformed compared to KNN with AIRM.
Visualization: t-SNE visualization of the learned features showed limited linear separability in 3D space, reinforcing the conclusion that discriminative information is embedded in the non-Euclidean manifold structure rather than a standard Euclidean feature space.

Significance and Claims

The authors claim that the integration of geometric analysis based on Ricci flow and manifold learning is an effective approach for diagnosing Alzheimer's disease. The study demonstrates that preserving the intrinsic Riemannian geometry of covariance descriptors is crucial for accurate similarity measurement and classification.

Specifically, the paper asserts that:

The combination of discrete Ricci flow and SPD manifold learning outperforms traditional Euclidean-based analysis and other metric approaches in this context.
The Affine-Invariant Riemannian Metric (AIRM) is superior for capturing the intrinsic manifold geometry of the SPD matrices, leading to the most discriminative representation.
The proposed method offers a robust alternative to deep learning approaches by utilizing interpretable geometric features and requiring smaller datasets, while maintaining high classification accuracy (96%).

The authors position this work as a validation of geometry-aware methods for capturing subtle neurodegenerative changes, suggesting that future research could expand to multi-modal data and deep Riemannian learning models.