Entropy-Enhanced Conformal Features from Ricci Flow for Robust Alzheimer's Disease Classification

TL;DR

This work addresses Alzheimer’s disease diagnosis from hippocampal surface morphology by introducing a landmark-free pipeline that uses discrete Ricci flow to achieve planar conformal parameterization. It derives three geometric signatures—area distortion, conformal factor, and Gaussian curvature—and encodes them with Shannon entropy to form compact, discriminative features. A broad classifier suite demonstrates state-of-the-art performance, with MLP and Logistic Regression achieving a mean accuracy of 98.62% on the ADNI dataset, validating the approach’s robustness and clinical potential. The method offers scalable, automated biomarkers for cortical morphometry and can extend to other neurodegenerative conditions and multi-class tasks.

Abstract

Background and Objective: In brain imaging, geometric surface models are essential for analyzing the 3D shapes of anatomical structures. Alzheimer's disease (AD) is associated with significant cortical atrophy, making such shape analysis a valuable diagnostic tool. The objective of this study is to introduce and validate a novel local surface representation method for the automated and accurate diagnosis of AD. Methods: The study utilizes T1-weighted MRI scans from 160 participants (80 AD patients and 80 healthy controls) from the Alzheimer's Disease Neuroimaging Initiative (ADNI). Cortical surface models were reconstructed from the MRI data using Freesurfer. Key geometric attributes were computed from the 3D meshes. Area distortion and conformal factor were derived using Ricci flow for conformal parameterization, while Gaussian curvature was calculated directly from the mesh geometry. Shannon entropy was applied to these three features to create compact and informative feature vectors. The feature vectors were used to train and evaluate a suite of classifiers (e.g. XGBoost, MLP, Logistic Regression, etc.). Results: Statistical significance of performance differences between classifiers was evaluated using paired Welch's t-test. The method proved highly effective in distinguishing AD patients from healthy controls. The Multi-Layer Perceptron (MLP) and Logistic Regression classifiers outperformed all others, achieving an accuracy and F$_1$ Score of 98.62%. Conclusions: This study confirms that the entropy of conformally-derived geometric features provides a powerful and robust metric for cortical morphometry. The high classification accuracy underscores the method's potential to enhance the study and diagnosis of Alzheimer's disease, offering a straightforward yet powerful tool for clinical research applications.

Figures (14)

• Figure 1: Circle packing schemes. (a): Thurston’s circle packing, (b): Tangential circle packing, (c): Inversive distance circle packing.
• Figure 2: Power circle. (a): The edge $e_{ij}$ is adjacent to the two faces $f_{ijk}$ and $f_{jil}$, (b): The edge $e_{ij}$ is attached to a single face $f_{ijk}$.
• Figure 3: (a): The region surrounding a vertex in the initial stage (3-dimensional), (b): The region surrounding a vertex in the final stage (2-dimensional) of Ricci energy optimization.
• Figure 4: Block diagram of the proposed method in this paper.
• Figure 5: Depiction of the functional regions of the left hemisphere of the cerebral cortex.