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

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.