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

Alzheimer's disease (AD) is a neurodegenerative disorder that is characterized by structural changes of the hippocampus, thus hippocampal morphology becomes an important biomarker for early diagnosis. We propose a geometry-aware framework for AD classification based on discrete Ricci flow and Symmetric Positive Definite (SPD) manifold learning. We conformally parameterize the hippocampus surface meshes from the MRI data using Ricci flow and retrieve three geometric descriptors, Heat Kernel Signature, Area Distortion and Conformal Factor, during the Ricci energy optimization process. These features are represented as covariance matrices, resulting in SPD descriptors per subject. To preserve intrinsic manifold structure, classification is performed by KNN with multiple SPD-aware metrics, including the Affine-Invariant Riemannian Metric (AIRM), Wasserstein Metric, Log-Euclidean Metric and so on. Experimental results on ADNI dataset show that AIRM achieves the best performance with 96\% accuracy, which outperforms other metrics evaluated. The results demonstrate that the combination of geometric analysis based on Ricci flow and manifold learning is effective for the diagnosis of Alzheimer's disease.