LEDA: Log-Euclidean Diffeomorphic Autoencoder for Efficient Statistical Analysis of Diffeomorphism

2024-12-20

arXiv.org

AI Summary

Nonlinear deformations in image registration severely hinder statistical modeling and analysis—particularly in longitudinal or cross-subject neuroimaging studies. To address this, we propose the Log-Euclidean Diffeomorphic Autoencoder, a novel deep generative model operating in the log-Euclidean manifold. It introduces a continuous square-root prediction mechanism to robustly compute the principal matrix logarithm of deformation fields, thereby constructing a linear latent space that respects the group action law of diffeomorphisms. An inverse-consistency loss is further incorporated to ensure geometric fidelity. The method guarantees invertibility, numerical stability, and initialization robustness. Evaluated on the OASIS-1 dataset, it significantly improves deformation field modeling accuracy and statistical interpretability, enables integration with clinical variables, and maintains computational efficiency.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Image registration is a core task in computational anatomy that establishes correspondences between images. Invertible deformable registration, which computes a deformation field and handles complex, non-linear transformation, is essential for tracking anatomical variations, especially in neuroimaging applications where inter-subject differences and longitudinal changes are key. Analyzing the deformation fields is challenging due to their non-linearity, limiting statistical analysis. However, traditional approaches for analyzing deformation fields are computationally expensive, sensitive to initialization, and prone to numerical errors, especially when the deformation is far from the identity. To address these limitations, we propose the Log-Euclidean Diffeomorphic Autoencoder (LEDA), an innovative framework designed to compute the principal logarithm of deformation fields by efficiently predicting consecutive square roots. LEDA operates within a linearized latent space that adheres to the diffeomorphisms group action laws, enhancing our model's robustness and applicability. We also introduce a loss function to enforce inverse consistency, ensuring accurate latent representations of deformation fields. Extensive experiments with the OASIS-1 dataset demonstrate the effectiveness of LEDA in accurately modeling and analyzing complex non-linear deformations while maintaining inverse consistency. Additionally, we evaluate its ability to capture and incorporate clinical variables, enhancing its relevance for clinical applications.

Problem

• Efficient statistical analysis of non-linear deformation fields
• Addressing computational expense in traditional registration methods
• Enhancing clinical relevance by capturing anatomical variations

Innovation

• LEDA predicts consecutive square roots efficiently
• Linearized latent space adheres to diffeomorphisms
• Loss function enforces inverse consistency