Variational Geometry-aware Neural Network based Method for Solving High-dimensional Diffeomorphic Mapping Problems

TL;DR

This paper introduces a scalable, mesh-free neural network framework that combines variational principles and quasi-conformal theory to solve high-dimensional diffeomorphic mapping problems with improved accuracy and robustness.

Contribution

It presents a novel, flexible neural network-based method that effectively handles high-dimensional diffeomorphic mappings by regulating conformality and volume distortions.

Findings

Accurately maps high-dimensional data in synthetic experiments.

Demonstrates robustness in real-world medical image registration.

Scales effectively to higher dimensions.

Abstract

Traditional methods for high-dimensional diffeomorphic mapping often struggle with the curse of dimensionality. We propose a mesh-free learning framework designed for $n$-dimensional mapping problems, seamlessly combining variational principles with quasi-conformal theory. Our approach ensures accurate, bijective mappings by regulating conformality distortion and volume distortion, enabling robust control over deformation quality. The framework is inherently compatible with gradient-based optimization and neural network architectures, making it highly flexible and scalable to higher-dimensional settings. Numerical experiments on both synthetic and real-world medical image data validate the accuracy, robustness, and effectiveness of the proposed method in complex registration scenarios.