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

2025-10-31

AI Summary

To address performance degradation of conventional methods in high-dimensional diffeomorphic mapping caused by the curse of dimensionality, this paper proposes a mesh-free, geometry-aware deep learning framework. Methodologically, it integrates variational principles with quasiconformal theory to construct an end-to-end differentiable neural network architecture that jointly regulates conformal and volumetric distortions, enabling explicit control over bijectivity, regularity, and deformation quality. Its key innovation lies in being the first to embed quasiconformal energy into a mesh-free variational learning paradigm, thereby circumventing mesh dependency and the computational bottleneck of Jacobian determinant evaluation. Evaluated on synthetic data and multi-modal medical image registration tasks, the method consistently outperforms mainstream voxel-based and flow-field-based approaches. It demonstrates superior accuracy, robustness, and scalability in high-dimensional settings (e.g., 3D/4D), establishing a new state-of-the-art for geometrically principled diffeomorphic registration.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Solving high-dimensional diffeomorphic mapping problems
• Overcoming curse of dimensionality in traditional methods
• Ensuring accurate bijective mappings with controlled deformation

Innovation

• Mesh-free learning framework for high-dimensional mapping
• Combines variational principles with quasi-conformal theory
• Regulates conformality and volume distortion for bijective mappings