提出无网格方法解决高维形变映射难题，兼顾形变质量与计算效率。
Variational Geometry-aware Neural Network based Method for Solving High-dimensional Diffeomorphic Mapping Problems
- 融合变分原理与拟共形理论，通过调控共形畸变和体积畸变保证映射双射性
- 在合成与真实医学图像上验证了高精度、强鲁棒性，适用于复杂配准场景
- 兼容梯度优化与神经网络，可扩展至高维问题，适合医学影像研究者
高维形变映射的传统方法常受维度灾难困扰。本文提出一种针对n维映射问题的无网格学习框架，将变分原理与拟共形理论无缝结合。通过调节共形畸变和体积畸变，确保映射的精确性和双射性，实现对形变质量的稳健控制。该框架天然兼容基于梯度的优化与神经网络结构，具备高度灵活性与可扩展性，适用于更高维度场景。在合成数据及真实医学影像上的数值实验验证了该方法在复杂配准任务中的准确性、鲁棒性与有效性。
原文摘要 · Abstract (English)
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.
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