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

Zhiwen Li, Cheuk Hin Ho, Lok Ming Lui

TL;DR

This work tackles high-dimensional diffeomorphic mapping for registration by introducing a mesh-free, variational learning framework that blends quasi-conformal geometry with neural networks. The method employs a differentiable bijectivity loss, a conformality distortion term, and a volumetric-prior to enforce diffeomorphic, volume-preserving deformations within a Deep Ritz–style objective, while hard boundary constraints are embedded in the network design for stable optimization. It demonstrates strong performance on synthetic 3D deformations and real 3D medical imaging tasks (e.g., 4DCT lung registration), with ablations confirming the benefits of hard boundary enforcement and the hybrid data formulations. The approach scales to high dimensions without mesh discretization, enabling robust, bijective, and distortion-controlled mappings in complex geometric registration problems with practical impact for medical imaging and computational geometry.

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.

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

TL;DR

This work tackles high-dimensional diffeomorphic mapping for registration by introducing a mesh-free, variational learning framework that blends quasi-conformal geometry with neural networks. The method employs a differentiable bijectivity loss, a conformality distortion term, and a volumetric-prior to enforce diffeomorphic, volume-preserving deformations within a Deep Ritz–style objective, while hard boundary constraints are embedded in the network design for stable optimization. It demonstrates strong performance on synthetic 3D deformations and real 3D medical imaging tasks (e.g., 4DCT lung registration), with ablations confirming the benefits of hard boundary enforcement and the hybrid data formulations. The approach scales to high dimensions without mesh discretization, enabling robust, bijective, and distortion-controlled mappings in complex geometric registration problems with practical impact for medical imaging and computational geometry.

Abstract

Traditional methods for high-dimensional diffeomorphic mapping often struggle with the curse of dimensionality. We propose a mesh-free learning framework designed for -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.

Paper Structure

This paper contains 29 sections, 1 theorem, 31 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Quasi-conformal Mapppings
  4. Volumetric Prior
  5. Deep Ritz Method
  6. Proposed Method
  7. Problem Setting
  8. Variational Formulations of $3$D Diffeomorphic Registration Problem
  9. Diffeomorphic Loss
  10. Conformality Loss
  11. Volumetric-prior Loss
  12. Data Loss
  13. Boundary Conditions: Hard Constraint v.s. Soft Constraint
  14. Network Architecture
  15. Numerical Experiments
  16. ...and 14 more sections

Key Result

theorem 1

Suppose $\Omega$ is bounded and simply connected, $S$, $T$ are continuous functions from $\Omega \subset \mathbb{R}^n \rightarrow \mathbb{R}$, and $\alpha_i > 0$ for $i = 1, \cdots, 6$. Let for some $c_j > 0$ for $j = 1, \cdots, 3$. Then the proposed model eqn:hybrid_formulation admits a minimizer in $\mathcal{A}$.

Figures (10)

  • Figure 1: An illustration for the quasi-conformal maps. (a) Under a $2$D quasi-conformal map, infinitesimal circles are mapped to infinitesimal ellipses. (b) Under an $n$-D quasi-conformal map, infinitesimal spheres of dimension $n-1$ are mapped to infinitesimal ellipsoids of dimension $n-1$.
  • Figure 2: Results of landmark registration \ref{['eqn:landmark_formulation']} on the Twisted Landmark Pairs. (a) The eight landmarks with their target positions indicated by the arrows. (b) The $3$D transformation $f_\theta$ obtained by the trained network. The points are colored by $\tanh(\frac{\ln 3}{2} \cdot \det \nabla f_\theta)$ so that the color$(p)=0.5$ when $\det \nabla f_\theta(p) = 1$. (c) Histogram of $\det\nabla f_\theta(p)$ with $100000$ uniformly random sampled points. (d) Two cross-sectional views $f_\theta(x=0.2)$ and $f_\theta(x=0.8)$. (e) Two cross-sectional views $f_\theta(y=0.2)$ and $f_\theta(y=0.8)$. (f) Two cross-sectional views $f_\theta(z=0.2)$ and $f_\theta(z=0.8)$.
  • Figure 3: Results of our proposed framework on the rotated sphere example in Section \ref{['sec:rotate']}. (a) Visualization of the rotated sphere. (b) The $3$D transformation $f_\theta$ obtained by the trained network. The color at each point is defined as in Fig. \ref{['fig:lm_8']}. (c) Histogram of $\det\nabla f_\theta(p)$. (d) Cross-sectional view $f_\theta(x=0.5)$. (e) Cross-sectional view $f_\theta(y=0.5)$. (f) Cross-sectional view $f_\theta(z=0.5)$.
  • Figure 4: Results of the Large Distortion Mapping example. (a)-(c) visualize the resulting mappings $f_\theta$, with two cross-sectional views $f_\theta(x=0.2),\, f_\theta(x=0.8)$ based on the three formulations in Section \ref{['sec:three_formulations']}. The color at each point is defined the same as in Fig. \ref{['fig:lm_8']}. (d)-(f) show the histograms of $\det \nabla f_\theta$ obtained from the three formulations. (g)-(i) display the conformality loss, intensity loss and landmark loss during training of the three formulations, respectively.
  • Figure 5: The Large Distortion Mapping example. Visualization of the registration results via three views, i.e. $x=0.5,\,y=0.5,\,z=0.5$. (a)-(b) Three views (slices) of the source image $S$ and target images $T$ respectively. (c) Three views of the absolute difference between $S$ and $T$. (d)-(e) The landmark matching registration results. (f)-(g) The intensity matching registration results. (h)-(i) The hybrid matching registration results.
  • ...and 5 more figures

Theorems & Definitions (3)

  • theorem 1
  • remark thmcounterremark
  • proof