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Stationary Velocity Fields on Matrix Groups for Deformable Image Registration

Johannes Bostelmann, Ole Gildemeister, Jan Lellmann

TL;DR

A decomposition condition is proved that allows us to apply a scaling-and-squaring approach for efficient numerical integration of the flow equation and move Euclidean transformations into the low-frequency part, toward which network architectures are often naturally biased, so that larger motions can be recovered more easily.

Abstract

The stationary velocity field (SVF) approach allows to build parametrizations of invertible deformation fields, which is often a desirable property in image registration. Its expressiveness is particularly attractive when used as a block following a machine learning-inspired network. However, it can struggle with large deformations. We extend the SVF approach to matrix groups, in particular $\SE(3)$. This moves Euclidean transformations into the low-frequency part, towards which network architectures are often naturally biased, so that larger motions can be recovered more easily. This requires an extension of the flow equation, for which we provide sufficient conditions for existence. We further prove a decomposition condition that allows us to apply a scaling-and-squaring approach for efficient numerical integration of the flow equation. We numerically validate the approach on inter-patient registration of 3D MRI images of the human brain.

Stationary Velocity Fields on Matrix Groups for Deformable Image Registration

TL;DR

A decomposition condition is proved that allows us to apply a scaling-and-squaring approach for efficient numerical integration of the flow equation and move Euclidean transformations into the low-frequency part, toward which network architectures are often naturally biased, so that larger motions can be recovered more easily.

Abstract

The stationary velocity field (SVF) approach allows to build parametrizations of invertible deformation fields, which is often a desirable property in image registration. Its expressiveness is particularly attractive when used as a block following a machine learning-inspired network. However, it can struggle with large deformations. We extend the SVF approach to matrix groups, in particular . This moves Euclidean transformations into the low-frequency part, towards which network architectures are often naturally biased, so that larger motions can be recovered more easily. This requires an extension of the flow equation, for which we provide sufficient conditions for existence. We further prove a decomposition condition that allows us to apply a scaling-and-squaring approach for efficient numerical integration of the flow equation. We numerically validate the approach on inter-patient registration of 3D MRI images of the human brain.

Paper Structure

This paper contains 16 sections, 2 theorems, 93 equations, 16 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Image registration using velocity flow on matrix group valued fields
  4. Extended flow equation and existence
  5. Numerical integration by scaling and squaring
  6. Similarity term and regularizer
  7. Network architecture and optimization
  8. Numerical Results
  9. Datasets
  10. Benchmark Metrics
  11. Results
  12. Conclusion
  13. Proofs
  14. Proof of Theorem \ref{['thm:existence']}
  15. Proof of Theorem \ref{['thm:decomp']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\Omega$ be compact, let $\mathfrak{g}$ denote the space of right-invariant vector fields on $G \subset \mathop{\mathrm{GL}}\nolimits (\mathbb{R}, 4)$, and let $\nu: \Omega \to \mathfrak{g}$ be such that its evaluation at the identity $\nu_{{id}}$ is Lipschitz continuous. Then the solution $M$ o

Figures (16)

  • Figure 1: Comparison of the stationary velocity field (SVF) approach and our proposed matrix group-valued approach using the $\mathop{\mathrm{SE}}\nolimits(3)$ group on synthetically deformed human brain MRI data. While both methods manage to align the 3D volumes under small deformations (top row), the SVF approach struggles when the input images are not pre-aligned (bottom row). The resulting deformation field shows clear alignment issues (center), which are alleviated by the proposed matrix group-valued approach (right).
  • Figure 2: 2D slices of the 3D deformation fields from the bottom row of Fig. \ref{['fig:alignments-iso']}. The arrow colors indicate the displacement in the direction orthogonal to the slice. While SVF roughly aligns the images judging in terms of visual quality in Fig. \ref{['fig:alignments-iso']}, inspecting the deformation field (b) shows that it is far from the ground truth (a). The proposed parametrization with matrix groups (c) yields a result that closely matches the ground truth.
  • Figure 3: Process of generating a deformation field $\phi_\theta$ from the parameter vector $\theta$. The network-based generator$G$ transforms the parameter vector into a velocity field$\nu_\theta$. The solution operator$S$ solves an associated flow equation, resulting in a final deformation $\phi_\theta$. Classically, the velocity field and the flow equation are formulated in Euclidean space; we extend the approach to the matrix group setting.
  • Figure 4: Components used to parametrize deformations in recent approaches. In this work, we introduce a neural fields based combination of matrix groups with a flow equation.
  • Figure 5: (a) Visualization of a right-invariant vector field on $SO(\mathbb{R},2)$, homeomorphically identified with a circle. (b) Visualization of a non-invariant vector field on $SO(\mathbb{R},2)$. Right-invariant fields on $SO(\mathbb{R},2)$ can be parametrized with a single value; this is, in general, not possible for non-invariant vector fields.
  • ...and 11 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • proof
  • proof