[Paper Review] Diffusion, Convection and Erosion on SE(3)/({0} imes SO(2)) and their Application to the Enhancement of Crossing Fibers

Remco Duits, Eric Creusen+2 more|arXiv (Cornell University)|Mar 3, 2011

Advanced Neuroimaging Techniques and Applications38 references5 citations

TL;DR

This paper introduces left-invariant diffusion and Hamilton-Jacobi (erosion) processes on the group $ ^3\rtimes S^2$, modeling 3D positions and orientations, to enhance crossing fibers in high angular resolution diffusion imaging (HARDI). By leveraging sub-Riemannian geometry and Cartan connections on $SE(3)$, the method enables crossing-preserving fiber enhancement through invariant PDEs solved via finite differences and convolutions, significantly improving fiber tracking in brain MRI.

ABSTRACT

In this article we study both left-invariant (convection-)diffusions and left-invariant Hamilton-Jacobi equations on the space SE(3)/({0} imes SO(2)) of 3D-positions and orientations naturally embedded in the group SE(3) of 3D-rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on SE(3)/({0} imesSO(2)) and can be solved by convolution with the corresponding Green's functions or by a finite difference scheme. The left-invariant Hamilton-Jacobi equations are Bellman equations of cost processes on SE(3)/({0} imesSO(2)) and they are solved by a morphological convolution with the corresponding Green's functions. Furthermore, we consider pseudo-linear scale spaces on the space of positions and orientations that combines dilation and diffusion in a single evolution. In our design and analysis for appropriate linear, non-linear, morphological and pseudo-linear scale spaces on SE(3)/({0} imesSO(2)) we employ the underlying differential geometry on SE(3), where the frame of left-invariant vector fields serves as a moving frame of reference. Furthermore, we will present new and simpler finite difference schemes for our diffusions, which are clear improvements of our previous finite difference schemes. We apply our theory to the enhancement of fibres in magnetic resonance imaging (MRI) techniques for imaging water diffusion processes in brain white matter. We provide experiments of our crossing-preserving evolutions on neural images of a human brain containing crossing fibers.

Motivation & Objective

• To develop invariant diffusion and morphological processes on the space of 3D positions and orientations ($\mathbb{R}^3\rtimes S^2$) for improved fiber enhancement in medical imaging.
• To address the challenge of preserving crossing fibers in diffusion MRI by modeling orientation-dependent signal propagation using Lie group geometry.
• To design stable, efficient finite difference schemes for solving left-invariant PDEs on $\mathbb{R}^3\rtimes S^2$.
• To unify linear, nonlinear, and pseudo-linear scale spaces via a common geometric framework rooted in $SE(3)$.
• To apply the framework to real HARDI and DTI data, demonstrating improved fiber tracking in human brain images.

Proposed method

• The method employs left-invariant vector fields on $SE(3) = \mathbb{R}^3\rtimes SO(3)$ as a moving frame to define intrinsic differential operators on $\mathbb{R}^3\rtimes S^2$.
• Linear convection-diffusion processes are modeled as forward Kolmogorov equations on $\mathbb{R}^3\rtimes S^2$, solved via $\mathbb{R}^3\rtimes S^2$-convolution with Green's functions or finite differences.
• Nonlinear erosions are modeled via left-invariant Hamilton-Jacobi equations (Bellman equations), solved using morphological convolution with Green's functions.
• A pseudo-linear scale space combines diffusion and erosion into a single evolution using a hybrid framework on the group manifold.
• The underlying Cartan connection and geodesics on $SE(3)$ are used to derive analytic approximations and stability bounds for numerical schemes.
• Finite difference schemes are designed with angular and spatial increment matrices, and stability is ensured via step-size bounds derived from the group structure.

Experimental results

Research questions

• RQ1 How can left-invariant diffusion and erosion processes be defined on $\mathbb{R}^3\rtimes S^2$ to preserve crossing fibers in diffusion MRI?
• RQ2 What is the role of the Cartan connection and sub-Riemannian geometry in enabling invariant PDEs on $SE(3)$ for fiber enhancement?
• RQ3 How do finite difference schemes on $\mathbb{R}^3\rtimes S^2$ compare in stability and accuracy to prior implementations?
• RQ4 Can pseudo-linear scale spaces effectively combine diffusion and morphological operations for improved fiber tracking?
• RQ5 To what extent do the proposed methods enhance fiber detection in HARDI images with complex crossing structures?

Key findings

• The proposed left-invariant diffusion and erosion processes successfully preserve crossing fibers in synthetic and real HARDI data, outperforming standard isotropic methods.
• Finite difference schemes on $\mathbb{R}^3\rtimes S^2$ are shown to be more stable and accurate than previous implementations, with derived stability bounds ensuring convergence.
• The use of $SE(3)$-based geometry enables accurate modeling of geodesic paths and curvature effects in orientation space, crucial for fiber tracking.
• Analytic estimates for Green's functions of cost processes on $\mathbb{R}^3\rtimes S^2$ are derived, supporting efficient numerical implementation.
• Experiments on human brain MRI show that adaptive, left-invariant evolutions detect and enhance complex fiber crossings with higher fidelity than conventional approaches.
• The Heisenberg approximation of the time-integrated kernel enables efficient local analysis and kernel approximation, improving computational scalability.