Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D

TL;DR

This paper introduces a fast, accurate method for grouping blood vessels in 2D and 3D images using nilpotent approximations of sub-Riemannian distances in roto-translation groups, improving perceptual grouping performance.

Contribution

It presents a novel approach using nilpotent group norms to approximate sub-Riemannian distances, enabling efficient vessel grouping in medical imaging.

Findings

Approximations are accurate compared to true sub-Riemannian distances.

Sub-Riemannian geometry significantly improves grouping performance.

Fast analytic approximations match or outperform data-adaptive methods.

Abstract

We propose an efficient approach for the grouping of local orientations (points on vessels) via nilpotent approximations of sub-Riemannian distances in the 2D and 3D roto-translation groups $SE(2)$ and $SE(3)$. In our distance approximations we consider homogeneous norms on nilpotent groups that locally approximate $SE(n)$, and which are obtained via the exponential and logarithmic map on $SE(n)$. In a qualitative validation we show that the norms provide accurate approximations of the true sub-Riemannian distances, and we discuss their relations to the fundamental solution of the sub-Laplacian on $SE(n)$. The quantitative experiments further confirm the accuracy of the approximations. Quantitative results are obtained by evaluating perceptual grouping performance of retinal blood vessels in 2D images and curves in challenging 3D synthetic volumes. The results show that 1) sub-Riemannian geometry is essential in achieving top performance and 2) that grouping via the fast analytic approximations performs almost equally, or better, than data-adaptive fast marching approaches on $\mathbb{R}^n$ and $SE(n)$.