Connected Components on Lie Groups and Applications to Multi-Orientation Image Analysis

TL;DR

A new algorithm to find the connected components of a compact set I from a Lie group G endowed with a left-invariant Riemannian distance is developed and it is illustrated that the method can efficiently identify branches in complex vascular trees on retinal images.

Abstract

We develop and analyze a new algorithm to find the connected components of a compact set $I$ from a Lie group $G$ endowed with a left-invariant Riemannian distance. For a given $δ>0$, the algorithm finds the largest cover of $I$ such that all sets in the cover are separated by at least distance $δ$. We call the sets in the cover the $δ$-connected components of I (closely related to $\check{\text{C}}$ech complexes of radius $δ/2$). The grouping relies on an iterative procedure involving morphological dilations with Hamilton-Jacobi-Bellman kernels on $G$ and notions of $δ$-thickened sets. We prove that the algorithm converges in finitely many iteration steps. We find the optimal value for $δ$ using persistence diagrams. We also propose specific affinity matrices that allow for grouping of $δ$-connected components based on their local proximity and alignment. Among the many different applications of the algorithm, in this article, we focus on illustrating that the method can efficiently identify (possibly overlapping) branches in complex vascular trees on retinal images. This is done by applying an orientation score transform to the images that allows us to view them as functions from $\mathbb{L}_2(G)$ where $G=SE(2)$, the Lie group of roto-translations. By applying our algorithm in this Lie group, we illustrate that we obtain $δ$-connected components that differentiate between crossing structures and that group well-aligned, nearby structures. This contrasts standard connected component algorithms in $\mathbb{R}^2$.

Figures (8)

• Figure 1: Visualization of the connected component algorithm in $\mathbb{R}^2$ and in the Lie group $SE(2)$ on an image of straight lines. The classical connected component algorithm in $\mathbb{R}^2$ cannot differentiate between the different line structures, unlike the extension to $SE(2)$. Here we applied the algorithm in Sec. \ref{['sec:20230822:Algorithm']} using parameters $(w_1,w_2,w_3)=(0.1,1,4)$ for the left-invariant metric \ref{['eq:MTFweights']}.
• Figure 2: Visualization of the effect of performing the connected component algorithm in $\mathbb{R}^2$ and in the Lie group $SE(2)$ on an image of ovals and lines. The classical connected component algorithm in $\mathbb{R}^2$ is not able to differentiate between the different line structures, unlike the extension to $SE(2)$. Here we applied the algorithm in Sec. \ref{['sec:20230822:Algorithm']} using parameters $(w_1,w_2,w_3)=(0.2,1,4)$ for the left-invariant metric \ref{['eq:MTFweights']}.
• Figure 3: Visualization of the effect of the connected component algorithm in $\mathbb{R}^2$ and in the Lie group $SE(2)$. The classical connected component algorithm is not able to differentiate between crossing vessels, and additionally breaks up not perfectly connected vessels, into different components. The algorithm presented in Sec. \ref{['sec:20230822:Algorithm']} using parameters $(w_1,w_2,w_3)=(0.2,1,4)$ for the left-invariant metric \ref{['eq:MTFweights']}, can better differentiate between different crossing structures and can additionally group well-aligned structures resulting in a more intuitive result.
• Figure 4: Visualization of the effect of the metric tensor field on the connected component algorithm in the Lie group $SE(2)$. The output heavily relies on the chosen distance metric $d_\mathcal{G}$ introduced in \ref{['eq:distance']}, with metric tensor parameters as defined in \ref{['weights']}.
• Figure 5: Visualization of distance balls and their logarithmic approximations in G=$SE(2)$ and G=SO(3). We see isocontours of $d(p_0,\cdot)$ in $G$, and on the bottom, we see the min-projection over the orientation $\theta$ of these contours. The visible contours are $d=0.5,1,1.5,2,2.5$ and the metric parameters are $(w_1,w_2,w_3)=(\tilde{w}_1,\tilde{w}_2,\tilde{w}_3)=(1,4,1)$. Parameter $w_1$ controls costs for tangential motion, $w_2$ controls costs for lateral motion on the base manifold (resp. $\mathbb{R}^2$ and $S^2$) and $w_3$ controls costs for changing the orientation along a geodesic. Thereby, together, they control the anisotropic shape of the Riemannian ball and the identification of connected components, recall white balls in column 2 of Fig. \ref{['fig:InfluenceMetricTensorField']}.

Theorems & Definitions (37)

• remark thmcounterremark: $\delta$-connected components and Čech/Vietoris-Rips complexes of radius $\delta/2$
• remark thmcounterremark
• remark thmcounterremark: Explanation weights of metric tensor field
• remark thmcounterremark
• remark thmcounterremark
• remark thmcounterremark: Inclusion of Symmetries
• remark thmcounterremark
• definition thmcounterdefinition: Riemannian Ball
• definition thmcounterdefinition: $\delta$-covering
• definition thmcounterdefinition: Covering number