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

## TL;DR

This paper introduces a new algorithm to identify connected components in images using Lie groups, improving analysis of complex structures like retinal vascular trees.

## Contribution

A novel algorithm for finding δ-connected components on Lie groups, with application to multi-orientation image analysis.

## Key findings

- The algorithm efficiently identifies branches in complex vascular trees using orientation score transforms.
- δ-connected components differentiate crossing structures and group aligned structures effectively.
- The method outperforms standard connected component algorithms in 2D Euclidean space.

## 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 \documentclass[12pt]{minimal}
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\begin{document}$$\delta >0$$\end{document}δ>0, the algorithm finds the largest cover of I such that all sets in the cover are separated by at least distance \documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}δ. We call the sets in the cover the \documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}δ-connected components of I (closely related to \documentclass[12pt]{minimal}
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\begin{document}$$\check{\text {C}}$$\end{document}Cˇech complexes of radius \documentclass[12pt]{minimal}
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\begin{document}$$\delta /2$$\end{document}δ/2). The grouping relies on an iterative procedure involving morphological dilations with Hamilton-Jacobi-Bellman kernels on G and notions of \documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}δ-thickened sets. We prove that the algorithm converges in finitely many iteration steps. We find the optimal value for \documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}δ using persistence diagrams. We also propose to use specific affinity matrices. This allows for grouping of \documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {L}_2(G)$$\end{document}L2(G) where \documentclass[12pt]{minimal}
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\begin{document}$$G=SE(2)$$\end{document}G=SE(2), the Lie group of roto-translations. By applying our algorithm in this Lie group, we illustrate that we obtain \documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}δ-connected components that differentiate between crossing structures and that group well-aligned, nearby structures. This contrasts standard connected component algorithms in \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^2$$\end{document}R2.