Connected Components on Lie Groups and Applications to Multi-Orientation Image Analysis
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 delta>0, the algorithm finds the largest cover of I such that all sets in the cover are separated by at least distance delta...
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Main Authors | , , , |
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Format | Journal Article |
Language | English |
Published |
26.09.2024
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Subjects | |
Online Access | Get full text |
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Summary: | 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 delta>0, the algorithm finds the largest cover of I such
that all sets in the cover are separated by at least distance delta. We call
the sets in the cover the delta-connected components of I. The grouping relies
on an iterative procedure involving morphological dilations with
Hamilton-Jacobi-Bellman-kernels on G and notions of delta-thickened sets. We
prove that the algorithm converges in finitely many iteration steps, and we
propose a strategy to find an optimal value for delta based on persistence
homology arguments. We also introduce the concept of affinity matrices. This
allows grouping delta-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 L_2(G) where G=SE(2), the Lie group of positions and
orientations. By applying our algorithm in this Lie group, we illustrate that
we obtain delta-connected components that differentiate between crossing
structures and that group well-aligned, nearby structures. This contrasts
standard connected component algorithms in R^2. |
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DOI: | 10.48550/arxiv.2409.18002 |