Leanne Vis | Searcharxiv

arXiv2026

Sub-Riemannian Snakes on the Projective Line Bundle with Applications to Segmentation of SEM Images

Geodesic tracking on the projective line bundle $\R^2 \times P^1 $ has many uses, including the segmentation of objects in images. However, global tracking requires expensive distance map computations. We provide a practical solution to this problem by introducing a snake model on $\R^2 \times P^1$, where we only compute the distance map where needed. Our method introduces a geometric criterion for switching between fast spatial snakes and computing minimizing geodesics of a new projective line bundle model. The new pseudo-distance underlying our geometric model is both symmetric and cusp-free, in contrast to previous geodesic sub-Riemannian models on $\R^2 \times P^1$. Our pseudo-distance satisfies the triangle inequality on a large set that we characterize, and includes a connected-component-informed cost function, which is highly advantageous in applications. Experiments on Scanning Electron Microscopy (SEM) images demonstrate our method's robust, automatic segmentation of overlapping electronic structures.

math.DG↗

arXiv2024

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 $\delta$. We call the sets in the cover the $\delta$-connected components of I (closely related to $\check{\text{C}}$ech complexes of radius $\delta/2$). 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. We find the optimal value for $\delta$ using persistence diagrams. We also propose specific affinity matrices that allow for grouping of $\delta$-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 $\mathbb{L}_2(G)$ where $G=SE(2)$, the Lie group of roto-translations. By applying our algorithm in this Lie group, we illustrate that we obtain $\delta$-connected components that differentiate between crossing structures and that group well-aligned, nearby structures. This contrasts standard connected component algorithms in $\mathbb{R}^2$.

math.DG↗