arXiv subjects

Nicky J. van den Berg

Publications and source records attributed to Nicky J. van den Berg.

3 recordsLinked to original sources

arXiv2026

Equivariant Eikonal Neural Networks: Grid-Free, Scalable Travel-Time Prediction on Homogeneous Spaces

We introduce Equivariant Neural Eikonal Solvers, a novel framework that integrates Equivariant Neural Fields (ENFs) with Neural Eikonal Solvers. Our approach employs a single neural field where a unified shared backbone is conditioned on signal-specific latent variables - represented as point clouds in a Lie group - to model diverse Eikonal solutions. The ENF integration ensures equivariant mapping from these latent representations to the solution field, delivering three key benefits: enhanced representation efficiency through weight-sharing, robust geometric grounding, and solution steerability. This steerability allows transformations applied to the latent point cloud to induce predictable, geometrically meaningful modifications in the resulting Eikonal solution. By coupling these steerable representations with Physics-Informed Neural Networks (PINNs), our framework accurately models Eikonal travel-time solutions while generalizing to arbitrary Riemannian manifolds with regular group actions. This includes homogeneous spaces such as Euclidean, position-orientation, spherical, and hyperbolic manifolds. We validate our approach through applications in seismic travel-time modeling of 2D, 3D, and spherical benchmark datasets. Experimental results demonstrate superior performance, scalability, adaptability, and user controllability compared to existing Neural Operator-based Eikonal solver methods.

cs.LG↗

arXiv2025

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 $δ>0$, 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 $\mathbb{R}^2$.

math.DG↗

arXiv2025

Crossing-Preserving Geodesic Tracking on Spherical Images

In image analysis one often encounters spherical images, for instance in retinal imaging. The behavior of the vessels in the retina is an indicator of several diseases. To automate disease diagnosis using retinal images, it is necessary to develop an algorithm that automatically identifies and tracks vessels. To deal with crossings due to projected blood vessels in the image it is common to lift retinal images to the space of planar positions and orientations $\mathbb{M} := \mathbb{R}^2 \times S^1$. This implicitly assumes that the flat image accurately represents the geometry of the retina. As the eyeball is a sphere (and not a plane), we propose to compute the cusp-free, crossing-preserving geodesics in the space of spherical positions and orientations $\mathbb{W}$ on wide-field images. We clarify how to relate both manifolds and compare the calculated geodesics. The results show clear advantages of crossing-preserving tracking in $\mathbb{W}$ over non-crossing-preserving tracking in $\mathbb{W}$ and are comparable to tracking results in $\mathbb{M}$.

math.DG↗