相关论文: Sub-Riemannian Fast Marching in SE(2)

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external…

We propose an efficient approach for the grouping of local orientations (points on vessels) via nilpotent approximations of sub-Riemannian distances in the 2D and 3D roto-translation groups $SE(2)$ and $SE(3)$. In our distance…

We introduce a modification of the Fast Marching Algorithm, which solves the generalized eikonal equation associated to an arbitrary continuous riemannian metric, on a two or three dimensional domain. The algorithm has a logarithmic…

Leveraging geodesic distances and the geometrical information they convey is key for many data-oriented applications in imaging. Geodesic distance computation has been used for long for image segmentation using Image based metrics. We…

Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient…

In order to detect salient lines in spherical images, we consider the problem of minimizing the functional $\int \limits_0^l C(\gamma(s)) \sqrt{\xi^2 + k_g^2(s)} \, {\rm d}s$ for a curve $\gamma$ on a sphere with fixed boundary points and…

The real symplectic Stiefel manifold is the manifold of symplectic bases of symplectic subspaces of a fixed dimension. It features in a large variety of applications in physics and engineering. In this work, we study this manifold with the…

In many robot motion planning problems, task objectives and physical constraints induce non-Euclidean geometry on the configuration space, yet many planners operate using Euclidean distances that ignore this structure. We address the…

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…

The fast marching algorithm, and its variants, solves numerically the generalized eikonal equation associated to an underlying riemannian metric. A major challenge for these algorithms is the non-isotropy of the riemannian metric.…

Many geometric machine learning and image analysis applications, require a left-invariant metric on the 5D homogeneous space of 3D positions and orientations SE(3)/SO(2). This is done in Equivariant Neural Networks (G-CNNs), or in PDE-Based…

We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on $SL(2, \mathbb{R})$. Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two…

We present Geodesic Semantic Search (GSS), a retrieval system that learns node-specific Riemannian metrics on citation graphs to enable geometry-aware semantic search. Unlike standard embedding-based retrieval that relies on fixed Euclidean…

We describe homogeneous sub-Riemannian geodesics for the standard sub-Riemannian structure on the group of proper motions of the plane SE(2). We show that this structure is not geodesically orbital, although the cut time is invariant w.r.t.…

Over the past few years, symmetric positive definite (SPD) matrices have been receiving considerable attention from computer vision community. Though various distance measures have been proposed in the past for comparing SPD matrices, the…

In this paper we study geodesics of left-invariant sub-Riemannian metrics on SO(3) and almost-Riemannian metrics on $S^2$. These structures are connected with each other, and it is possible to use information about one of them to obtain…

We develop an accelerated gradient descent algorithm on the Grassmann manifold to compute the subspace spanned by a number of leading eigenvectors of a symmetric positive semi-definite matrix. This has a constant cost per iteration and a…

We study random walks on sub-Riemannian manifolds using the framework of retractions, i.e., approximations of normal geodesics. We show that such walks converge to the correct horizontal Brownian motion if normal geodesics are approximated…

We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full $H^2$-metric without zero…

Distance functions are crucial in robotics for representing spatial relationships between a robot and its environment. They provide an implicit, continuous, and differentiable representation that integrates seamlessly with control,…