Geodesic Methods in Computer Vision and Graphics

Abstract

This paper reviews both the theory and practice of the numerical computation of geodesic distances on Riemannian manifolds. The notion of Riemannian manifold allows one to define a local metric (a symmetric positive tensor field) that encodes the information about the problem one wishes to solve. This takes into account a local isotropic cost (whether some point should be avoided or not) and a local anisotropy (which direction should be preferred). Using this local tensor field, the geodesic distance is used to solve many problems of practical interest such as segmentation using geodesic balls and Voronoi regions, sampling points at regular geodesic distance or meshing a domain with geodesic Delaunay triangles. The shortest paths for this Riemannian distance, the so-called geodesics, are also important because they follow salient curvilinear structures in the domain. We show several applications of the numerical computation of geodesic distances and shortest paths to problems in surface and shape processing, in particular segmentation, sampling, meshing and comparison of shapes.

Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in $${\text {SO(3)}}$$ SO(3)

“…Here we consider a simplified model, where the optical axis (the best approximation of a line passing through the optical center of the cornea, lens, and fovea) coincides with the visual axis (the line connecting fixation point and the fovea). 1 The average radius of a human eye is R ≈ 10.5 mm, and the maximum distance between nodal point N and the central point C is a max = 17 mm − 10.5 mm = 6.5 mm. Now we switch to mathematical object coordinates of the retina where we use the eyeball radius to express lengths, i.e., we have R = 1 and a max = 6.5 mm 10.5 mm ·R = 13 21 in dimensionless coordinates.…”

“…In computer vision, it is common to extract salient curves in flat images via data-driven minimal paths or geodesics [1,3,5,2,4]. The minimizing geodesic is defined as the curve that minimizes the length functional, which is typically weighted by a cost function with high values at image locations with high curve saliency.…”

From active contours to minimal geodesic paths: New solutions to active contours problems by Eikonal equations

“…The minimal path model introduced by Cohen and Kimmel (1997) is a powerful tool in the fields of image analysis and medical imaging (Cohen, 2006;Peyré et al, 2010). It is designed to search for the global minimum of the weighted curve length energy (Caselles et al, 1997;Yezzi et al, 1997), measured along a curve C ∈ Lip([0, 1], Ω) as follows:…”

“…A large number of well-established Eikonal solvers have been exploited such as the fast marching methods (Sethian, 1999;Mirebeau, 2014aMirebeau, ,b, 2018. The global optimality and the efficient solutions lead to a series of successful minimal path-based applications as reviewed in (Peyré et al, 2010). However, the original minimal geodesic path model (Cohen and Kimmel, 1997) depends only on the first-order derivative term of a curve.…”

FLIC: Fast linear iterative clustering with active search

“…Inspired by SLIC, Wang et al (2013a) implemented an algorithm SSS that considered the structural information within images. It uses the geodesic distance (Peyré et al 2010) computed by the geometric flows instead of the simple Euclidean distance. However, efficiency is poor because of the bottleneck caused by the high computational cost of measuring geodesic distances.…”