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

Abstract

In order to detect salient lines in spherical images, we consider the problem of minimizing the functional l 0 C(γ (s)) ξ 2 + k 2 g (s) ds for a curve γ on a sphere with fixed boundary points and directions. The total length l is free, s denotes the spherical arclength, and k g denotes the geodesic curvature of γ . Here the smooth external cost C ≥ δ > 0 is obtained from spherical data. We lift this problem to the sub-Riemannian (SR) problem in Lie group SO(3) and show that the spherical projection of certain SR geodesics provides a solution to our curve optimization problem. In fact, this holds only for the geodesics whose spherical projection does not exhibit a cusp. The problem is a spherical extension of a well-known contour perception model, where we extend the model by Boscain and Rossi to the general case ξ > 0, C = 1. For C = 1, we derive SR geodesics and evaluate the first cusp time. We show that these curves have a simpler expression when they are parameterized by spherical arclength rather than by sub-Riemannian arclength. For case C = 1 (data-driven SR geodesics), we solve via a SR Fast Marching method. Finally, we show an experiment of vessel tracking in a spherical image of the retina and study the effect of including the spherical geometry in analysis of vessels curvature.

Geometrical optical illusion via sub-Riemannian geodesics in the roto-translation group

“…SR geodesics and their application to image analysis were also studied in [9,34,43], e.g. to retinal vessel tracking by Bekkers et al [8,42,7]. Explicit formulas for SR geodesics and optimal synthesis in SE 2 are obtained by Sachkov [56].…”

Hamiltonian Fast Marching: A Numerical Solver for Anisotropic and Non-Holonomic Eikonal PDEs

“…The Hamiltonian appearing in the eikonal equation is represented internally in the HFM software in a sum of squares form (33), involving weights and offsets, denoted by α i (p) ≥ 0 andė i (p) ∈ Z d , where p ∈ X and 1 ≤ i ≤ I. Our differentiation techniques assume that the offsets remain constant, but that the weights are proportional to (the inverse square of) one or several 16 user provided cost functions c(p, l), 1 ≤ l ≤ L, which themselves are subject to a linear perturbation εξ(p, l).…”

“…Consider a point p ∈ X tagged Trial, and which value U (p) must be updated with respect to the Accepted points, as specified in the last line of Algorithm 1. In view of the Hamiltonian's expression (33), this means that the currently stored value U (p) must be replaced with the largest solution λ ∈ R to the following univariate quadratic equation…”

“…Our objective is to relate the first order Taylor expansions of the boundary conditions σ ε , weights α ε , and solution U ε . Before we begin, let us mention that the simplified numerical scheme (33) considered in this section makes U ε differentiable, but that it is only almost everywhere differentiable in the general case (18). That is because a ∈ R → max{0, a} 2 is everywhere differentiable, but not (a, b) ∈ R 2 → max{a, b} 2 .…”

“…Tubular structure segmentation in medical images is one of the main applications intended for the HFM library. Early versions, and related software of the first author, are used for that purpose in [12,52,15,22,14,33,3]. Vessel extraction in images of the retina is illustrated in Figure 24, using two popular models involving respectively radius and orientation lifting.…”