Computing geodesic paths encoding a curvature prior for curvilinear structure tracking

Chen, Da; Mirebeau, Jean-Marie; Shu, Minglei; Cohen, Laurent D.

doi:10.1073/pnas.2218869120

Cited by 2 publications

(1 citation statement)

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“…The principles of biological visual systems are actively used in computer vision. Based on these principles, effective image processing methods are created, e.g., image reconstruction [17] and detection of salient lines in images [18,19]. In [18], optimal trajectories of a System (1) are used to detect salient lines in images.…”

Section: Introductionmentioning

confidence: 99%

Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector

Mashtakov,

Sachkov

2023

Mathematics

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We study a time-optimal problem in the roto-translation group with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and turn with a given minimum turning radius. Our work generalizes the sub-Riemannian problem by adding a restriction on the velocity vector to lie in a circular sector. The sub-Riemannian problem is given by a special case when the sector is the full disc. The trajectories of the system are applicable in image processing to detect salient lines. We study the local and global controllability of the system and the existence of a solution for given arbitrary boundary conditions. In a general case of the sector opening angle, the system is globally but not small-time locally controllable. We show that when the angle is obtuse, a solution exists for any boundary conditions, and when the angle is reflex, a solution does not exist for some boundary conditions. We apply the Pontryagin maximum principle and derive a Hamiltonian system for extremals. Analyzing a phase portrait of the Hamiltonian system, we introduce the rectified coordinates and obtain an explicit expression for the extremals in Jacobi elliptic functions. We show that abnormal extremals are of circular type, and they correspond to motions of a car along circular arcs of minimal possible radius. The normal extremals in a general case are given by concatenation of segments of sub-Riemannian geodesics in SE2 and arcs of circular extremals. We show that, in a general case, the vertical (momentum) part of the extremals is periodic. We partially study the optimality of the extremals and provide estimates for the cut time in terms of the period of the vertical part.

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Section: Introductionmentioning

confidence: 99%

Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector

Mashtakov,

Sachkov

2023

Mathematics

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Computing geodesic paths encoding a curvature prior for curvilinear structure tracking

Chen

Mirebeau²,

Shu

et al. 2023

Proc. Natl. Acad. Sci. U.S.A.

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In this paper, we introduce an efficient method for computing curves minimizing a variant of the Euler–Mumford elastica energy, with fixed endpoints and tangents at these endpoints, where the bending energy is enhanced with a user-defined and data-driven scalar-valued term referred to as the curvature prior. In order to guarantee that the globally optimal curve is extracted, the proposed method involves the numerical computation of the viscosity solution to a specific static Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE). For that purpose, we derive the explicit Hamiltonian associated with this variant model equipped with a curvature prior, discretize the resulting HJB PDE using an adaptive finite difference scheme, and solve it in a single pass using a generalized fast-marching method. In addition, we also present a practical method for estimating the curvature prior values from image data, designed for the task of accurately tracking curvilinear structure centerlines. Numerical experiments on synthetic and real-image data illustrate the advantages of the considered variant of the elastica model with a prior curvature enhancement in complex scenarios where challenging geometric structures appear.

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Computing geodesic paths encoding a curvature prior for curvilinear structure tracking

Cited by 2 publications

References 61 publications

Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector

Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector

Computing geodesic paths encoding a curvature prior for curvilinear structure tracking

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