Parallel Algorithm and Software for Image Inpainting via Sub-Riemannian Minimizers on the Group of Rototranslations

Abstract

The paper is devoted to an approach for image inpainting developed on the basis of neurogeometry of vision and sub-Riemannian geometry. Inpainting is realized by completing damaged isophotes (level lines of brightness) by optimal curves for the left-invariant sub-Riemannian problem on the group of rototranslations (motions) of a plane SE(2). The approach is considered as anthropomorphic inpainting since these curves satisfy the variational principle discovered by neurogeometry of vision. A parallel algorithm and software to restore monochrome binary or halftone images represented as series of isophotes were developed. The approach and the algorithm for computation of completing arcs are presented in detail.

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

“…Many results using sub-Riemannian (SR) geodesics as model for subjective contours in SE 2 are already present in literature. 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].…”

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

“…in order to avoid confusion with the chain law for differentiation. Recall (26) and note that s max for a given geodesics is determined by its initial momentum. We write s max (h(0)) when we want to stress this dependence.…”

Relation between Euler’s elasticae and sub-Riemannian geodesics on SE(2)

“…The problem can be seen as an optimal motion planning problem for the Reeds-Shepp car, which can move forward and backward and rotate on a place [6]. It has important relations to vision [13,14] and image processing [11]. SR-geodesics are critical points of the SR-length functional.…”

“…In this paper we consider two classical geometric control problems [1,2]: the problem of Euler's elasticae and the problem of sub-Riemannian (SR) geodesics on SE (2). Solution curves to both problems have many applications in mechanics [3,4,5], robotics [6,8], image processing [9,10,11,12] and modelling of human visual system [13,14]. Although solutions are well known in geometric control community, the authors have noticed a common confusion in applied societies where people sometimes mix these two curves.…”