Well-composed images and rigid transformations

Abstract

We study the conditions under which the topological properties of a 2D well-composed binary image are preserved under arbitrary rigid transformations. This work initiates a more global study of digital image topological properties under such transformations, which is a crucial but under-considered problem in the context of image processing, e.g., for image registration and warping.

Digital shapes, digital boundaries and rigid transformations: A topological discussion

“…In this paper -that is mainly related to the works published in [7,8] -we expose some preliminary results devoted to this question. More precisely, we focus on the specific case of digital shapes defined on Z 2 , and on their behaviour under rigid transformations.…”

A Tutorial on Well-Composedness

“…In that sense, they are "topologically equivalent". Making regular any image is then straightforward, using for example a super-resolution strategy [137] able to make any well-composed image regular. Figure 50 shows letters whose topology is lost under rigid transformation due to the local critical patterns depicted in red: 4-connected components are decomposed into several other 4-connected components, and the 8-components corresponding to the holes are merged with the background.…”

“…Fortunately, Ngo et al [137,138,136] studied under which conditions the topology of a 2D digital image is preserved under discrete rigid transformation (DRT) and proved that if the initial image is regular (a criterion based on some forbidden patterns described in Figure 49) including the usual critical configurations of Latecki [104]), then the resulting DRT is well-composed and the adjacency trees of the initial and final im- ages are isomorphic. In that sense, they are "topologically equivalent".…”