Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties

Abstract

In this paper we provide an analytical description of various classes of digital circles, spheres and in some cases hyperspheres, defined in a morphological framework. The topological properties of these objects, especially the separation of the digital space, are discussed according to the shape of the structuring element. The proposed framework is generic enough so that it encompasses most of the digital circle definitions that appear in the literature and extends them to dimension 3 and sometimes dimension n.

Implicit Digital Surfaces in Arbitrary Dimensions

“…It is easy to see that that the F 0 -digitization of a hyperplane corresponds to the supercover of a hyperplane [14] and that the digitizations of hyperspheres are particular cases of those described in [10]. See Figure 3 for some examples of implicit surfaces.…”

“…Alternative models have been introduced to overcome some limitations of the closed centered model [10] (open or semi-open models, exterior or interior Gaussian models, etc.). For the sake of clarity, we here only focus on the closed centered model.…”

“…In section two, we will present the digitization models and present a somewhat simplified flake family than the one proposed in [10]. In section three, we will discuss the conditions under which topological properties are preserved by the digitization process as we propose it.…”

“…They have been introduced in [10] in order to analytically characterize minimal (with respect to set inclusion) and k-separating digital hyperspheres. Using adjacency flakes as structuring elements in the digitization scheme provides the offset region defining the digital object with quite simple analytical characterization, while preserving important topological properties.…”

Optimal Consensus Set for nD Fixed Width Annulus Fitting

“…Most annuli fitting methods try to minimize the thickness of the fitted annuli [1,5,17]. In our case, we are interested in digital circles and spheres and more specifically Andres digital circles and hyperspheres [3] or k-Flake digital circles-spheres [18]. In those cases, the thickness is directly linked with topological properties.…”