Jean-Luc Toutant

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

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.

Discrete circles: an arithmetical approach with non-constant thickness

In the present paper, we introduce an arithmetical definition of discrete circles with a non-constant thickness and we exhibit different classes of them depending on the arithmetical discrete lines. On the one hand, it results in the characterization of regular discrete circles with integer parameters as well as J. Bresenhams circles. As far as we know, it is the first arithmetical definition of the latter one. On the other hand, we introduce new discrete circles, actually the thinnest ones for the usual discrete connectedness relations.

Minimal arithmetic thickness connecting discrete planes

While connected arithmetic discrete lines are entirely characterized, only partial results exist for the more general case of arithmetic discrete hyperplanes. In the present paper, we focus on the three-dimensional case, that is on arithmetic discrete planes. Thanks to arithmetic reductions on a vector n, we provide algorithms either to determine whether a given arithmetic discrete plane with n as normal vector is connected, or to compute the minimal thickness for which an arithmetic discrete plane with normal vector n is connected.

Implicit Digital Surfaces in Arbitrary Dimensions

In this paper we introduce a notion of digital implicit surface in arbitrary dimensions. The digital implicit surface is the result of a morphology inspired digitization of an implicit surface {x ∈ ℝn : f(x) = 0} which is the boundary of a given closed subset of ℝ n , {x ∈ ℝn : f(x) ≤ 0}. Under some constraints, the digital implicit surface has some interesting properties, such as k-tunnel freeness. Furthermore, for a large class of the digital implicit surfaces, there exists a very simple analytical characterization.

On the connecting thickness of arithmetical discrete planes

While connected rational arithmetical discrete lines and connected rational arithmetical discrete planes are entirely characterized, only partial results exist for the irrational arithmetical discrete planes. In the present paper, we focus on the connectedness of irrational arithmetical discrete planes, namely the arithmetical discrete planes with a normal vector of which the coordinates are not Q-linear dependent. Given v ∈ R3, we compute the lower bound of the thicknesses 2-connecting the arithmetical discrete planes with normal vector v. In particular, we show how the translation parameter operates in the connectedness of the arithmetical discrete planes.

Arithmetic Characterization of Polynomial based Discrete Curves

In the present paper, we investigate discretization of curves based on polynomials in the 2-dimensional space. Under some assumptions, we propose an arithmetic characterization of thin and connected discrete approximations of such curves. In fact, we reach usual discretization models, that is, GIQ, OBQ and BBQ but with a generic arithmetic definition.

Arc recognition on irregular isothetic grids and its application to reconstruction of noisy digital contours

In the present paper, we introduced an arc recognition technique suitable for irregular isothetic object. It is based on the digital inter-pixel (DIP) circle model, a pixel-based representation of the Kovalevskys circle. The adaptation to irregular image structurations allows us to apply DIP models for circle recognition in noisy digital contours. More precisely, the noise detector from Kerautret and Lachaud (2009) provides a multi-scale representation of the input contour with boxes of various size. We convert them into an irregular isothetic object and, thanks to the DIP model, reduce the recognition of arcs of circles in this object to a simple problem of point separation.

Characterization of simple closed surfaces in Z 3 : a new proposition with a graph-theoretical approach

In the present paper, we propose a topological characterization of digital surfaces. We introduce simple local conditions on the neighborhood of a voxel. If each voxel of a 26-connected digital set satisfies them, we prove a Jordan theorem and ensure that this set is strong 6-separating in Z3. Thus, we consider it as a digital surface.