Rémy Malgouyres

Homotopy in two-dimensional digital images

We recall the basic definitions concerning homotopy in 2D Digital Topology, and we set and prove several results concerning homotopy of subsets. Then we introduce an explicit isomorphism between the fundamental group and a free group. As a consequence, we provide an algorithm for deciding whether two closed path are homotopic.

Some topological properties of discrete surfaces

A basic property of a simple closed surface is the Jordans property: the complement of the surface has two connected components. We call backcomponent any such component, and the union of a backcomponent and the surface is called the closure of this back-component. We introduce the notion of strong surface as a surface which satisfies a strong homotopy property: the closure of a back-component is strongly homotopic to that back-component. This means that we can homotopically remove any subset of a strong surface from the closure of a backcomponent. On the basis of some results on homotopy ([2]), and strong homotopy ([3], [4], [5]), we have proved that the simple closed 26-surfaces defined by Morgenthaler and Rosenfeld ([19]), and the simple closed 18-surfaces defined by Malgouyres ([15]) are both strong surfaces. Thus, strong surfaces appear as an interesting generalization of these two notions of a surface.

Topology Preservation Within Digital Surfaces

Given two connected subsets Y?X of the set of the surfels of a connected digital surface, we propose three equivalent ways to express Y being homotopic to X. The first characterization is based on sequential deletion of simple surfels. This characterization enables us to define thinning algorithms within a digital Jordan surface. The second characterization is based on the Euler characteristics of sets of surfels. This characterization enables us, given two connected sets Y?X of surfels, to decide whether Y is n-homotopic to X. The third characterization is based on the (digital) fundamental group.

Some Topological Properties of Surfaces in Z3

A basic property of a simple closed surface is the Jordan property: the complement of the surface has two connected components. We call back-component any such component, and the union of a back-component and the surface is called the closure of this back-component. We introduce the notion of strong surface as a surface which satisfies a strong homotopy property: the closure of a back-component is strongly homotopic to that back-component. This means that we can homotopically remove any subset of a strong surface from the closure of a back-component. On the basis of some results on homotopy, and strong homotopy, we have proved that the simple closed 26-surfaces defined by Morgenthaler and Rosenfeld, and the simple closed 18-surfaces defined by one of the authors are both strong surfaces. Thus, strong surfaces appear as an interesting generalization of these two notions of a surface.

Fast computation of the normal vector field of the surface of a 3-D discrete object

The shape description of the surface of three-dimensional discrete objects is widely used for displaying these objects, or measuring some useful parameters. Elementary components of discrete surfaces, called surfels, contain some geometric information, but at a scale that is too small with respect to the scale at which we actually want to describe objects. We present here a fast computational technique to compute the normal vector field of a discrete object at a given scale. Its time cost is proportional to the number of surfels at and little dependent on the scale. We prove that our algorithm converges toward the right value in the case of a plane surface. We also give some experimental results on families of curved surfaces.

A definition of surfaces of Z 3 . A new 3D discrete Jordan theorem

We provide a definition of surfaces in Z3 which generalizes the surfaces of Morgenthaler and Rosenfeld (1981) when these are analyzed with the 26-connectivity. The surfaces thus defined which are finite satisfy a 3D Jordan property (i.e. the complement of a finite surface has two connected components) and we provide an algorithmic characterization of interior and exterior points. Besides, each point of a finite surface is 6-adjacent to both an interior and an exterior point.

A new local property of strong n -surfaces

Abstract In Bertrand and Malgouyres (1996), two characterizations of discrete surfaces of Z 3 are proposed which are called strong 18- surfaces and strong 26- surfaces . However, strong surfaces are defined by global properties and the question of their local characterization remains. We propose a new local characterization of those of the separating and thin objects which are strong surfaces.

COMPLETE LOCAL CHARACTERIZATION OF STRONG 26-SURFACES: CONTINUOUS ANALOGS FOR STRONG 26-SURFACES

In Ref. 6, two similar characterizations of discrete surfaces of ℤ3 are proposed which are called strong 18-surfaces and strong 26-surfaces. The proposed characterizations consist in some natural global properties of surfaces. In this paper, we first give local necessary conditions for an object to be a strong 26-surface. An object satisfying these local properties is called a near strong 26-surface. Then we construct continuous analogs for near strong 26-surfaces and, using the continuous Jordan Theorem, we prove that the necessary local conditions previously introduced in fact give a complete local characterization of strong 26-surfaces: the class of near strong 26-surfaces coincides with the class of strong 26-surfaces.

Convergence of binomial-based derivative estimation for C2 noisy discretized curves

We present a derivative estimator for discrete curves and discretized functions which uses convolutions with integer-only binomial masks. The convergence results work for C^2 functions, and as a consequence we obtain a complete uniform convergence result for parameterized C^2 curves for derivatives of any order.

There is no local characterization of separating and thin objects in Z 3

Abstract We prove that there is no local characterization of the class of objects S of Z 3 which separate Z 3 in two 6-connected components and such that any point of S is 6-adjacent to both 6-components of Z 3 S .