Dimension on Discrete Spaces

In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by means of axioms, and the axioms are based on an obvious geometrical background. This work presents some discrete models of n-dimensional Euclidean spaces, n-dimensional spheres, a torus and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities It also analyzes the topology of (3+1)-spacetime. We are also discussing the question by R. Sorkin [19] about how to derive the system of simplicial complexes from a system of open covering of a topological space S.Comment: 16 pages, 8 figures, Latex. Figures are not included, available from the author upon request. Preprint SU-GP-93/1-1. To appear in "International Journal of Theoretical Physics

Sufficient conditions for topological invariance of 2D images under rigid transformations

International audienceIn ℝ^2, rigid transformations are topology-preserving operations. However, this property is generally no longer true when considering digital images instead of continuous ones, due to digitization effects. In this article, we investigate this issue by studying discrete rigid transformations (DRTs) on ℤ^2. More precisely, we define conditions under which digital images preserve their topological properties under any arbitrary DRTs. Based on the recently introduced notion of DRT graph and the classical notion of simple point, we first identify a family of local patterns that authorize topological invariance under DRTs. These patterns are then involved in a local analysis process that guarantees topological invariance of whole digital images in linear time

On the equivalence between hierarchical segmentations and ultrametric watersheds

We study hierarchical segmentation in the framework of edge-weighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical segmentations. We end this paper by showing how to use the proposed framework in practice in the example of constrained connectivity; in particular it allows to compute such a hierarchy following a classical watershed-based morphological scheme, which provides an efficient algorithm to compute the whole hierarchy.Comment: 19 pages, double-colum

Digitally Continuous Multivalued Functions, Morphological Operations and Thinning Algorithms

In a recent paper (Escribano et al. in Discrete Geometry for Computer Imagery 2008. Lecture Notes in Computer Science, vol. 4992, pp. 81–92, 2008) we have introduced a notion of continuity in digital spaces which extends the usual notion of digital continuity. Our approach, which uses multivalued functions, provides a better framework to define topological notions, like retractions, in a far more realistic way than by using just single-valued digitally continuous functions. In this work we develop properties of this family of continuous functions, now concentrating on morphological operations and thinning algorithms. We show that our notion of continuity provides a suitable framework for the basic operations in mathematical morphology: erosion, dilation, closing, and opening. On the other hand, concerning thinning algorithms, we give conditions under which the existence of a retraction F:X⟶X∖D guarantees that D is deletable. The converse is not true, in general, although it is in certain particular important cases which are at the basis of many thinning algorithms

A Topological Method of Surface Representation

Abstract. A new method of representing a surface in the 3D space as a single digitally continuous sequence of faces is described. The method is based on topological properties of quasi-manifolds. It is realized as tracing the boundary of a growing set of labeled faces. As the result the surface is encoded as a single sequence of mutually adjacent faces. Each face is encoded by one byte. The code of the surface of a three-dimensional object takes much less memory space then the raster representation of the object. The object may be exactly reconstructed from the code. Surfaces of a genus greater that zero (e.g. that of a torus) may also be encoded by a single continuous sequence. The traversal algorithm recognizes the genus of the surface.

Tesselations by Connection in Orders

The watershed transformation is a powerful tool for segmenting images, but its precise definition in discrete spaces raises difficult problems. We propose a new approach in the framework of orders. We introduce the tesselation by connection, which is a transformation that preserves the connectivity, and can be implemented by a parallel algorithm