Alternate product adjacencies in digital topology

[EN] We study properties of Cartesian products of digital images, using a variety of adjacencies that have appeared in the literature.Boxer, L. (2018). Alternate product adjacencies in digital topology. Applied General Topology. 19(1):21-53. doi:10.4995/agt.2018.7146SWORD215319

Chain Homotopies for Object Topological Representations

This paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (non-unique) algebraic-topological format called AM-model. An AM-model for a given object K is determined by a concrete chain homotopy and it provides, in particular, integer (co)homology generators of K and representative (co)cycles of these generators. An algorithm for computing an AM-model and the cohomological invariant HB1 (derived from the rank of the cohomology ring) with integer coefficients for a finite simplicial complex in any dimension is designed here. A concept of generators which are "nicely" representative cycles is also presented. Moreover, we extend the definition of AM-models to 3D binary digital images and we design algorithms to update the AM-model information after voxel set operations (union, intersection, difference and inverse)

On the Cohomology of 3D Digital Images

We propose a method for computing the cohomology ring of three--dimensional (3D) digital binary-valued pictures. We obtain the cohomology ring of a 3D digital binary--valued picture $I$, via a simplicial complex K(I)topologically representing (up to isomorphisms of pictures) the picture I. The usefulness of a simplicial description of the "digital" cohomology ring of 3D digital binary-valued pictures is tested by means of a small program visualizing the different steps of the method. Some examples concerning topological thinning, the visualization of representative (co)cycles of (co)homology generators and the computation of the cup product on the cohomology of simple pictures are showed.Comment: Special Issue: Advances in Discrete Geometry and Topolog

Coding cells of digital spaces: a framework to write generic digital topology algorithms

This paper proposes a concise coding of the cells of n-dimensional finite regular grids. It induces a simple, generic and efficient framework for implementing classical digital topology data structures and algorithms. Discrete subsets of multidimensional images (e.g. regions, digital surfaces, cubical cell complexes) have then a common and compact representation. Moreover, algorithms have a straightforward and efficient implementation, which is independent from the dimension or sizes of digital images. We illustrate that point with generic hypersurface boundary extraction algorithms by scanning or tracking. This framework has been implemented and basic operations as well as the presented applications have been benchmarked

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

Towards digital cohomology

We propose a method for computing the Z 2–cohomology ring of a simplicial complex uniquely associated with a three–dimensional digital binary–valued picture I. Binary digital pictures are represented on the standard grid Z 3, in which all grid points have integer coordinates. Considering a particular 14–neighbourhood system on this grid, we construct a unique simplicial complex K(I) topologically representing (up to isomorphisms of pictures) the picture I. We then compute the cohomology ring on I via the simplicial complex K(I). The usefulness of a simplicial description of the digital Z 2–cohomology ring of binary digital pictures is tested by means of a small program visualizing the different steps of our method. Some examples concerning topological thinning, the visualization of representative generators of cohomology classes and the computation of the cup product on the cohomology of simple 3D digital pictures are showed

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

Reusing integer homology information of binary digital images

In this paper, algorithms for computing integer (co)homology of a simplicial complex of any dimension are designed, extending the work done in [1,2,3]. For doing this, the homology of the object is encoded in an algebraic-topological format (that we call AM-model). Moreover, in the case of 3D binary digital images, having as input AM-models for the images I and J, we design fast algorithms for computing the integer homology of I ∪J, I ∩J and I ∖J

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