Encoding Specific 3D Polyhedral Complexes Using 3D Binary Images

We build upon the work developed in [4] in which we presented a method to “locally repair” the cubical complex Q(I) associated to a 3D binary image I, to obtain a “well-composed” polyhedral complex P(I), homotopy equivalent to Q(I). There, we developed a new codification system for P(I), called ExtendedCubeMap (ECM) representation, that encodes: (1) the (geometric) information of the cells of P(I) (i.e., which cells are presented and where), under the form of a 3D grayscale image gP ; (2) the boundary face relations between the cells of P(I), under the form of a set BP of structuring elements. In this paper, we simplify ECM representations, proving that geometric and topological information of cells can be encoded using just a 3D binary image, without the need of using colors or sets of structuring elements. We also outline a possible application in which well-composed polyhedral complexes can be useful.Junta de Andalucía FQM-369Ministerio de Economía y Competitividad MTM2012-32706Ministerio de Economía y Competitividad MTM2015-67072-

One More Step Towards Well-Composedness of Cell Complexes over nD Pictures

An nD pure regular cell complex K is weakly well-composed (wWC) if, for each vertex v of K, the set of n-cells incident to v is face-connected. In previous work we proved that if an nD picture I is digitally well composed (DWC) then the cubical complex Q(I) associated to I is wWC. If I is not DWC, we proposed a combinatorial algorithm to “locally repair” Q(I) obtaining an nD pure simplicial complex PS(I) homotopy equivalent to Q(I) which is always wWC. In this paper we give a combinatorial procedure to compute a simplicial complex PS(¯I) which decomposes the complement space of |PS(I)| and prove that PS(¯I) is also wWC. This paper means one more step on the way to our ultimate goal: to prove that the nD repaired complex is continuously well-composed (CWC), that is, the boundary of its continuous analog is an (n − 1)- manifold.Ministerio de Economía y Competitividad MTM2015-67072-

How to Make nD Functions Digitally Well-Composed in a Self-dual Way

International audienceLatecki et al. introduced the notion of 2D and 3D well-composed images, i.e., a class of images free from the " connectivities paradox " of digital topology. Unfortunately natural and synthetic images are not a priori well-composed. In this paper we extend the notion of " digital well-composedness " to nD sets, integer-valued functions (gray-level images), and interval-valued maps. We also prove that the digital well-composedness implies the equivalence of connectivities of the level set components in nD. Contrasting with a previous result stating that it is not possible to obtain a discrete nD self-dual digitally well-composed function with a local interpolation, we then propose and prove a self-dual discrete (non-local) interpolation method whose result is always a digitally well-composed function. This method is based on a sub-part of a quasi-linear algorithm that computes the morphological tree of shapes