3D Well-composed Polyhedral Complexes

A binary three-dimensional (3D) image I is well-composed if the boundary surface of its continuous analog is a 2D manifold. In this paper, we present a method to locally “repair” the cubical complex Q(I) (embedded in R3) associated to I to obtain a polyhedral complex P (I) homotopy equivalent to Q(I) such that the boundary surface of P (I) is a 2D manifold (and, hence, P (I) is a well-composed polyhedral complex). For this aim, we develope a new codification system for a complex K, called ExtendedCubeMap (ECM) representation of K, that codifies: (1) the information of the cells ofK (including geometric information), under the form of a 3D grayscale image gP ; and (2) the boundary face relations between the cells of K, under the form of a set BP of structuring elements that can be stored as indexes in a look-up table. We describe a procedure to locally modify the ECM representation EQ of the cubical complex Q(I) to obtain an ECM representation of a well-composed polyhedral complex P (I) that is homotopy equivalent to Q(I). The construction of the polyhedral complex P (I) is accomplished for proving the results though it is not necessary to be done in practice, since the image gP (obtained by the repairing process on EQ) together with the set BP codify all the geometric and topological information of P (I).