Associating Cell Complexes to Four Dimensional Digital Objects

Abstract

Abstract. The goal of this paper is to construct an algebraic-topological representation of a 80-adjacent doxel-based 4-dimensional digital object. Such that representation consists on associating a cell complex homologically equivalent to the digital object. To determine the pieces of this cell complex, algorithms based on weighted complete graphs and integral operators are shown. We work with integer coefficients, in order to compute the integer homology of the digital object.

An Efficient Algorithm to Compute Subsets of Points in ℤ n

“…A similar proof of Theorem 1 in [8] shows that two isomorphic subsets with at least 2 n−1 points are isometric. The converse implication is obvious, taking into account that isometry is a stronger concept than isomorphism.…”

“…Below, we show the extension of the method shown in [8] to ignore congruent subsets that differ by isometries of the n-dimensional space. This method consisted in associating each subset with a multi-graph.…”

“…By considering previous results, Algorithm 1 in [8] (whose pseudocode is extended in Algorithm 3.1 for any dimension) was implemented. …”

“…In this paper, we have shown an algorithm more efficient than the extension of that in [8] to compute subsets of points non-congruent by isometries of the n-dimensional space. By using this algorithm for n = 3 and n = 4, 22 and 402 non-isometric subsets (see Figures 4 and 5 for some examples of these subsets), respectively, have been computed by using a low CPU time.…”

“…In this paper we show a more efficient algorithm than that in [8] to compute subsets of points non-congruent by isometries. This algorithm can be used to reconstruct the object from the digital image.…”