This paper develops theoretical contributions in discrete mathematics (digital topology) with important practical applications in image analysis (thinning and skeletonization) for pattern recognition and computer vision. The concepts of point, m-cell, (proper) d-face, subcomplex, and free pair are used to define operations such as attachment, detachment, and collapse. The simplicity of points in two-, three-, and four-dimensional (4D) discrete spaces is introduced, using the collapse operation.

The paper has seven sections and one appendix. Couprie and Bertrand’s excellent description uses formalization and suitable examples. The reader will appreciate Section 1’s well-written introduction, Section 2’s compact description of notions related to cubical complexes, and Section 3’s clear presentation of the basic properties of the collapse operation and simple sets. Section 4 proves the upstream/downstream confluence properties related to acyclicity, connectedness, and collapsibility. The fifth section presents the new results concerning the characterization of simple cells with less than four dimensions, and Section 6 addresses the analysis for a higher number of dimensions. Section 7 is the conclusion.

The authors propose and discuss two algorithms running in linear time, for checking simplicity. The proofs for most assertions are presented in the corresponding sections; only a few of the results are described in the appendix. The paper is consistent and well documented, with very good examples--Bing’s house, Schlegel diagrams, and dunce hats--and inspired comments. It includes 23 definitions, lemmas, and propositions.