Ascending and descending regions of a discrete Morse function

Abstract

We present an algorithm which produces a decomposition of a regular cellular complex with a discrete Morse function analogous to the Morse-Smale decomposition of a smooth manifold with respect to a smooth Morse function. The advantage of our algorithm compared to similar existing results is that it works, at least theoretically, in any dimension. Practically, there are dimensional restrictions due to the size of cellular complexes of higher dimensions, though. We prove that the algorithm is correct in the sense that it always produces a decomposition into descending and ascending regions of the critical cells in a finite number of steps, and that, after a finite number of subdivisions, all the regions are topological discs. The efficiency of the algorithm is discussed and its performance on several examples is demonstrated.Comment: 23 pages, 12 figure

Collapses and Watersheds in Pseudomanifolds of Arbitrary Dimension

“…Within this applicative context, the segmentation of simplicial complexes was the subject of many papers in the last decade. L. De Floriani et al [28,16] tackled the problem as a Smale-like decomposition in discrete Morse theory (see also [29]) where the simplicial complex is segmented into ascending and descending subcomplexes. Based on the same theory, H. Edelsbrunner and J. Harer [27] proposed another decomposition algorithm and they informally discuss some links with watershed algorithms.…”

Min-max theory for cell complexes

“…The discrete analogues of these objects were constructed by Jerše and Mramor Kosta in [4]. The idea is that descending discs should be unions of V -paths, just as in the smooth case (where descending discs are foliated by integral curves of a gradient-like vector field).…”

Morse theory and persistent homology for topological analysis of 3D images of complex materials

“…The connections between discrete Morse theory (or simple homotopy), persistent homology, and the use of stable and unstable manifolds to partition scalar functions on cell complexes have also been explored recently by other authors [18,19,20,21,22,7].…”