A combinatorial approach based on forman theory

Abstract

© The Author(s) 2014. In this chapter, we review algorithms based on Forman theory. These algorithms are combinatorial in nature and, thus, dimension-independent. Forman-based algorithms start from a Forman gradient vector field computed as a preliminary step. We discuss how to encode a Forman gradient over irregular simplicial complexes, by taking into account the relation between the input simplicial complex and its dual, and between the descending and ascending Morse complexes (Sect. 5.1). We describe the main algorithms for computing a Forman gradient on 2D and 3D scalar fields (Sect. 5.3). Finally we present algorithms which, given a Forman gradient, extract the collection of cells of the ascending and descending Morse complexes, the Morse-Smale cells, or the vertices and edges of the Morse-Smale complex (Sect. 5.4).