Smale-Like Decomposition and Forman Theory for Discrete Scalar Fields

Abstract

Abstract. Forman theory, which is a discrete alternative for cell complexes to the well-known Morse theory, is currently finding several applications in areas where the data to be handled are discrete, such as image processing and computer graphics. Here, we show that a discrete scalar field f , defined on the vertices of a triangulated multidimensional domain Σ, and its gradient vector field Grad f through the Smale-like decomposition of f [6], are both the restriction of a Forman function F and its gradient field Grad F that extends f over all the simplexes of Σ. We present an algorithm that gives an explicit construction of such an extension. Hence, the scalar field f inherits the properties of Forman gradient vector fields and functions from field Grad F and function F .

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].…”