Critical sets of PL and discrete Morse theory: A correspondence. (August 2020)

• Record Type: Journal Article
• Authors: Fugacci, Ulderico; Landi, Claudia; Varlı, Hanife
• Is Part Of: Computers & graphics. Volume 90(2020)
• Page Start: 43
• Page End: 50
• Publication Date: 2020-08
• Subjects: Critical point -- Gradient vector field -- Relative perfectness; Computer graphics -- Periodicals; 006.6
• DOI: 10.1016/j.cag.2020.05.020
• Languages: English
• ISSNs: 0097-8493

Abstract

Highlights: Discretization of Morse theory has led to different notions of critical sets. The various notions of PL critical points given in the literature are equivalent. A correspondence exists between PL critical points and discrete Morse cells. The correspondence requires an optimality condition called relative perfectness. An algorithm builds relative perfect gradient vector fields up to dimension 3. Graphical abstract: Abstract: Piecewise-linear (PL) Morse theory and discrete Morse theory are used in shape analysis tasks to investigate the topological features of discretized spaces. In spite of their common origin in smooth Morse theory, various notions of critical points have been given in the literature for the discrete setting, making a clear understanding of the relationships occurring between them not obvious. This paper aims at providing equivalence results about critical points of the two discretized Morse theories. First of all, we prove the equivalence of the existing notions of PL critical points. Next, under an optimality condition called relative perfectness, we show a dimension agnostic correspondence between the set of PL critical points and that of discrete critical simplices of the combinatorial approach. Finally, we show how a relatively perfect discrete gradient vector field can be algorithmically built up to dimension 3. This way, we guarantee a formal and operative connection between critical sets in the PL and discrete theories.