Edge contraction in persistence-generated discrete Morse vector fields. (August 2018)

• Record Type: Journal Article
• Authors: Dey, Tamal K.; Slechta, Ryan
• Is Part Of: Computers & graphics. Volume 74(2018)
• Page Start: 33
• Page End: 43
• Publication Date: 2018-08
• Subjects: Discrete Morse theory -- Edge contraction -- Computational topology -- Persistent homology -- Topological data analysis; Computer graphics -- Periodicals; 006.6
• DOI: 10.1016/j.cag.2018.05.002
• Languages: English
• ISSNs: 0097-8493

Abstract

Highlights: Defining a contraction operator for discrete Morse vector fields on surfaces. Proving a commutation result for Algorithm 1 and our contraction operator. Experimentally demonstrating that our operator preserves vector field structure. Graphical abstract: Abstract: Recently, discrete Morse vector fields have been shown to be useful in various applications. Analogous to the simplification of large meshes using edge contractions, one may want to simplify the cell complex K on which a discrete Morse vector field V ( K ) is defined. To this end, we define a gradient aware edge contraction operator for triangulated 2-manifolds with the following guarantee. If V ( K ) was generated by a specific persistence-based method, then the vector field that results from our contraction operator is exactly the same as the vector field produced by applying the same persistence-based method to the contracted complex. An implication of this result is that local operations on V ( K ) are sufficient to produce the persistence-based vector field on the contracted complex. Furthermore, our experiments show that the structure of the vector field is largely preserved by our operator. For example, 1-unstable manifolds remain largely unaffected by the contraction. This suggests that for some applications of discrete Morse theory, it is sufficient to use a contracted complex.