Generating Discrete Morse Functions from Point Data

Abstract

If K is a finite simplicial complex and h is an injective map from the vertices of K to R, we show how to extend h to a discrete Morse function in the sense of Forman [Forman 02] in a reasonably efficient manner so that the resulting discrete Morse function mirrors the large-scale behavior of h. A concrete algorithm is given for the case where K is a subcomplex of R 3 .

Birth and death in discrete Morse theory

“…Let M be a finite cell decomposition of a space X and suppose that for each 0 = t 0 < t 1 < · · · < t r = 1, we have a discrete Morse function F ti : M → R. (Note we always use the same cellulation M for each i.) In applications, we would probably have the values of each F ti only on the zero cells of M , but we can always extend this to a discrete Morse function on all of M via the algorithm we presented in [King, et al (2005)]. There is no canonical choice of such a function, but it may be taken to be arbitrarily close to the function assigning to each cell the maximum of the values among its vertices.…”

“…Indeed, this case is important for applications where the sample of points taken from an object may change over time, thereby yielding different cell decompositions. Since these samples form the basis of our algorithm for generating discrete Morse functions [King, et al (2005)], we must consider this possibility. However we ask that the cellulations M i and M i+1 have a common subdivision M (2i+1)/2 .…”

“…Algorithm 1 below returns for each pair i, i + 1 all pairs of connected critical cells. The input to the algorithm are four lists obtained from the algorithm in [King, et al (2005)]: the lists C i and C i+1 of critical cells in slices i and i + 1, and the lists of pairs of V i and V i+1 .…”

“…A V -path is then a directed path in the modified graph. Given a finite cell complex M and a real valued function on its 0-cells, the algorithm in [King, et al (2005)] gives a discrete gradient vector field which is meant to be a combinatorial analogue of the gradient vector field of a function sampled at the 0-cells. (Though explicitly stated for simplicial complexes, it easily extends to cell complexes.)…”

“…Choose some distinct values for the 0-cells of N so that d(κ) has the minimum value by far. Apply the algorithm in [King, et al (2005)] to get a discrete gradient vector field U for this function, doing as much cancellation of critical cells as possible using the algorithm.…”