Abstract
We address the basic question in discrete Morse theory of combining discrete gradient fields that are partially defined on subsets of the given complex. This is a well-posed question when the discrete gradient field V is generated using a fixed algorithm which has a local nature. One example is ProcessLowerStars, a widely used algorithm for computing persistent homology associated to a grey-scale image in 2D or 3D. While the algorithm for V may be inherently local, being computed within stars of vertices and so embarrassingly parallelizable, in practical use, it is natural to want to distribute the computation over patches Pi, apply the chosen algorithm to compute the fields Vi associated to each patch, and then assemble the ambient field V from these. Simply merging the fields from the patches, even when that makes sense, gives a wrong answer. We develop both very general merging procedures and leaner versions designed for specific, easy-to-arrange covering patterns.
Highlights
Discrete Morse theory [1] is a fairly new and powerful tool, created by Robin Forman in the 1990s, that has many applications in many fields
In discrete Morse theory, the discrete vector field determines the maps in the simplified chain complex of the Morse complex that is created from the critical cells of the discrete Morse function
Given a general algorithm α applied to cells in a regular CW complex K, which takes data from k-neighborhoods of cells in K and uses those data to either pair cells together in a list, which is to become the discrete vector field V, or places the cells singly in the list of critical cells C
Summary
Discrete Morse theory [1] is a fairly new and powerful tool, created by Robin Forman in the 1990s, that has many applications in many fields. Given a general algorithm α applied to cells in a regular CW complex K, which takes data from k-neighborhoods of cells in K and uses those data to either pair cells together in a list, which is to become the discrete vector field V, or places the cells singly in the list of critical cells C. We need the discrete vector field of the overlaps of the horizontal strips in the part of our algorithm when we start applying our formula vertically. We apply our formula to these last two patches and end up with the discrete vector field of the whole image We make this precise with a pseudo-code for the distributed ProcessLowerStars algorithm. It is clear that the finer covering of the plane by individual patches has the same property as above: its δ-enlargement has the same nerve as the covering itself
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