Optimal topological simplification of discrete functions on surfaces

Details Optimal topological simplification of discrete functions on surfaces Optimal topological simplification of discrete functions on surfaces

2010-01-08

arxiv.org/abs/1001.1269v2

Discrete Comput Geom (2012) 47:347-377

Computer Science

Computational Geometry

27 pages, 8 figures

Scientific paper

We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2{\delta}, constructively proving the tightness of a lower bound on the number of critical points given by the stability theorem of persistent homology in dimension two for any input function. We also show that an optimal solution can be computed in linear time after persistence pairs have been computed.

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