Abstract
Given a function f on a surface and a tolerance δ>0, we construct a function f δ subject to ‖f δ −f‖∞≤δ such that f δ has a minimum number of critical points. Our construction relies on a connection between discrete Morse theory and persistent homology and completely removes homological noise with persistence ≤2δ from the input function f. The number of critical points of the resulting simplified function f δ achieves the lower bound dictated by the stability theorem of persistent homology. We show that the simplified function can be computed in linear time after persistence pairs have been computed.
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Bauer, U., Lange, C. & Wardetzky, M. Optimal Topological Simplification of Discrete Functions on Surfaces. Discrete Comput Geom 47, 347–377 (2012). https://doi.org/10.1007/s00454-011-9350-z
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DOI: https://doi.org/10.1007/s00454-011-9350-z