NDLI: Homological reconstruction and simplification in $R^{3}$

Content Provider	ACM Digital Library
Author	Glisse, Marc Bauer, Ulrich Attali, Dominique Lieutier, André Devillers, Olivier
Abstract	We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L) can be realized as the homology of some complex $H_{*}(X)$ with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in $R^{3}.$ As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on $S^{3}$ within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.
Starting Page	117
Ending Page	126
Page Count	10
File Format	PDF
ISBN	9781450320313
DOI	10.1145/2462356.2462373
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2013-06-17
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Persistence Np-hard problems Homology
Content Type	Text
Resource Type	Article

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Homological reconstruction and simplification in $R^{3}$