Homological Reconstruction and Simplification in R3

Dominique Attali 1 Ulrich Bauer 2 Olivier Devillers 3 Marc Glisse 3 André Lieutier 4
GIPSA-DIS - Département Images et Signal
3 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : We consider the problem of deciding whether the persistent homology group of a simplicial pair (K, L) can be realized as the homology H∗(X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on S3 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.
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Submitted on : Thursday, June 13, 2013 - 3:08:22 PM
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Dominique Attali, Ulrich Bauer, Olivier Devillers, Marc Glisse, André Lieutier. Homological Reconstruction and Simplification in R3. 29th International Symposium on Computational Geometry (SoCG 2013), Jun 2013, Rio de Janeiro, Brazil. pp.117-125, ⟨10.1145/2462356.2462373⟩. ⟨hal-00833791⟩



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