Homological Reconstruction and Simplification in R3

Dominique Attali 1 Ulrich Bauer 2 Olivier Devillers 3 Marc Glisse 3 André Lieutier 4
1 GIPSA-AGPIG - AGPIG
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 contained in X and X contained in 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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Computational Geometry, Elsevier, 2015, 48 (8), pp.606-621. 〈10.1016/j.comgeo.2014.08.010〉
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Dominique Attali, Ulrich Bauer, Olivier Devillers, Marc Glisse, André Lieutier. Homological Reconstruction and Simplification in R3. Computational Geometry, Elsevier, 2015, 48 (8), pp.606-621. 〈10.1016/j.comgeo.2014.08.010〉. 〈hal-01132440〉

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