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Homological Reconstruction and Simplification in R3

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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https://hal.inria.fr/hal-00833791
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Submitted on : Thursday, June 13, 2013 - 3:08:22 PM
Last modification on : Tuesday, October 19, 2021 - 11:22:25 PM
Long-term archiving on: : Tuesday, April 4, 2017 - 9:42:16 PM

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Dominique Attali, Ulrich Bauer, Olivier Devillers, Marc Glisse, André Lieutier. Homological Reconstruction and Simplification in R3. SoCG 2013 - 29th Annual Symposium on Computational Geometry, Jun 2013, Rio de Janeiro, Brazil. pp.117-125, ⟨10.1145/2462356.2462373⟩. ⟨hal-00833791⟩

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