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Conference Papers Year : 2013

Homological Reconstruction and Simplification in R3

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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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Dates and versions

hal-00833791 , version 1 (13-06-2013)

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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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