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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 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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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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