Homological Reconstruction and Simplification in R3

Dominique Attali; Ulrich Bauer; Olivier Devillers; Marc Glisse; André Lieutier

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

2 IST Austria - Institute of Science and Technology [Austria]

3 GEOMETRICA - Geometric computing

CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France

4 Dassault Systèmes

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.

Document type :

Conference papers

Domain :

Computer Science [cs] / Computational Geometry [cs.CG]

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https://hal.inria.fr/hal-00833791
Contributor : Olivier Devillers <>
Submitted on : Thursday, June 13, 2013 - 3:08:22 PM
Last modification on : Thursday, February 7, 2019 - 5:54:13 PM
Long-term archiving on : Tuesday, April 4, 2017 - 9:42:16 PM

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