Homological reconstruction and simplification in R3 Attali, Dominique Bauer, Ulrich ; https://orcid.org/0000-0002-9683-0724 Devillers, Olivier Glisse, Marc Lieutier, André 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 double-struck R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on double-struck 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. Elsevier 2015 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/1805 Attali D, Bauer U, Devillers O, Glisse M, Lieutier A. Homological reconstruction and simplification in R3. <i>Computational Geometry: Theory and Applications</i>. 2015;48(8):606-621. doi:<a href="https://doi.org/10.1016/j.comgeo.2014.08.010">10.1016/j.comgeo.2014.08.010</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1016/j.comgeo.2014.08.010 info:eu-repo/semantics/altIdentifier/wos/000357353200006 info:eu-repo/semantics/openAccess