article published yes Dominique Attali author Ulrich Bauer author 2ADD483A-F248-11E8-B48F-1D18A9856A870000-0002-9683-0724 Olivier Devillers author Marc Glisse author André Lieutier author HeEd department Topological Complex Systems project 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. Elsevier2015 eng 00035735320000610.1016/j.comgeo.2014.08.010 488606 - 621 https://research-explorer.ista.ac.at/record/2812 Attali, Dominique, Ulrich Bauer, Olivier Devillers, Marc Glisse, and André Lieutier. “Homological Reconstruction and Simplification in R3.” <i>Computational Geometry: Theory and Applications</i>. Elsevier, 2015. <a href="https://doi.org/10.1016/j.comgeo.2014.08.010">https://doi.org/10.1016/j.comgeo.2014.08.010</a>. Attali, Dominique, et al. “Homological Reconstruction and Simplification in R3.” <i>Computational Geometry: Theory and Applications</i>, vol. 48, no. 8, Elsevier, 2015, pp. 606–21, doi:<a href="https://doi.org/10.1016/j.comgeo.2014.08.010">10.1016/j.comgeo.2014.08.010</a>. 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> D. Attali, U. Bauer, O. Devillers, M. Glisse, A. Lieutier, Computational Geometry: Theory and Applications 48 (2015) 606–621. Attali D, Bauer U, Devillers O, Glisse M, Lieutier A. 2015. Homological reconstruction and simplification in R3. Computational Geometry: Theory and Applications. 48(8), 606–621. Attali, D., Bauer, U., Devillers, O., Glisse, M., &#38; Lieutier, A. (2015). Homological reconstruction and simplification in R3. <i>Computational Geometry: Theory and Applications</i>. Elsevier. <a href="https://doi.org/10.1016/j.comgeo.2014.08.010">https://doi.org/10.1016/j.comgeo.2014.08.010</a> D. Attali, U. Bauer, O. Devillers, M. Glisse, and A. Lieutier, “Homological reconstruction and simplification in R3,” <i>Computational Geometry: Theory and Applications</i>, vol. 48, no. 8. Elsevier, pp. 606–621, 2015. 18052018-12-11T11:54:06Z2025-09-23T09:18:30Z