{"date_updated":"2023-02-23T10:59:19Z","date_published":"2015-06-03T00:00:00Z","abstract":[{"text":"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.","lang":"eng"}],"department":[{"_id":"HeEd"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","doi":"10.1016/j.comgeo.2014.08.010","intvolume":"        48","author":[{"first_name":"Dominique","full_name":"Attali, Dominique","last_name":"Attali"},{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9683-0724","last_name":"Bauer","full_name":"Bauer, Ulrich","first_name":"Ulrich"},{"last_name":"Devillers","first_name":"Olivier","full_name":"Devillers, Olivier"},{"last_name":"Glisse","first_name":"Marc","full_name":"Glisse, Marc"},{"first_name":"André","full_name":"Lieutier, André","last_name":"Lieutier"}],"oa_version":"None","issue":"8","type":"journal_article","day":"03","scopus_import":1,"status":"public","volume":48,"_id":"1805","publication":"Computational Geometry: Theory and Applications","date_created":"2018-12-11T11:54:06Z","page":"606 - 621","title":"Homological reconstruction and simplification in R3","language":[{"iso":"eng"}],"publication_status":"published","month":"06","quality_controlled":"1","publisher":"Elsevier","publist_id":"5305","ec_funded":1,"project":[{"name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","grant_number":"318493","call_identifier":"FP7"}],"citation":{"ama":"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>","short":"D. Attali, U. Bauer, O. Devillers, M. Glisse, A. Lieutier, Computational Geometry: Theory and Applications 48 (2015) 606–621.","mla":"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>.","ieee":"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.","chicago":"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>.","ista":"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.","apa":"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>"},"year":"2015","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"2812"}]}}