{"doi":"10.1016/j.comgeo.2014.08.010","abstract":[{"lang":"eng","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."}],"_id":"1805","scopus_import":"1","citation":{"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.","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>","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.","short":"D. Attali, U. Bauer, O. Devillers, M. Glisse, A. Lieutier, Computational Geometry: Theory and Applications 48 (2015) 606–621.","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>","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>.","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>."},"OA_type":"free access","page":"606 - 621","article_processing_charge":"No","article_type":"original","type":"journal_article","year":"2015","title":"Homological reconstruction and simplification in R3","department":[{"_id":"HeEd"}],"day":"03","quality_controlled":"1","publication_status":"published","external_id":{"isi":["000357353200006"]},"corr_author":"1","oa_version":"Published Version","main_file_link":[{"url":"https://doi.org/10.1016/j.comgeo.2014.08.010","open_access":"1"}],"volume":48,"date_created":"2018-12-11T11:54:06Z","date_updated":"2025-09-23T09:18:30Z","month":"06","author":[{"full_name":"Attali, Dominique","first_name":"Dominique","last_name":"Attali"},{"full_name":"Bauer, Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","last_name":"Bauer","first_name":"Ulrich","orcid":"0000-0002-9683-0724"},{"first_name":"Olivier","last_name":"Devillers","full_name":"Devillers, Olivier"},{"full_name":"Glisse, Marc","first_name":"Marc","last_name":"Glisse"},{"last_name":"Lieutier","first_name":"André","full_name":"Lieutier, André"}],"ec_funded":1,"related_material":{"record":[{"id":"2812","status":"public","relation":"earlier_version"}]},"OA_place":"publisher","oa":1,"publisher":"Elsevier","project":[{"call_identifier":"FP7","grant_number":"318493","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems"}],"language":[{"iso":"eng"}],"intvolume":"        48","publication":"Computational Geometry: Theory and Applications","isi":1,"issue":"8","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_published":"2015-06-03T00:00:00Z","status":"public","publist_id":"5305"}