{"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 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.","lang":"eng"}],"status":"public","_id":"2812","page":"117 - 125","main_file_link":[{"open_access":"1","url":"http://hal.archives-ouvertes.fr/hal-00833791/"}],"day":"01","acknowledgement":"Some of the authors were partially supported by the GIGA ANR grant (contract ANR-09-BLAN-0331-01) and the European project CG-Learning (contract 255827).","publisher":"ACM","author":[{"first_name":"Dominique","full_name":"Attali, Dominique","last_name":"Attali"},{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9683-0724","first_name":"Ulrich","last_name":"Bauer","full_name":"Bauer, Ulrich"},{"last_name":"Devillers","full_name":"Devillers, Olivier","first_name":"Olivier"},{"first_name":"Marc","full_name":"Glisse, Marc","last_name":"Glisse"},{"first_name":"André","last_name":"Lieutier","full_name":"Lieutier, André"}],"publication":"Proceedings of the 29th annual symposium on Computational Geometry","related_material":{"record":[{"id":"1805","relation":"later_version","status":"public"}]},"type":"conference","department":[{"_id":"HeEd"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-11T11:59:44Z","conference":{"end_date":"2013-06-20","location":"Rio de Janeiro, Brazil","start_date":"2013-06-17","name":"SoCG: Symposium on Computational Geometry"},"date_updated":"2023-02-23T10:15:15Z","oa":1,"year":"2013","title":"Homological reconstruction and simplification in R3","language":[{"iso":"eng"}],"publist_id":"4072","scopus_import":1,"quality_controlled":"1","oa_version":"Submitted Version","date_published":"2013-06-01T00:00:00Z","publication_status":"published","month":"06","citation":{"chicago":"Attali, Dominique, Ulrich Bauer, Olivier Devillers, Marc Glisse, and André Lieutier. “Homological Reconstruction and Simplification in R3.” In <i>Proceedings of the 29th Annual Symposium on Computational Geometry</i>, 117–25. ACM, 2013. <a href=\"https://doi.org/10.1145/2462356.2462373\">https://doi.org/10.1145/2462356.2462373</a>.","ieee":"D. Attali, U. Bauer, O. Devillers, M. Glisse, and A. Lieutier, “Homological reconstruction and simplification in R3,” in <i>Proceedings of the 29th annual symposium on Computational Geometry</i>, Rio de Janeiro, Brazil, 2013, pp. 117–125.","ama":"Attali D, Bauer U, Devillers O, Glisse M, Lieutier A. Homological reconstruction and simplification in R3. In: <i>Proceedings of the 29th Annual Symposium on Computational Geometry</i>. ACM; 2013:117-125. doi:<a href=\"https://doi.org/10.1145/2462356.2462373\">10.1145/2462356.2462373</a>","short":"D. Attali, U. Bauer, O. Devillers, M. Glisse, A. Lieutier, in:, Proceedings of the 29th Annual Symposium on Computational Geometry, ACM, 2013, pp. 117–125.","ista":"Attali D, Bauer U, Devillers O, Glisse M, Lieutier A. 2013. Homological reconstruction and simplification in R3. Proceedings of the 29th annual symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 117–125.","mla":"Attali, Dominique, et al. “Homological Reconstruction and Simplification in R3.” <i>Proceedings of the 29th Annual Symposium on Computational Geometry</i>, ACM, 2013, pp. 117–25, doi:<a href=\"https://doi.org/10.1145/2462356.2462373\">10.1145/2462356.2462373</a>.","apa":"Attali, D., Bauer, U., Devillers, O., Glisse, M., &#38; Lieutier, A. (2013). Homological reconstruction and simplification in R3. In <i>Proceedings of the 29th annual symposium on Computational Geometry</i> (pp. 117–125). Rio de Janeiro, Brazil: ACM. <a href=\"https://doi.org/10.1145/2462356.2462373\">https://doi.org/10.1145/2462356.2462373</a>"},"doi":"10.1145/2462356.2462373"}