{"id":343745,"date":"2016-12-29T21:04:48","date_gmt":"2016-12-30T05:04:48","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=343745"},"modified":"2018-10-16T21:35:18","modified_gmt":"2018-10-17T04:35:18","slug":"computing-inversion-free-mappings-simplex-assembly","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/computing-inversion-free-mappings-simplex-assembly\/","title":{"rendered":"Computing Inversion-Free Mappings by Simplex Assembly"},"content":{"rendered":"<p><span class=\"tx\">We present a novel method, called\u00a0<\/span><span class=\"tx f865\">Simplex Assembly<\/span><span class=\"tx\">, to compute<\/span><span class=\"tx\">inversion-free mappings with low or bounded distortion on simpli-<\/span><span class=\"tx\">cial meshes. Our method involves two steps: simplex disassembly<\/span><span class=\"tx\">and simplex assembly. Given a simplicial mesh and its initial piece-<\/span><span class=\"tx\">wise af\ufb01ne mapping, we project the af\ufb01ne transformation associated<\/span><span class=\"tx\">with each simplex into the inversion-free and distortion-bounded<\/span><span class=\"tx\">space.\u00a0The projection disassembles the input mesh into disjoint<\/span><span class=\"tx\">simplices. The disjoint simplices are then assembled to recover the<\/span><span class=\"tx\">original connectivity by minimizing the mapping distortion and the<\/span><span class=\"tx\">difference of the disjoint vertices with respect to the piecewise af\ufb01ne<\/span><span class=\"tx\">transformations, while the piecewise af\ufb01ne mapping is restricted<\/span><span class=\"tx\">inside the feasible space. Due to the use of af\ufb01ne transformations as<\/span><span class=\"tx\">variables, our method explicitly guarantees that no inverted simplex<\/span><span class=\"tx\">occurs, and that the mapping distortion is below the bound during<\/span><span class=\"tx\">the optimization. Compared with existing methods, our method is<\/span><span class=\"tx\">robust to an initialization with many inverted elements and positional<\/span><span class=\"tx\">constraints.\u00a0We demonstrate the ef\ufb01ciency and robustness of our<\/span><span class=\"tx\">method through a variety of geometric processing tasks<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We present a novel method, called\u00a0Simplex Assembly, to computeinversion-free mappings with low or bounded distortion on simpli-cial meshes. Our method involves two steps: simplex disassemblyand simplex assembly. Given a simplicial mesh and its initial piece-wise af\ufb01ne mapping, we project the af\ufb01ne transformation associatedwith each simplex into the inversion-free and distortion-boundedspace.\u00a0The projection disassembles the input mesh [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13562,13547],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-343745","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computer-vision","msr-research-area-systems-and-networking","msr-locale-en_us"],"msr_publishername":"ACM","msr_edition":"ACM Transactions on Graphics (TOG) - 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