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After constructing an as smooth as possible symmetric cross field satisfying a sparse set of directional constraints (to capture the geometric structure of the surface), the mesh is cut open in order to enable a low distortion unfolding. Then a seamless globally smooth parametrization is computed whose iso-parameter lines follow the cross field directions. In contrast to previous methods, sparsely distributed directional constraints are sufficient to automatically determine the appropriate number, type and position of singularities in the quadrangulation. Both steps of the algorithm (cross field and parametrization) can be formulated as a mixed-integer problem which we solve very efficiently by an adaptive greedy solver. We show several complex examples where high quality quad meshes are generated in a fully automatic manner.<\/jats:p>","DOI":"10.1145\/1531326.1531383","type":"journal-article","created":{"date-parts":[[2009,7,28]],"date-time":"2009-07-28T08:43:55Z","timestamp":1248770635000},"page":"1-10","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":391,"title":["Mixed-integer quadrangulation"],"prefix":"10.1145","volume":"28","author":[{"given":"David","family":"Bommes","sequence":"first","affiliation":[{"name":"RWTH Aachen University"}]},{"given":"Henrik","family":"Zimmer","sequence":"additional","affiliation":[{"name":"RWTH Aachen University"}]},{"given":"Leif","family":"Kobbelt","sequence":"additional","affiliation":[{"name":"RWTH Aachen University"}]}],"member":"320","published-online":{"date-parts":[[2009,7,27]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/882262.882296"},{"key":"e_1_2_2_2_1","unstructured":"Alliez P. Ucelli G. Gotsman C. and Attene M. 2005. Recent advances in remeshing of surfaces. Research report AIM@SHAPE Network of Excellence.  Alliez P. Ucelli G. Gotsman C. and Attene M. 2005. Recent advances in remeshing of surfaces. Research report AIM@SHAPE Network of Excellence."},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2008.01142.x"},{"key":"e_1_2_2_4_1","doi-asserted-by":"crossref","unstructured":"Bommes D. Vossemer T. and Kobbelt L. 2009. Quadrangular parameterization for reverse engineering. Lecture Notes in Computer Science to appear.  Bommes D. Vossemer T. and Kobbelt L. 2009. Quadrangular parameterization for reverse engineering. 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In SGP '05: Proceedings of the third Eurographics symposium on Geometry processing, Eurographics Association, Aire-la-Ville, Switzerland, Switzerland, 85."},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/1508044.1508091"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/1409060.1409100"},{"key":"e_1_2_2_16_1","first-page":"3","article-title":"Quadcover - surface parameterization using branched coverings","volume":"26","author":"Nieser M.","year":"2007","unstructured":"K&amp;#228;lberer, F., Nieser , M. , and Polthier , K. 2007 . Quadcover - surface parameterization using branched coverings . Computer Graphics Forum 26 , 3 (Sept.), 375--384. K&amp;#228;lberer, F., Nieser, M., and Polthier, K. 2007. Quadcover - surface parameterization using branched coverings. 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