{"id":51,"date":"2013-11-26T12:42:22","date_gmt":"2013-11-26T10:42:22","guid":{"rendered":"http:\/\/www.eng.biu.ac.il\/weberof\/?page_id=51"},"modified":"2013-11-26T17:56:07","modified_gmt":"2013-11-26T15:56:07","slug":"complex-coordinates","status":"publish","type":"page","link":"https:\/\/www.eng.biu.ac.il\/weberof\/publications\/complex-coordinates\/","title":{"rendered":"Complex-Coordinates"},"content":{"rendered":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
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Complex Barycentric Coordinates with Applications to Planar Shape Deformation<\/font><\/td>\n<\/tr>\n

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\n\t\tOfir Weber<\/a>
\n\t\tMirela Ben-Chen<\/a>
\n\t\tCraig Gotsman<\/A><\/font>\n\t\t<\/td>\n<\/tr>\n

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\n\t\tEurographics 2009 - Computer Graphics Forum 28, 2 (2009)<\/a>
\n\t\tG\u00fcnter Enderle Best Paper Award
\n\t\tUS Patent Number: 2010\/0214312<\/a>
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\n\t\tImage deformation using Point-to-Point complex barycentric coordinates.
\n\t\tClick on the image for hi-res version\n\t\t<\/td>\n<\/tr>\n
Abstract:<\/font><\/td>\n<\/tr>\n
Barycentric coordinates are heavily used in computer graphics applications to generalize a set of given data values. Traditionally, the coordinates are required to satisfy a number of key properties, the first being that they are real and positive. In this paper we relax this requirement, allowing the barycentric coordinates to be complex numbers. This allows us to generate new families of barycentric coordinates, which have some powerful advantages over traditional ones. Applying complex barycentric coordinates to data which is itself complex-valued allows to manipulate functions from the complex plane to itself, which may be interpreted as planar mappings. These mappings are useful in shape and image deformation applications. We use Cauchy\u2019s theorem from complex analysis to construct complex barycentric coordinates on (not necessarily convex) polygons, which are shown to be equivalent to planar Green coordinates. These generate conformal mappings from a given source region to a given target region, such that the image of the source region is close to the target region. We then show how to improve the Green coordinates in two ways. The first provides a much better fit to the polygonal target region, and the second allows to generate deformations based on positional constraints, which provide a more intuitive user interface than the conventional cage-based approach. These define two new types of complex barycentric coordinates, which are shown to be very effective in interactive deformation and animation scenarios.<\/td>\n<\/tr>\n
Paper - PDF:<\/font><\/td>\n<\/tr>\n
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Supplementary Material - PDF:<\/font><\/td>\n<\/tr>\n
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Video - QuickTime (H264):<\/font><\/td>\n<\/tr>\n
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BibTeX entry:<\/font><\/td>\n<\/tr>\n
\n\t\t<\/a> @article{Complex_Coordinates:2009,
\n\t\tauthor = {Ofir Weber and Mirela Ben-Chen and Craig Gotsman},
\n\t\ttitle = {Complex Barycentric Coordinates with Applications to Planar Shape Deformation},
\n\t\tjournal = {Computer Graphics Forum (Proceedings of Eurographics)},
\n\t\tvolume = {28},
\n\t\tnumber = {2},
\n\t\tyear = {2009},
\n\t\t}\n\t<\/td>\n<\/tr>\n
More examples:<\/font><\/td>\n<\/tr>\n
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\n\t\tComparison of the Point-to-Point Cauchy-Green complex barycentric coordinates and the MLS similarity coordinates.
\n\t\t(left to right) Original image; Deformation using Point-to-Point coordinates; Deformation using MLS coordinates.
\n\t\tThe Point-to-Point coordinates better handle control points whose Euclidean distance is small, yet their geodesic distance within the cage is large.
\n\t\tClick on the image for hi-res version\n\t\t<\/td>\n<\/tr>\n

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\n\t\tDeformation of a bird using Szego coordinates. The cage has 21 vertices.
\n\t\tClick on the image for hi-res version\n\t\t<\/td>\n<\/tr>\n

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\n\t\tA giraffe (left), and its deformation (right) using Point-to-point Cauchy-Green coordinates, with 16 control points. The cage has 113 vertices.
\n\t\tClick on the image for hi-res version\n\t\t<\/td>\n<\/tr>\n

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\n\t\tColor-coded visualization of one P2P coordinate function.
\n\t\t(left to right) Real part; Imaginary part; Absolute value.
\n\t\tClick on the image for hi-res version\n\t\t<\/td>\n<\/tr>\n<\/table>\n","protected":false},"excerpt":{"rendered":"

Complex Barycentric Coordinates with Applications to Planar Shape Deformation Ofir Weber Mirela Ben-Chen Craig Gotsman Eurographics 2009 – Computer Graphics Forum 28, 2 (2009) G\u00fcnter Enderle Best Paper Award US Patent Number: 2010\/0214312 Image deformation using Point-to-Point complex barycentric coordinates. Click on the image for hi-res version Abstract: Barycentric coordinates are heavily used in computer … Continue reading Complex-Coordinates<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":19,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-51","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.eng.biu.ac.il\/weberof\/wp-json\/wp\/v2\/pages\/51"}],"collection":[{"href":"https:\/\/www.eng.biu.ac.il\/weberof\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.eng.biu.ac.il\/weberof\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.eng.biu.ac.il\/weberof\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.eng.biu.ac.il\/weberof\/wp-json\/wp\/v2\/comments?post=51"}],"version-history":[{"count":9,"href":"https:\/\/www.eng.biu.ac.il\/weberof\/wp-json\/wp\/v2\/pages\/51\/revisions"}],"predecessor-version":[{"id":89,"href":"https:\/\/www.eng.biu.ac.il\/weberof\/wp-json\/wp\/v2\/pages\/51\/revisions\/89"}],"up":[{"embeddable":true,"href":"https:\/\/www.eng.biu.ac.il\/weberof\/wp-json\/wp\/v2\/pages\/19"}],"wp:attachment":[{"href":"https:\/\/www.eng.biu.ac.il\/weberof\/wp-json\/wp\/v2\/media?parent=51"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}