Bijective Composite Mean Value Mappings
We introduce the novel concept of composite barycentric mappings and give theoretical conditions under which they are guaranteed to be bijective. We then focus on mean value mappings and derive a simple procedure for computing their Jacobians, leading to an efficient GPU‐assisted implementation for...
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| Vydáno v: | Computer graphics forum Ročník 32; číslo 5; s. 137 - 146 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Oxford, UK
Blackwell Publishing Ltd
01.08.2013
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | We introduce the novel concept of composite barycentric mappings and give theoretical conditions under which they are guaranteed to be bijective. We then focus on mean value mappings and derive a simple procedure for computing their Jacobians, leading to an efficient GPU‐assisted implementation for interactively designing composite mean value mappings which are bijective up to pixel resolution. We provide a number of examples of 2D image deformation and an example of 3D shape deformation based on a natural extension of the concept to spatial mappings. |
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| AbstractList | We introduce the novel concept of composite barycentric mappings and give theoretical conditions under which they are guaranteed to be bijective. We then focus on mean value mappings and derive a simple procedure for computing their Jacobians, leading to an efficient GPU-assisted implementation for interactively designing composite mean value mappings which are bijective up to pixel resolution. We provide a number of examples of 2D image deformation and an example of 3D shape deformation based on a natural extension of the concept to spatial mappings. We introduce the novel concept of composite barycentric mappings and give theoretical conditions under which they are guaranteed to be bijective. We then focus on mean value mappings and derive a simple procedure for computing their Jacobians, leading to an efficient GPU-assisted implementation for interactively designing composite mean value mappings which are bijective up to pixel resolution. We provide a number of examples of 2D image deformation and an example of 3D shape deformation based on a natural extension of the concept to spatial mappings. [PUBLICATION ABSTRACT] |
| Author | Schneider, T. Hormann, K. Floater, M.S. |
| Author_xml | – sequence: 1 givenname: T. surname: Schneider fullname: Schneider, T. organization: Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland – sequence: 2 givenname: K. surname: Hormann fullname: Hormann, K. organization: Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland – sequence: 3 givenname: M.S. surname: Floater fullname: Floater, M.S. organization: Department of Mathematics, University of Oslo, Norway |
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| Cites_doi | 10.1145/2070781.2024171 10.1145/1183287.1183295 10.1111/j.1467-8659.2012.03163.x 10.1080/10586458.1993.10504266 10.1007/s10444-005-9008-6 10.1145/1141911.1141920 10.1007/s10444-008-9098-z 10.1145/166117.166118 10.1145/1073204.1073229 10.1145/2185520.2185604 10.1111/j.1467-8659.2011.02028.x 10.1016/S0167-8396(03)00002-5 10.1109/2945.965344 10.1111/j.1467-8659.2008.01292.x 10.1145/2010324.1964973 10.1145/1778765.1778815 10.1215/S0012-7094-63-03008-4 10.1007/s10444-012-9282-z 10.1145/1276377.1276466 10.1109/38.365004 10.1111/j.1467-8659.2011.02027.x 10.1016/j.cagd.2005.06.004 10.1111/j.1467-8659.2010.01760.x 10.1111/j.1467-8659.2009.01600.x 10.1145/1360612.1360677 10.1109/TVCG.2012.109 10.1111/cgf.12023 10.1111/j.1467-8659.2009.01399.x 10.1145/344779.344859 |
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Solomon J., Ben-Chen M., Butscher A., Guibas L.: As-killing-as-possible vector fields for planar deformation. Computer Graphics Forum 30, 5 (Aug. 2011), 1543-1552. Proceedings of SGP. 8. Lipman Y., Levin D., Cohen-Or D.: Green coordinates. ACM Transactions on Graphics 27, 3 (Aug. 2008), Article 78, 10 pages. Proceedings of SIGGRAPH. 2. Joshi P., Meyer M., DeRose T., Green B., Sanocki T.: Harmonic coordinates for character articulation. ACM Transactions on Graphics 26, 3 (July 2007), Article 71, 9 pages. Proceedings of SIGGRAPH. 2. Floater M. S., Kós G., Reimers M.: Mean value coordinates in 3D. Computer Aided Geometric Design 22, 7 (Oct. 2005), 623-631. 2. Warren J., Schaefer S., Hirani A. N., Desbrun M.: Barycentric coordinates for convex sets. Advances in Computational Mathematics 27, 3 (Oct. 2007), 319-338. 2. Floater M. S., Kosinka J.: On the injectivity of Wachspress and mean value mappings between convex polygons. Advances in Computational Mathematics 32, 2 (Feb. 2010), 163-174. 2, 4. Weber O., Gotsman C.: Controllable conformal maps for shape deformation and interpolation. ACM Transactions on Graphics 29, 4 (July 2010), Article 78, 11 pages. Proceedings of SIGGRAPH. 2. Meisters G. H., Olech C.: Locally one-to-one mappings and a classical theorem on the schlicht functions. Duke Mathematical Journal 30, 1 (Mar. 1963), 63-80. 2. Wachspress E. L.: A Rational Finite Element Basis, vol. 114 of Mathematics in Science and Engineering. Academic Press, New York , 1975. 2. Winkler T., Drieseberg J., Alexa M., Hormann K.: Multi-scale geometry interpolation. Computer Graphics Forum 29, 2 (May 2010), 309-318. Proceedings of Eurographics. 8. Ruprecht D., Müller H.: Image warping with scattered data interpolation. IEEE Computer Graphics and Applications 15, 2 (Mar. 1995), 37-43. 8. Jacobson A., Baran I., Popović J., Sorkine O.: Bounded biharmonic weights for real-time deformation. ACM Transactions on Graphics 30, 4 (July 2011), Article 78, 8 pages. Proceedings of SIGGRAPH. 2. Manson J., Schaefer S.: Moving least squares coordinates. Computer Graphics Forum 29, 5 (July 2010), 1517-1524. Proceedings of SGP. 2. Schaefer S., McPhail T., Warren J.: Image deformation using moving least squares. ACM Transactions on Graphics 25, 3 (July 2006), 533-540. Proceedings of SIGGRAPH. 8. Harmon D., Panozzo D., Sorkine O., Zorin D.: Interference-aware geometric modeling. ACM Transactions on Graphics 30, 6 (Dec. 2011), Article 137, 10 pages. Proceedings of SIGGRAPH Asia. 8. Hormann K., Sukumar N.: Maximum entropy coordinates for arbitrary polytopes. Computer Graphics Forum 27, 5 (July 2008), 1513-1520. Proceedings of SGP. 2. Pinkall U., Polthier K.: Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2, 1 (1993), 15-36. 2. Kaufmann P., Wang O., Sorkine-Hornung A., Sorkine-Hornung O., Smolic A., Gross M.: Finite element image warping. Computer Graphics Forum 32, 2 (May 2013), 31-39. Proceedings of Eurographics. 8. Weber O., Ben-Chen M., Gotsman C.: Complex barycentric coordinates with applications to planar shape deformation. Computer Graphics Forum 28, 2 (Apr. 2009), 587-597. Proceedings of Eurographics. 2. Li X.-Y., Hu S.-M.: Poisson coordinates. IEEE Transactions on Visualization and Computer Graphics 19, 2 (Feb. 2013), 344-352. 2. Jacobson A., Weinkauf T., Sorkine-Hornung O.: Smooth shape-aware functions with controlled extrema. Computer Graphics Forum 31, 5 (Aug. 2012), 1577-1586. Proceedings of SGP. 8. Lipman Y.: Bounded distortion mapping spaces for triangular meshes. ACM Transactions on Graphics 31, 4 (July 2012), Article 108, 13 pages. Proceedings of SIGGRAPH. 2. July 2000 Feb. 2013; 19 July 2010; 29 Aug. 1993 July 2012; 31 July 2006; 25 2012 July 2011; 30 July 2007 Oct.Dec. 2001; 7 Mar. 1963; 30 Aug. 1995 Aug. 2011; 30 June 2006 1975 July 2005; 24 Oct. 2006; 25 1993; 2 Aug. 2012; 31 Apr. 2009; 28 Mar. 1995; 15 Oct. 2005; 22 Aug. 2008; 27 Feb. 2010; 32 Dec. 2011; 30 Oct. 2007; 27 July 2008; 27 July 2007; 26 May 2013; 32 May 2010; 29 Mar. 2003; 20 e_1_2_7_5_2 e_1_2_7_4_2 e_1_2_7_3_2 e_1_2_7_2_2 e_1_2_7_9_2 e_1_2_7_8_2 e_1_2_7_7_2 e_1_2_7_6_2 e_1_2_7_19_2 e_1_2_7_18_2 e_1_2_7_16_2 e_1_2_7_15_2 e_1_2_7_14_2 e_1_2_7_13_2 Jacobson A. (e_1_2_7_12_2) 2011; 30 e_1_2_7_10_2 e_1_2_7_26_2 e_1_2_7_27_2 e_1_2_7_28_2 e_1_2_7_29_2 Lipman Y. (e_1_2_7_20_2) 2012; 31 Jacobson A. (e_1_2_7_11_2) 2012 Langer T. (e_1_2_7_17_2) 2006 e_1_2_7_25_2 e_1_2_7_24_2 e_1_2_7_30_2 e_1_2_7_23_2 e_1_2_7_22_2 e_1_2_7_32_2 e_1_2_7_21_2 e_1_2_7_33_2 e_1_2_7_34_2 e_1_2_7_35_2 Wachspress E. L. (e_1_2_7_31_2) 1975 e_1_2_7_36_2 |
| References_xml | – reference: Ju T., Schaefer S., Warren J.: Mean value coordinates for closed triangular meshes. ACM Transactions on Graphics 24, 3 (July 2005), 561-566. Proceedings of SIGGRAPH. 2. – reference: Ruprecht D., Müller H.: Image warping with scattered data interpolation. IEEE Computer Graphics and Applications 15, 2 (Mar. 1995), 37-43. 8. – reference: Weber O., Gotsman C.: Controllable conformal maps for shape deformation and interpolation. ACM Transactions on Graphics 29, 4 (July 2010), Article 78, 11 pages. Proceedings of SIGGRAPH. 2. – reference: Weber O., Ben-Chen M., Gotsman C.: Complex barycentric coordinates with applications to planar shape deformation. Computer Graphics Forum 28, 2 (Apr. 2009), 587-597. Proceedings of Eurographics. 2. – reference: Manson J., Schaefer S.: Moving least squares coordinates. Computer Graphics Forum 29, 5 (July 2010), 1517-1524. Proceedings of SGP. 2. – reference: Gain J. E., Dodgson N. A.: Preventing self-intersection under free-form deformation. IEEE Transactions on Visualization and Computer Graphics 7, 4 (Oct.Dec. 2001), 289-298. 2. – reference: Joshi P., Meyer M., DeRose T., Green B., Sanocki T.: Harmonic coordinates for character articulation. ACM Transactions on Graphics 26, 3 (July 2007), Article 71, 9 pages. Proceedings of SIGGRAPH. 2. – reference: Pinkall U., Polthier K.: Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2, 1 (1993), 15-36. 2. – reference: Winkler T., Drieseberg J., Alexa M., Hormann K.: Multi-scale geometry interpolation. Computer Graphics Forum 29, 2 (May 2010), 309-318. Proceedings of Eurographics. 8. – reference: Warren J., Schaefer S., Hirani A. N., Desbrun M.: Barycentric coordinates for convex sets. Advances in Computational Mathematics 27, 3 (Oct. 2007), 319-338. 2. – reference: Weber O., Ben-Chen M., Gotsman C., Hormann K.: A complex view of barycentric mappings. Computer Graphics Forum 30, 5 (Aug. 2011), 1533-1542. Proceedings of SGP. 2. – reference: Li X.-Y., Hu S.-M.: Poisson coordinates. IEEE Transactions on Visualization and Computer Graphics 19, 2 (Feb. 2013), 344-352. 2. – reference: Harmon D., Panozzo D., Sorkine O., Zorin D.: Interference-aware geometric modeling. ACM Transactions on Graphics 30, 6 (Dec. 2011), Article 137, 10 pages. Proceedings of SIGGRAPH Asia. 8. – reference: Kaufmann P., Wang O., Sorkine-Hornung A., Sorkine-Hornung O., Smolic A., Gross M.: Finite element image warping. Computer Graphics Forum 32, 2 (May 2013), 31-39. Proceedings of Eurographics. 8. – reference: Floater M. S.: Mean value coordinates. Computer Aided Geometric Design 20, 1 (Mar. 2003), 19-27. 2, 3. – reference: Lipman Y., Levin D., Cohen-Or D.: Green coordinates. ACM Transactions on Graphics 27, 3 (Aug. 2008), Article 78, 10 pages. Proceedings of SIGGRAPH. 2. – reference: Solomon J., Ben-Chen M., Butscher A., Guibas L.: As-killing-as-possible vector fields for planar deformation. Computer Graphics Forum 30, 5 (Aug. 2011), 1543-1552. Proceedings of SGP. 8. – reference: Wachspress E. L.: A Rational Finite Element Basis, vol. 114 of Mathematics in Science and Engineering. Academic Press, New York , 1975. 2. – reference: Hormann K., Sukumar N.: Maximum entropy coordinates for arbitrary polytopes. Computer Graphics Forum 27, 5 (July 2008), 1513-1520. Proceedings of SGP. 2. – reference: Jacobson A., Baran I., Popović J., Sorkine O.: Bounded biharmonic weights for real-time deformation. ACM Transactions on Graphics 30, 4 (July 2011), Article 78, 8 pages. Proceedings of SIGGRAPH. 2. – reference: Jacobson A.: Bijective mappings with generalized barycentric coordinates: a counterexample. Tech. rep., Department of Computer Science, ETH Zurich , 2012. 1. – reference: Schaefer S., McPhail T., Warren J.: Image deformation using moving least squares. ACM Transactions on Graphics 25, 3 (July 2006), 533-540. Proceedings of SIGGRAPH. 8. – reference: Lipman Y.: Bounded distortion mapping spaces for triangular meshes. ACM Transactions on Graphics 31, 4 (July 2012), Article 108, 13 pages. Proceedings of SIGGRAPH. 2. – reference: Floater M. S., Kosinka J.: On the injectivity of Wachspress and mean value mappings between convex polygons. Advances in Computational Mathematics 32, 2 (Feb. 2010), 163-174. 2, 4. – reference: Jacobson A., Weinkauf T., Sorkine-Hornung O.: Smooth shape-aware functions with controlled extrema. Computer Graphics Forum 31, 5 (Aug. 2012), 1577-1586. Proceedings of SGP. 8. – reference: Floater M. S., Kós G., Reimers M.: Mean value coordinates in 3D. Computer Aided Geometric Design 22, 7 (Oct. 2005), 623-631. 2. – reference: Meisters G. H., Olech C.: Locally one-to-one mappings and a classical theorem on the schlicht functions. Duke Mathematical Journal 30, 1 (Mar. 1963), 63-80. 2. – reference: Hormann K., Floater M. S.: Mean value coordinates for arbitrary planar polygons. ACM Transactions on Graphics 25, 4 (Oct. 2006), 1424-1441. 2, 3, 4. – volume: 29 start-page: 309 issue: 2 year: May 2010 end-page: 318 article-title: Multi‐scale geometry interpolation publication-title: Computer Graphics Forum – volume: 31 issue: 4 year: July 2012 article-title: Bounded distortion mapping spaces for triangular meshes publication-title: ACM Transactions on Graphics – volume: 27 start-page: 319 issue: 3 year: Oct. 2007 end-page: 338 article-title: Barycentric coordinates for convex sets publication-title: Advances in Computational Mathematics – volume: 29 start-page: 1517 issue: 5 year: July 2010 end-page: 1524 article-title: Moving least squares coordinates publication-title: Computer Graphics Forum – volume: 26 issue: 3 year: July 2007 article-title: Harmonic coordinates for character articulation publication-title: ACM Transactions on Graphics – volume: 20 start-page: 19 issue: 1 year: Mar. 2003 end-page: 27 article-title: Mean value coordinates publication-title: Computer Aided Geometric Design – start-page: 157 year: July 2000 end-page: 164 – volume: 25 start-page: 1424 issue: 4 year: Oct. 2006 end-page: 1441 article-title: Mean value coordinates for arbitrary planar polygons publication-title: ACM Transactions on Graphics – start-page: 173 year: Aug. 1995 end-page: 182 – volume: 30 issue: 4 year: July 2011 article-title: Bounded biharmonic weights for real‐time deformation publication-title: ACM Transactions on Graphics – start-page: 439 year: Aug. 1995 end-page: 448 – volume: 30 start-page: 1543 issue: 5 year: Aug. 2011 end-page: 1552 article-title: As‐killing‐as‐possible vector fields for planar deformation publication-title: Computer Graphics Forum – year: 1975 – volume: 25 start-page: 533 issue: 3 year: July 2006 end-page: 540 article-title: Image deformation using moving least squares publication-title: ACM Transactions on Graphics – volume: 32 start-page: 163 issue: 2 year: Feb. 2010 end-page: 174 article-title: On the injectivity of Wachspress and mean value mappings between convex polygons publication-title: Advances in Computational Mathematics – volume: 15 start-page: 37 issue: 2 year: Mar. 1995 end-page: 43 article-title: Image warping with scattered data interpolation publication-title: IEEE Computer Graphics and Applications – volume: 30 start-page: 1533 issue: 5 year: Aug. 2011 end-page: 1542 article-title: A complex view of barycentric mappings publication-title: Computer Graphics Forum – volume: 29 issue: 4 year: July 2010 article-title: Controllable conformal maps for shape deformation and interpolation publication-title: ACM Transactions on Graphics – volume: 28 start-page: 587 issue: 2 year: Apr. 2009 end-page: 597 article-title: Complex barycentric coordinates with applications to planar shape deformation publication-title: Computer Graphics Forum – year: 2012 – volume: 32 start-page: 31 issue: 2 year: May 2013 end-page: 39 article-title: Finite element image warping publication-title: Computer 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| Snippet | We introduce the novel concept of composite barycentric mappings and give theoretical conditions under which they are guaranteed to be bijective. We then focus... |
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| SubjectTerms | Computation Computer graphics Computer science Deformation G.1.1 [Numerical Analysis]: Interpolation-Interpolation formulas I.3.3 [Computer Graphics]: Picture/Image Generation-Line and curve generation Jacobians Mapping Pixels Studies Three dimensional Two dimensional |
| Title | Bijective Composite Mean Value Mappings |
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