Shape-Up: Shaping Discrete Geometry with Projections
We introduce a unified optimization framework for geometry processing based on shape constraints. These constraints preserve or prescribe the shape of subsets of the points of a geometric data set, such as polygons, one‐ring cells, volume elements, or feature curves. Our method is based on two key c...
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| Vydané v: | Computer graphics forum Ročník 31; číslo 5; s. 1657 - 1667 |
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| Hlavní autori: | , , , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Oxford, UK
Blackwell Publishing Ltd
01.08.2012
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | We introduce a unified optimization framework for geometry processing based on shape constraints. These constraints preserve or prescribe the shape of subsets of the points of a geometric data set, such as polygons, one‐ring cells, volume elements, or feature curves. Our method is based on two key concepts: a shape proximity function and shape projection operators. The proximity function encodes the distance of a desired least‐squares fitted elementary target shape to the corresponding vertices of the 3D model. Projection operators are employed to minimize the proximity function by relocating vertices in a minimal way to match the imposed shape constraints. We demonstrate that this approach leads to a simple, robust, and efficient algorithm that allows implementing a variety of geometry processing applications, simply by combining suitable projection operators. We show examples for computing planar and circular meshes, shape space exploration, mesh quality improvement, shape‐preserving deformation, and conformal parametrization. Our optimization framework provides a systematic way of building new solvers for geometry processing and produces similar or better results than state‐of‐the‐art methods. |
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| AbstractList | We introduce a unified optimization framework for geometry processing based on shape constraints. These constraints preserve or prescribe the shape of subsets of the points of a geometric data set, such as polygons, one-ring cells, volume elements, or feature curves. Our method is based on two key concepts: a shape proximity function and shape projection operators. The proximity function encodes the distance of a desired least-squares fitted elementary target shape to the corresponding vertices of the 3D model. Projection operators are employed to minimize the proximity function by relocating vertices in a minimal way to match the imposed shape constraints. We demonstrate that this approach leads to a simple, robust, and efficient algorithm that allows implementing a variety of geometry processing applications, simply by combining suitable projection operators. We show examples for computing planar and circular meshes, shape space exploration, mesh quality improvement, shape-preserving deformation, and conformal parametrization. Our optimization framework provides a systematic way of building new solvers for geometry processing and produces similar or better results than state-of-the-art methods. [PUBLICATION ABSTRACT] We introduce a unified optimization framework for geometry processing based on shape constraints. These constraints preserve or prescribe the shape of subsets of the points of a geometric data set, such as polygons, one‐ring cells, volume elements, or feature curves. Our method is based on two key concepts: a shape proximity function and shape projection operators. The proximity function encodes the distance of a desired least‐squares fitted elementary target shape to the corresponding vertices of the 3D model. Projection operators are employed to minimize the proximity function by relocating vertices in a minimal way to match the imposed shape constraints. We demonstrate that this approach leads to a simple, robust, and efficient algorithm that allows implementing a variety of geometry processing applications, simply by combining suitable projection operators. We show examples for computing planar and circular meshes, shape space exploration, mesh quality improvement, shape‐preserving deformation, and conformal parametrization. Our optimization framework provides a systematic way of building new solvers for geometry processing and produces similar or better results than state‐of‐the‐art methods. |
| Author | Pauly, Mark Deuss, Mario Bouaziz, Sofien Weise, Thibaut Schwartzburg, Yuliy |
| Author_xml | – sequence: 1 givenname: Sofien surname: Bouaziz fullname: Bouaziz, Sofien organization: École Polytechnique Fédérale de Lausanne, Switzerland – sequence: 2 givenname: Mario surname: Deuss fullname: Deuss, Mario organization: École Polytechnique Fédérale de Lausanne, Switzerland – sequence: 3 givenname: Yuliy surname: Schwartzburg fullname: Schwartzburg, Yuliy organization: École Polytechnique Fédérale de Lausanne, Switzerland – sequence: 4 givenname: Thibaut surname: Weise fullname: Weise, Thibaut organization: École Polytechnique Fédérale de Lausanne, Switzerland – sequence: 5 givenname: Mark surname: Pauly fullname: Pauly, Mark organization: École Polytechnique Fédérale de Lausanne, Switzerland |
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Graph. 30 (2011), 140:1-140:10. 2, 7. – reference: Mitra N. J., Guibas L. J., Pauly M.: Symmetrization. ACM Trans. Graph. 26 (2007). 10. – reference: Besl P. J., McKay N. D.: A method for registration of 3-d shapes. IEEE Trans. Pattern Anal. Mach. Intell. 14 (1992), 239-256. 3. – reference: Nocedal J., Wright S. J.: Numerical Optimization. Springer, 2000. 9. – reference: Horn B.: Closed-form solution of absolute orientation using unit quaternions. J. of the Opt. Society of America A 4 (1987), 629-642. 5. – reference: Shi L., Yu Y., Bell N., Feng W.-W.: A fast multi-grid algorithm for mesh deformation. ACM Trans. Graph. 25 (2006), 1108-1117. 10. – reference: Eigensatz M., Pauly M.: Positional, metric, and curvature control for constraint-based surface deformation. Comput. Graph. Forum 28, 2 (2009), 551-558. 2, 8. – reference: Tournois J., Wormser C., Alliez P., Desbrun M.: Interleaving delaunay refinement and optimization for practical isotropic tetrahedron mesh generation. ACM Trans. Graph. 28 (2009), 75:1-75:9. 8. – reference: Gander W., Hrebicek J.: Solving Problems in Scientific Computing Using Maple and MATLAB. Springer-Verlag New York , 1995. 6. – reference: Alexa M., Wardetzky M.: Discrete laplacians on general polygonal meshes. ACM Trans. Graph. 30 (2011), 102:1-102:10. 2, 7. – reference: Yang Y.-L., Yang Y.-J., Pottmann H., Mitra N. J.: Shape space exploration of constrained meshes. ACM Trans. Graph. 30(2011), 124:1-124:12. 7. – reference: Lévy B., Petitjean S., Ray N., Maillot J.: Least squares conformal maps for automatic texture atlas generation. ACM Trans. Graph. 21 (2002), 362-371. 2. – reference: Umeyama S.: Least-squares estimation of transformation parameters between two point patterns. PAMI 13 (1991), 376-380. 5, 10. – reference: Zadravec M., Schiftner A., Wallner J.: Designing quad-dominant meshes with planar faces. CGF 29 (2010), 1671-1679. 2, 7. – reference: Pottmann H., Wallner J.: The focal geometry of circular and conical meshes. Adv. Comp. Math 29 (2008), 249-268. 7. – reference: Igarashi T., Moscovich T., Hughes J. F.: As-rigid-as-possible shape manipulation. ACM Trans. Graph. 24 (2005), 1134-1141. 2, 8. – reference: Desbrun M., Meyer M., Alliez P.: Intrinsic parameterizations of surface meshes. CGF 21 (2002), 209-218. 2. – reference: Müller M., Heidelberger B., Teschner M., Gross M.: Meshless deformations based on shape matching. ACM Trans. Graph. 24 (2005), 471-478. 2. – reference: Liu Y., Pottmann H., Wallner J., Yang Y.-L., Wang W.: Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graph. 25 (2006), 681-689. 2, 7, 10. – reference: Zhang Y., Bajaj C., Xu G.: Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow. Comm. in num. meth. in eng. 25 (2009), 1-18. 2. – reference: Botsch M., Kobbelt L., Pauly M., Alliez P., Levy B.: Polygon Mesh Processing. 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Graph. – start-page: 6 year: 2010 – volume: 30 start-page: 140:1 year: 2011 end-page: 140:10 article-title: General planar quadrilateral mesh design using conjugate direction field publication-title: ACM Trans. Graph. – volume: 39 start-page: 361 year: 2007 end-page: 368 article-title: Preserving form features in interactive mesh deformation publication-title: Comput. Aided Des. – volume: 28 start-page: 33:1 year: 2009 end-page: 33:10 article-title: iWIRES: An analyze‐and‐edit approach to shape manipulation publication-title: ACM Trans. Graph. – volume: 4 start-page: 629 year: 1987 end-page: 642 article-title: Closed‐form solution of absolute orientation using unit quaternions publication-title: J. of the Opt. Society of America A – volume: 21 start-page: 362 year: 2002 end-page: 371 article-title: Least squares conformal maps for automatic texture atlas generation publication-title: ACM Trans. Graph. – start-page: 6 year: 1995 – volume: 42 start-page: 2955 year: 1994 end-page: 2966 article-title: Inconsistent signal feasibility problems: Least‐squares solutions in a product space publication-title: IEEE Transactions on Signal Processing – volume: 29 start-page: 249 year: 2008 end-page: 268 article-title: The focal geometry of circular and conical meshes publication-title: Adv. Comp. Math – volume: 30 start-page: 102:1 year: 2011 end-page: 102:10 article-title: Discrete laplacians on general polygonal meshes publication-title: ACM Trans. Graph. – start-page: 9 year: 2000 – year: 2002 – volume: 26 year: 2007 article-title: Symmetrization publication-title: ACM Trans. Graph. – year: 2008 – year: 2006 – year: 2004 – volume: 25 start-page: 681 year: 2006 end-page: 689 article-title: Geometric modeling with conical meshes and developable surfaces publication-title: ACM Trans. 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Programming – volume: 45 start-page: 362 year: 1989 end-page: 370 article-title: A simple approach for the estimation of circular arc center and its radius publication-title: Computer Vision, Graphics, and Image Processing – volume: 25 start-page: 1 year: 2009 end-page: 18 article-title: Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow publication-title: Comm. in num. meth. in eng. – volume: 29 start-page: 1671 year: 2010 end-page: 1679 article-title: Designing quad‐dominant meshes with planar faces publication-title: CGF – ident: e_1_2_8_3_2 doi: 10.1145/2010324.1964997 – ident: e_1_2_8_40_2 doi: 10.1109/34.88573 – ident: e_1_2_8_11_2 doi: 10.1145/1268517.1268522 – ident: e_1_2_8_18_2 doi: 10.1145/1073204.1073323 – ident: e_1_2_8_4_2 doi: 10.1145/1057432.1057457 – ident: e_1_2_8_32_2 – ident: e_1_2_8_43_2 doi: 10.1111/j.1467-8659.2010.01776.x – ident: e_1_2_8_35_2 – start-page: 6 volume-title: Polygon Mesh Processing year: 2010 ident: e_1_2_8_5_2 doi: 10.1201/b10688 – ident: e_1_2_8_25_2 doi: 10.1145/1073204.1073216 – ident: e_1_2_8_19_2 doi: 10.1145/1618452.1618521 – ident: e_1_2_8_28_2 doi: 10.1016/j.cad.2007.02.010 – ident: e_1_2_8_34_2 doi: 10.1145/1015706.1015736 – ident: e_1_2_8_22_2 doi: 10.1145/2070781.2024174 – ident: e_1_2_8_10_2 doi: 10.1145/2010324.1964999 – ident: e_1_2_8_20_2 doi: 10.1145/566654.566590 – ident: e_1_2_8_21_2 doi: 10.1145/1141911.1141941 – ident: e_1_2_8_37_2 – ident: e_1_2_8_23_2 – ident: e_1_2_8_17_2 doi: 10.1364/JOSAA.4.000629 – ident: e_1_2_8_27_2 doi: 10.1007/s10107-006-0014-3 – ident: e_1_2_8_26_2 – ident: e_1_2_8_15_2 doi: 10.1007/978-3-642-97619-3 – ident: e_1_2_8_8_2 doi: 10.1145/1576246.1531383 – ident: e_1_2_8_36_2 doi: 10.1145/1141911.1142001 – ident: e_1_2_8_14_2 doi: 10.1111/j.1467-8659.2009.01395.x – ident: e_1_2_8_24_2 doi: 10.1145/1276377.1276456 – ident: e_1_2_8_31_2 – start-page: 9 volume-title: Numerical Optimization year: 2000 ident: e_1_2_8_29_2 – ident: e_1_2_8_12_2 doi: 10.1111/1467-8659.00580 – ident: e_1_2_8_6_2 doi: 10.1109/34.121791 – ident: e_1_2_8_38_2 doi: 10.1016/0734-189X(89)90088-1 – ident: e_1_2_8_42_2 doi: 10.1002/cnm.1067 – ident: e_1_2_8_7_2 – ident: e_1_2_8_33_2 – ident: e_1_2_8_30_2 doi: 10.1007/s10444-007-9045-4 – ident: e_1_2_8_41_2 doi: 10.1145/2070781.2024158 – ident: e_1_2_8_39_2 doi: 10.1145/1531326.1531381 – ident: e_1_2_8_2_2 doi: 10.1145/566570.566588 – ident: e_1_2_8_13_2 – ident: e_1_2_8_9_2 doi: 10.1109/78.330356 – ident: e_1_2_8_16_2 doi: 10.1145/1531326.1531339 |
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| Snippet | We introduce a unified optimization framework for geometry processing based on shape constraints. These constraints preserve or prescribe the shape of subsets... |
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| SubjectTerms | 3-D graphics and systems Computer graphics Finite element method Geometry I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems languages Mathematical analysis Mathematical models Operators Optimization Preserves Projection Proximity Studies |
| Title | Shape-Up: Shaping Discrete Geometry with Projections |
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