A Biorthogonal Wavelet Approach based on Dual Subdivision
In this paper a biorthogonal wavelet approach based on Doo‐Sabin subdivision is presented. In the dual subdivision like Doo‐Sabin scheme, all the old control vertices disappear after one subdivision step, which is a big challenge to the biorthogonal wavelet construction. In our approach, the barycen...
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| Vydáno v: | Computer graphics forum Ročník 27; číslo 7; s. 1815 - 1822 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Oxford, UK
Blackwell Publishing Ltd
01.10.2008
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | In this paper a biorthogonal wavelet approach based on Doo‐Sabin subdivision is presented. In the dual subdivision like Doo‐Sabin scheme, all the old control vertices disappear after one subdivision step, which is a big challenge to the biorthogonal wavelet construction. In our approach, the barycenters of the V‐faces corresponding to the old vertices are selected as the vertices associated with the scaling functions to construct the scaling space. The lifting scheme is used to guarantee the fitting quality of the wavelet transform, and a local orthogonalization is introduced with a discrete inner product operation to improve the computation efficiency. Sharp feature modeling based on extended Doo‐Sabin subdivision rules is also discussed in the framework of our wavelet construction. The presented wavelet construction is proven to be stable and effective by the experimental results. |
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| AbstractList | In this paper a biorthogonal wavelet approach based on Doo‐Sabin subdivision is presented. In the dual subdivision like Doo‐Sabin scheme, all the old control vertices disappear after one subdivision step, which is a big challenge to the biorthogonal wavelet construction. In our approach, the barycenters of the V‐faces corresponding to the old vertices are selected as the vertices associated with the scaling functions to construct the scaling space. The lifting scheme is used to guarantee the fitting quality of the wavelet transform, and a local orthogonalization is introduced with a discrete inner product operation to improve the computation efficiency. Sharp feature modeling based on extended Doo‐Sabin subdivision rules is also discussed in the framework of our wavelet construction. The presented wavelet construction is proven to be stable and effective by the experimental results. |
| Author | Qin, Guiming Zhang, Hui Sun, Hanqiu Qin, Kaihuai |
| Author_xml | – sequence: 1 givenname: Hui surname: Zhang fullname: Zhang, Hui organization: Department of Computer Science & Technology, Tsinghua University, Beijing 100084, China – sequence: 2 givenname: Guiming surname: Qin fullname: Qin, Guiming organization: Department of Computer Science & Engineering, The Chinese University of Hong Kong, Hong Kong, China – sequence: 3 givenname: Kaihuai surname: Qin fullname: Qin, Kaihuai organization: Department of Computer Science & Technology, Tsinghua University, Beijing 100084, China – sequence: 4 givenname: Hanqiu surname: Sun fullname: Sun, Hanqiu organization: Department of Computer Science & Engineering, The Chinese University of Hong Kong, Hong Kong, China |
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| References_xml | – reference: Bartels R. H., Samavati F. F.: Reversing subdivision rules: local linear conditions and observations on inner products. Journal of Computational and Applied Mathematics 119, 1-2 (2000), 29-67. – reference: Nasri A. H.: A non-uniform Doo-Sabin subdivision scheme with boundary control. Tech. Rep. TR97/1, American University of Beirut, 1997. – reference: Bertram M.: Biorthogonal Loop-subdivision wavelets. Computing 72, 1-2 (2004), 29-39. – reference: Sweldens W.: The lifting scheme: A construction of second generation wavelets. SIAM J. Comput. 29, 2 (1997), 511-546. – reference: Catmull E., Clark J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design 10, 6 (1978), 350-355. – reference: Doo D., Sabin M.: Behaviors of recursive division surfaces near extraordinary points. Computer-Aided Design 10, 6 (1978), 356-360. – reference: Wang H., Qin K., Tang K.: Efficient wavelet construction with Catmull-Clark subdivision. The Visual Computer 22, 9-11 (2006), 874-884. – reference: Bertram M., Duchaineau M. A., Hamann B., Joy K. I.: Generalized B-spline subdivision-surface wavelets for geometry compression. IEEE Transactions on Visualization and Computer Graphics 10, 3 (2004), 326-338. – reference: Valette S., Prost R.: Wavelet-based multiresolution analysis of irregular surface meshes. IEEE Transactions on Visualization and Computer Graphics 10, 2 (2004), 113-122. – reference: Samavati F. F., Mahdavi-Amiri N., Bartels R. H.: Multiresolution curve and surface editing: reversing subdivision rules by least-squares data fitting. Computer Graphics Forum 18, 2 (1999), 97-119. – reference: Wang H., Qin K., Sun H.: √3-subdivision-based biorthogonal wavelets. IEEE Transactions on Visualization and Computer Graphics 13, 5 (2007), 914-924. – reference: Nasri A. H.: Polyhedral subdivision methods for free-form surfaces. ACM Trans. on Graphics 6, 1 (1987), 29-73. – reference: Chaikin G. M.: An algorithm for high-speed curve generation. Computer Graphics and Image Processing 3 (1974), 346-349. – reference: Sweldens W.: The lifting scheme: A custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal 3, 2 (1996), 186-200. – reference: Valette S., Prost R.: Wavelet-based progressive compression scheme for triangle meshes: Wavemesh. IEEE Transactions on Visualization and Computer Graphics 10, 2 (2004), 123-129. – reference: Bonneau G.-P.: Multiresolution analysis on irregular surface meshes. IEEE Transactions on Visualization and Computer Graphics 4, 4 (1998), 365-378. – reference: Ma W.: Subdivision surfaces for CAD - an overview. Computer-Aided Design 37, 7 (2005), 693-709. – reference: Lounsbery J. M., DeRose T., Warren J.: Multiresolution analysis for surfaces of arbitrary topological type. ACM Trans. on Graphics 16, 1 (1997), 34-73. – reference: Samavati F. F., Mahdavi-Amiri N., Bartels R. H.: Multiresolution surfaces having arbitrary topologies by a reverse Doo subdivision method. Computer Graphics Forum 21, 2 (2002), 121-136. – volume: 29 start-page: 511 issue: 2 year: 1997 end-page: 546 article-title: The lifting scheme: A construction of second generation wavelets publication-title: SIAM J. Comput. – start-page: 387 year: 1998 end-page: 394 – volume: 21 start-page: 121 issue: 2 year: 2002 end-page: 136 article-title: Multiresolution surfaces having arbitrary topologies by a reverse Doo subdivision method publication-title: Computer Graphics Forum – volume: 10 start-page: 326 issue: 3 year: 2004 end-page: 338 article-title: Generalized B‐spline subdivision‐surface wavelets for geometry compression publication-title: IEEE Transactions on Visualization and Computer Graphics – volume: 4 start-page: 365 issue: 4 year: 1998 end-page: 378 article-title: Multiresolution analysis on irregular surface meshes publication-title: IEEE Transactions on Visualization and Computer Graphics – start-page: 173 year: 1995 end-page: 182 – volume: 10 start-page: 123 issue: 2 year: 2004 end-page: 129 article-title: Wavelet‐based progressive compression scheme for triangle meshes: Wavemesh publication-title: IEEE Transactions on Visualization and Computer Graphics – volume: 10 start-page: 113 issue: 2 year: 2004 end-page: 122 article-title: Wavelet‐based multiresolution analysis of irregular surface meshes publication-title: IEEE Transactions on Visualization and Computer Graphics – volume: 18 start-page: 97 issue: 2 year: 1999 end-page: 119 article-title: Multiresolution curve and surface editing: reversing subdivision rules by least‐squares data fitting publication-title: Computer Graphics Forum – volume: 72 start-page: 29 issue: 1‐2 year: 2004 end-page: 39 article-title: Biorthogonal Loop‐subdivision wavelets publication-title: Computing – year: 1994 – volume: 119 start-page: 29 issue: 1‐2 year: 2000 end-page: 67 article-title: Reversing subdivision rules: local linear conditions and observations on inner products publication-title: Journal of Computational and Applied Mathematics – volume: 10 start-page: 356 issue: 6 year: 1978 end-page: 360 article-title: Behaviors of recursive division surfaces near extraordinary points publication-title: Computer-Aided Design – volume: 37 start-page: 693 issue: 7 year: 2005 end-page: 709 article-title: Subdivision surfaces for CAD ‐ an overview publication-title: Computer-Aided Design – volume: 3 start-page: 346 year: 1974 end-page: 349 article-title: An algorithm for high‐speed curve generation publication-title: Computer Graphics and Image Processing – start-page: 25 year: 2004 end-page: 33 – volume: 22 start-page: 874 issue: 9‐11 year: 2006 end-page: 884 article-title: Efficient wavelet construction with Catmull‐Clark subdivision publication-title: The Visual Computer – start-page: 161 year: 1995 end-page: 172 – volume: 13 start-page: 914 issue: 5 year: 2007 end-page: 924 article-title: √3‐subdivision‐based biorthogonal wavelets publication-title: IEEE Transactions on Visualization and Computer Graphics – year: 1997 – volume: 6 start-page: 29 issue: 1 year: 1987 end-page: 73 article-title: Polyhedral subdivision methods for free‐form surfaces publication-title: ACM Trans. on Graphics – volume: 3 start-page: 186 issue: 2 year: 1996 end-page: 200 article-title: The lifting scheme: A custom‐design construction of biorthogonal wavelets publication-title: Appl. Comput. Harmon. Anal – volume: 10 start-page: 350 issue: 6 year: 1978 end-page: 355 article-title: Recursively generated B‐spline surfaces on arbitrary topological meshes publication-title: Computer-Aided Design – volume: 16 start-page: 34 issue: 1 year: 1997 end-page: 73 article-title: Multiresolution analysis for surfaces of arbitrary topological type publication-title: ACM Trans. on Graphics – start-page: 389 year: 2000 end-page: 396 – ident: e_1_2_7_5_2 doi: 10.1109/2945.765329 – ident: e_1_2_7_13_2 – ident: e_1_2_7_22_2 doi: 10.1109/TVCG.2004.1260763 – ident: e_1_2_7_27_2 – ident: e_1_2_7_19_2 doi: 10.1145/218380.218439 – ident: e_1_2_7_18_2 doi: 10.1111/1467-8659.00572 – ident: e_1_2_7_10_2 doi: 10.1145/218380.218440 – ident: e_1_2_7_3_2 doi: 10.1109/TVCG.2004.1272731 – ident: e_1_2_7_15_2 doi: 10.1145/27625.27628 – ident: e_1_2_7_8_2 doi: 10.1016/0146-664X(74)90028-8 – ident: e_1_2_7_12_2 – ident: e_1_2_7_24_2 doi: 10.1109/TVCG.2007.1031 – ident: e_1_2_7_25_2 doi: 10.1007/s00371-006-0074-7 – ident: e_1_2_7_20_2 doi: 10.1006/acha.1996.0015 – ident: e_1_2_7_14_2 doi: 10.1016/j.cad.2004.08.008 – ident: e_1_2_7_23_2 doi: 10.1109/TVCG.2004.1260764 – ident: e_1_2_7_4_2 doi: 10.1007/s00607-003-0044-0 – ident: e_1_2_7_6_2 doi: 10.1016/S0377-0427(00)00370-8 – volume-title: A non‐uniform Doo‐Sabin subdivision scheme with boundary control year: 1997 ident: e_1_2_7_16_2 – volume: 29 start-page: 511 issue: 2 year: 1997 ident: e_1_2_7_21_2 article-title: The lifting scheme: A construction of second generation wavelets publication-title: SIAM J. Comput. – ident: e_1_2_7_7_2 doi: 10.1016/0010-4485(78)90110-0 – ident: e_1_2_7_2_2 doi: 10.1109/VISUAL.2000.885720 – ident: e_1_2_7_26_2 doi: 10.1145/280814.280942 – ident: e_1_2_7_17_2 doi: 10.1111/1467-8659.00361 – ident: e_1_2_7_11_2 doi: 10.1145/237748.237750 – ident: e_1_2_7_9_2 doi: 10.1016/0010-4485(78)90111-2 |
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| SubjectTerms | and object representations Hierarchy and geometric transformations I.3.5 [Computational Geometry and Object Modeling]: Curve I.3.5 [Computational Geometry and Object Modeling]: Curve, surface, solid, and object representations solid surface Wavelet transforms |
| Title | A Biorthogonal Wavelet Approach based on Dual Subdivision |
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