Optimal Interpolatory Wavelets Transform for Multiresolution Triangular Meshes

ABSTRACT

In recent years, several matrix-valued subdivisions have been proposed for triangular meshes. The ma-trix-valued subdivisions can simulate and extend the traditional scalar-valued subdivision, such as loop and subdivision. In this paper, we study how to construct the matrix-valued subdivision wavelets, and propose the novel biorthogonal wavelet based on matrix-valued subdivisions on multiresolution triangular meshes. The new wavelets transform not only inherits the advantages of subdivision, but also offers more resolutions of complex models. Based on the matrix-valued wavelets proposed, we further optimize the wavelets transform for interactive modeling and visualization applications, and develop the efficient interpolatory loop subdivision wavelets transform. The transform simplifies the computation, and reduces the memory usage of matrix-valued wavelets transform. Our experiments showed that the novel wavelets transform is sufficiently stable, and performs well for noise reduction and fitting quality especially for multiresolution semi-regular triangular meshes.

In recent years, several matrix-valued subdivisions have been proposed for triangular meshes. The ma-trix-valued subdivisions can simulate and extend the traditional scalar-valued subdivision, such as loop and subdivision. In this paper, we study how to construct the matrix-valued subdivision wavelets, and propose the novel biorthogonal wavelet based on matrix-valued subdivisions on multiresolution triangular meshes. The new wavelets transform not only inherits the advantages of subdivision, but also offers more resolutions of complex models. Based on the matrix-valued wavelets proposed, we further optimize the wavelets transform for interactive modeling and visualization applications, and develop the efficient interpolatory loop subdivision wavelets transform. The transform simplifies the computation, and reduces the memory usage of matrix-valued wavelets transform. Our experiments showed that the novel wavelets transform is sufficiently stable, and performs well for noise reduction and fitting quality especially for multiresolution semi-regular triangular meshes.

Cite this paper

nullC. Zhao and H. Sun, "Optimal Interpolatory Wavelets Transform for Multiresolution Triangular Meshes,"*Applied Mathematics*, Vol. 2 No. 3, 2011, pp. 369-378. doi: 10.4236/am.2011.23044.

nullC. Zhao and H. Sun, "Optimal Interpolatory Wavelets Transform for Multiresolution Triangular Meshes,"

References

[1] M. Lounsbery, T. DeRose and J. Warren, “Multiresolution Analysis for Surfaces of Arbitrary Topological Type,” ACM Transaction of Graphics, Vol. 16, No. 1, 1997, pp. 34–73.

[2] S. Valette and R. ProstS, “Wavelet-Based Multiresolution Analysis of Irregular Surface Meshes,” IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 2, 2004, pp. 113-122. doi:10.1109/TVCG.2004.1260763

[3] S. Valette and R. ProstS, “Wavelet-Based Progressive Compression Scheme for Triangle Meshes: Wavemesh,” IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 2, 2004, pp. 123-129. doi:10.1109/TVCG.2004.1260764

[4] F. F. Samavati and R. H. Bartels, “Multiresolution Curve and Surface Editing: Reversing Subdivision Rules by Least-Squares Data Fitting,” Computer Graphics Forum, Vol. 18, No. 2, 1999, pp. 97-119. doi:10.1111/1467-8659.00361

[5] F. F. Samavati, N. Mahdavi-Amiri and R. H. Bartels, “Multiresolution Surfaces Having Arbitrary Topologies by a Reverse Doo Subdivision Method,” Computer Graphics Forum, Vol. 21, No. 2, 2002, pp. 121-136. doi:10.1111/1467-8659.00572

[6] W. Sweldens, “The Lifting Scheme: A Custom-Design Construction of Biorthogonal Wavelets,” Applied and Computational Harmonic Analysis, Vol. 3, No. 2, 1996, pp. 186-200. doi:10.1006/acha.1996.0015

[7] P. Schroder and W. Sweldens, “Spherical Wavelets: Efficiently Representing Functions on the Sphere,” In SIGGRAPH’95: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, Los Angeles, 6-11 August 1995, pp. 161-172.

[8] M. Bertram, M. Duchaineau, B. Hamann and K. Joy, “Bicubic Subdivision-Surface Wavelets for Large-Scale Isosurface Representation and Visualization,” In VIS’00: Proceedings of the Conference on Visualization’00, Salt Lake City, 8-13 October 2000, pp. 389-396.

[9] M. Bertram, M. Duchaineau, B. Hamann and K. Joy, “Generalized b-Spline Subdivision-Surface Wavelets for Geometry Compression,” IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 3, 2004, pp. 326-338. doi:10.1109/TVCG.2004.1272731

[10] M. Bertram, “Biorthogonal Loop-Subdivision Wavelets,” Computing, Vol. 72, No. 1-2, 2004, pp. 29-39. doi:10.1007/s00607-003-0044-0

[11] D. Li, K. Qin and H. Sun, “Unlifted Loop Subdivision Wavelets,” In PG’04: Proceedings of the Computer Graphics and Applications, 12th Pacific Conference, IEEE Computer Society, 2004, pp. 25-33.

[12] H. Wang, K. Qin and K. Tang, “Efficient Wavelet Construction with Catmull-Clark Subdivision,” Visual Computer, Vol. 22, No. 9, 2006, pp. 874-884. doi:10.1007/s00371-006-0074-7

[13] H. Wang, K. Qin and H. Sun, “ -Subdivision-Based Biorthogonal Wavelets,” IEEE Transaction on Visualization and Computer Graphics, Vol. 13, No. 5, 2007, pp. 914-925. doi:10.1109/TVCG.2007.1031

[14] H. Wang, K. Tang and K. Qin, “Biorthogonal Wavelets Based on Gradual Subdivision of Quadrilateral Meshes,” Computer Aided Geometric Design, Vol. 25, No. 9, 2008, pp. 816-836. doi:10.1016/j.cagd.2007.11.002

[15] H. Wang, K. Tang and K. Qin, “Biorthogonal Wavelets Based on Interpolatory p2 Subdivision,” Computer Graphics Forum, Vol. 28, No. 6, 2009, pp. 1572-1585. doi:10.1111/j.1467-8659.2009.01349.x

[16] H. Zhang, G. Qin, K. Qin and H. Sun, “A Biorthogonal Wavelet Approach Based on Dual Subdivision,” Computer Graphics Forum, Vol. 27, No. 7, 2009, pp. 1815-1822. doi:10.1111/j.1467-8659.2008.01327.x

[17] C. Zhao, H. Sun and K. Qin, “Computing Efficient Matrix-Valued Wavelets for Meshes,” Pacific Conference on Computer Graphics and Applications, Hangzhou, Vol. 0, 25-27 September 2010, pp. 32–38.

[18] C. K. Chui and Q. Jiang, “Surface Subdivision Schemes Generated by Refinable Bivariate Spline Function Vectors,” Applied and Computational Harmonic Analysis, Vol. 15, 2003, pp. 147-162. doi:10.1016/S1063-5203(03)00062-9

[19] C. K. Chui and Q. Jiang, “Matrix-Valued Symmetric Templates for Interpolatory Surface Subdivisions, i: Regular Vertices,” Applied and Computational Harmonic Analysis, Vol. 19, 2005, pp. 303-339. doi:10.1016/j.acha.2005.03.004

[20] C. K. Chui and Q. Jiang, “From Extension of Loop’s Approximation Scheme to Interpolatory Subdivisions,” Computer Aided Geometric Design, Vol. 25, No. 2, 2008, pp. 96-115. doi:10.1016/j.cagd.2007.05.004

[1] M. Lounsbery, T. DeRose and J. Warren, “Multiresolution Analysis for Surfaces of Arbitrary Topological Type,” ACM Transaction of Graphics, Vol. 16, No. 1, 1997, pp. 34–73.

[2] S. Valette and R. ProstS, “Wavelet-Based Multiresolution Analysis of Irregular Surface Meshes,” IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 2, 2004, pp. 113-122. doi:10.1109/TVCG.2004.1260763

[3] S. Valette and R. ProstS, “Wavelet-Based Progressive Compression Scheme for Triangle Meshes: Wavemesh,” IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 2, 2004, pp. 123-129. doi:10.1109/TVCG.2004.1260764

[4] F. F. Samavati and R. H. Bartels, “Multiresolution Curve and Surface Editing: Reversing Subdivision Rules by Least-Squares Data Fitting,” Computer Graphics Forum, Vol. 18, No. 2, 1999, pp. 97-119. doi:10.1111/1467-8659.00361

[5] F. F. Samavati, N. Mahdavi-Amiri and R. H. Bartels, “Multiresolution Surfaces Having Arbitrary Topologies by a Reverse Doo Subdivision Method,” Computer Graphics Forum, Vol. 21, No. 2, 2002, pp. 121-136. doi:10.1111/1467-8659.00572

[6] W. Sweldens, “The Lifting Scheme: A Custom-Design Construction of Biorthogonal Wavelets,” Applied and Computational Harmonic Analysis, Vol. 3, No. 2, 1996, pp. 186-200. doi:10.1006/acha.1996.0015

[7] P. Schroder and W. Sweldens, “Spherical Wavelets: Efficiently Representing Functions on the Sphere,” In SIGGRAPH’95: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, Los Angeles, 6-11 August 1995, pp. 161-172.

[8] M. Bertram, M. Duchaineau, B. Hamann and K. Joy, “Bicubic Subdivision-Surface Wavelets for Large-Scale Isosurface Representation and Visualization,” In VIS’00: Proceedings of the Conference on Visualization’00, Salt Lake City, 8-13 October 2000, pp. 389-396.

[9] M. Bertram, M. Duchaineau, B. Hamann and K. Joy, “Generalized b-Spline Subdivision-Surface Wavelets for Geometry Compression,” IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 3, 2004, pp. 326-338. doi:10.1109/TVCG.2004.1272731

[10] M. Bertram, “Biorthogonal Loop-Subdivision Wavelets,” Computing, Vol. 72, No. 1-2, 2004, pp. 29-39. doi:10.1007/s00607-003-0044-0

[11] D. Li, K. Qin and H. Sun, “Unlifted Loop Subdivision Wavelets,” In PG’04: Proceedings of the Computer Graphics and Applications, 12th Pacific Conference, IEEE Computer Society, 2004, pp. 25-33.

[12] H. Wang, K. Qin and K. Tang, “Efficient Wavelet Construction with Catmull-Clark Subdivision,” Visual Computer, Vol. 22, No. 9, 2006, pp. 874-884. doi:10.1007/s00371-006-0074-7

[13] H. Wang, K. Qin and H. Sun, “ -Subdivision-Based Biorthogonal Wavelets,” IEEE Transaction on Visualization and Computer Graphics, Vol. 13, No. 5, 2007, pp. 914-925. doi:10.1109/TVCG.2007.1031

[14] H. Wang, K. Tang and K. Qin, “Biorthogonal Wavelets Based on Gradual Subdivision of Quadrilateral Meshes,” Computer Aided Geometric Design, Vol. 25, No. 9, 2008, pp. 816-836. doi:10.1016/j.cagd.2007.11.002

[15] H. Wang, K. Tang and K. Qin, “Biorthogonal Wavelets Based on Interpolatory p2 Subdivision,” Computer Graphics Forum, Vol. 28, No. 6, 2009, pp. 1572-1585. doi:10.1111/j.1467-8659.2009.01349.x

[16] H. Zhang, G. Qin, K. Qin and H. Sun, “A Biorthogonal Wavelet Approach Based on Dual Subdivision,” Computer Graphics Forum, Vol. 27, No. 7, 2009, pp. 1815-1822. doi:10.1111/j.1467-8659.2008.01327.x

[17] C. Zhao, H. Sun and K. Qin, “Computing Efficient Matrix-Valued Wavelets for Meshes,” Pacific Conference on Computer Graphics and Applications, Hangzhou, Vol. 0, 25-27 September 2010, pp. 32–38.

[18] C. K. Chui and Q. Jiang, “Surface Subdivision Schemes Generated by Refinable Bivariate Spline Function Vectors,” Applied and Computational Harmonic Analysis, Vol. 15, 2003, pp. 147-162. doi:10.1016/S1063-5203(03)00062-9

[19] C. K. Chui and Q. Jiang, “Matrix-Valued Symmetric Templates for Interpolatory Surface Subdivisions, i: Regular Vertices,” Applied and Computational Harmonic Analysis, Vol. 19, 2005, pp. 303-339. doi:10.1016/j.acha.2005.03.004

[20] C. K. Chui and Q. Jiang, “From Extension of Loop’s Approximation Scheme to Interpolatory Subdivisions,” Computer Aided Geometric Design, Vol. 25, No. 2, 2008, pp. 96-115. doi:10.1016/j.cagd.2007.05.004