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Mean Square Error Approximation for Wavelet-Based Semiregular Mesh Compression
July/August 2006 (vol. 12 no. 4)
pp. 649-657

Abstract—The objective of this paper is to propose an efficient model-based bit allocation process optimizing the performances of a wavelet coder for semiregular meshes. More precisely, this process should compute the best quantizers for the wavelet coefficient subbands that minimize the reconstructed mean square error for one specific target bitrate. In order to design a fast and low complex allocation process, we propose an approximation of the reconstructed mean square error relative to the coding of semiregular mesh geometry. This error is expressed directly from the quantization errors of each coefficient subband. For that purpose, we have to take into account the influence of the wavelet filters on the quantized coefficients. Furthermore, we propose a specific approximation for wavelet transforms based on lifting schemes. Experimentally, we show that, in comparison with a "naïve” approximation (depending on the subband levels), using the proposed approximation as distortion criterion during the model-based allocation process improves the performances of a wavelet-based coder for any model, any bitrate, and any lifting scheme.

[1] “Information Technology— JPEG 2000, Image Coding System— Part 1: Core Coding System,” ISO/IEC 15444-1, 2000.
[2] M. Lounsbery, T. DeRose, and J. Warren, “Multiresolution Analysis for Surfaces of Arbitrary Topological Type,” Trans. Graphics, vol. 16, no. 1, 1997.
[3] A. Khodakovsky, P. Schröder, and W. Sweldens, “Progressive Geometry Compression,” Proc. SIGGRAPH, 2000.
[4] A. Khodakovsky and I. Guskov, “Normal Mesh Compression,” Geometric Modeling for Scientific Visualization, Springer-Verlag, 2002.
[5] F. Payan and M. Antonini, “MSE Approximation for Model-Based Compression of Multiresolution Semiregular Meshes,” Proc. IEEE EUSIPCO 2005 (13th European Conf. Signal Processing), Sept. 2005.
[6] F. Payan and M. Antonini, “An Efficient Bit Allocation for Compressing Normal Meshes with an Error-Driven Quantization,” Computer Aided Geometric Design, special issue on geometric mesh processing, vol. 22, pp. 466-486, July 2005.
[7] J. Woods and T. Naveen, “A Filter Based Bit Allocation Scheme for Subband Compression of HDTV,” IEEE Trans. Image Processing, 1992.
[8] C. Cheong, K. Aizawa, T. Saito, and M. Hatori, “Subband Image Coding with Biorthogonal Wavelets,” IEICE Trans. Fundamentals, vol. E75-A, no. 7, July 1992.
[9] K. Park and R.A. Haddad, “Modeling, Analysis, and Optimum Design of Quantized M-Band Filter Banks,” IEEE Trans. Signal Processing, vol. 43, no. 11, Nov. 1995.
[10] P. Moulin, “A Multiscale Relaxation Algorithm for SNR Maximization in Nonorthogonal Subband Coding,” IEEE Trans. Image Processing, vol. 4, no. 9, 1995.
[11] B. Usevitch, “Optimal Bit Allocation for Biorthogonal Wavelet Coding,” Proc. IEEE Data Compression Conf., Apr. 1996.
[12] C. Touma and C. Gotsman, “Triangle Mesh Compression,” Proc. Conf. Graphics Interface '98, pp. 26-34, 1998.
[13] J. Kovacevic and W. Sweldens, “Wavelet Families of Increasing Order in Arbitrary Dimensions,” IEEE Trans. Image Processing, 1999.
[14] N.S. Jayant and P. Noll, Digital Coding of Waveforms, E. Cliffs, ed., New Jersey: Prentice Hall, 1984.
[15] R. Gray and T. Stockham, “Dithered Quantizers,” IEEE Trans. Information Theory, vol. 39, no. 3, May 1993.
[16] A. Gersho and R. Gray, Vector Quantization and Signal Compression. Norwell, Mass.: Kluwer Academic Publishers, 1992.
[17] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding. Engelwood Cliffs, N.J.: Prentice Hall PTR, 1995.
[18] D. Li, K. Qin, and H. Sun, “Unlifted Loop Subdivision Wavelets,” Proc. 12th Pacific Conf. Computer Graphics and Applications, 2004.
[19] M. Bertram, “Biorthogonal Loop-Subdivision Wavelets,” Computing, vol. 72, nos. 1-2, pp. 29-39, 2004.
[20] P. Schröder and W. Sweldens, “Spherical Wavelets: Efficiently Representing Functions on the Sphere,” Proc. SIGGRAPH '95, pp. 161-172, 1995.
[21] A. Lee, W. Sweldens, P. Schröder, P. Cowsar, and D. Dobkin, “MAPS: Multiresolution Adaptive Parametrization of Surfaces,” Proc. SIGGRAPH, 1998.
[22] I. Guskov, K. Vidimce, W. Sweldens, and P. Schröder, “Normal Meshes,” Siggraph 2000, Computer Graphics Proc., pp. 95-102, 2000.
[23] S. Lavu, H. Choi, and R. Baraniuk, “Geometry Compression of Normal Meshes Using Rate-Distortion Algorithms,” Proc. Eurographics/ACM SIGGRAPH Symp. Geometry Processing, 2003.
[24] N. Aspert, D. Santa-Cruz, and T. Ebrahimi, “Mesh: Measuring Errors between Surfaces Using the Hausdorff Distance,” Proc. IEEE Int'l Conf. Multimedia and Expo, vol. 1, pp. 705-708, 2002.

Index Terms:
Weighted mean square error (MSE), biorthogonal wavelet, lifting scheme, butterfly scheme, bit allocation, geometry coding, semiregular meshes.
Fr?d?ric Payan, Marc Antonini, "Mean Square Error Approximation for Wavelet-Based Semiregular Mesh Compression," IEEE Transactions on Visualization and Computer Graphics, vol. 12, no. 4, pp. 649-657, July-Aug. 2006, doi:10.1109/TVCG.2006.73
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