Christian Lessig
Eugene Fiume
Skip Abstract Section
Abstract
We propose the SOHO wavelet basis—the first spherical Haar wavelet basis that is both orthogonal and symmetric, making it particularly well suited for the approximation and processing of all-frequency signals on the sphere. We obtain the basis with a novel spherical subdivision scheme that defines a partition acting as the domain of the basis functions. Our construction refutes earlier claims doubting the existence of a basis that is both orthogonal and symmetric. Experimental results for the representation of spherical signals verify that the superior theoretical properties of the SOHO wavelet basis are also relevant in practice.
- Alfeld, P., Neamtu, M., and Schumaker, L. L. 1996a. Bernstein-Bézier polynomials on spheres and sphere-like surfaces. Comput. Aid. Geom. Des. 13, 4, 333--349. Google Scholar
Digital Library
- Alfeld, P., Neamtu, M., and Schumaker, L. L. 1996b. Fitting scattered data on sphere-like surfaces using spherical splines. J. Comput. Appl. Math. 73, 1-2, 5--43. Google ScholarDigital Library
- Bonneau, G.-P. 1999. Optimal triangular Haar bases for spherical data. In Proceedings of the Conference on Visualization (VIS'99). IEEE Computer Society Press, Los Alamitos, CA, 279--284. Google ScholarDigital Library
- Cabral, B., Max, N., and Springmeyer, R. 1987. Bidirectional reflection functions from surface bump maps. In Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH'87). ACM Press, New York, NY, 273--281. Google ScholarDigital Library
- Clarke, P. J., Lavalee, D. A., Blewitt, G., and van Dam, T. 2004. Choice of basis functions for the representation of seasonal surface loading signals in geodetic time series. AGU Fall Meeting Abstracts, A121+.Google Scholar
- DeVore, R. A., Jawerth, B., and Lucier, B. J. 1992. Image compression through wavelet transform coding. IEEE Trans. Inform. Theory 38, 2, 719--746.Google ScholarDigital Library
- Donoho, D. L. 1993. Unconditional bases are optimal bases for data compression and statistical estimation. Appl. Comp. Harm. Anal. 1, 100--115.Google ScholarCross Ref
- Finkelstein, A. and Salesin, D. H. 1994. Multiresolution curves. In Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH'94). ACM Press, New York, NY, 261--268. Google ScholarDigital Library
- Fisher, N. I., Lewis, T., and Embleton, B. J. J. 1993. Statistical Analysis of Spherical Data. Cambridge University Press, Cambridge, UK.Google Scholar
- Freeden, W. 1999. Multiscale Modelling of Spaceborne Geodata. B.G. Teubner, Stuttgart, Leipzig.Google Scholar
- Freeden, W., Gervens, T., and Schreiner, M. 1998. Constructive Approximation on the Sphere (With Applications to Geomathematics). Oxford Sciences Publication. Clarendon Press, Oxford University.Google Scholar
- Gautron, P., Křivánek, J., Pattanaik, S. N., and Bouatouch, K. 2004. A novel hemispherical basis for accurate and efficient rendering. In Proceedings of the Eurographics Symposium on Rendering. 321--330. Google ScholarDigital Library
- Girardi, M. and Sweldens, W. 1997. A new class of unbalanced Haar wavelets that form an unconditional basis for Lp on general measure spaces. J. Fourier Anal. Appl. 3, 4.Google ScholarCross Ref
- Green, P., Kautz, J., Matusik, W., and Durand, F. 2006. View-dependent precomputed light transport using nonlinear Gaussian function approximations. In Proceedings of the Symposium on Interactive 3D Graphics and Games (SI3D'06). ACM Press, New York, NY, 7--14. Google ScholarDigital Library
- Gross, M. H. 1996. L2 Optimal Oracles and Compression Strategies for Semiorthogonal Wavelets.Google Scholar
- Kajiya, J. T. 1986. The rendering equation. In Proceedings of the 13th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH'86). ACM Press, New York, NY, 143--150. Google ScholarDigital Library
- Katsuyuki, T., Gengsheng, L. Z., and Gullberg, G. T. 2001. Cone-beam image reconstruction using spherical harmonics. Phys. Medicine Biol. 46, N127--N138(1).Google ScholarCross Ref
- Kautz, J., Sloan, P.-P., and Snyder, J. 2002. Fast, arbitrary BRDF shading for low-frequency lighting using spherical harmonics. In Proceedings of the 13th Eurographics Workshop on Rendering (EGRW'02). Eurographics Association, Aire-la-Ville, Switzerland, 291--296. Google ScholarDigital Library
- Koenderink, J. J., van Doorn, A. J., and Stavridi, M. 1996. Bidirectional reflection distribution function expressed in terms of surface scattering modes. In Proceedings of the 4th European Conference on Computer Vision-Volume II (ECCV'96). Springer-Verlag, 28--39. Google ScholarDigital Library
- Lessig, C. 2007. Orthogonal and symmetric wavelets on the sphere. MSc thesis. University of Toronto.Google Scholar
- Lounsbery, M., DeRose, T. D., and Warren, J. 1997. Multiresolution analysis for surfaces of arbitrary topological type. ACM Trans. Graph. 16, 1, 34--73. Google ScholarDigital Library
- Ma, W.-C., Hsiao, C.-T., Lee, K.-Y., Chuang, Y.-Y., and Chen, B.-Y. 2006. Real-time triple product relighting using spherical local-frame parameterization. Vis. Comput. 9-11, 682--692. Pacific Graphics 2006 Conference Proceedings. Google ScholarDigital Library
- MacRobert, T. M. 1948. Spherical Harmonics; An Elementary Treatise on Harmonic Functions, with Applications. Dover Publications.Google Scholar
- Makhotkin, O. A. 1996. Analysis of radiative transfer between surfaces by hemispherical harmonics. J. Quant. Spectros. Radiat. Tran. 56, 869--879.Google ScholarCross Ref
- Narcowich, F. J. and Ward, J. D. 1996. Nonstationary wavelets on the m-sphere for scattered data. App. Comput. Harm. Anal. 3, 324--336.Google ScholarCross Ref
- Ng, R., Ramamoorthi, R., and Hanrahan, P. 2003. All-frequency shadows using non-linear wavelet lighting approximation. ACM Trans. Graph. 22, 3, 376--381. Google ScholarDigital Library
- Ng, R., Ramamoorthi, R., and Hanrahan, P. 2004. Triple product wavelet integrals for all-frequency relighting. ACM Trans. Graph. 23, 3, 477--487. Google ScholarDigital Library
- Nielson, G. M., Jung, I.-H., and Sung, J. 1997. Haar wavelets over triangular domains with applications to multiresolution models for flow over a sphere. In Proceedings of the 8th Conference on Visualization (VIS'97). IEEE Computer Society Press, Los Alamitos, CA 143--ff. Google ScholarDigital Library
- Pratt, W. K. 1991. Digital Image Processing 2nd Ed. John Wiley & Sons, Inc., New York, NY. Google ScholarDigital Library
- Ramamoorthi, R. and Hanrahan, P. 2002. Frequency space environment map rendering. In ACM SIGGRAPH 2002 Papers. ACM Press, New York, NY, 517--526. Google ScholarDigital Library
- Roşca, D. 2005. Haar wavelets on spherical triangulations. In Advances in Multiresolution for Geometric Modelling, N. A. Dodgson, M. S. Floater, and M. A. Sabin, Eds. Mathematics and Visualization. Springer, 405--417.Google Scholar
- Schröder, P. and Sweldens, W. 1995. Spherical wavelets: Efficiently representing functions on the sphere. In Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH'95). ACM Press, New York, NY, 161--172. Google ScholarDigital Library
- Sillion, F., Arvo, J., Westin, S., and Greenberg, D. P. 1991. A global illumination solution for general reflectance distributions. In Proceedings of the 18th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH'91). ACM Press, New York, NY, 187--196. Google ScholarDigital Library
- Sloan, P.-P., Kautz, J., and Snyder, J. 2002. Precomputed radiance transfer for real-time rendering in dynamic, low-frequency lighting environments. In ACM SIGGRAPH 2002 Papers. ACM Press, New York, NY, 527--536. Google ScholarDigital Library
- Sloan, P.-P., Luna, B., and Snyder, J. 2005. Local, deformable precomputed radiance transfer. In ACM SIGGRAPH 2005 Papers. ACM Press, New York, NY, 1216--1224. Google ScholarDigital Library
- Stollnitz, E. J., Derose, T. D., and Salesin, D. H. 1996. Wavelets for Computer Graphics: Theory and Applications. Morgan Kaufmann Publishers Inc., San Francisco, CA. Google ScholarDigital Library
- Sun, W. and Mukherjee, A. 2006. Generalized wavelet product integral for rendering dynamic glossy objects. ACM Trans. Graph. 25, 3, 955--966. Google ScholarDigital Library
- Sweldens, W. 1996. The lifting scheme: A custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3, 2, 186--200.Google ScholarCross Ref
- Todhunter, I. 1901. Spherical Trigonometry, for the Use of Colleges and Schools. Macmillan, New York.Google Scholar
- Tsai, Y.-T. and Shih, Z.-C. 2006. All-frequency precomputed radiance transfer using spherical radial basis functions and clustered tensor approximation. In ACM SIGGRAPH 2006 Papers. ACM Press, New York, NY, 967--976. Google ScholarDigital Library
- Wang, R., Ng, R., Luebke, D., and Humphreys, G. 2006. Efficient wavelet rotation for environment map rendering. In Proceedings of the Eurographics Symposium on Rendering. Springer-Verlag, Vienna. Published as Rendering Techniques 2006. Google ScholarDigital Library
- Westin, S. H., Arvo, J. R., and Torrance, K. E. 1992. Predicting reflectance functions from complex surfaces. In ACM SIGGRAPH 1992 Papers. ACM Press, New York, NY, 255--264. Google ScholarDigital Library
- Zhou, K., Hu, Y., Lin, S., Guo, B., and Shum, H.-Y. 2005. Precomputed shadow fields for dynamic scenes. In ACM SIGGRAPH 2005 Papers. ACM Press, New York, NY, 1196--1201. Google ScholarDigital Library
Index Terms
- SOHO: Orthogonal and symmetric Haar wavelets on the sphere
Recommendations
New features for speech enhancement using bivariate shrinkage based on redundant wavelet filter-banks
There are some dependencies among wavelet coefficients.Bivariate shrinkage was proposed based on parent-child coefficients dependencies.Two-channel critically sampled filter-bank is not suitable for bivariate shrinkage.Multi-channel redundant wavelet ...
Research on Random Noise Attenuation in Transform Domain
ICDMA '11: Proceedings of the 2011 Second International Conference on Digital Manufacturing & AutomationRadom noise attenuation method by contourlet transform is researChed in this paper. Compared with wavelet transform, the contourlet transform proposed by Do and VetterLi offers a high degree of directionaLity and anisotropy, and it represents images ...
Combining curvelet transform and wavelet transform for image denoising
ICIC'10: Proceedings of the Advanced intelligent computing theories and applications, and 6th international conference on Intelligent computingWavelet transform has the good characteristic of time-frequency locality and many researches show that it can perform well for denoising in smooth and singular areas. But it isn't suitable for describing the signals, which have high dimensional ...
Comments