“Spherical wavelets: efficiently representing functions on the sphere” by Schröder and Sweldens

Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classical constructions have been limited to simple domains such as intervals and rectangles. In this paper we present a wavelet construction for scalar functions defined on the sphere. We show how biorthogonal wavelets with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allow fully adaptive subdivisions. We give examples of functions defined on the sphere, such as topographic data, bidirectional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets.

1. ALFELD, P., NEAMTU, M., AND SCHUMAKER, L. L. Bernstein- Bdzier polynomials on circles, sphere, and sphere-like surfaces. Preprint.
2. CARNICER, J. M., DAHMEN, W., AND PElqA, J. M. Local decompositions of refinable spaces. Tech. rep., Insitut far Geometrie und angewandete Mathematik, RWTH Aachen, 1994.
3. CHRISTENSEN, E H., STOLLNITZ, E. J., SALESIN, D. H., AND DEROSE, T. D. Wavelet Radiance. In Proceedings of the 5th Eurographics Workshop on Rendering, 287-302, June 1994.
4. COHEN, A., DAUBECHIES, I., AND FEAUVEAU, J. Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45 (1992), 485-560.
5. COHEN, A., DAUBECHIES, I., JAWERTH, B., AND VIAL, P. Multiresolution analysis, wavelets and fast algorithms on an interval. C. R. Acad. Sci. Paris Sdr. I Math. I, 316 (1993), 417-421.
6. DAHLKE, S., DAHMEN, W., SCHMITT, E., AND WEINREICH, I. Multiresolution analysis and wavelets on S2 and S3. Tech. Rep. 104, Institut far Geometrie und angewandete Mathematik, RWTH Aachen, 1994.
7. DAHMEN, W. Stability of multiscale transformations. Tech. rep., Institut ftir Geometrie und angewandete Mathematik, RWTH Aachen, 1994.
8. DAHMEN, W., PROSSDORF, S., AND SCHNEIDER, R. Multiscale methods for pseudo-differential equations on smooth manifolds. In Conference on Wavelets: Theory, Algorithms, and Applications, C. K. C. et al., Ed. Academic Press, San Diego, CA, 1994, pp. 385-424.
9. DAUBECHIES, I. Ten Lectures on Wavelets. CBMS-NSF Regional Conf. Series in Appl. Math., Vol. 61. Society for Industrial and Applied Mathematics, Philadelphia, PAL, 1992.
10. DUTTON, G. Locational Properties of Quaternary Triangular Meshes. In Proceedings of the Fourth International Symposium on Spatial Data Handling, 901-910, July 1990.
11. DYN, N., LEVlN, D., AND GREGORY, J. A Butterfly Subdivision Scheme for Surface Interpolation with Tension Control. Transactions on Graphics 9, 2 (April 1990), 160-169.
12. FEKETE, G. Rendering and Managing Spherical Data with Sphere Quadtrees. In Proceedings of Visualization 90, 1990.
13. FREEDEN, W., AND WINDHEUSER, U. Spherical Wavelet Transform and its Discretization. Tech. Rep. 125, Universit,it Kaiserslautern, Fachbereich Mathematik, 1994.
14. GIRARDI, M., AND SWELDENS, W. A new class of unbalanced Haar wavelets that form an unconditional basis for Lv on general measure spaces. Tech. Rep. 1995:2, Industrial Mathematics Initiative, Department of Mathematics, University of South Carolina, 1995. (ftp ://ftp. math. scarolina, edu/pub/imi_95/imi95_2, ps).
15. GONDEK, J. S., MEYER, G. W., AND NEWMAN, J. G. Wavelength Dependent Reflectance Functions. In Computer Graphics Proceedings, Annual Conference Series, 213-220, 1994.
16. GORTLER, S., SCHRODER, R, COHEN, M., AND HANRAHAN, R Wavelet Radiosity. In Computer Graphics Proceedings, Annual Conference Series, 221-230, August 1993.
17. GORTLER, S. J., AND COHEN, M. F. Hierarchical and Variational Geometric Modeling with Wavelets. In Proceedings Symposium on Interactive 3D Graphics, 35-42, April 1995.
18. LIU, Z., GORTLER, S. J., AND COHEN, M. F. Hierarchical Spacetime Control. Computer Graphics Proceedings, Annual Conference Series, 35-42, July 1994.
19. LOUNSBERY, M. Multiresolution Analysis for Smfaces of Arbitrary Topological Type. PhD thesis, University of Washington, 1994.
20. LOUNSBERY, M., DEROSE, T. D., AND WARREN, J. Multiresolution Surfaces of Arbitrary Topological Type. Department of Computer Science and Engineering 93-10-05, University of Washington, October 1993. Updated version available as 93-10-05b, January, 1994.
21. MITREA, M. Singular integrals, Ha~ffy spaces and Clifford wavelets. No. 1575 in Lecture Notes in Math. 1994.
22. NIELSON, G. M. Scattered Data Modeling. IEEE Computer Graphics and Applications 13, 1 (January 1993), 60-70.
23. SCHLICK, C. A customizable reflectance model for everyday rendering. In Fourth Eurographics Workshop on Rendering, 73-83, June 1993.
24. SCHRODER, P., AND HANRAHAN, P. Wavelet Methods for Radiance Computations. In Proceedings 5th Eurographics Workshop on Rendering, June 1994.
25. SCHRODER, P., AND SWELDENS, W. Spherical wavelets: Texture processing. Tech. Rep. 1995:4, Industrial Mathematics Initiative, Department of Mathematics, University of South Carolina, 1995. (ftp ://ftp. math. scarolina, edu/pub/imi_95/imi95_4, ps ).
26. SILLION, F. X., ARVO, J. R., WESTIN, S. H., AND GREENBERG, D. E A global illumination solution for general reflectance distributions. Computer Graphics (SIGGRAPH ’91 Proceedings), Vol. 25, No. 4, pp. 187-196, July 1991.
27. SWELDENS, W. The lifting scheme: A construction of second generation wavelets. Department of Mathematics, University of South Carolina.
28. SWELDENS, W. The lifting scheme: A customdesign construction of biorthogonal wavelets. Tech. Rep. 1994:7, Industrial Mathematics Initiative, Department of Mathematics, University of South Carolina, 1994. (ftp ://ftp. math. scarolina, edu/pub/imi_94/imi94_7, ps ).
29. WESTERMAN, R. A Multiresolution Framework for Volume Rendering. In Proceedings ACM Workshop on Volume Visualization, 51-58, October 1994.
30. WESTIN, S. H., ARVO, J. R., AND TORRANCE, K. E. Predicting reflectance functions from complex surfaces. Computer Graphics (SIGGRAPH ’92 Proceedings), Vol. 26, No. 2, pp. 255-264, July 1992.