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Wavelet representation of functions defined on tetrahedrical grids
Keywords:
Volume modeling, multiresolution, waveletsAbstract
In this paper, a method for representing scalar functions on volumes is presented. The method is based on wavelets and it can be used for representing volumetric data (geometric or scalar) defifined on non structured grids. The basic contribution is the extension of wavelets to represent scalar functions on volumetric domains of arbitrary topological type. This extension is made by constructing a wavelet basis defifined on any tetrahedrized volume. This basis construction is achieved using multiresolution analysis and the lifting scheme
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References
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[15] Nielson, G., Brunet, P., Gross, M., Hagen, H., and Klimenko, S. (1994). Research issues in data modeling for scientific visualization. IEEE Computer Graphics and Applications, pages 70–73.
[16] Nielson, G., Jung, I. H., and Sung, J. (1997a). Haar wavelets over triangular domains with applications to multiresolution models for flow over a sphere. In IEEE Visualization ’97, Phoenix, Arizona.
[17] Nielson, G. M., Hagen, H., and Heinrich Muller, ¨ E. (1997b). Scientific Visualization. Overviews. Methodologies. Techniques. IEEE Computer Society.
[18] Ranjan, V. and Fournier, A. (1994). Volume models for volumetric data. IEEE Computer Graphics and Applications, pages 28–36.
[19] Schroeder, P. and Sweldens, W. (1995). Spherical wavelets: Efficiently representing functions on the sphere. ACM Proceedings of SIGGRAPH’95, pages 161–172.
[20] Stollnitz, E., DeRose, T., and Salesin, D. (1996). Wavelets for Computer Graphics: Theory and Applications. Morgan Kaufmann Publishers, Inc.
[21] Sweldens, W. (1995). The lifting scheme: A new philosophy in biorthogonal wavelet constructions. Proceedings of the SPIE, 2569:68–79.
[22] Sweldens, W. (1996). The lifting scheme: A custom-design construction of biorthogonal wavelets. Applied and Computational Harmonic Analysis, 3:186–200.
[23] Westermann, R. (1994). A multiresolution framework for volume rendering. 1994 Symposium on Volume Visualization, pages 51–57.
[24] Zorin, D., Schroeder, P., and Sweldens, W. (1996). Interpolating subdivision for meshes with arbitrary topology. ACM Proceedings of SIGGRAPH’96, pages 189–192.
[2] Chui, C. (1992). An introduction to wavelets. In Charles Chui, S. E., editor, Wavelet Analysis and its Applications. Academic Press.
[3] Chui, C. and Quak, E. (1992). Wavelets on a bounded interval. Numerical Methods of Approximation Theory, 9:53–75.
[4] Cignoni, P., DeFloriani, L., Montani, C., Puppo, E., and Scopigno, R. (1994). Multiresolution modeling and visualization of volume data based on simplial complexes. 1994 Symposium on Volume Visualization, pages 19–26.
[5] Cohen, A., Daubechies, I., and Vial, P. (1993). Wavelets on the interval and fast wavelet transforms. Applied and Computational Harmonic Analysis, 1:54 – 81.
[6] Daubechies, I. (1992). Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics-SIAM, Philadelphia, Pennsylvania.
[7] DeFloriani, L. and Puppo, E. (1995). Hierarchical triangulation for multiresolution surface description. ACM Transactions on Graphics, 14:363–411.
[8] DeVore, R., Jawerth, B., and Lucier, B. (1992). Image compression through wavelet transform coding. IEEE Transactions on Information Theory, 38:719–746.
[9] Gross, M., Lippert, L., Dittrich, R., and H ring, S. (1997). Two methods for wavelet-based volume rendering. Technical Report CS Dept. Internal Report 247, Institute for Information Systems - CS Dept.-Swiss Federal Institute of Technology.
[10] Kaufman, A., Yagel, R., and Cohen, D. (1993). Volume graphics. IEEE Computer, pages 51–64.
[11] Lounsbery, J. M. (1994). Multiresolution Analysis for Surfaces of Arbitrary Topological Type. PhD thesis, University of Washington, Washington, Seattle.
[12] Mallat, S. (1989). A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Machine Intelligence, 11:674–693.
[13] Muraki, S. (1992). Approximation and rendering of volume data using wavelet transforms. Proceedings os Visualization ’92, pages 21–28.
[14] Muraki, S. (1995). Multiscale volume representation by a DoG wavelet. IEEE Transactions on Visualization and Computer Graphics, 1:109–116.
[15] Nielson, G., Brunet, P., Gross, M., Hagen, H., and Klimenko, S. (1994). Research issues in data modeling for scientific visualization. IEEE Computer Graphics and Applications, pages 70–73.
[16] Nielson, G., Jung, I. H., and Sung, J. (1997a). Haar wavelets over triangular domains with applications to multiresolution models for flow over a sphere. In IEEE Visualization ’97, Phoenix, Arizona.
[17] Nielson, G. M., Hagen, H., and Heinrich Muller, ¨ E. (1997b). Scientific Visualization. Overviews. Methodologies. Techniques. IEEE Computer Society.
[18] Ranjan, V. and Fournier, A. (1994). Volume models for volumetric data. IEEE Computer Graphics and Applications, pages 28–36.
[19] Schroeder, P. and Sweldens, W. (1995). Spherical wavelets: Efficiently representing functions on the sphere. ACM Proceedings of SIGGRAPH’95, pages 161–172.
[20] Stollnitz, E., DeRose, T., and Salesin, D. (1996). Wavelets for Computer Graphics: Theory and Applications. Morgan Kaufmann Publishers, Inc.
[21] Sweldens, W. (1995). The lifting scheme: A new philosophy in biorthogonal wavelet constructions. Proceedings of the SPIE, 2569:68–79.
[22] Sweldens, W. (1996). The lifting scheme: A custom-design construction of biorthogonal wavelets. Applied and Computational Harmonic Analysis, 3:186–200.
[23] Westermann, R. (1994). A multiresolution framework for volume rendering. 1994 Symposium on Volume Visualization, pages 51–57.
[24] Zorin, D., Schroeder, P., and Sweldens, W. (1996). Interpolating subdivision for meshes with arbitrary topology. ACM Proceedings of SIGGRAPH’96, pages 189–192.
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Published
2002-05-01
How to Cite
Castro, S. M., Castro, L. R., Boscardín, L. B., & De Giusti, A. E. (2002). Wavelet representation of functions defined on tetrahedrical grids. Journal of Computer Science and Technology, 1(06), 13 p. Retrieved from https://journal.info.unlp.edu.ar/JCST/article/view/968
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