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Haar-LikeWavelets over Tetrahedra
DOI:
https://doi.org/10.24215/16666038.17.e13Keywords:
subdivision methods, volumetric data, multiresolution analysis, tetrahedral meshesAbstract
In this paper we define a Haar-like wavelets basis that form a basis for L2(T,S,μ), μ being the Lebesgue measure and S the σ -algebra of all tetrahedra generated from a subdivision method of the T tetrahedron. As 3D objects are, in general, modeled by tetrahedral grids, this basis allows the multiresolution representation of scalar functions defined on polyhedral volumes, like colour, brightness, density and other properties of an 3D object.
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References
[1] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics SIAM, 1992.
[2] S. Mallat, “Multiresolution approximations and wavelet orthonormal basis of L 2 (R),” Trans. of the American Math. Soc., vol. 315, no. 1, pp. 69–88, 1989.
[3] C. Chui, An Introduction to Wavelets. Charles Chui, Series Editor, Academic Press, 1992.
[4] J. M. Lounsbery, Multiresolution Analysis for Surfaces of Arbitrary Topological Type. PhD thesis, University of Washington, Washington, Seattle, 1994.
[5] P. Schröeder and W. Sweldens, “Spherical wavelets: Efficiently representing functions on the sphere,” ACM Proceedings of SIGGRAPH’95, pp. 161–172, Aug. 1995.
[6] G. Nielson, P. Brunet, M. Gross, H. Hagen, and S. Klimenko, “Research issues in data modeling for scientific visualization,” IEEE Computer Graphics and Applications, pp. 70–73, Mar. 1994.
[7] J. Krommweh and G. Plonka, “Directional haar wavelet frames on triangles,” Applied Computational Harmonic Analysis, vol. 27, pp. 215–234, 2009.
[8] V. Pop and D. Rosca, “Generalized piecewise constant orthogonal wavelet bases on 2D domains,” Applied Analysis, vol. 90, no. 3-4, pp. 715–723, 2011.
[9] L. Boscardin, L. Castro, S. Castro, and A. De Giusti, “Wavelets defined over tetrahedra,” Journal of Computer Science and Technology, vol. 6, no. 1, pp. 46–52, 2006.
[10] M. Girardi and W. Sweldens, “A new class of unbalanced Haar wavelets that form an unconditional basis for L p on general masure spaces,” J. Fourier Anal. Appl., vol. 3, pp. 457–474, 1997.
[11] V. Ranjan and A. Fournier, “Volume models for volumetric data,” IEEE Computer Graphics and Applications, pp. 28–36, July 1994.
[12] S. Muraki, “Approximation and rendering of volume data using wavelet transforms,” Proceedings os Visualization ’92, pp. 21–28, 1992.
[13] A. Kaufman, R. Yagel, and D. Cohen, “Volume graphics,” IEEE Computer, pp. 51–64, July 1993.
[14] J. Bey, “Tetrahedral grid refinement,” Computing, vol. 55, no. 4, pp. 355–378, 1995.
[2] S. Mallat, “Multiresolution approximations and wavelet orthonormal basis of L 2 (R),” Trans. of the American Math. Soc., vol. 315, no. 1, pp. 69–88, 1989.
[3] C. Chui, An Introduction to Wavelets. Charles Chui, Series Editor, Academic Press, 1992.
[4] J. M. Lounsbery, Multiresolution Analysis for Surfaces of Arbitrary Topological Type. PhD thesis, University of Washington, Washington, Seattle, 1994.
[5] P. Schröeder and W. Sweldens, “Spherical wavelets: Efficiently representing functions on the sphere,” ACM Proceedings of SIGGRAPH’95, pp. 161–172, Aug. 1995.
[6] G. Nielson, P. Brunet, M. Gross, H. Hagen, and S. Klimenko, “Research issues in data modeling for scientific visualization,” IEEE Computer Graphics and Applications, pp. 70–73, Mar. 1994.
[7] J. Krommweh and G. Plonka, “Directional haar wavelet frames on triangles,” Applied Computational Harmonic Analysis, vol. 27, pp. 215–234, 2009.
[8] V. Pop and D. Rosca, “Generalized piecewise constant orthogonal wavelet bases on 2D domains,” Applied Analysis, vol. 90, no. 3-4, pp. 715–723, 2011.
[9] L. Boscardin, L. Castro, S. Castro, and A. De Giusti, “Wavelets defined over tetrahedra,” Journal of Computer Science and Technology, vol. 6, no. 1, pp. 46–52, 2006.
[10] M. Girardi and W. Sweldens, “A new class of unbalanced Haar wavelets that form an unconditional basis for L p on general masure spaces,” J. Fourier Anal. Appl., vol. 3, pp. 457–474, 1997.
[11] V. Ranjan and A. Fournier, “Volume models for volumetric data,” IEEE Computer Graphics and Applications, pp. 28–36, July 1994.
[12] S. Muraki, “Approximation and rendering of volume data using wavelet transforms,” Proceedings os Visualization ’92, pp. 21–28, 1992.
[13] A. Kaufman, R. Yagel, and D. Cohen, “Volume graphics,” IEEE Computer, pp. 51–64, July 1993.
[14] J. Bey, “Tetrahedral grid refinement,” Computing, vol. 55, no. 4, pp. 355–378, 1995.
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Published
2017-10-01
How to Cite
Boscardín, L. B., Castro, L. R., & Castro, S. M. (2017). Haar-LikeWavelets over Tetrahedra. Journal of Computer Science and Technology, 17(02), e13. https://doi.org/10.24215/16666038.17.e13
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