Wavelets Defined Over Non Nested Tetrahedral Grids: A Theoretical Approach
The main contribution of this paper is the definition of wavelets over non nested tetrahedral grids, allowing the representation of functions defined on an a irregular tetrahedralization. In this way, it is possible to represent different attributes of a 3D object such as its color, brightness, density, etc. This representation consists of a set of coefficients corresponding to a coarse resolution followed by a set of detail coefficients that measures the error between two successive approximations. In this work the analysis matrix that allows going from a fine to a coarser resolution and the synthesis matrix needed for going from a coarse resolution to a finer one, are presented. All this is within the framework of non nested tetrahedral grids.
S. Mallat, “Multiresolution approximations and wavelet orthonormal basis of L2 (R),” Trans. of the American Math. Soc., vol. 315, no. 1, pp. 69–88, 1989.
I. Daubechies, Ten Lectures on Wavelets. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics SIAM, 1992.
C. Chui, An Introduction to Wavelets. Charles Chui, Series Editor, Academic Press, 1992.
W. Sweldens, “The lifting scheme: a construction of second generation wavelets,” Tech. Rep. 1994:7, Industrial Mathematics Initiative, Dept. of Mathematics, University of South Caroline, 1994.
D. Donoho, “Interpolating wavelet transforms,” Preprint, Departament of Statistics, Stanford University, 1992.
D. Donoho, “On minimum entropy segmentation,” Preprint, Departament of Statistics, Stanford University, 1993.
D. Donoho, Smooth wavelets descompositions with blocky coefficient kernels. Recent advances in wavelet analysis, New York: Academic Press, 1993.
J. M. Lounsbery, Multiresolution Analysis for Surfaces of Arbitrary Topological Type. PhD thesis, University of Washington, Washington, Seattle, 1994.
M. Girardi and W. Sweldens, “A new class of unbalanced Haar wavelets that form an unconditional basis for Lp on general masure spaces,” J. Fourier Anal. Appl., vol. 3, pp. 457–474, 1997.
P. Schr¨oeder and W. Sweldens, “Spherical wavelets: Efficiently representing functions on the sphere,” ACM Proceedings of SIGGRAPH’95, pp. 161–172, Aug. 1995.
A. Gerussi, Analyse Multir´esolution Non Emboˆıt´ee. Applications `a la visualisation Scientifique. PhD thesis, Universit´e Joseph Fourier, Grenoble, France, 2000.
G.Bonneau, S. Hahmann, and G. Nielson, “BLac wavelets: a multiresolution analysis with non nested spaces,” IEEE Visualization’96, pp. 43–48, 1996.
J. M. Maubach, “Local bisection refinement for n-simplicial grids generated by reflection,” SIAM J. Sci. Comput., vol. 16, pp. 210–227, 1995.
L. B. Boscardin, “Wavelets definidas sobre vol´umenes.” Tesis de Magister, Universidad Nacional del Sur, Bah´ıa Blanca, Argentina, 2001.
L. Boscardin, S. Castro, and L. Castro, “Haar like wavelets over tetrahedra,” Journal of Computer Science & Technology, vol. 17, no. 2, pp. 92–99, 2017.
J. Bey, “Tetrahedral grid refinement,” Computing, vol. 55, no. 4, pp. 355–378, 1995.
L. B. S. Castro, L. Castro and A. D. Giusti, “Multiresolution wavelet based model for large irregular volume data sets,” in Proceedings of the 14-th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision 2006, WSCG’2006, pp. 101–108, 2006.
E. Danovaro, L. De Floriani, M. Lee, and H. Samet, “Multiresolution tetrahedral meshes: an analysis and a comparison,” in IEEE SMI’02, pp. 83–94, 2002.
E. Danovaro and L.De Floriani, “Half-edge multitessellation: a compact representation for multiresolution tetrahedral meshes,” in Proceedings of the First International Symposium on 3D Data Processing Visualization and Transmission, (IEEE, 3DPVT’02), 2002.
P. Cignoni, D. Costanza, C. Montani, C. Rocchini, and R. Scopigno, “Simplification of tetrahedral volume with accurate error evaluation,” in Proceedings IEEE Visualization 2000, pp. 85–92, 2000.
Copyright (c) 2019 Liliana B. Boscardin, Silvia M. Castro, Liliana R. Castro
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.