The construction of nonseparable and compactly supported orthonormal wavelet bases of L 2(R n); n ≥ 2, is still a challenging and an open research problem. In this paper, we provide a special method for the construction of such wavelet bases. The wavelets constructed by this method are dyadic wavelets. Also, we show that our proposed method can be adapted for an eventual construction of multidimensional orthogonal multiwavelet matrix masks, candidates for generating multidimensional multiwavelet bases.
@article{bwmeta1.element.doi-10_2478_s11533-008-0052-6,
author = {Abderrazek Karoui},
title = {A general construction of nonseparable multivariate orthonormal wavelet bases},
journal = {Open Mathematics},
volume = {6},
year = {2008},
pages = {504-525},
zbl = {1151.42009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0052-6}
}
Abderrazek Karoui. A general construction of nonseparable multivariate orthonormal wavelet bases. Open Mathematics, Tome 6 (2008) pp. 504-525. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0052-6/
[1] Cohen A., Daubechies I., Non-separable bidimensional wavelet bases, Rev. Mat. Iberoamericana, 1993, 9, 51–137 http://dx.doi.org/10.1146/annurev.ms.09.080179.000411 | Zbl 0792.42021
[2] Daubechies I., Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 61, SIAM, Philadelphia, PA, 1992
[3] Ji H., Riemenschneider S.D., Shen Z., Multivariate compactly supported fundamental refinable functions, duals, and biorthogonal wavelets, Stud. Appl. Math., 1999, 102, 173–204 http://dx.doi.org/10.1111/1467-9590.00108 | Zbl 1005.42019
[4] Karoui A., A technique for the construction of compactly supported biorthogonal wavelets of L 2(Rn); n ≥ 2, J. Math. Anal. Appl., 2000, 249, 367–392 http://dx.doi.org/10.1006/jmaa.2000.6867
[5] Karoui A., A note on the construction of nonseparable wavelet bases and multiwavelet matrix filters of L 2(R n); where n ≥ 2, Electron. Res. Announc. Amer. Math. Soc., 2003, 9, 32-39 | Zbl 1020.65110
[6] Kovačević J., Vetterli M., Nonseparable two-and three-dimensional wavelets, IEEE Trans. Signal Process., 1995, 43, 1269–1273 http://dx.doi.org/10.1109/78.382414
[7] Lai M.-J., Methods for Constructing Nonseparable Wavelets, In: Wavelet Analysis: twenty year’s development, D.X. Zhou (Ed.), World Scientific, 2002, 231–251
[8] Lawton W., Lee S.L., Shen Z., Stability and orthonormality of multivariate refinable functions, SIAM J. Math. Anal., 1997, 28, 999–1014 http://dx.doi.org/10.1137/S003614109528815X | Zbl 0872.41003
[9] Lin E.B., Ling Y., Image compression and denoising via nonseparable wavelet approximation, J. Comput. Appl. Math., 2003, 155, 131–152 http://dx.doi.org/10.1016/S0377-0427(02)00896-8 | Zbl 1022.65147
[10] Maass P., Families of orthogonal two-dimensional wavelets, SIAM J. Math. Anal., 1996, 27, 1454–1481 http://dx.doi.org/10.1137/S003614109324649X | Zbl 0881.42022
[11] Shen Z., Refinable function vectors, SIAM J. Math. Anal., 1998, 29, 235–250 http://dx.doi.org/10.1137/S0036141096302688 | Zbl 0913.42028
[12] Wang J.W., Chen C.H., Chien W.M., Tsai C.M., Texture classification using non-separable two-dimensional wavelets, Pattern Recognition Letters, 1998, 19, 1225–1234 http://dx.doi.org/10.1016/S0167-8655(98)00105-6 | Zbl 0913.68185