Applied and Computational Mathematics
Volume 8, Issue 3, June 2019, Pages: 65-69
Received: Jul. 8, 2019;
Published: Aug. 27, 2019
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Zhaofeng Li, College of Science, China Three Gorges University, Yichang, China; Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, China
Hongying Xiao, College of Science, China Three Gorges University, Yichang, China; Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, China
Wavelet analysis is a newly rapidly developing subject in the late twentieth century. As a time-frequency analysis tool, wavelet analysis has many advantages over other time-frequency tools, such as in signal processing, image processing, speech processing, pattern recognition, quantum physics and other fields. Multiresolution analysis (MRA for short) is an important method for studying wavelet orthonormal wavelet bases with rational dilation 2. However, p/q-band wavelet is known to have advantages over 2-band wavelet in some aspects such as in signal processing and attracted more and more interest in recent years. But there are relatively less results for the case of p/q-band. This paper studies the orthonormal wavelet bases with rational dilation factor p/q based on multiresolution analysis by a polyphase decomposition technique. First, we gave the concept of Multiresolution analysis with rational dilation p/q and deduced an identity of the masks matrix. Also, a perfect reconstruction condition in terms of masks was presented. Further, we gave the refinement and wavelet matrices respectively and derived the characteristic roots and the corresponding orthonormal characteristic vectors of the wavelet matrix, and then a method with characteristic vectors was reduced to achieve the orthonormal wavelet bases with rational dilation factor p/q. In the end, an example is offered to verify this theory.
Characteristic Vectors in p/q-Channel Orthonormal Wavelet, Applied and Computational Mathematics.
Vol. 8, No. 3,
2019, pp. 65-69.
S. Mallat, “Multiresolution approximations and wavelet orthonormal bases of L2 (R),” Transactions of the American mathematical society, vol. 315, pp. 69-87.
Meyer, Y., “Principe d'incertitude bases hibertiennes et algebres d'oerateurs,” Sem. Bourbaki., 662 (1986).
I. Daubechies, “Ten lectures on wavelets,” CBMF conference series in applied mathematics 61, SIAM, Philadelphia, 1992.
P. Auscher, “Wavelet bases for L2 (R) with rational dilation factor Wavelets and Their Applications," Jones and Barlett, Boston, 439-451, 1992.
Ilker Bayram, Ivan W. Selesnick, “Design of orthonormal and overcomplete wavelet transforms based on rational sampling factors,” Proc. SPIE 6763, 67630H (2007).
Marcin Bownik, Darrin Speegle, “The wavelet dimension function for real dilations and dilations admitting non-MSF wavelets,” Approximation Theory X: Wavelets, Splines, and Applications, 63-85, Vanderbilt University Press, 2002.
Sun Qiyu, Bi Ning and Huang Daren, “An introduction to multiband wavelets,” Zhejiang university press, 2001.
M. K. Mihcak, I. Kozintsev, K. Ramchandran and P. Moulin, “Low-complexity image denoising based on statistical modeling of wavelet coefficients,” IEEE Signal Processing Letters, 6 (1999), 300-303.
A. K. Soman, P. P. Vaidyanathan and T. Q. Nguyen, “Linear phase paraunitary filter banks: theory, factorizations, and applications,” IEEE Trans. Signal Processing, 41 (1993), 3480-3496.
L. Gan and K. K. Ma, “A simplified lattice factorization for linear-phase perfect reconstruction filter bank,” IEEE Signal Processing Letters, 8 (2001), 207-209.
Chao, Zhang, et al. "Optimal scale of crop classification using unmanned aerial vehicle remote sensing imagery based on wavelet packet transform." Transactions of the Chinese Society of Agricultural Engineering (2016).
Shleymovich M. P., M. V. Medvedev, and S. A. Lyasheva. "Object detection in the images in industrial process control systems based on salient points of wavelet transform analysis." International Conference on Industrial Engineering, Applications and Manufacturing IEEE, (2017): 1-6.
A. Ron and Z. Shen, “Affine systems in L2 (Rd): the analysis of the analysis operator,” Journal of functional analysis, 148 (1997), 408-447.
Y. D. Huang and Z. X. Cheng, “Explicit construction of wavelet tight frames with dilation factor a,” Acta Mathematica Scientia, 2007, 27A (1), 7-18.