The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA
Applied and Computational Mathematics
Volume 7, Issue 3, June 2018, Pages: 161-166
Received: Aug. 9, 2018;
Published: Aug. 13, 2018
Views 801 Downloads 60
Yuanyuan Zhang, College of Sciences, China Three Gorges University, Yichang, China; Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, China
Zhaofeng Li, College of Sciences, China Three Gorges University, Yichang, China; Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, China
In 1974, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing. In 1986, Y. Meryer created a real small wave base, and the wavelet analysis began to flourish after a multi scale analysis of the same method of constructing the small wave base with S. Mallat. In order to analyze and deal with non-stationary signals, a series of new signal analysis theories are proposed.: Short Time Fourier Transform, time-frequency analysis, wavelet transform, and fractional Fourier transform and so on. In this paper, an explicit algorithm is given to construct the minimum-energy frames based on frame multiresolution analysis via characteristic vectors of the mask matrix. In section 2, we show the structure of minimum-energy wavelet frames in terms of their masks (Lemma 1) and discuss that we should eliminate the correlation of the rows of the mask matrix by the polyphase decomposition technique. Based on FMRA, an explicit algorithm is given to construct this frames. By this method, all the minimum-energy wavelet frames can be obtained. As an application, several examples are showed to explain this method in section 3. This method can also be applied in other fields of wavelet analysis.
The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA, Applied and Computational Mathematics.
Vol. 7, No. 3,
2018, pp. 161-166.
S. Mallat, “Multiresolution approximations and orthonormal bases of wavelets for L2 (R),” Trans. Amer. Math. Soc., vol. 315, 1989, pp. 69-87.
P. Steffen, P. N. Heller, “R. A. Gopinath and C. S. Burrus, Theory of regular M-band wavelet bases,” IEEE Trans. Signal Processing, vol. 41 (1993), 3497-3510.
C. Chaux and L. Duval, “Image analysis using a dual-tree M-band wavelet transform,” IEEE Transactions on Image Processing, vol. 15, 2006, pp. 2397-2412.
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, vol. 6, 1999, pp. 300- 303.
L. Gan and K. K. Ma, A simplified lattice factorization for linear-phase perfect reconstruction filter bank, IEEE Signal Processing Letters, vol. 8, 2001, pp. 207-209.
B. Han, Symmetric orthonormal scaling functions and wavelets with dilation factor 4, Adv. Comput. Math. Vol. 8, 1998, pp. 221-247.
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.
I. Daubechies, “Ten lectures on wavelets,” CBMF conference series in applied mathematics 61, SIAM, Philadelphia, 1992.
J. J. Benedetto and S. Li, “The theorem of multiresolution analysis frames and application to filter banks,” Appl. Comput. Harmon. Anal., vol. 5, 1998, pp. 398-427.
C. K. Chui and W. He, “Compactly supported tight frames associated with refinable functions,” Appl. Comp. Harmonic Anal., vol. 8, 2000, pp. 293-319.
W. Lawton, S. L. Lee and Z. Shen, “An algorithm for matrix extension and wavelet construction,” Math. Comp., vol. 65, 1996, pp. 723-737.
Cui Lihong, Cheng Zhengxing and Yang Shouzhi, “Explicit Structuer of Wavelet Tight Frames,” Acta Mathematica Scientia, vol. 24 (1), 2004, pp. 94-104.
Sun Qiyu, Bi Ning and Huang Daren, An introduction to multiband wavelets, Zhejiang university press, 2001.