Applied and Computational Mathematics

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An Explicit Algorithm for the Construction of 3-Band Wavelet Frames Based on FMRA

Received: 09 August 2018    Accepted:     Published: 13 August 2018
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Abstract

In 1974, an French engineer, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing and in 1986, the famous mathematician, Y. Meyer, created a real small wave base. From then on, Wavelet transform is a rapidly developing new subfield in mathematics and is used in more and more fields, such as signal analysis, image processing, quantum mechanics and theoretical physics etc. Multiresolution analysis is a systematic method for constructing orthonormal wavelet bases and most of the current dilation is M=2 With the development of wavelet transform, M>2 -band wavelet is known to have advantages over 2-band wavelet in some aspects such as in signal processing. However, there are relatively less results for the case of M>2. Based on this fact and inspired by other similar papers, this paper studies the 3-band wavelets and wavelet frames associated with a given refinable function based on frame multiresolution analysis. In this paper, firstly, a sufficient and necessary condition which the refinable function should satisfy for the existence of wavelet frames is showed. Further, an explicit algorithm to construct this frames is worked out and finally, several designed examples are constructed to illustrate this algorithm.

DOI 10.11648/j.acm.20180703.21
Published in Applied and Computational Mathematics (Volume 7, Issue 3, June 2018)
Page(s) 155-160
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Frame Multiresolution Analysis, Polyphase Decomposition, Unitary Matrix Extension

References
[1] S. Mallat, “Multiresolution approximations and wavelet orthonormal bases of L2 (R),” Transactions of the American mathematical society, vol. 315, pp. 69-87.
[2] Daubechies, “Ten lectures on wavelets,” CBMF conference series in applied mathematics 61, SIAM, Philadelphia, 1992.
[3] Ron and Z. Shen, “Affine systems in L2 (Rd): the analysis of the analysis operator,” Journal of functional analysis, 148 (1997), 408-447.
[4] J. J. Benedetto and S. Li, “The theorem of multiresolution analysis frames and application to filter banks,” Appl. Comput. Harmon. Anal., 5 (1998), 398-427.
[5] 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.
[6] P. Steffen, P. N. Heller, R. A. Gopinath and C. S. Burrus, “Theory of regular M-band wavelet bases,” IEEE Trans. Signal Processing, 41 (1993), 3497-3510.
[7] C. Chaux and L. Duval, “Image analysis using a dual-tree M-band wavelet transform,” IEEE Transactions on Image Processing, 15 (2006), 2397-2412.
[8] 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.
[9] 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.
[10] 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).
[11] 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.
[12] C. Chui and W. He, “Compactly supported tight frames associated with refinable functions,” Appl. Comp. Harmonic Anal., 8 (2000), 293–319.
[13] W. Lawton, S. L. Lee and Z. Shen, “An algorithm for matrix extension and wavelet construction,” Math. Comp., 65 (1996), 723-737.
[14] Y. D. Huang and Z. X. Cheng, “Explicit construction of wavelet tight frames with dilation factor α,” Acta Mathematica Scientia, 2007, 27A (1), 7-18.
[15] J. J. Sun, Y. Huang, S. Y. Sun and L. H. Cui, “Parameterizations of masks for 3-band tight wavelet frames by symmetric extension of polyphase matrix,” Applied Mathematics and Computation, 225 (2013), 461-474.
[16] W. Lawton, S. L. Lee and Z. Shen, “An algorithm for matrix extension and wavelet construction,” Math. Comp., 65 (1996), 723-737.
[17] Sun Qiyu, “An Algorithm for the construction of symmetric and anti-symmetric M band wavelets,” In Wavelet Applications in Signal and Image Processing VIII, Proceedings of SPIE, 4119 (2000).
[18] B. Han, “Symmetric orthonormal scaling functions and wavelets with dilation factor 4,” Adv. Comput. Math., 8 (1998), 221-247.
Author Information
  • College of Science, China Three Gorges University, Yichang, China; Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, China

  • College of Science, China Three Gorges University, Yichang, China; Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, China

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    Zhaofeng Li, Yuanyuan Zhang. (2018). An Explicit Algorithm for the Construction of 3-Band Wavelet Frames Based on FMRA. Applied and Computational Mathematics, 7(3), 155-160. https://doi.org/10.11648/j.acm.20180703.21

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    Zhaofeng Li; Yuanyuan Zhang. An Explicit Algorithm for the Construction of 3-Band Wavelet Frames Based on FMRA. Appl. Comput. Math. 2018, 7(3), 155-160. doi: 10.11648/j.acm.20180703.21

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    AMA Style

    Zhaofeng Li, Yuanyuan Zhang. An Explicit Algorithm for the Construction of 3-Band Wavelet Frames Based on FMRA. Appl Comput Math. 2018;7(3):155-160. doi: 10.11648/j.acm.20180703.21

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  • @article{10.11648/j.acm.20180703.21,
      author = {Zhaofeng Li and Yuanyuan Zhang},
      title = {An Explicit Algorithm for the Construction of 3-Band Wavelet Frames Based on FMRA},
      journal = {Applied and Computational Mathematics},
      volume = {7},
      number = {3},
      pages = {155-160},
      doi = {10.11648/j.acm.20180703.21},
      url = {https://doi.org/10.11648/j.acm.20180703.21},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.acm.20180703.21},
      abstract = {In 1974, an French engineer, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing and in 1986, the famous mathematician, Y. Meyer, created a real small wave base. From then on, Wavelet transform is a rapidly developing new subfield in mathematics and is used in more and more fields, such as signal analysis, image processing, quantum mechanics and theoretical physics etc. Multiresolution analysis is a systematic method for constructing orthonormal wavelet bases and most of the current dilation is M=2 With the development of wavelet transform, M>2 -band wavelet is known to have advantages over 2-band wavelet in some aspects such as in signal processing. However, there are relatively less results for the case of M>2. Based on this fact and inspired by other similar papers, this paper studies the 3-band wavelets and wavelet frames associated with a given refinable function based on frame multiresolution analysis. In this paper, firstly, a sufficient and necessary condition which the refinable function should satisfy for the existence of wavelet frames is showed. Further, an explicit algorithm to construct this frames is worked out and finally, several designed examples are constructed to illustrate this algorithm.},
     year = {2018}
    }
    

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    AU  - Zhaofeng Li
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    AB  - In 1974, an French engineer, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing and in 1986, the famous mathematician, Y. Meyer, created a real small wave base. From then on, Wavelet transform is a rapidly developing new subfield in mathematics and is used in more and more fields, such as signal analysis, image processing, quantum mechanics and theoretical physics etc. Multiresolution analysis is a systematic method for constructing orthonormal wavelet bases and most of the current dilation is M=2 With the development of wavelet transform, M>2 -band wavelet is known to have advantages over 2-band wavelet in some aspects such as in signal processing. However, there are relatively less results for the case of M>2. Based on this fact and inspired by other similar papers, this paper studies the 3-band wavelets and wavelet frames associated with a given refinable function based on frame multiresolution analysis. In this paper, firstly, a sufficient and necessary condition which the refinable function should satisfy for the existence of wavelet frames is showed. Further, an explicit algorithm to construct this frames is worked out and finally, several designed examples are constructed to illustrate this algorithm.
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