ZHANG Shiqiang, ZHANG Shufang, WANG Yan. Biorthogonal Wavelet Construction Using Homotopy Method[J]. Chinese Journal of Electronics, 2015, 24(4): 772-775. DOI: 10.1049/cje.2015.10.018
Citation: ZHANG Shiqiang, ZHANG Shufang, WANG Yan. Biorthogonal Wavelet Construction Using Homotopy Method[J]. Chinese Journal of Electronics, 2015, 24(4): 772-775. DOI: 10.1049/cje.2015.10.018

Biorthogonal Wavelet Construction Using Homotopy Method

Funds: This work is supported by the Key Project of National Natural Science Foundation of China (No.61231006)
More Information
  • Corresponding author:

    WANG Yan (corresponding author)was born in 1974, Shanxi Province.He received the Ph.D. degree in communicationand information systems fromDalian Maritime University in 2007.He is now a supervisor of postgraduate,associate professor of Dalian MaritimeUniversity. His research interests includesignal processing, image processingand wavelet analysis. (Email: dlmuwangyang@dlmu.edu.cn)

  • Received Date: February 02, 2015
  • Revised Date: March 09, 2015
  • Published Date: October 09, 2015
  • The biorthogonal wavelets families are used widely because they have compact support, complete symmetry and linear phase. According to Bézout's theorem, the biorthogonal wavelets available now are only some particular examples of total solutions. The quantity of solutions is decided jointly by the scaling function vanishing moment N and dual vanishing moment Ñ. The relationship of N, Ñ and solutions' quantity is discussed in detail. According to the constraint conditions which the compact biorthogonal wavelets satisfy, a number of biorthogonal wavelets are constructed in which the global convergent homotopy method is used for different N and Ñ. The filter coefficients and plots of scaling function, dual scaling function, wavelet function and dual wavelet function are given.
  • I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, USA, 1992.
    S. Mallat, "Characterization of signals from multiscale edges", IEEE Transactions on PAMI, Vol.14, No.7, pp.710-732, 1992.
    S. Mallat, "Singularity detection and processing with wavelets", IEEE Transactions on IT, Vol.38, No.2, pp.617-642, 1992.
    Wang Yan, "Wavelets in edges detection", Journal of Liaoning Normal University, Vol.26, No.3, pp.244-250, 2003.
    S. Mallat, A Wavelet Tour of Signal Processing-The Sparse Way (Third Edition), China Machine Press, Beijing, China, 2012.
    A. Cohen, I. Daubechies and J.C. Feauveau, "Biorthogonal bases of compactly supported wavelets", Comm. Pure Appl. Math., Vol.45, pp.485-560, 1992.
    J. Verschelde and Ronald Cools, "Symbolic homotopy construction", Applicable Algebra in Engineering, Communication and Computing, Vol.4, No.3, pp.169-183, 1993.
    J. Verschelde and Ronald Cools, "Symmetric homotopy construction", J. Comput. Appl. Math., Vol.50, No.1-3. pp.575- 592, 1994.
    J. Verschelde and Ann Haegemans, "Homotopies for solving polynomial systems within a bounded domain", Theoretical Comp. Sci. A., Vol.133, No.3, pp.165-185, 1994.
    J. Verschelde, "A General-purpose solver for polynomial systems by homotopy continuation", ACM Transactions on Mathematical Software, Vol.25, No.2, pp.251-276, 1999.
    J. Verschelde, "Polynomial homotopy continuation with PHCpack", ACM Communications in Computer Algebra., Vol.44, No.4, pp.217-220, 2010.
    Wang Yan, "Coifman wavelet construction using homotopy method", Chinese Journal of Electronics, Vol.15, No.3, pp.451- 454, 2006.
    Wang Yan, "Daubechies wavelet construction using homotopy method", Chinese Journal of Electronics, Vol.16, No.1, pp.93- 96, 2007.
    T.Y. Li, "Numerical solution of multivariate polynomial systems by homotopy continuation methods", Acta Numerica, Vol.6, No.1, pp.399-436, 1997.
  • Related Articles

    [1]LI Shanshan, ZHOU Jian, WANG Xuan. A New Online Dynamic Testing Method for Nonlinear Distortion of Broadcast Transmitter[J]. Chinese Journal of Electronics, 2020, 29(2): 385-390. DOI: 10.1049/cje.2020.02.003
    [2]XU Yuwei, LIU Feng, WU Chuankun. Preimage Distributions of Perfect Nonlinear Functions and Vectorial Plateaued Functions[J]. Chinese Journal of Electronics, 2019, 28(5): 933-937. DOI: 10.1049/cje.2019.06.013
    [3]WANG Zhongxiao, XU Hong, QI Wenfeng. On the Cycle Structure of Some Nonlinear Feedback Shift Registers[J]. Chinese Journal of Electronics, 2014, 23(4): 801-804.
    [4]YANG Jing, AN Ning, WANG Kunxia, WANG Aiguo, LI Lian. An Efficient Causal Structure Learning Algorithm Based on Recursive Simultaneous Equations Model[J]. Chinese Journal of Electronics, 2013, 22(3): 553-557.
    [5]RAO Bin, XIAO Shunping, WANG Xuesong, LI Yongzhen. Nonlinear Kalman Filtering with Numerical Integration[J]. Chinese Journal of Electronics, 2011, 20(3): 452-456.
    [6]LIU Jun, YU Jinshou. A New Method to Fault Diagnosis for a Class of Nonlinear Systems[J]. Chinese Journal of Electronics, 2011, 20(2): 217-222.
    [7]ZHANG Yongjie, SUN Qin. Preconditioned Bi-conjugate Gradient Method of Large-scale Sparse Complex Linear Equation Group[J]. Chinese Journal of Electronics, 2011, 20(1): 192-194.
    [8]HUANG Cheng, WU Jianhui, SHI Longxing. On the Use of Multi-Tone for the Estimation andMeasurement of Noise Power Ratio inThird-Order Nonlinear System[J]. Chinese Journal of Electronics, 2010, 19(4): 763-768.
    [9]Xuan Hengnong, Sun Mingming, He Tao. Symbolic Computation and Lie Symmetry Groups for Two Nonlinear Differential-Difference Equations[J]. Chinese Journal of Electronics, 2010, 19(3): 495-498.
    [10]Cao Shaozhong, Li Yang, Tu Xuyan. Arbitrary-Order Approximate Solution to Integral State Equation for Generalized Affine Nonlinear Systems[J]. Chinese Journal of Electronics, 2010, 19(3): 441-445.

Catalog

    Article Metrics

    Article views (404) PDF downloads (540) Cited by()
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return