Algorithms for matrix extension and orthogonal wavelet filter banks over algebraic number fields

As a finite dimensional linear space over the rational number field Q, an algebraic number field is of particular importance and interest in mathematics and engineering. Algorithms using algebraic number fields can be efficiently implemented involving only integer arithmetics. We observe that all known finitely supported orthogonal wavelet low-pass filters in the literature have coefficients coming from an algebraic number field. Therefore, it is of theoretical and practical interest for us to consider orthogonal wavelet filter banks over algebraic number fields. In this paper, we formulate the matrix extension problem over any general subfield of C (including an algebraic number field as a special case), and we provide step-by-step algorithms to implement our main results. As an application, we obtain a satisfactory algorithm for constructing orthogonal wavelet filter banks over algebraic number fields. Several examples are provided to illustrate the algorithms proposed in this paper.

[1]  Zuowei Shen,et al.  An algorithm for matrix extension and wavelet construction , 1996, Math. Comput..

[2]  Alexander Petukhov,et al.  Construction of symmetric orthogonal bases of wavelets and tight wavelet frames with integer dilation factor , 2004 .

[3]  B. Han.  Pairs of frequency-based nonhomogeneous dual wavelet frames in the distribution space , 2009, 0907.3501.

[4]  Bin Han,et al.  Compactly supported orthonormal complex wavelets with dilation 4 and symmetry , 2009 .

[5]  Zuowei Shen,et al.  Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions , 1999, Adv. Comput. Math..

[6]  Bin Han,et al.  Matrix Extension with Symmetry and Its Application to Symmetric Orthonormal Multiwavelets , 2010, SIAM J. Math. Anal..

[7]  Bin Han,et al.  Symmetric orthonormal scaling functions and wavelets with dilation factor 4 , 1998, Adv. Comput. Math..

[8]  B. Han,et al.  Generalized interpolating refinable function vectors , 2009 .

[9]  Bin Han,et al.  Analysis and Construction of Multivariate Interpolating Refinable Function Vectors , 2009 .

[10]  B. Boston.  Matrix Extension with Symmetry and Applications to Symmetric Orthonormal Complex M-wavelets , 2009 .

[11]  Song Li,et al.  Symmetric tight wavelet frames with rational coefficients , 2011 .

[12]  P. Vaidyanathan.  Multirate Systems And Filter Banks , 1992 .

[13]  C. Chui,et al.  Construction of Compactly Supported Symmetric and Antisymmetric Orthonormal Wavelets with Scale = 3 , 1995 .

[14]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

[15]  Bin Han.  Symmetric orthogonal filters and wavelets with linear-phase moments , 2011, J. Comput. Appl. Math..

[16]  Bin Han,et al.  Examples of refinable componentwise polynomials , 2007 .

[17]  Dante C. Youla,et al.  Notes on n-Dimensional System Theory , 1979 .

[18]  Andreas Klappenecker.  Algebraic wavelet filters , 1996, Int. J. Imaging Syst. Technol..

[19]  D. Hardin,et al.  Fractal Functions and Wavelet Expansions Based on Several Scaling Functions , 1994 .