Construction of two-direction tight wavelet frames

Abstract

We investigate the construction of two-direction tight wavelet frames. First, a sufficient condition for a two-direction refinable function generating two-direction tight wavelet frames is derived. Second, a simple constructive method of two-direction tight wavelet frames is given. Third, based on the obtained two-direction tight wavelet frames, one can construct a symmetric multiwavelet frame easily. Finally, some examples are given to illustrate the results.

Keywords

Two-direction refinable function / two-direction tight wavelet frame / two-direction quadrature mirror filter (TQMF) condition / multiwavelet / symmetry

References

[1] Charina M, Chui C, He W. Tight frames of compactly supported multivariate multiwavelets. J Comput Appl Math, 2010, 233: 2044-2061

[2] Chui C, He W. Compactly supported tight frames associated with refinable functions. Appl Comput Harmon Anal, 2000, 8: 293-391

[3] Chui C, He W, Stöckler J, Sun Q. Compactly suppo<?Pub Caret?>rted tight affine frames with integer dilations and maximum vanishing moments. Adv Comput Math, 2003, 18: 159-187

[4] Cvetkovic Z, Vetterli M. Oversampled filter banks. IEEE Trans Signal Process, 1998, 46: 1245-1255

[5] Daubechies I, Han B, Ron A, Shen Z. Framelets: MRA-based constructions of wavelet frames. Appl Comput Harmon Anal, 2003, 14: 1-46

[6] Guo W, Peng L. The construct theory of multiwavelet frames. Sci Sin Math, 2010, 40: 1115-1128 (in Chinese)

[7] Han B, Mo Q. Multiwavelet frames from refinable function vectors. Adv Comput Math, 2003, 18: 211-245

[8] Krommweh J. Tight frame characterization of multiwavelet vector functions in terms of the polyphase matrix. Int J Wavelets Multiresolut Inf Process, 2009, 7: 9-21

[9] Lai M, Stöchler J. Construction of multivariate compactly supported tight wavelet frames. Appl Comput Harmon Anal, 2006, 21: 324-348

[10] Li Y, Yang S. Construction of nonseparable dual Ω-wavelet frames in L2(ℝr).Appl Math Comput, 2009, 215: 2082-2094

[11] Li Y, Yang S. Dual multiwavelet frames with symmetry from two-direction refinable functions. Bull Iranian Math Soc, 2011, 37: 199-214

[12] Petukhov A. Symmetric framelets. Constr Approx, 2003, 19: 309-328

[13] Ron A, Shen Z. Affine systems in L2(ℝd): The analysis of the analysis operator. J Funct Anal, 1997, 148: 408-447

[14] Skopina M. On construction of multivariate wavelet frames. Appl Comput Harmon Anal, 2009, 27: 55-72

[15] Yang S, Li Y. Two-direction refinable functions and two-direction wavelets with high approximation order and regularity. Sci China Ser A, 2007, 50: 1687-1704

[16] Yang S, Li Y. Two-direction refinable functions and two-direction wavelets with dilation factor m.Appl Math Comput, 2007, 188: 1908-1920

[17] Yang S, Xie C. A class of orthogonal two-direction refinable functions and two-direction wavelets. Int J Wavelets Multiresolut Inf Process, 2008, 6: 883-894

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg