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Properties of Discrete Framelet Transforms

Published online by Cambridge University Press:  28 January 2013

Bin Han
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Abstract

As one of the major directions in applied and computational harmonic analysis, theclassical theory of wavelets and framelets has been extensively investigated in thefunction setting, in particular, in the function spaceL2(ℝd). A discrete wavelettransform is often regarded as a byproduct in wavelet analysis by decomposing andreconstructing functions in L2(ℝd)via nested subspaces of L2(ℝd) ina multiresolution analysis. However, since the input/output data and all filters in adiscrete wavelet transform are of discrete nature, to understand better the performance ofwavelets and framelets in applications, it is more natural and fundamental to directlystudy a discrete framelet/wavelet transform and its key properties. The main topic of thispaper is to study various properties of a discrete framelet transform purely in thediscrete/digital setting without involving the function spaceL2(ℝd). We shall develop acomprehensive theory of discrete framelets and wavelets using an algorithmic approach bydirectly studying a discrete framelet transform. The connections between our algorithmicapproach and the classical theory of wavelets and framelets in the function setting willbe addressed. Using tensor product of univariate complex-valued tight framelets, we shallalso present an example of directional tight framelets in this paper.

Keywords

discrete framelet transformperfect reconstructionsparsitystabilitydual framelet filter banksdiscrete affine systemslinear-phase momentsvanishing momentssum rules
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 18 - 47
Copyright
© EDP Sciences, 2013

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References

Références

Candès, E. J., Donoho, D. L.. New tight frames of curvelets and optimal representations of objects with C2 singularities, Comm. Pure Appl. Math. 56 (2004), 219–266. CrossRefGoogle Scholar
C. K. Chui. An introduction to wavelets. Academic Press, Inc., Boston, MA, 1992.
Chui, C. K., He, W.. Compactly supported tight frames associated with refinable functions. Appl. Comput. Harmon. Anal. 8 (2000), 293–319. CrossRefGoogle Scholar
Chui, C. K., He, W., Stöckler, J.. Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comput. Harmon. Anal. 13 (2002), 224–262. CrossRefGoogle Scholar
Cohen, A., Daubechies, I.. A stability criterion for biorthogonal wavelet bases and their related subband coding scheme. Duke Math. J. 68 (1992), 313–335. CrossRefGoogle Scholar
Cohen, A., Daubechies, I., Feauveau, J.-C.. Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45 (1992), 485–560. CrossRefGoogle Scholar
Daubechies, I.. Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909–996. CrossRefGoogle Scholar
I. Daubechies. Ten lectures on wavelets. SIAM, CBMS Series, 1992.
Daubechies, I., Han, B.. Pairs of dual wavelet frames from any two refinable functions, Constr. Approx. 20 (2004), 325–352. CrossRefGoogle Scholar
Daubechies, I., Han, B., Ron, A., Shen, Z.. Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14 (2003), 1–46. CrossRefGoogle Scholar
Ehler, M.. On multivariate compactly supported bi-frames, J. Fourier Anal. Appl. 13 (2007), 511–532. CrossRefGoogle Scholar
Ehler, M., Han, B.. Wavelet bi-frames with few generators from multivariate refinable functions, Appl. Computat. Harmon. Anal. 25 (2008), 407–414. CrossRefGoogle Scholar
Guo, K., Labate, D.. Optimally sparse multidimensional representation using shearlets. SIAM J. Math. Anal. 39 (2007), 298–318. CrossRefGoogle Scholar
Guo, K., Labate, D., Lim, W.-Q, Weiss, G., Wilson, E.. Wavelets with composite dilations and their MRA properties. Appl. Comput. Harmon. Anal. 20 (2006), 202–236. CrossRefGoogle Scholar
Han, B.. On dual wavelet tight frames. Appl. Comput. Harmon. Anal. 4 (1997), 380–413. CrossRefGoogle Scholar
Han, B.. Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, J. Approx. Theory. 110 (2001), 18–53. CrossRefGoogle Scholar
Han, B.. Symmetry property and construction of wavelets with a general dilation matrix, Linear Algebra and its Applications. 353 (2002), 207–225 CrossRefGoogle Scholar
Han, B.. Computing the smoothness exponent of a symmetric multivariate refinable function, SIAM J. Matrix Anal. Appl. 24 (2003), 693–714. CrossRefGoogle Scholar
Han, B.. Vector cascade algorithms and refinable function vectors in Sobolev spaces, J. Approx. Theory. 124 (2003), 44–88. CrossRefGoogle Scholar
Han, B.. Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix, J. Comput. Appl. Math. 155 (2003), 43–67. CrossRefGoogle Scholar
B. Han. Classification and construction of bivariate subdivision schemes, Proceedings on Curves and Surfaces Fitting : Saint-Malo 2002, A. Cohen, J.-L. Merrien, and L. L. Schumaker eds., (2003), 187–197.
Han, B.. Symmetric multivariate orthogonal refinable functions, Appl. Comput. Harmon. Anal.. 17 (2004), 277–292. CrossRefGoogle Scholar
Han, B.. Matrix extension with symmetry and applications to symmetric orthonormal complex M-wavelets. J. Fourier Anal. Appl. 15 (2009), 684–705. CrossRefGoogle Scholar
Han, B.. Dual multiwavelet frames with high balancing order and compact fast frame transform. Appl. Comput. Harmon. Anal. 26 (2009), 14–42. CrossRefGoogle Scholar
Han, B.. The structure of balanced multivariate biorthogonal multiwavelets and dual multiframelets. Math. Comp. 79 (2010), 917–951. CrossRefGoogle Scholar
Han, B.. Symmetric orthonormal complex wavelets with masks of arbitrarily high linear-phase moments and sum rules. Adv. Comput. Math. 32 (2010), 209–237. CrossRefGoogle Scholar
Han, B.. Pairs of frequency-based nonhomogeneous dual wavelet frames in the distribution space, Appl. Comput. Harmon. Anal. 29 (2010), 330–353. CrossRefGoogle Scholar
Han, B.. Nonhomogeneous wavelet systems in high dimensions, Appl. Comput. Harmon. Anal. 32 (2012), 169–196. CrossRefGoogle Scholar
Han, B., Mo, Q.. Splitting a matrix of Laurent polynomials with symmetry and its application to symmetric framelet filter banks, SIAM J. Matrix Anal. Appl. 26 (2004), 97–124. CrossRefGoogle Scholar
Han, B., Mo, Q.. Symmetric MRA tight wavelet frames with three generators and high vanishing moments, Appl. Comput. Harmon. Anal. 18 (2005), 67–93. CrossRefGoogle Scholar
Han, B., Shen, Z.. Dual wavelet frames and Riesz bases in Sobolev spaces, Constr. Approx. 29 (2009), 369–406. CrossRefGoogle Scholar
Han, B., Zhuang, X. S.. Analysis and construction of multivariate interpoalting refinable function vectors, Acta Appl. Math. 107 (2009), 143–171. CrossRefGoogle Scholar
Jia, R. Q.. Approximation properties of multivariate wavelets, Math. Comp. 67 (1998), 647–665. CrossRefGoogle Scholar
Lai, M. J., Stöckler, J.. Construction of multivariate compactly supported tight wavelet frames, Appl. Comput. Harmon. Anal. 21 (2006), 324–348. CrossRefGoogle Scholar
S. Mallat. A wavelet tour of signal processing. Third edition. Elsevier/Academic Press, Amsterdam, 2009.
Y. Meyer. Wavelets and operators. Cambridge University Press, Cambridge, 1992.
Ron, A., Shen, Z.. Affine systems in L2(ℝd): the analysis of the analysis operator, J. Funct. Anal. 148 (1997), 408–447. CrossRefGoogle Scholar
Selesnick, I. W., Baraniuk, R. G., Kingsbury, N. G.. The dual-tree complex wavelet transform. IEEE Signal Proc. Magazine, 123 (2005), 123–151. CrossRefGoogle Scholar
G. Strang, T. Nguyen. Wavelets and filter banks. Wellesley College, 2nd edition, 1996.
M. Vetterli, J. Kovacĕvić. Wavelets and subband coding. Prentice Hall Signal Processing Series, 1995.