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Symmetric MRA tight wavelet frames with three generators and high vanishing moments
"... Abstract. Let φ be a compactly supported symmetric realvalued refinable function in L2(R) with a finitely supported symmetric realvalued mask on Z. Under the assumption that the shifts of φ are stable, in this paper we prove that one can always construct three wavelet functions ψ1, ψ2 and ψ3 such ..."
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Cited by 15 (8 self)
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Abstract. Let φ be a compactly supported symmetric realvalued refinable function in L2(R) with a finitely supported symmetric realvalued mask on Z. Under the assumption that the shifts of φ are stable, in this paper we prove that one can always construct three wavelet functions ψ1, ψ2 and ψ3 such that (i) All the wavelet functions ψ1, ψ2 and ψ3 are compactly supported, realvalued and finite linear combinations of the functions φ(2 · −k), k ∈ Z; (ii) Each of the wavelet functions ψ1, ψ2 and ψ3 is either symmetric or antisymmetric; (iii) {ψ1, ψ2, ψ3} generates a tight wavelet frame in L2(R), that is,
PAIRS OF FREQUENCYBASED NONHOMOGENEOUS DUAL WAVELET FRAMES IN THE DISTRIBUTION SPACE
"... Abstract. In this paper, we study stationary and nonstationary nonhomogeneous dual wavelet frames with an arbitrary real dilation factor in the frequency domain by introducing and investigating a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space. This notion of a p ..."
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Cited by 6 (5 self)
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Abstract. In this paper, we study stationary and nonstationary nonhomogeneous dual wavelet frames with an arbitrary real dilation factor in the frequency domain by introducing and investigating a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space. This notion of a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space enables us to completely separate its perfect reconstruction property from its stability property in function spaces. The results in this paper lead to a natural explanation for the oblique extension principle for constructing dual wavelet frames from refinable functions without any a priori condition on the generating wavelet functions and refinable functions. A pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space, that is not necessarily derived from refinable functions via a multiresolution analysis, has a natural multiresolutionlike structure, which is closely linked to the fast wavelet frame transform. Moreover, nonhomogeneous dual wavelet frames in the distribution space play a basic role in understanding dual wavelet frames in various function spaces and have a close relation to nonstationary dual wavelet frames, which are of interest in applications. To illustrate the flexibility and generality of the results in this paper, we characterize a pair of fully nonstationary dual wavelet frames in the distribution space. Our results naturally link a nonstationary dual wavelet frame filter bank having the perfect reconstruction property to a pair of nonstationary dual wavelet frames in the distribution space.
Tight and sibling frames originated from discrete splines, Signal Process. 86(7
, 2006
"... We present a new family of frames, which are generated by perfect reconstruction filter banks. The filter banks are based on the discrete interpolatory splines and are related to the Butterworth filters. Each filter bank comprises one interpolatory symmetric lowpass filter one bandpass and one hig ..."
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Cited by 4 (4 self)
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We present a new family of frames, which are generated by perfect reconstruction filter banks. The filter banks are based on the discrete interpolatory splines and are related to the Butterworth filters. Each filter bank comprises one interpolatory symmetric lowpass filter one bandpass and one highpass filters. In the sibling frames case all the filters are linear phase and generate symmetric scaling functions and analysis and synthesis pairs of framelets. In the tight frame case all the analysis waveforms coincide with their synthesis counterparts. In the sibling frame we can vary the framelets making them different for the synthesis and the analysis cases. This enables us to swap the vanishing moments between the synthesis and the analysis framelets or to add smoothness to the synthesis framelet. We constructed dual pairs of frames, where all the waveforms are symmetric and all the framelets have the same number of vanishing moments. Although most of the designed filters are IIR, they allow fast implementation via recursive procedures. The waveforms are well localized in time domain despite their infinite support. key words: frames, filter banks, sibling frames, IIR filters
MATRIX EXTENSION WITH SYMMETRY AND APPLICATIONS TO SYMMETRIC ORTHONORMAL COMPLEX MWAVELETS
"... Abstract. Matrix extension with symmetry is to find a unitary square matrix P of 2πperiodic trigonometric polynomials with symmetry such that the first row of P is a given row vector p of 2πperiodic trigonometric polynomials with symmetry satisfying pp T = 1. Matrix extension plays a fundamental ro ..."
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Cited by 4 (2 self)
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Abstract. Matrix extension with symmetry is to find a unitary square matrix P of 2πperiodic trigonometric polynomials with symmetry such that the first row of P is a given row vector p of 2πperiodic trigonometric polynomials with symmetry satisfying pp T = 1. Matrix extension plays a fundamental role in many areas such electronic engineering, system sciences, wavelet analysis, and applied mathematics. In this paper, we shall solve matrix extension with symmetry by developing a stepbystep simple algorithm to derive a desired square matrix P from a given row vector p of 2πperiodic trigonometric polynomials with complex coefficients and symmetry. As an application of our algorithm for matrix extension with symmetry, for any dilation factor M, we shall present two families of compactly supported symmetric orthonormal complex Mwavelets with arbitrarily high vanishing moments. Wavelets in the first family have the shortest possible supports with respect to their orders of vanishing moments; their existence relies on the establishment of nonnegativity on the real line of certain associated polynomials. Wavelets in the second family have increasing orders of linearphase moments and vanishing moments, which are desirable properties in numerical algorithms.
Compactly supported symmetric C ∞ wavelets with spectral approximation order, preprint
, 2006
"... Abstract. In this paper, we obtain symmetric C ∞ realvalued tight wavelet frames in L2(R) with compact support and the spectral frame approximation order. Furthermore, we present a family of symmetric compactly supported C ∞ orthonormal complex wavelets in L2(R). A complete analysis of nonstationar ..."
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Cited by 3 (3 self)
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Abstract. In this paper, we obtain symmetric C ∞ realvalued tight wavelet frames in L2(R) with compact support and the spectral frame approximation order. Furthermore, we present a family of symmetric compactly supported C ∞ orthonormal complex wavelets in L2(R). A complete analysis of nonstationary tight wavelet frames and orthonormal wavelet bases in L2(R) is given. 1.
Overcomplete discrete wavelet transforms with rational dilation factors
 IEEE Trans. Signal Processing
, 2009
"... Abstract—This paper develops an overcomplete discrete wavelet transform (DWT) based on rational dilation factors for discretetime signals. The proposed overcomplete rational DWT is implemented using selfinverting FIR filter banks, is approximately shiftinvariant, and can provide a dense sampling ..."
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Cited by 3 (1 self)
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Abstract—This paper develops an overcomplete discrete wavelet transform (DWT) based on rational dilation factors for discretetime signals. The proposed overcomplete rational DWT is implemented using selfinverting FIR filter banks, is approximately shiftinvariant, and can provide a dense sampling of the timefrequency plane. A straightforward algorithm is described for the construction of minimallength perfect reconstruction filters with a specified number of vanishing moments; whereas, in the nonredundant rational case, no such algorithm is available. The algorithm is based on matrix spectral factorization. The analysis/synthesis functions (discretetime wavelets) can be very smooth and can be designed to closely approximate the derivatives of the Gaussian function. Index Terms—Filter bank, frame, matrix spectral factorization, rational dilation factor, wavelet transforms. I.
MATRIX SPLITTING WITH SYMMETRY AND SYMMETRIC TIGHT FRAMELET FILTER BANKS WITH TWO HIGHPASS FILTERS
"... Abstract. The oblique extension principle introduced in [3, 5] is a general procedure to construct tight wavelet frames and their associated filter banks. Symmetric tight framelet filter banks with two highpass filters have been studied in [13, 16, 17]. Tight framelet filter banks with or without s ..."
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Cited by 2 (2 self)
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Abstract. The oblique extension principle introduced in [3, 5] is a general procedure to construct tight wavelet frames and their associated filter banks. Symmetric tight framelet filter banks with two highpass filters have been studied in [13, 16, 17]. Tight framelet filter banks with or without symmetry have been constructed in [1]–[21] and references therein. This paper is largely motivated by several results in [11, 13, 17] to further study tight wavelet frames and their associated filter banks with symmetry and two highpass filters. Our study not only leads to a simpler algorithm for the construction of tight framelet filter banks with symmetry and two highpass filters, but also allows us to obtain a wider class of tight wavelet frames with symmetry which are not available in the current literature. The key ingredient in our investigation is a complete characterization of splitting positive semidefinite 2 × 2 matrices of Laurent polynomials with symmetry. Several examples are provided to illustrate the results and algorithms in this paper. 1.
Design of orthonormal and overcomplete wavelet transforms based on rational sampling factors
 In Proc. Fifth SPIE Conference on Wavelet Applications in Industrial Processing
, 2007
"... Most wavelet transforms used in practice are based on integer sampling factors. Wavelet transforms based on rational sampling factors offer in principle the potential for timescale signal representations having a finer frequency resolution. Previous work on rational wavelet transforms and filter ba ..."
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Cited by 1 (1 self)
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Most wavelet transforms used in practice are based on integer sampling factors. Wavelet transforms based on rational sampling factors offer in principle the potential for timescale signal representations having a finer frequency resolution. Previous work on rational wavelet transforms and filter banks includes filter design methods and frequency domain implementations. We present several specific examples of Daubechiestype filters for a discrete orthonormal rational wavelet transform (FIR filters having a maximum number of vanishing moments) obtained using Gröbner bases. We also present the design of overcomplete rational wavelet transforms (tight frames) with FIR filters obtained using polynomial matrix spectral factorization.
Math. Model. Nat. Phenom. Properties of Discrete Framelet Transforms
"... Abstract. As one of the major directions in applied and computational harmonic analysis, the classical theory of wavelets and framelets has been extensively investigated in the function setting, in particular, in the function space L2(R d). A discrete wavelet transform is often regarded as a byprodu ..."
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Abstract. As one of the major directions in applied and computational harmonic analysis, the classical theory of wavelets and framelets has been extensively investigated in the function setting, in particular, in the function space L2(R d). A discrete wavelet transform is often regarded as a byproduct in wavelet analysis by decomposing and reconstructing functions in L2(R d) via nested subspaces of L2(R d) in a multiresolution analysis. However, since the input/output data and all filters in a discrete wavelet transform are of discrete nature, to understand better the performance of wavelets and framelets in applications, it is more natural and fundamental to directly study a discrete framelet/wavelet transform and its key properties. The main topic of this paper is to study various properties of a discrete framelet transform purely in the discrete/digital setting without involving the function space L2(R d). We shall develop a comprehensive theory of discrete framelets and wavelets using an algorithmic approach by directly studying a discrete framelet transform. The connections between our algorithmic approach and the classical theory of wavelets and framelets in the function setting will be addressed. Using tensor product of univariate complexvalued tight framelets, we shall also present an example of directional tight framelets in this paper. Key words: discrete framelet transform, perfect reconstruction, sparsity, stability, dual framelet filter banks, discrete affine systems, vanishing moments, sum rules. AMS subject classification: 42C40, 42C15