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41
Factoring wavelet transforms into lifting steps
 J. Fourier Anal. Appl
, 1998
"... ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This dec ..."
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Cited by 443 (7 self)
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ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is wellknown to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a selfcontained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, nonunitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a waveletlike transform that maps integers to integers. 1.
Frametheoretic analysis of oversampled filter banks
 IEEE Trans. Sign. Proc
"... Abstract—We provide a frametheoretic analysis of oversampled finite impulse response (FIR) and infinite impulse response (IIR) uniform filter banks (FB’s). Our analysis is based on a new relationship between the FB’s polyphase matrices and the frame operator corresponding to an FB. For a given over ..."
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Cited by 74 (5 self)
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Abstract—We provide a frametheoretic analysis of oversampled finite impulse response (FIR) and infinite impulse response (IIR) uniform filter banks (FB’s). Our analysis is based on a new relationship between the FB’s polyphase matrices and the frame operator corresponding to an FB. For a given oversampled analysis FB, we present a parameterization of all synthesis FB’s providing perfect reconstruction. We find necessary and sufficient conditions for an oversampled FB to provide a frame expansion. A new frametheoretic procedure for the design of paraunitary FB’s from given nonparaunitary FB’s is formulated. We show that the frame bounds of an FB can be obtained by an eigenanalysis of the polyphase matrices. The relevance of the frame bounds as a characterization of important numerical properties of an FB is assessed by means of a stochastic sensitivity analysis. We consider special cases in which the calculation of the frame bounds and synthesis filters is simplified. Finally, simulation results are presented. Index Terms — Filter banks, frames, oversampling, polyphase representation.
Wavelet Families Of Increasing Order In Arbitrary Dimensions
, 1997
"... . We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its ..."
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Cited by 45 (0 self)
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. We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, inplace calculation, and integerto integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. Submitted to IEEE Transactions on Image Processing Over the last decade several constructions of compactly supported wavelets have originated both from signal processing and mathematical analysis. In signal processing, critically sampled wavelet transforms are known as filter banks or subband transforms [32, 43, 54, 56]. In mathematical analysis, wavelets are defined as translates and dilates of one fixed function and ar...
Waveletbased image coding: An overview
 Applied and Computational Control, Signals, and Circuits
, 1998
"... ABSTRACT This paper presents an overview of waveletbased image coding. We develop the basics of image coding with a discussion of vector quantization. We motivate the use of transform coding in practical settings,and describe the properties of various decorrelating transforms. We motivate the use o ..."
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Cited by 37 (3 self)
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ABSTRACT This paper presents an overview of waveletbased image coding. We develop the basics of image coding with a discussion of vector quantization. We motivate the use of transform coding in practical settings,and describe the properties of various decorrelating transforms. We motivate the use of the wavelet transform in coding using ratedistortion considerations as well as approximationtheoretic considerations. Finally,we give an overview of current coders in the literature. 1
NearPerfectReconstruction PseudoQMF
 IEEE Trans. Signal Processing
, 1994
"... A novel approach to the design of Mchannel pseudoquadrature mirror filter (QMF) banks is presented. In this approach, the prototype filter is constrained to be a linearphase spectralfactor of a 2_1Ith band filter. As a result, the overall transfer function of the analysis/synthesis system is a d ..."
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Cited by 36 (2 self)
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A novel approach to the design of Mchannel pseudoquadrature mirror filter (QMF) banks is presented. In this approach, the prototype filter is constrained to be a linearphase spectralfactor of a 2_1Ith band filter. As a result, the overall transfer function of the analysis/synthesis system is a delay. Moreover, the aliasing cancellation {AC) constraint is derived such that all the significant aliasing terms are canceled. Consequently, the aliasing level at the output is comparable to the stopband attenuation of the prototype filter. In other words, the only error at the output of the analysis/synthesis system is the aliasing error which is at the level of stopband attenuation. Using this approazh, it is possible to design a pseudoQMF bank where the stopband attenuation of the analysis land thus synthesis) filters is on the order of100 dB. Moreover, the resulting reconstruction error is also on the order of100 riB. Several examples are included.
Perfect reconstruction filter banks with rational sampling factors
 IEEE Trans. Signal Process
, 1993
"... ..."
Image Processing with Multiscale Stochastic Models
, 1993
"... In this thesis, we develop image processing algorithms and applications for a particular class of multiscale stochastic models. First, we provide background on the model class, including a discussion of its relationship to wavelet transforms and the details of a twosweep algorithm for estimation. A ..."
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Cited by 29 (3 self)
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In this thesis, we develop image processing algorithms and applications for a particular class of multiscale stochastic models. First, we provide background on the model class, including a discussion of its relationship to wavelet transforms and the details of a twosweep algorithm for estimation. A multiscale model for the error process associated with this algorithm is derived. Next, we illustrate how the multiscale models can be used in the context of regularizing illposed inverse problems and demonstrate the substantial computational savings that such an approach offers. Several novel features of the approach are developed including a technique for choosing the optimal resolution at which to recover the object of interest. Next, we show that this class of models contains other widely used classes of statistical models including 1D Markov processes and 2D Markov random fields, and we propose a class of multiscale models for approximately representing Gaussian Markov random fields...
The Theory and Design of ArbitraryLength CosineModulated Filter Banks and Wavelets, Satisfying Perfect Reconstruction
 IEEE Trans. Signal Processing
, 1996
"... It is well known that FIR filter banks that satisfy the perfectreconstruction (PR) property can be obtained by cosine modulation of a linearphase prototype filter of length N = 2rnM, where M is the number of channels. In this paper, we present a PR cosinemodulated filter bank where the length of ..."
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Cited by 20 (4 self)
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It is well known that FIR filter banks that satisfy the perfectreconstruction (PR) property can be obtained by cosine modulation of a linearphase prototype filter of length N = 2rnM, where M is the number of channels. In this paper, we present a PR cosinemodulated filter bank where the length of the prototype filter is arbitrary. The design is formulated as a quadraticconstrained leastsquares optimization problem, where the optimized parameters are the prototype filler coefficients. Additional regularity conditions are imposed on the filter bank to obtain the cosinemodulated orthonormal bases of compactly supported wavelets. Design examples are given.
Orthogonal complex filter banks and wavelets: some properties and design
 IEEE TRANS. ON SIGNAL PROC
, 1999
"... Recent wavelet research has primarily focused on realvalued wavelet bases. However, complex wavelet bases offer a number of potential advantageous properties. For example, it has been recently suggested that the complex Daubechies wavelet can be made symmetric. However, these papers always imply th ..."
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Cited by 15 (0 self)
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Recent wavelet research has primarily focused on realvalued wavelet bases. However, complex wavelet bases offer a number of potential advantageous properties. For example, it has been recently suggested that the complex Daubechies wavelet can be made symmetric. However, these papers always imply that if the complex basis has a symmetry property, then it must exhibit linear phase as well. In this paper, we prove that a linearphase complex orthogonal wavelet does not exist. We study the implications of symmetry and linear phase for both complex and realvalued orthogonal wavelet bases. As a byproduct, we propose a method to obtain a complex orthogonal wavelet basis having the symmetry property and approximately linear phase. The numerical analysis of the phase response of various complex and real Daubechies wavelets is given. Both real and complexsymmetric orthogonal wavelet can only have symmetric amplitude spectra. It is often desired to have asymmetric amplitude spectra for processing general complex signals. Therefore, we propose a method to design general complex orthogonal perfect reconstruct filter banks (PRFB’s) by a parameterization scheme. Design examples are given. It is shown that the amplitude spectra of the general complex conjugate quadrature filters (CQF’s) can be asymmetric with respect the zero frequency. This method can be used to choose optimal complex orthogonal wavelet basis for processing complex signals such as in radar and sonar.