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44
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 573 (8 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.
The Lifting Scheme: A Construction Of Second Generation Wavelets
, 1997
"... . We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to ..."
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Cited by 541 (16 self)
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. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, inplace calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...
Theory Of Regular MBand Wavelet Bases
 IEEE TRANS. ON SIGNAL PROCESSING
, 1993
"... This paper constructs Kregular Mband orthonormal wavelet bases. Kregularity of the wavelet basis is known to be useful in numerical analysis applications and in image coding using wavelet techniques. Several characterizations of Kregularity and their importance are described. An explicit formula ..."
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Cited by 87 (6 self)
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This paper constructs Kregular Mband orthonormal wavelet bases. Kregularity of the wavelet basis is known to be useful in numerical analysis applications and in image coding using wavelet techniques. Several characterizations of Kregularity and their importance are described. An explicit formula is obtained for all minimal length Mband scaling filters. A new statespace approach to constructing the wavelet filters from the scaling filters is also described. When Mband wavelets are constructed from unitary filter banks they give rise to wavelet tight frames in general (not orthonormal bases). Conditions on the scaling filter so that the wavelet bases obtained from it is orthonormal is also described.
Theory and Design of SignalAdapted FIR Paraunitary Filter Banks
 IEEE TRANS. SIGNAL PROCESSING
, 1998
"... We study the design of signaladapted FIR paraunitary filter banks, using energy compaction as the adaptation criterion. We present some important properties that globally optimal solutions to this optimization problem satisfy. In particular, we show that the optimal filters in the first channel ..."
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Cited by 41 (6 self)
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We study the design of signaladapted FIR paraunitary filter banks, using energy compaction as the adaptation criterion. We present some important properties that globally optimal solutions to this optimization problem satisfy. In particular, we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semiinfinite programming (SIP) problem. The remaining filters are related to the first through a matrix eigenvector decomposition. We discuss
Design of Hybrid Filter Banks for Analog/Digital Conversion
 IEEE Trans. on Signal Processing
, 1998
"... This paper presents design algorithms for hybrid filter banks (HFB's) for highspeed, highresolution conversion between analog and digital signals. The HFB is an unconventional class of filter bank that employs both analog and digital filters. When used in conjunction with an array of slower s ..."
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Cited by 36 (2 self)
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This paper presents design algorithms for hybrid filter banks (HFB's) for highspeed, highresolution conversion between analog and digital signals. The HFB is an unconventional class of filter bank that employs both analog and digital filters. When used in conjunction with an array of slower speed converters, the HFB improves the speed and resolution of the conversion compared with the standard timeinterleaved array conversion technique. The analog and digital filters in the HFB must be designed so that they adequately isolate the channels and do not introduce reconstruction errors that limit the resolution of the system. To design continuoustime analog filters for HFB's, a discretetimetocontinuoustime ("ZtoS") transform is developed to convert a perfect reconstruction (PR) discretetime filter bank into a nearPR HFB; a computationally efficient algorithm based on the fast Fourier transform (FFT) is developed to design the digital filters for HFB's. A twochannel HFB is designed with sixthorder continuoustime analog filters and length 64 FIR digital filters that yield 086 dB average aliasing error. To design discretetime analog filters (e.g., switchedcapacitors or chargecoupled devices) for HFB's, a lossless factorization of a PR discretetime filter bank is used so that reconstruction error is not affected by filter coefficient quantization. A gain normalization technique is developed to maximize the dynamic range in the finiteprecision implementation. A fourchannel HFB is designed with 9bit (integer) filter coefficients. With internal aliasing error is 070 dB, and with the equivalent of 20 bits internal precision, maximum aliasing is 0100 dB. The 9bit filter coefficients degrade the stopband attenuation (compared with unquantized coefficients)...
Wavelets through a looking glass: The world of the spectrum
 Applied and Numerical Harmonic Analysis, Birkhäuser
, 2002
"... ..."
Theory And Design Of Optimum FIR Compaction Filters
 IEEE TRANS. SIGNAL PROCESSING
, 1998
"... The problem of optimum FIR energy compaction filter design for a given number of channels M and a filter order N is considered. The special cases where N ! M and N = 1 have analytical solutions that involve eigenvector decomposition of the autocorrelation matrix and the power spectrum matrix respec ..."
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Cited by 29 (11 self)
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The problem of optimum FIR energy compaction filter design for a given number of channels M and a filter order N is considered. The special cases where N ! M and N = 1 have analytical solutions that involve eigenvector decomposition of the autocorrelation matrix and the power spectrum matrix respectively. In this paper, we deal with the more difficult case of M ! N ! 1. For the twochannel case and for a restricted but important class of random processes, we give an analytical solution for the compaction filter which is characterized by its zeros on the unitcircle. This also corresponds to the optimal twochannel FIR filter bank that maximizes the coding gain under the traditional quantization noise assumptions. This can also be used to generate optimal wavelets. For the arbitrary M \Gammachannel case, we provide a very efficient suboptimal design method called the window method. The method involves two stages that are associated with the above two special cases. As the order incre...
High Performance Compression of Visual Information  A Tutorial Review  Part I: Still Pictures
, 1999
"... Digital images have become an important source of information in the modern world of communication systems. In their raw form, digital images require a tremendous amount of memory. Many research efforts have been devoted to the problem of image compression in the last two decades. Two different comp ..."
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Cited by 26 (0 self)
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Digital images have become an important source of information in the modern world of communication systems. In their raw form, digital images require a tremendous amount of memory. Many research efforts have been devoted to the problem of image compression in the last two decades. Two different compression categories must be distinguished: lossless and lossy. Lossless compression is achieved if no distortion is introduced in the coded image. Applications requiring this type of compression include medical imaging and satellite photography. For applications such as videotelephony or multimedia applications some loss of information is usually tolerated in exchange for a high compression ratio.
The DoubleDensity DualTree DWT
, 2004
"... This paper introduces the doubledensity dualtree discrete wavelet transform (DWT), which is a DWT that combines the doubledensity DWT and the dualtree DWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on ..."
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Cited by 25 (1 self)
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This paper introduces the doubledensity dualtree discrete wavelet transform (DWT), which is a DWT that combines the doubledensity DWT and the dualtree DWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets. One pair of the four wavelets are designed to be offset from the other pair of wavelets so that the integer translates of one wavelet pair fall midway between the integer translates of the other pair. Simultaneously, one pair of wavelets are designed to be approximate Hilbert transforms of the other pair of wavelets so that two complex (approximately analytic) wavelets can be formed. Therefore, they can be used to implement complex and directional wavelet transforms. The paper develops a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed. This design procedure employs a fractionaldelay allpass filter, spectral factorization, and filterbank completion. The solutions have vanishing moments, compact support, a high degree of smoothness, and are nearly shiftinvariant.
The Role of Linear SemiInfinite Programming in SignalAdapted QMF Bank Design
, 1995
"... The design of an orthogonal FIR quadraturemirror filter (QMF) bank (H; G) adapted to input signal statistics is considered. The adaptation criterion is maximization of the coding gain and has so far been viewed as a difficult nonlinear constrained optimization problem. In this paper, it is shown t ..."
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Cited by 23 (6 self)
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The design of an orthogonal FIR quadraturemirror filter (QMF) bank (H; G) adapted to input signal statistics is considered. The adaptation criterion is maximization of the coding gain and has so far been viewed as a difficult nonlinear constrained optimization problem. In this paper, it is shown that in fact the coding gain depends only upon the product filter P (z) = H(z)H(z \Gamma1 ), and this transformation leads to a stable class of linear optimization problems having finitely many variables and infinitely many constraints, termed linear semi infinite programming (SIP) problems. The soughtfor, original filter, H(z), is obtained by deflation and spectral factorization of P (z). With the SIP formulation, every locally optimal solution is also globally optimal and can be computed using reliable numerical algorithms. The natural regularity properties inherent in the SIP formulation enhance the performance of these algorithms. We present a comprehensive theoretical analysis of ...