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189
Complex wavelets for shift invariant analysis and filtering of signals
 J. Applied and Computational Harmonic Analysis
, 2001
"... This paper describes a form of discrete wavelet transform, which generates complex coefficients by using a dual tree of wavelet filters to obtain their real and imaginary parts. This introduces limited redundancy (2m: 1 for mdimensional signals) and allows the transform to provide approximate shift ..."
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Cited by 384 (40 self)
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This paper describes a form of discrete wavelet transform, which generates complex coefficients by using a dual tree of wavelet filters to obtain their real and imaginary parts. This introduces limited redundancy (2m: 1 for mdimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good wellbalanced frequency responses. Here we analyze why the new transform can be designed to be shift invariant and describe how to estimate the accuracy of this approximation and design suitable filters to achieve this. We discuss two different variants of the new transform, based on odd/even and quartersample shift (Qshift) filters, respectively. We then describe briefly how the dual tree may be extended for images and other multidimensional signals, and finally summarize a range of applications of the transform that take advantage of its unique properties. 2001 Academic Press 1.
The DualTree Complex Wavelet Transform  A coherent framework for multiscale signal and image processing
, 2005
"... The dualtree complex wavelet transform (CWT) is a relatively recent enhancement to the discrete wavelet transform (DWT), with important additional properties: It is nearly shift invariant and directionally selective in two and higher dimensions. It achieves this with a redundancy factor of only 2 ..."
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Cited by 270 (29 self)
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The dualtree complex wavelet transform (CWT) is a relatively recent enhancement to the discrete wavelet transform (DWT), with important additional properties: It is nearly shift invariant and directionally selective in two and higher dimensions. It achieves this with a redundancy factor of only 2 d for ddimensional signals, which is substantially lower than the undecimated DWT. The multidimensional (MD) dualtree CWT is nonseparable but is based on a computationally efficient, separable filter bank (FB). This tutorial discusses the theory behind the dualtree transform, shows how complex wavelets with good properties can be designed, and illustrates a range of applications in signal and image processing. We use the complex number symbol C in CWT to
Image Processing with Complex Wavelets
 Phil. Trans. Royal Society London A
, 1997
"... this paper we consider how wavelets may be used for image processing. To date, there has been considerable interest in wavelets for image compression, and they are now commonly used by researchers for this purpose, even though the main international standards still use the discrete cosine transform ..."
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Cited by 162 (5 self)
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this paper we consider how wavelets may be used for image processing. To date, there has been considerable interest in wavelets for image compression, and they are now commonly used by researchers for this purpose, even though the main international standards still use the discrete cosine transform (dct). However for image processing tasks, other than compression, the takeup of wavelets has been less enthusiastic. Here we analyse possible reasons for this and present some new ways to use wavelets which offer significant advantages. A good review of wavelets and their application to compression may be found in Rioul & Vetterli (1991) and indepth coverage is given in the book by Vetterli & Kovacevic (1995). An issue of the Proceedings of the IEEE (Kovacevic & Daubechies 1996) has been devoted to wavelets and includes many very readable articles by leading experts. In x 2 of this paper we introduce the basic discrete wavelet filter tree and show how it may be used to decompose multidimensional signals. In x 3 we show some typical wavelets and illustrate the similar shapes of those which all satisfy the perfect reconstruction constraints. Unfortunately, as explained in x 4, discrete
Hilbert Transform Pairs of Wavelet Bases
, 2001
"... This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an a ..."
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Cited by 82 (10 self)
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This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an alternative derivation and explanation for the result by Kingsbury, that the dualtree DWT is (nearly) shiftinvariant when the scaling filters satisfy the same offset.
Life Beyond Bases: The Advent of Frames (Part I)
, 2007
"... Redundancy is a common tool in our daily lives. Before we leave the house, we double and triplecheck that we turned off gas and lights, took our keys, and have money (at least those worrywarts among us do). When an important date is coming up, we drive our loved ones crazy by confirming “just onc ..."
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Cited by 72 (8 self)
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Redundancy is a common tool in our daily lives. Before we leave the house, we double and triplecheck that we turned off gas and lights, took our keys, and have money (at least those worrywarts among us do). When an important date is coming up, we drive our loved ones crazy by confirming “just once more” they are on top of it. Of course, the reason we are doing that is to avoid a disaster by missing or forgetting something, not to drive our loved ones crazy. The same idea of removing doubt is present in signal representations. Given a signal, we represent it in another system, typically a basis, where its characteristics are more readily apparent in the transform coefficients. However, these representations are typically nonredundant, and thus corruption or loss of transform coefficients can be serious. In comes redundancy; we build a safety net into our representation so that we can avoid those disasters. The redundant counterpart of a basis is called a frame [no one seems to know why they are called frames, perhaps because of the bounds in (25)?]. It is generally acknowledged (at least in the signal processing and harmonic analysis communities) that frames were born in 1952 in the paper by Duffin and Schaeffer [32]. Despite being over half a century old, frames gained popularity only in the last decade, due mostly to the work of the three wavelet pioneers—Daubechies, Grossman, and Meyer [29]. Framelike ideas, that is, building redundancy into a signal expansion, can be found in pyramid
Nonlinear Approximation Based Image Recovery Using Adaptive Sparse Reconstructions and Iterated Denoising: Part I  Theory
 IEEE Trans. Image Process
, 2004
"... We study the robust estimation of missing regions in images and video using adaptive, sparse reconstructions. Our primary application is on missing regions of pixels containing textures, edges, and other image features that are not readily handled by prevalent estimation and recovery algorithms. ..."
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Cited by 64 (7 self)
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We study the robust estimation of missing regions in images and video using adaptive, sparse reconstructions. Our primary application is on missing regions of pixels containing textures, edges, and other image features that are not readily handled by prevalent estimation and recovery algorithms. We assume that we are given a linear transform that is expected to provide sparse decompositions over missing regions such that a portion of the transform coe#cients over missing regions are zero or close to zero. We adaptively determine these small magnitude coe#cients through thresholding, establish sparsity constraints, and estimate missing regions in images using information surrounding these regions. Unlike prevalent algorithms, our approach does not necessitate any complex preconditioning, segmentation, or edge detection steps, and it can be written as a sequence of denoising operations. We show that the region types we can e#ectively estimate in a mean squared error sense are those for which the given transform provides a close approximation using sparse nonlinear approximants. We show the nature of the constructed estimators and how these estimators relate to the utilized transform and its sparsity over regions of interest. The developed estimation framework is general, and can readily be applied to nonstationary signals with a suitable choice of linear transforms. Part I discusses fundamental issues, and Part II is devoted to adaptive algorithms with extensive simulation examples that demonstrate the power of the proposed techniques.
Sandler: Complex domain onset detection for musical signals
 Proceedings of the 6th International Conference on Digital Audio E ff ects (DAFx03
, 2003
"... We present a novel method for onset detection in musical signals. It improves over previous energybased and phasebased approaches by combining both types of information in the complex domain. It generates a detection function that is sharp at the position of onsets and smooth everywhere else. Resu ..."
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Cited by 59 (1 self)
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We present a novel method for onset detection in musical signals. It improves over previous energybased and phasebased approaches by combining both types of information in the complex domain. It generates a detection function that is sharp at the position of onsets and smooth everywhere else. Results on a handlabelled dataset show that high detection rates can be achieved at very low error rates. The approach is more robust than its predecessors both theoretically and practically. 1.
The Design of Approximate Hilbert Transform Pairs of Wavelet Bases
, 2002
"... Several authors have demonstrated that significant improvements can be obtained in waveletbased signal processing by utilizing a pair of wavelet transforms where the wavelets form a Hilbert transform pair. This paper describes design procedures, based on spectral factorization, for the design of p ..."
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Cited by 55 (9 self)
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Several authors have demonstrated that significant improvements can be obtained in waveletbased signal processing by utilizing a pair of wavelet transforms where the wavelets form a Hilbert transform pair. This paper describes design procedures, based on spectral factorization, for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate Hilbert transform pair. Both orthogonal and biorthogonal FIR solutions are presented, as well as IIR solutions. In each case, the solution depends on an allpass filter having a flat delay response. The design procedure allows for an arbitrary number of vanishing wavelet moments to be specified. A Matlab program for the procedure is given, and examples are also given to illustrate the results.
Shift Invariant Properties Of The DualTree Complex Wavelet Transform
 In Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP
, 1999
"... We discuss the shift invariant properties of a new implementation of the Discrete Wavelet Transform, which employs a dual tree of wavelet filters to obtain the real and imaginary parts of complex wavelet coefficients. This introduces limited redundancy (2 m :1 for mdimensional signals) and allows ..."
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Cited by 41 (6 self)
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We discuss the shift invariant properties of a new implementation of the Discrete Wavelet Transform, which employs a dual tree of wavelet filters to obtain the real and imaginary parts of complex wavelet coefficients. This introduces limited redundancy (2 m :1 for mdimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good wellbalanced frequency responses. 1. INTRODUCTION The Discrete Wavelet Transform (DWT) in its maximally decimated form (Mallat's dyadic filter tree [1]) has established an impressive reputation as a tool for signal compression, but its use for other signal analysis and reconstruction tasks has been hampered by two main disadvantages: ffl Lack of shift invariance, which means that small shifts in the input signal can cause major variations i...