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55
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
Video denoising using 2D and 3D dualtree complex wavelet transforms
 Wavelet Appl Signal Image Proc. X (Proc. SPIE 5207
, 2003
"... The denoising of video data should take into account both temporal and spatial dimensions, however, true 3D transforms are rarely used for video denoising. Separable 3D transforms have artifacts that degrade their performance in applications. This paper describes the design and application of the n ..."
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Cited by 71 (5 self)
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The denoising of video data should take into account both temporal and spatial dimensions, however, true 3D transforms are rarely used for video denoising. Separable 3D transforms have artifacts that degrade their performance in applications. This paper describes the design and application of the nonseparable oriented 3D dualtree wavelet transform for video denoising. This transform gives a motionbased multiscale decomposition for video — it isolates in its subbands motion along different directions. In addition, we investigate the denoising of video using the 2D and 3D dualtree oriented wavelet transforms, where the 2D transform is applied to each frame individually.
Image analysis using a dualtree Mband wavelet transform
 IEEE Trans. Image Process
, 2006
"... Abstract—We propose a twodimensional generalization to theband case of the dualtree decomposition structure (initially proposed by Kingsbury and further investigated by Selesnick) based on a Hilbert pair of wavelets. We particularly address: 1) the construction of the dual basis and 2) the result ..."
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Cited by 53 (26 self)
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Abstract—We propose a twodimensional generalization to theband case of the dualtree decomposition structure (initially proposed by Kingsbury and further investigated by Selesnick) based on a Hilbert pair of wavelets. We particularly address: 1) the construction of the dual basis and 2) the resulting directional analysis. We also revisit the necessary preprocessing stage in theband case. While several reconstructions are possible because of the redundancy of the representation, we propose a new optimal signal reconstruction technique, which minimizes potential estimation errors. The effectiveness of the proposedband decomposition is demonstrated via denoising comparisons on several image types (natural, texture, seismics), with variousband wavelets and thresholding strategies. Significant improvements in terms of both overall noise reduction and direction preservation are observed. Index Terms—Direction selection, dualtree, Hilbert transform, image denoising,band filter banks, wavelets.
The phaselet transform  an integral redundancy nearly shiftinvariant wavelet transform
 IEEE Trans. on Signal Proc
, 2003
"... This paper introduces an approximately shift invariant redundant dyadic wavelet transform the phaselet transform that includes the popular dualtree complex wavelet transform of Kingsbury [1] as a special case. The main idea is to use a finite set of wavelets that are related to each other in a sp ..."
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Cited by 15 (1 self)
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This paper introduces an approximately shift invariant redundant dyadic wavelet transform the phaselet transform that includes the popular dualtree complex wavelet transform of Kingsbury [1] as a special case. The main idea is to use a finite set of wavelets that are related to each other in a special way and hence called phaseletsto achieve approximate shiftredundancy; bigger the set better the approximation. A sufficient condition on the associated scaling filters to achieve this is that they are fractional shifts of each other. Algorithms for the design of phaselets with a fixed number vanishing moments is presented building upon the work of Selesnick [2] for the design of wavelet pairs for Kingsbury’s dualtree complex wavelet transform. Construction of 2dimensional directional bases from tensor products of 1d phaselets is also described. Phaselets as a new approach to redundant wavelet transforms and their construction are both novel and should be interesting to the reader independently of the approximate shift invariance property that this paper argues they possess. 1
Construction of Hilbert transform pairs of wavelet bases and Gaborlike transforms
 IEEE TRANS. SIGNAL PROCESS
, 2009
"... We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximationtheoretic characterization of scaling functions—the Bspline factorization theorem. In particular, starting from welllocalized scaling functions, we construct HT pairs of b ..."
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Cited by 14 (9 self)
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We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximationtheoretic characterization of scaling functions—the Bspline factorization theorem. In particular, starting from welllocalized scaling functions, we construct HT pairs of biorthogonal wavelet bases of by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimallylocalized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a onesided spectrum. Based on the tensorproduct of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of, we then discuss a methodology for constructing 2D directionalselective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multidimensional extension of the HT—the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient fast Fourier transform (FFT)based filterbank algorithm for implementing the associated complex wavelet transform.
Multiresolution Monogenic Signal Analysis Using the Riesz–Laplace Wavelet Transform
, 2009
"... The monogenic signal is the natural 2D counterpart of the 1D analytic signal. We propose to transpose the concept to the wavelet domain by considering a complexified version of the Riesz transform which has the remarkable property of mapping a realvalued (primary) wavelet basis of into a complex ..."
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Cited by 12 (1 self)
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The monogenic signal is the natural 2D counterpart of the 1D analytic signal. We propose to transpose the concept to the wavelet domain by considering a complexified version of the Riesz transform which has the remarkable property of mapping a realvalued (primary) wavelet basis of into a complex one. The Riesz operator is also steerable in the sense that it give access to the Hilbert transform of the signal along any orientation. Having set those foundations, we specify a primary polyharmonic spline wavelet basis of that involves a single Mexicanhatlike mother wavelet (Laplacian of a Bspline). The important point is that our primary wavelets are quasiisotropic: they behave like multiscale versions of the fractional Laplace operator from which they are derived, which ensures steerability. We propose to pair these realvalued basis functions with their complex Riesz counterparts to specify a multiresolution monogenic signal analysis. This yields a representation where each wavelet index is associated with a local orientation, an amplitude and a phase. We give a corresponding waveletdomain method for estimating the underlying instantaneous frequency. We also provide a mechanism for improving the shift and rotationinvariance of the wavelet decomposition and show how to implement the transform efficiently using perfectreconstruction filterbanks. We illustrate the specific featureextraction capabilities of the representation and present novel examples of waveletdomain processing; in particular, a robust, tensorbased analysis of directional image patterns, the demodulation of interferograms, and the reconstruction of digital holograms.
An Investigation of 3D DualTree Wavelet Transform for Video Coding
 in Proceedings of International Conference on Image Processing (ICIP
, 2004
"... This paper examines the properties of a recently introduced 3D dualtree discrete wavelet transform (DDWT) for video coding. The 3D DDWT is an attractive video representation because it isolates motion along different directions in separate sub bands. However, it is an over complete transform w ..."
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Cited by 10 (4 self)
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This paper examines the properties of a recently introduced 3D dualtree discrete wavelet transform (DDWT) for video coding. The 3D DDWT is an attractive video representation because it isolates motion along different directions in separate sub bands. However, it is an over complete transform with 8:1 redundancy. We examine the effectiveness of the iterative projectivebased noiseshaping scheme proposed by Kingsbury [3] on reducing the number of coefficients. We also investigate the correlation between sub bands at the same spatial/temporal location, both in the significance map and in actual coefficient value.