Rational multiresolution analysis and fast wavelet transform: application to wavelet shrinkage denoising

Baussard, Alexandre; Nicolier, Fred; Truchetet, Frédéric

doi:10.1016/j.sigpro.2004.06.001

Cited by 50 publications

(36 citation statements)

References 6 publications

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“…Since the wavelet has infinite support in the time domain, there is no FIR solution of the analysis/synthesis filters for the tree-structured fast wavelet transform (FWT). An FWT implementation is available, however, in the frequency domain, as shown in [9], or by FIR-based approximation as the derivation of discrete Meyer (dMeyer) wavelet from Meyer wavelet. For a parallel filter bank structure, more efficient implementation is available via FFT-based algorithms and can benefit from advanced DSP techniques on parallel calculations.…”

Section: Discussionmentioning

confidence: 99%

Complex rational orthogonal wavelet and its application in communications

White

2006

IEEE Signal Process. Lett.

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Abstract-This letter proposes a generalized method to construct complex wavelets under the framework of rational multiresolution analysis, MRA( ), where is a rational number. Theorems and examples are given for the construction of complex rational orthogonal wavelets (CROWs) whose real and imaginary parts form an exact Hilbert transform pair. Since the classical Mallat's MRA is a special case of the rational MRA( ) with = 2, the theorems hold for the construction of complex dyadic wavelets. Based on a rational MRA( ) with 1 2, the constructed CROWs not only achieve the benefit by capturing the phase information that the real-valued wavelets are lacking but also have the unique fine rational orthogonal property that is suited for specific application scenarios where binary orthogonality achieved by dyadic wavelets is not sufficient for the scale resolution. The CROWs' application in communications as the modulation signal pulse for PSK/QAM signaling is discussed.

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Section: Discussionmentioning

confidence: 99%

Complex rational orthogonal wavelet and its application in communications

White

2006

IEEE Signal Process. Lett.

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“…For increased flexibility in defining the wavelet subbands, we also provide rational frame designs. Extensions to the rational case of Meyer and Shannon constructions for classical wavelets were originally proposed by Auscher [25] and have further been generalized by [26].…”

Section: B Rational Dilation Factorsmentioning

confidence: 99%

“…Since is the average graph, or "backbone" of the multislice graph, and only all-postiive or all-negative eigennetwork, must be . The weights can also be obtained by a constrained leastsquares fitting problem, which exploits the weights in associated with each eigennetwork : (26) where is a pre-defined variation between the slices of the network that we like to capture. This strategy will not be further explored here.…”

Section: B Graph Wavelet Designmentioning

confidence: 99%

Tight Wavelet Frames on Multislice Graphs

Leonardi

Ville

2013

IEEE Trans. Signal Process.

108

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Abstract-We present a framework for the design of wavelet transforms tailored to data defined on multislice graphs (i.e., multiplex or dynamic graphs). Graphs with multiple types of interactions are ubiquitous in real life, motivating the extension of wavelets to these complex domains. Our framework generalizes the recently proposed spectral graph wavelet transform (SGWT) [D. Hammond, P. Vandergheynst, and R. Gribonval, "Wavelets on Graphs via Spectral Graph Theory," Appl. Comput. Harmon. Anal., vol. 30, pp. 129-150, Mar. 2011], which is designed in the spectral (frequency) domain of an arbitrary finite weighted graph. We extend the SGWT to form a tight frame, which conserves energy in the wavelet domain, and define the relationship between conventional and spectral graph wavelets. We then propose a design for multislice graphs that is based on the higher-order singular value decomposition (HOSVD), a powerful tool from multilinear algebra. In particular, the multiple adjacency matrices are stacked to form a tensor and the HOSVD decomposition provides information about its third-order structure, analogous to that provided by matrix factorizations. We obtain a set of "eigennetworks" and from these graph wavelets, which exploit the variability across the graphs. We demonstrate the feasibility of our method 1) by capturing different orientations of a gray-scale image and 2) by decomposing brain signals from functional magnetic resonance imaging. We show its effectiveness to identify variability across graph edges and provide meaningful decompositions.Index Terms-Higher-order singular value decomposition (HOSVD), multislice graph, spectral graph theory, tensor decompositions, wavelet transform.

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“…This class of orthogonal rational wavelets was derived under the framework of a rational multiresolution analysis (MRA) proposed by Auscher in [11] and was generalized by Baussard et al in [12] to permit more general types of roll-off in the transition bands of the wavelet frequency spectrum. In [7], explicit formulas were derived for the special class of rational wavelets with a dilation factor of and whose spectrum has no constant passband between the two transition bands.…”

Section: B Rational Wavelet Signalingmentioning

confidence: 99%

Optimum Receiver Design for Broadband Doppler Compensation in Multipath/Doppler Channels With Rational Orthogonal Wavelet Signaling

White

2007

IEEE Trans. Signal Process.

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Rational multiresolution analysis and fast wavelet transform: application to wavelet shrinkage denoising

Cited by 50 publications

References 6 publications

Complex rational orthogonal wavelet and its application in communications

Complex rational orthogonal wavelet and its application in communications

Tight Wavelet Frames on Multislice Graphs

Optimum Receiver Design for Broadband Doppler Compensation in Multipath/Doppler Channels With Rational Orthogonal Wavelet Signaling

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