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Journal ArticleDOI

Wavelet Transform With Tunable Q-Factor

Ivan Selesnick
- 01 Aug 2011 - 
- Vol. 59, Iss: 8, pp 3560-3575
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TLDR
A discrete-time wavelet transform for which the Q-factor is easily specified and the transform can be tuned according to the oscillatory behavior of the signal to which it is applied, based on a real-valued scaling factor.
Abstract
This paper describes a discrete-time wavelet transform for which the Q-factor is easily specified. Hence, the transform can be tuned according to the oscillatory behavior of the signal to which it is applied. The transform is based on a real-valued scaling factor (dilation-factor) and is implemented using a perfect reconstruction over-sampled filter bank with real-valued sampling factors. Two forms of the transform are presented. The first form is defined for discrete-time signals defined on all of Z. The second form is defined for discrete-time signals of finite-length and can be implemented efficiently with FFTs. The transform is parameterized by its Q-factor and its oversampling rate (redundancy), with modest oversampling rates (e.g., three to four times overcomplete) being sufficient for the analysis/synthesis functions to be well localized.

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Citations
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TL;DR: In this article, a sparse feature extraction method based on the tunable Q-factor wavelet transform and overlapping group shrinkage is developed to extract weak features from signals with group property.
References
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TL;DR: This paper presents a meta-analyses of the wavelet transforms of Coxeter’s inequality and its applications to multiresolutional analysis and orthonormal bases.
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Ten Lectures on Wavelets

TL;DR: In this article, the regularity of compactly supported wavelets and symmetry of wavelet bases are discussed. But the authors focus on the orthonormal bases of wavelets, rather than the continuous wavelet transform.
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Atomic Decomposition by Basis Pursuit

TL;DR: Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions.
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Fast Image Recovery Using Variable Splitting and Constrained Optimization

TL;DR: A new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an l2 data-fidelity term and a nonsmooth regularizer is proposed.
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Calculation of a constant Q spectral transform

TL;DR: In this article, a constant Q transform with a constant ratio of center frequency to resolution has been proposed to obtain a constant pattern in the frequency domain for sounds with harmonic frequency components.
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