Sparse signal approximations have become a fundamental tool in signal processing with wide-ranging applications from source separation to signal acquisition. The ever-growing number of possible applications and, in particular, the ever-increasing problem sizes now addressed lead to new challenges in terms of computational strategies and the development of fast and efficient algorithms has become paramount. Recently, very fast algorithms have been developed to solve convex optimization problems that are often used to approximate the sparse approximation problem; however, it has also been shown, that in certain circumstances, greedy strategies, such as orthogonal matching pursuit, can have better performance than the convex methods. In this paper, improvements to greedy strategies are proposed and algorithms are developed that approximate orthogonal matching pursuit with computational requirements more akin to matching pursuit. Three different directional optimization schemes based on the gradient, the conjugate gradient, and an approximation to the conjugate gradient are discussed, respectively. It is shown that the conjugate gradient update leads to a novel implementation of orthogonal matching pursuit, while the gradient-based approach as well as the approximate conjugate gradient methods both lead to fast approximations to orthogonal matching pursuit, with the approximate conjugate gradient method being superior to the gradient method.

Gradient Pursuits

https://www.mat.univie.ac.at/~esiprpr/esi1667.pdf

G-frames and G-Riesz Bases ⁄

Foundations Of Time Frequency Analysis

G-frames are generalized frames which include ordinary frames, bounded invertible linear operators, as well as many recent generalizations of frames, e.g., bounded quasi-projectors and frames of subspaces. G-frames are natural generalizations of frames and provide more choices on analyzing functions from frame expansion coefficients. We give characterizations of g-frames and prove that g-frames share many useful properties with frames. We also give generalized version of Riesz bases and orthonormal bases. As an application, we get atomic resolutions for bounded linear operators.

G-frames and G-Riesz Bases

Constant Q cepstral coefficients

Theory, implementation and applications of nonstationary Gabor frames

Sparse and structured signal expansions on dictionaries can be obtained through explicit modeling in the coefficient domain. The originality of the present article lies in the construction and the study of generalized shrinkage operators, whose goal is to identify structured significance maps and give rise to structured thresholding. These generalize Group-Lasso and the previously introduced Elitist Lasso by introducing more flexibility in the coefficient domain modeling, and lead to the notion of social sparsity. The proposed operators are studied theoretically and embedded in iterative thresholding algorithms. Moreover, a link between these operators and a convex functional is established. Numerical studies on both simulated and real signals confirm the benefits of such an approach.

/pdf/social-sparsity-neighborhood-systems-enrich-structured-1imxelwjke.pdf

Social Sparsity! Neighborhood Systems Enrich Structured Shrinkage Operators

An efficient and perfectly invertible signal transform feat uring a constant-Q frequency resolution is presented. The proposed approach is based on the idea of the recently introduced nonstationary Gabor frames. Exploiting the properties of the operator corresponding to a family of analysis atoms, this approach overcomes the problems of the classical implementations of constant-Q transforms, in particular, computational intensity and lack of i nvertibility. Perfect reconstruction is guaranteed by using an easy t o calculate dual system in the synthesis step and computation time is kept low by applying FFT-based processing. The proposed method is applied to real-life signals and evaluated in comparison to a related approach, recently introduced specifically for audio signa ls.

/pdf/constructing-an-invertible-constant-q-transform-with-4bmcne2htz.pdf

Constructing an invertible constant-q transform with nonstationary gabor frames

Audio signal processing frequently requires time-frequency representations and in many applications, a non-linear spacing of frequency bands is preferable. This paper introduces a framework for efficient implementation of invertible signal transforms allowing for non-uniform frequency resolution. Non-uniformity in frequency is realized by applying nonstationary Gabor frames with adaptivity in the frequency domain. The realization of a perfectly invertible constant-Q transform is described in detail. To achieve real-time processing, independent of signal length, slice-wise processing of the full input signal is proposed and referred to as sliCQ transform. By applying frame theory and FFT-based processing, the presented approach overcomes computational inefficiency and lack of invertibility of classical constant-Q transform implementations. Numerical simulations evaluate the efficiency of the proposed algorithm and the method's applicability is illustrated by experiments on real-life audio signals .

https://www.univie.ac.at/nonstatgab/pdf_files/dogrhove12_amsart.pdf

A Framework for Invertible, Real-Time Constant-Q Transforms

In this paper, we propose a time-frequency representation where the frequency bins are distributed uniformly in log-frequency and their Q-factors obey a linear function of the bin center frequencies. The latter allows for time-frequency representations where the bandwidths can be e.g. constant on the log-frequency scale (constant Q) or constant on the auditory critical-band scale (smoothly varying Q). The proposed techniques are published as a Matlab toolbox that extends [3]. Besides the features that stem from [3] – perfect reconstruction and computational efficiency – we propose here a technique for computing coefficient phases in a way that makes their interpretation more natural. Other extensions include flexible control of the Qvalues and more regular sampling of the time-frequency plane in order to simplify signal processing in the transform domain.

Monika Dörfler

Papers

Theory, implementation and applications of nonstationary Gabor frames

Social Sparsity! Neighborhood Systems Enrich Structured Shrinkage Operators

Constructing an invertible constant-q transform with nonstationary gabor frames

A Framework for Invertible, Real-Time Constant-Q Transforms

A Matlab Toolbox for Efficient Perfect Reconstruction Time-Frequency Transforms with Log-Frequency Resolution