Beyond coherence: Recovering structured time–frequency representations

Borup, Lasse; Gribonval, Rémi; Nielsen, Morten

doi:10.1016/j.acha.2007.09.002

Cited by 15 publications

(19 citation statements)

References 18 publications

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“…This notion was later refined in [31] by considering the so-called structured p-Babel function, defined for some family S of subsets of I and some 1 ≤ p < ∞ by…”

Section: Coherence and Concentrationmentioning

confidence: 99%

Analysis of data separation and recovery problems using clustered sparsity

2011

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Data often have two or more fundamental components, like cartoon-like and textured elements in images; point, filament, and sheet clusters in astronomical data; and tonal and transient layers in audio signals. For many applications, separating these components is of interest. Another issue in data analysis is that of incomplete data, for example a photograph with scratches or seismic data collected with fewer than necessary sensors. There exists a unified approach to solving these problems which is minimizing the 1 norm of the analysis coefficients with respect to particular frame(s). This approach using the concept of clustered sparsity leads to similar theoretical bounds and results, which are presented here. Furthermore, necessary conditions for the frames to lead to sufficiently good solutions are also shown.

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“…This notion was later refined in [31] by considering the so-called structured p-Babel function, defined for some family S of subsets of I and some 1 ≤ p < ∞ by…”

Section: Coherence and Concentrationmentioning

confidence: 99%

Analysis of data separation and recovery problems using clustered sparsity

2011

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“…In the years since that work, two streams of research emerged: theoretical work, showing that, indeed, one could in certain settings obtain the sparsest possible representations to an underdetermined problem by`1 optimization; see, e.g., [9,13,14,15,17,47] for a selection of general work concerning`1 minimization, and [2,16,18,23,26] for work somewhat relevant to component separation; empirical work, showing that combined representations such as wavelets with curvelets or wavelets with sinusoids often gave very compelling separations of real signals and images (see, for instance, [1,10,24,25,29,37,42,43,44,45,46,50]). We have already mentioned the empirical successes of Starck and collaborators.…”

Section: Minimum`1 Decomposition and Perfect Separationmentioning

confidence: 99%

“…• Theoretical work, showing that, indeed, one could in certain settings obtain the sparsest possible representations to an underdetermined problem by ℓ 1 optimization; see, e.g., [9,13,14,15,17,45] for a selection of general work concerning ℓ 1 minimization, and [2,16,18,24,27] for work somewhat relevant to component separation.…”

Section: Minimum ℓ 1 Decomposition and Perfect Separationmentioning

confidence: 99%

Microlocal Analysis of the Geometric Separation Problem

Donoho

Kutyniok

2012

Comm Pure Appl Math

190

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Image data are often composed of two or more geometrically distinct constituents; in galaxy catalogs, for instance, one sees a mixture of pointlike structures (galaxy superclusters) and curvelike structures (filaments). It would be ideal to process a single image and extract two geometrically 'pure' images, each one containing features from only one of the two geometric constituents. This seems to be a seriously underdetermined problem, but recent empirical work achieved highly persuasive separations.We present a theoretical analysis showing that accurate geometric separation of point and curve singularities can be achieved by minimizing the ℓ 1 norm of the representing coefficients in two geometrically complementary frames: wavelets and curvelets. Driving our analysis is a specific property of the ideal (but unachievable) representation where each content type is expanded in the frame best adapted to it. This ideal representation has the property that important coefficients are clustered geometrically in phase space, and that at fine scales, there is very little coherence between a cluster of elements in one frame expansion and individual elements in the complementary frame. We formally introduce notions of cluster coherence and clustered sparsity and use this machinery to show that the underdetermined systems of linear equations can be stably solved by ℓ 1 minimization; microlocal phase space helps organize the calculations that cluster coherence requires.

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“…Mutual coherence is always lower-bounded by N−P P (N−1) , and equality holds if and only if (ϕ i ) 1 i N is an equiangular tight frame, see [210]. Finer variants based on cumulative coherences have been proposed in [120,24]. To take into account the influence of the support I = supp(x 0 ) of the vector x 0 to recover, Tropp introduced in [222] the Exact Recovery Condition (ERC), defined as…”

Section: Stronger Criteria For ℓmentioning

confidence: 99%

“…Though the forthcoming results can be stated for a large family of distributions, for the sake of concreteness, we only consider the white Gaussian model where W ∼ N (0, σ 2 Id P×P ), with known variance σ 2 . Under the observation model (24), the ideal choice of λ should be the one which minimizes the quadratic estimation risk E W ( x ⋆ (Y ) − x 0 2 ). This is obviously not realistic as x 0 is not available, and in practice, only one realization of Y is observed.…”

Section: Unbiased Risk Estimationmentioning

confidence: 99%

Low Complexity Regularization of Linear Inverse Problems

Vaiter

Peyré

Fadili

2015

Sampling Theory, a Renaissance

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Inverse problems and regularization theory is a central theme in imaging sciences, statistics and machine learning. The goal is to reconstruct an unknown vector from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown vector is to solve a convex optimization problem that enforces some prior knowledge about its structure. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of ℓ 2 -stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem.

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Beyond coherence: Recovering structured time–frequency representations

Cited by 15 publications

References 18 publications

Analysis of data separation and recovery problems using clustered sparsity

Analysis of data separation and recovery problems using clustered sparsity

Microlocal Analysis of the Geometric Separation Problem

Low Complexity Regularization of Linear Inverse Problems

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