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Exact Support Recovery for Sparse Spikes Deconvolution
, 2013
"... This paper studies sparse spikes deconvolution over the space of measures. For nondegenerate sums of Diracs, we show that, when the signaltonoise ratio is large enough, total variation regularization (which the natural extension of ℓ 1 norm of vector to the setting of measures) recovers the exact ..."
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This paper studies sparse spikes deconvolution over the space of measures. For nondegenerate sums of Diracs, we show that, when the signaltonoise ratio is large enough, total variation regularization (which the natural extension of ℓ 1 norm of vector to the setting of measures) recovers the exact same number of Diracs. We also show that both the locations and the heights of these Diracs converge toward those of the input measure when the noise drops to zero. The exact speed of convergence is governed by a specific dual certificate, which can be computed by solving a linear system. Finally we draw connections between the performances of sparse recovery on a continuous domain and on a discretized grid.
A 2D SPECTRAL ANALYSIS METHOD TO ESTIMATE THE MODULATION PARAMETERS IN STRUCTURED ILLUMINATION MICROSCOPY
"... Structured illumination microscopy is a recent imaging technique that aims at going beyond the classical optical resolution limits by reconstructing a highresolution image from several lowresolution images acquired through modulation of the transfer function of the microscope. A precise knowledge ..."
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Structured illumination microscopy is a recent imaging technique that aims at going beyond the classical optical resolution limits by reconstructing a highresolution image from several lowresolution images acquired through modulation of the transfer function of the microscope. A precise knowledge of the sinusoidal modulation parameters is necessary to enable the superresolution effect expected after reconstruction. In this work, we investigate the retrieval of these parameters directly from the acquired data, using a novel 2D spectral estimation method. Index Terms — Structured illumination microscopy, SIM, sinusoidal modulation, frequency estimation, spectral analysis, superresolution 1.
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"... Superresolution consists in recovering the fine details of a signal from lowresolution measurements. Here we consider the estimation of Dirac pulses with positive amplitudes at arbitrary locations, from noisy lowpassfiltered samples. Maximumlikelihood estimation of the unknown parameters amount ..."
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Superresolution consists in recovering the fine details of a signal from lowresolution measurements. Here we consider the estimation of Dirac pulses with positive amplitudes at arbitrary locations, from noisy lowpassfiltered samples. Maximumlikelihood estimation of the unknown parameters amounts to a difficult nonconvex matrix problem of structured low rank approximation. To solve it, we propose a new heuristic iterative algorithm, yielding stateoftheart results. Index Terms — Dirac pulses, sparse spike deconvolution, superresolution, structured low rank approximation
Robust Spike Train Recovery from Noisy Data by Structured Low Rank Approximation
"... Abstract—We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to solve a difficult, even believed NPhard, matrix problem of structured low rank approxima ..."
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Abstract—We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to solve a difficult, even believed NPhard, matrix problem of structured low rank approximation. We propose a new heuristic iterative optimization algorithm to solve it. Although it comes, in absence of convexity, with no convergence proof, it converges in practice to a local solution, and even to the global solution of the problem, when the noise level is not too high. Thus, our method improves upon the classical Cadzow denoising method, for same implementation ease and speed. I. INTRODUCTION AND PROBLEM FORMULATION Reconstruction of signals lying in linear spaces, including bandlimited signals and splines, has long been the dominant
2D Prony–Huang Transform: A New Tool for 2D Spectral Analysis
"... Abstract — This paper provides an extension of the 1D Hilbert Huang transform for the analysis of images using recent optimization techniques. The proposed method consists of: 1) adaptively decomposing an image into oscillating parts called intrinsic mode functions (IMFs) using a mode decomposition ..."
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Abstract — This paper provides an extension of the 1D Hilbert Huang transform for the analysis of images using recent optimization techniques. The proposed method consists of: 1) adaptively decomposing an image into oscillating parts called intrinsic mode functions (IMFs) using a mode decomposition procedure and 2) providing a local spectral analysis of the obtained IMFs in order to get the local amplitudes, frequencies, and orientations. For the decomposition step, we propose two robust 2D mode decompositions based on nonsmooth convex optimization: 1) a genuine 2D approach, which constrains the local extrema of the IMFs and 2) a pseudo2D approach, which separately constrains the extrema of lines, columns, and diagonals. The spectral analysis step is an optimization strategy based on Prony annihilation property and applied on small square patches of the IMFs. The resulting 2D Prony–Huang transform is validated on simulated and real data. Index Terms — Empirical mode decomposition, spectral analysis, convex optimization, nonstationary image analysis.
SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 1 Enhancing Sparsity and Resolution via Reweighted Atomic Norm Minimization
"... Abstract—The mathematical theory of superresolution developed recently by Candès and FernandesGranda states that a continuous, sparse frequency spectrum can be recovered with infinite precision via a (convex) atomic norm technique given a set of regularly spaced timespace samples. This theory w ..."
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Abstract—The mathematical theory of superresolution developed recently by Candès and FernandesGranda states that a continuous, sparse frequency spectrum can be recovered with infinite precision via a (convex) atomic norm technique given a set of regularly spaced timespace samples. This theory was then extended to the cases with partial/compressive samples and/or multiple measurement vectors via atomic norm minimization (ANM), known as offgrid/continuous compressed sensing. However, a major problem of existing atomic norm methods is that the frequencies can be recovered only if they are sufficiently separated, prohibiting commonly known high resolution. In this paper, a novel nonconvex optimization method is proposed which guarantees exact recovery under no resolution limit and hence achieves high resolution. A locally convergent iterative algorithm is implemented to solve the nonconvex problem. The algorithm iteratively carries out ANM with a sound reweighting strategy which enhances sparsity and resolution, and is termed as reweighted atomicnorm minimization (RAM). Extensive numerical simulations are carried out to demonstrate the performance of the proposed method with application to direction of arrival (DOA) estimation. Index Terms—Continuous compressed sensing (CCS), DOA estimation, frequency estimation, gridless sparse method, high resolution, reweighted atomic norm minimization (RAM). I.
A Quadratically Convergent Algorithm for Structured LowRank Approximation
"... Structured LowRank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix M, the goal is to compute a matrix M ′ of given rank r in a linear or affine subspace E of matrices (usually encoding a specific structure) suc ..."
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Structured LowRank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix M, the goal is to compute a matrix M ′ of given rank r in a linear or affine subspace E of matrices (usually encoding a specific structure) such that the Frobenius distance ‖M − M ′ ‖ is small. We propose a Newtonlike iteration for solving this problem, whose main feature is that it converges locally quadratically to such a matrix under mild transversality assumptions between the manifold of matrices of rank r and the linear/affine subspace E. We also show that the distance between the limit of the iteration and the optimal solution of the problem is quadratic in the distance between the input matrix and the manifold of rank r matrices in E. To illustrate the applicability of this algorithm, we propose a Maple implementation and give experimental results for several applicative problems that can be modeled by Structured LowRank Approximation: univariate approximate GCDs (Sylvester matrices), lowrank Matrix completion (coordinate spaces) and denoising procedures (Hankel matrices). Experimental results give evidence that this allpurpose algorithm is competitive with stateoftheart numerical methods dedicated to these problems.