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18
An analysis of block sampling strategies in compressed sensing
, 2013
"... Compressed sensing (CS) is a theory which guarantees the exact recovery of sparse signals from a few number of linear projections. The sampling schemes suggested by current CS theories are often of little relevance since they cannot be implemented on practical acquisition systems. In this paper, we ..."
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Compressed sensing (CS) is a theory which guarantees the exact recovery of sparse signals from a few number of linear projections. The sampling schemes suggested by current CS theories are often of little relevance since they cannot be implemented on practical acquisition systems. In this paper, we study a new random sampling approach that consists in selecting a set of blocks that are predefined by the application of interest. A typical example is the case where the blocks consist in horizontal lines in the 2D Fourier plane. We provide theoretical results on the number of blocks that are required for exact sparse signal reconstruction in a noise free setting. We illustrate this theory for various sensing matrices appearing in applications such as timefrequency bases. A typical result states that it is sufficient to acquire no more than O ( sln 2 (n) ) lines in the 2D Fourier domain for the perfect reconstruction of an ssparse image of size √ n × √ n. The proposed results have a large number of potential applications in systems such as magnetic resonance imaging, radiointerferometry or ultrasound imaging. Keywords: Compressed Sensing, blocks of measurements, sampling continuous trajectories, exact recovery, ℓ 1 minimization. 1
On stable reconstructions from nonuniform Fourier measurements
 SIAM J. Imaging Sci
, 2014
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Fundamental performance limits for ideal decoders in highdimensional linear inverse problems. arXiv:1311.6239
, 2013
"... The primary challenge in linear inverse problems is to design stable and robust “decoders” to reconstruct highdimensional vectors from a lowdimensional observation through a linear operator. Sparsity, lowrank, and related assumptions are typically exploited to design decoders which performance is ..."
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The primary challenge in linear inverse problems is to design stable and robust “decoders” to reconstruct highdimensional vectors from a lowdimensional observation through a linear operator. Sparsity, lowrank, and related assumptions are typically exploited to design decoders which performance is then bounded based on some measure of deviation from the idealized model, typically using a norm. This paper focuses on characterizing the fundamental performance limits that can be expected from an ideal decoder given a general model, i.e., a general subset of “simple ” vectors of interest. First, we extend the socalled notion of instance optimality of a decoder to settings where one only wishes to reconstruct some part of the original high dimensional vector from a lowdimensional observation. This covers practical settings such as medical imaging of a region of interest, or audio source separation when one is only interested in estimating the contribution of a specific instrument to a musical recording. We define instance optimality relatively to a model much beyond the traditional framework of sparse recovery, and characterize the existence of an instance optimal decoder in terms of joint properties of the model and the considered linear operator. Noiseless and noiserobust settings are both considered. We show somewhat surprisingly that the existence of noiseaware instance optimal decoders for all noise levels implies the existence of a noiseblind decoder. A consequence of our results is that for models that are rich enough to contain an orthonormal basis, the existence of an `2/`2 instance optimal decoder is only possible when the linear operator is not substantially dimensionreducing. This covers wellknown cases (sparse vectors, lowrank matrices) as well as a number of seemingly new situations (structured sparsity and sparse inverse covariance matrices for instance). We exhibit an operatordependent norm which, under a modelspecific generalization of the Restricted Isometry Property (RIP), always yields a feasible instance optimality property. This norm can be upper bounded by an atomic norm relative to the considered model. 1
The quest for optimal sampling: Computationally efficient, structureexploiting sampling strategies for compressed sensing. Compressed Sensing and Its Applications
, 2014
"... measurements for compressed sensing ..."
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A note on compressed sensing of structured sparse wavelet coefficients from subsampled Fourier measurements. arXiv
, 2014
"... This note complements the paper The quest for optimal sampling: Computationally efficient, structureexploiting measurements for compressed sensing [2]. Its purpose is to present a proof of a result stated therein concerning the recovery via compressed sensing of a signal that has structured sparsit ..."
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This note complements the paper The quest for optimal sampling: Computationally efficient, structureexploiting measurements for compressed sensing [2]. Its purpose is to present a proof of a result stated therein concerning the recovery via compressed sensing of a signal that has structured sparsity in a Haar wavelet basis when sampled using a multilevelsubsampled discrete Fourier transform. In doing so, it provides a simple exposition of the proof in the case of Haar wavelets and discrete Fourier samples of more general result recently provided in Breaking the coeherence barrier: A new theory for compressed sensing [1]. 1
CAN EVERYTHING BE COMPUTED? ON THE SOLVABILITY COMPLEXITY INDEX AND TOWERS OF ALGORITHMS
"... ABSTRACT. This paper addresses and establishes some of the fundamental barriers in the theory of computations and finally settles the long standing computational spectral problem. Due to the barriers presented in this paper, there are many problems, some of them at the heart of computational theory ..."
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ABSTRACT. This paper addresses and establishes some of the fundamental barriers in the theory of computations and finally settles the long standing computational spectral problem. Due to the barriers presented in this paper, there are many problems, some of them at the heart of computational theory, that do not fit into the classical frameworks of theory of computation. Hence, we are in need for a new extended theory capable of handling these new issues. Many computational problems can be solved as follows: a sequence of approximations is created by an algorithm, and the solution to the problem is the limit of this sequence (think about computing eigenvalues of a matrix for example). However, as we demonstrate, for several basic problems in computations (computing spectra of infinite dimensional operators, solutions to linear equations or roots of polynomials using rational maps) such a procedure based on one limit is impossible. Yet, one can compute solutions to these problems, but only by using several limits. This may come as a surprise, however, this touches onto the definite boundaries of computational mathematics. To analyze this phenomenon we use the Solvability Complexity Index (SCI). The SCI is the smallest number of limits needed in order to compute a desired quantity. In several cases (spectral problems, inverse problems) we provide sharp results on the SCI, thus we establish the barriers for what can be achieved computationally. For example, we show that the SCI of spectra and essential spectra of infinite matrices is equal to three, and that the SCI of spectra of selfadjoint
Generalized sampling reconstruction from Fourier measurements using compactly supported shearlets
"... Abstract In this paper we study the general reconstruction of a compactly supported function from its Fourier coefficients using compactly supported shearlet systems. We assume that only finitely many Fourier samples of the function are accessible and based on this finite collection of measurements ..."
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Abstract In this paper we study the general reconstruction of a compactly supported function from its Fourier coefficients using compactly supported shearlet systems. We assume that only finitely many Fourier samples of the function are accessible and based on this finite collection of measurements an approximation is sought in a finite dimensional shearlet reconstruction space. We analyse this sampling and reconstruction process by a recently introduced method called generalized sampling. In particular by studying the stable sampling rate of generalized sampling we then show stable recovery of the signal is possible using an almost linear rate. Furthermore, we compare the result to the previously obtained rates for wavelets.
On
, 2014
"... the role of total variation in compressed sensing structure dependence ..."
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On
, 2014
"... the role of total variation in compressed sensing uniform random sampling ..."
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the role of total variation in compressed sensing uniform random sampling