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Generalized sampling and infinitedimensional compressed sensing
"... We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finitedimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demo ..."
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Cited by 33 (20 self)
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We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finitedimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demonstrate that existing finitedimensional techniques are illsuited for solving a number of important problems. This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. The main conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. The key to these developments is the introduction of two new concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize the fundamentally infinitedimensional reconstruction problem.
Spread spectrum magnetic resonance imaging
 IEEE Trans. Med. Imag
"... Abstract—We propose a novel compressed sensing technique to accelerate the magnetic resonance imaging (MRI) acquisition process. The method, coined spread spectrum MRI or simply s MRI, consists of premodulating the signal of interest by a linear chirp before randomspace undersampling, and then rec ..."
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Cited by 16 (1 self)
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Abstract—We propose a novel compressed sensing technique to accelerate the magnetic resonance imaging (MRI) acquisition process. The method, coined spread spectrum MRI or simply s MRI, consists of premodulating the signal of interest by a linear chirp before randomspace undersampling, and then reconstructing the signal with nonlinear algorithms that promote sparsity. The effectiveness of the procedure is theoretically underpinned by the optimization of the coherence between the sparsity and sensing bases. The proposed technique is thoroughly studied by means of numerical simulations, as well as phantom and in vivo experiments on a 7T scanner. Our results suggest that s MRI performs better than stateoftheart variable densityspace undersampling approaches. Index Terms—Compressed sensing,magnetic resonance imaging (MRI), spread spectrum. I.
Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing
, 2013
"... In this paper we bridge the substantial gap between existing compressed sensing theory and its current use in realworld applications. 1 We do so by introducing a new mathematical framework for overcoming the socalled coherence ..."
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Cited by 13 (4 self)
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In this paper we bridge the substantial gap between existing compressed sensing theory and its current use in realworld applications. 1 We do so by introducing a new mathematical framework for overcoming the socalled coherence
Wavelet Shrinkage With Consistent Cycle Spinning Generalizes Total Variation Denoising
"... Abstract—We introduce a new waveletbased method for the implementation of TotalVariationtype denoising. The data term is leastsquares, while the regularization term is gradientbased. The particularity of our method is to exploit a link between the discrete gradient and wavelet shrinkage with cy ..."
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Cited by 11 (6 self)
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Abstract—We introduce a new waveletbased method for the implementation of TotalVariationtype denoising. The data term is leastsquares, while the regularization term is gradientbased. The particularity of our method is to exploit a link between the discrete gradient and wavelet shrinkage with cycle spinning, which we express by using redundant wavelets. The redundancy of the representation gives us the freedom to enforce additional constraints (e.g., normalization) on the solution to the denoising problem. We perform optimization in an augmentedLagrangian framework, which decouples the difficultdimensional constrainedoptimization problem into a sequence of easier scalar unconstrained problems that we solve efficiently via traditional wavelet shrinkage. Our method can handle arbitrary gradientbased regularizers. In particular, it can be made to adhere to the popular principle of least total variation. It can also be used as a maximum a posteriori estimator for a variety of priors. We illustrate the performance of our method for image denoising and for the statistical estimation of sparse stochastic processes. Index Terms—Signal denoising, total variation, wavelet regularization, cycle spinning, augmented Lagrangian. I.
On stable reconstructions from nonuniform Fourier measurements
 SIAM J. Imaging Sci
, 2014
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Epigraphical splitting for solving constrained convex formulations of inverse problems with proximal tools
, 2014
"... We propose a proximal approach to deal with a class of convex variational problems involving nonlinear constraints. A large family of constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower level set of a sum of convex functions evaluated over different ..."
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Cited by 2 (1 self)
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We propose a proximal approach to deal with a class of convex variational problems involving nonlinear constraints. A large family of constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower level set of a sum of convex functions evaluated over different, but possibly overlapping, blocks of the signal. For such constraints, the associated projection operator generally does not have a simple form. We circumvent this difficulty by splitting the lower level set into as many epigraphs as functions involved in the sum. A closed halfspace constraint is also enforced, in order to limit the sum of the introduced epigraphical variables to the upper bound of the original lower level set. In this paper, we focus on a family of constraints involving linear transforms of distance functions to a convex set or `1,p norms with p ∈ {1, 2,+∞}. In these cases, the projection onto the epigraph of the involved function has a closed form expression. The proposed approach is validated in the context of image restoration with missing samples, by making use of constraints based on NonLocal Total Variation. Experiments show that our method leads to significant improvements in term of convergence speed over existing algorithms for solving similar constrained problems. A second application to a pulse shape design problem is provided in order to illustrate the flexibility of the proposed approach.
Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples
, 2014
"... In this paper, we consider the problem of recovering a compactlysupported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sample points give ..."
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In this paper, we consider the problem of recovering a compactlysupported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sample points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sample points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Gröchenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing this dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm – based on the existing generalized sampling framework – that allows for stable and quasioptimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice – jittered, radial and spiral sampling schemes – and provide several examples illustrating the effectiveness of our approach when tested on these schemes. 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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3D Steerable Wavelets in Practice
"... Abstract — We introduce a systematic and practical design for steerable wavelet frames in 3D. Our steerable wavelets are obtained by applying a 3D version of the generalized Riesz transform to a primary isotropic wavelet frame. The novel transform is selfreversible (tight frame) and its elementary ..."
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Abstract — We introduce a systematic and practical design for steerable wavelet frames in 3D. Our steerable wavelets are obtained by applying a 3D version of the generalized Riesz transform to a primary isotropic wavelet frame. The novel transform is selfreversible (tight frame) and its elementary constituents (Riesz wavelets) can be efficiently rotated in any 3D direction by forming appropriate linear combinations. Moreover, the basis functions at a given location can be linearly combined to design custom (and adaptive) steerable wavelets. The features of the proposed method are illustrated with the processing and analysis of 3D biomedical data. In particular, we show how those wavelets can be used to characterize directional patterns and to detect edges by means of a 3D monogenic analysis. We also propose a new inverseproblem formalism along with an optimization algorithm for reconstructing 3D images from a sparse set of waveletdomain edges. The scheme results in highquality image reconstructions which demonstrate the featurereduction ability of the steerable wavelets as well as their potential for solving inverse problems. Index Terms — 3D wavelet transform, edge detection, image reconstruction, monogenic signal, riesz transform, steerability. I.