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Generalized sampling and infinite-dimensional 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 finite-dimensional 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 finite-dimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demonstrate that existing finite-dimensional techniques are ill-suited 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.
Stable and robust sampling strategies for compressive imaging
, 2013
"... In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained ..."
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Cited by 8 (1 self)
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In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained through variable density sampling strategies that concentrate on lower frequencies. The wavelet and Fourier transform domains are not incoherent because low-order wavelets and low-order frequencies are correlated, so compressive sensing theory does not immediately imply sampling strategies and reconstruction guarantees. In this paper we turn to a more refined notion of coherence – the so-called local coherence – measuring for each sensing vector separately how correlated it is to the sparsity basis. For Fourier measurements and Haar wavelet sparsity, the local coherence can be controlled and bounded explicitly, so for matrices comprised of frequencies sampled from a suitable inverse square power-law density, we can prove the restricted isometry property with near-optimal embedding dimensions. Consequently, the variable-density sampling strategy we provide allows for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by ℓ1-minimization and by total variation minimization. The local coherence framework developed in this paper should be of independent interest in sparse recovery problems more generally, as it implies that for optimal sparse recovery results, it suffices to have bounded average coherence from sensing basis to sparsity basis – as opposed to bounded maximal coherence – as long as the sampling strategy is adapted accordingly. 1
On stable reconstructions from univariate nonuniform Fourier measurements. arXiv:1310.7820
, 2013
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Gradient waveform design for variable density sampling in Magnetic Resonance Imaging
, 2014
"... Fast coverage of k-space is a major concern to speed up data acquisition in Magnetic Resonance Imaging (MRI) and limit image distortions due to long echo train durations. The hardware gradient constraints (magnitude, slew rate) must be taken into account to collect a sufficient amount of samples in ..."
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Cited by 2 (1 self)
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Fast coverage of k-space is a major concern to speed up data acquisition in Magnetic Resonance Imaging (MRI) and limit image distortions due to long echo train durations. The hardware gradient constraints (magnitude, slew rate) must be taken into account to collect a sufficient amount of samples in a minimal amount of time. However, sampling strategies (e.g., Compressed Sensing) and optimal gradient waveform design have been developed separately so far. The major flaw of existing methods is that they do not take the sampling density into account, the latter being central in sampling theory. In particular, methods using optimal control tend to agglutinate samples in high curvature areas. In this paper, we develop an iterative algorithm to project any parameterization of k-space trajectories onto the set of feasible curves that fulfills the gradient constraints. We show that our projection algorithm provides a more efficient alternative than existing approaches and that it can be a way of reducing acquisition time while maintaining sampling density for piece-wise linear trajectories.
A consistent and stable approach to generalized sampling
- Journal of Fourier Analysis and Applications
, 2014
"... Abstract We consider the problem of generalized sampling, in which one seeks to obtain reconstructions in arbitrary finite dimensional spaces from finitely samples taken with respect to an arbitrary orthonormal basis. Typical approaches to this problem consider solutions obtained via the consistent ..."
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Cited by 1 (1 self)
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Abstract We consider the problem of generalized sampling, in which one seeks to obtain reconstructions in arbitrary finite dimensional spaces from finitely samples taken with respect to an arbitrary orthonormal basis. Typical approaches to this problem consider solutions obtained via the consistent reconstruction technique of Eldar et al and also solutions of overcomplete linear systems. However, the consistent reconstruction technique is known to be non-convergent and ill-posed in important cases, such as the recovery of wavelet coefficients from Fourier samples, and whilst the latter approach presents solutions which are convergent and numerically stable when the system is sufficiently overcomplete, the solution becomes inconsistent with the original measurements. In this paper, we consider generalized sampling via a non-linear minimization problem and prove that the minimizers present solutions which are convergent, numerically stable and consistent with the original measurements. We also provide analysis in the case of reconstructing in compactly supported wavelets from Fourier samples. We show that for wavelets of sufficient smoothness, there is a linear relationship between the number of wavelet coefficients which can be accurately recovered and the number of Fourier samples.
A projection algorithm for gradient waveforms design in Magnetic Resonance Imaging
, 2014
"... Collecting the maximal amount of useful information in a given scanning time is a major concern in Magnetic Resonance Imaging (MRI) to speed up image acquisition. The hardware constraints (gradient magnitude, slew rate,...), physical distortions (e.g., off-resonance effects) and sampling theorems (S ..."
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Collecting the maximal amount of useful information in a given scanning time is a major concern in Magnetic Resonance Imaging (MRI) to speed up image acquisition. The hardware constraints (gradient magnitude, slew rate,...), physical distortions (e.g., off-resonance effects) and sampling theorems (Shannon, compressed sensing) must be taken into account simultaneously, which makes this problem extremely challenging. To date, the main approach to design gradient waveform has consisted of selecting an initial shape (e.g. spiral, radial lines,...) and then traversing it as fast as possible. In this paper, we propose an alternative solution: instead of reparameterizing an initial trajectory, we propose to project it onto the convex set of admissible curves. This method has various advantages. First, it better preserves the density of the input curve which is critical in sampling theory. Second, it allows to smooth high curvature areas making the acquisition time shorter in some cases. We develop an efficient iterative algorithm based on convex programming and propose comparisons between the two approaches. For piecewise linear trajectories, our approach generates a gain of scanning time ranging from 20 % (echo planar imaging) to 300% (travelling salesman problem) without degrading image quality in terms of signal-to-noise ratio (SNR). For smoother trajectories such as spirals, our method better preserves the sampling density of the input curve, making the sampling pattern relevant for compressed sensing, contrarily to the reparameterization based approaches.
VARIABLE DENSITY SAMPLING BASED ON PHYSICALLY PLAUSIBLE GRADIENT WAVEFORM. APPLICATION TO 3D MRI ANGIOGRAPHY.
"... Performing k-space variable density sampling is a popular way of reducing scanning time in Magnetic Resonance Imaging (MRI). Un-fortunately, given a sampling trajectory, it is not clear how to traverse it using gradient waveforms. In this paper, we actually show that ex-isting methods [1, 2] can yie ..."
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Performing k-space variable density sampling is a popular way of reducing scanning time in Magnetic Resonance Imaging (MRI). Un-fortunately, given a sampling trajectory, it is not clear how to traverse it using gradient waveforms. In this paper, we actually show that ex-isting methods [1, 2] can yield large traversal time if the trajectory contains high curvature areas. Therefore, we consider here a new method for gradient waveform design which is based on the pro-jection of unrealistic initial trajectory onto the set of hardware con-straints. Next, we show on realistic simulations that this algorithm allows implementing variable density trajectories resulting from the piecewise linear solution of the Travelling Salesman Problem in a reasonable time. Finally, we demonstrate the application of this ap-proach to 2D MRI reconstruction and 3D angiography in the mouse brain. Index Terms — MRI, Compressive sensing, Variable density sampling, gradient waveform design, hardware constraints, angiog-raphy. 1.
