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33
A generalized sampling theorem for stable reconstructions in arbitrary bases
 J. Fourier Anal. Appl
"... We introduce a generalized framework for sampling and reconstruction in separable Hilbert spaces. Specifically, we establish that it is always possible to stably reconstruct a vector in an arbitrary Riesz basis from sufficiently many of its samples in any other Riesz basis. This framework can be vie ..."
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Cited by 30 (19 self)
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We introduce a generalized framework for sampling and reconstruction in separable Hilbert spaces. Specifically, we establish that it is always possible to stably reconstruct a vector in an arbitrary Riesz basis from sufficiently many of its samples in any other Riesz basis. This framework can be viewed as an extension of the wellknown consistent reconstruction technique (Eldar et al). However, whilst the latter imposes stringent assumptions on the reconstruction basis, and may in practice be unstable, our framework allows for recovery in any (Riesz) basis in a manner that is completely stable. Whilst the classical Shannon Sampling Theorem is a special case of our theorem, this framework allows us to exploit additional information about the approximated vector (or, in this case, function), for example sparsity or regularity, to design a reconstruction basis that is better suited. Examples are presented illustrating this procedure.
Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon
 Appl. Comput. Harmon. Anal
"... We introduce a simple and efficient method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we establish, provided the dimension of the reco ..."
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Cited by 27 (18 self)
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We introduce a simple and efficient method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we establish, provided the dimension of the reconstruction space is chosen suitably in relation to the number of samples, this procedure can be implemented in a completely numerically stable manner. Moreover, the accuracy of the resulting approximation is determined solely by the choice of reconstruction basis, meaning that reconstruction vectors can be readily tailored to the particular problem at hand. An important example of this approach is the accurate recovery of a piecewise analytic function from its first few Fourier coefficients. Whilst the standard Fourier projection suffers from the Gibbs phenomenon, by reconstructing in a piecewise polynomial basis we obtain an approximation with rootexponential accuracy in terms of the number of Fourier samples and exponential accuracy in terms of the degree of the reconstruction. Numerical examples illustrate the advantage of this approach over other existing methods. 1
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
On optimal wavelet reconstructions from fourier samples: linearity and universality of the stable sampling rate.
, 2012
"... Abstract In this paper we study the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples. For this, we use the recently introduced framework of generalized sampling. Our first result demonstrates that using generalized sampling one obtains a stable a ..."
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Cited by 11 (9 self)
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Abstract In this paper we study the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples. For this, we use the recently introduced framework of generalized sampling. Our first result demonstrates that using generalized sampling one obtains a stable and accurate reconstruction, provided the number of Fourier samples grows linearly in the number of wavelet coefficients recovered. For the class of Daubechies wavelets we derive the exact constant of proportionality. Our second result concerns the optimality of generalized sampling for this problem. Under some mild assumptions we show that generalized sampling cannot be outperformed in terms of approximation quality by more than a constant factor. Moreover, for the class of socalled perfect methods, any attempt to lower the sampling ratio below a certain critical threshold necessarily results in exponential illconditioning. Thus generalized sampling provides a nearlyoptimal solution to this problem.
Beyond incoherence: stable and robust sampling strategies for compressive imaging,” preprint
, 2012
"... 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 9 (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 loworder wavelets and loworder frequencies are correlated, so compressed sensing theory does not immediately imply sampling strategies and reconstruction guarantees. In this paper we turn to a more refined notion of coherence – the socalled 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, so for matrices comprised of frequencies sampled from suitable powerlaw densities, we can prove the restricted isometry property with nearoptimal embedding dimensions. Consequently, the variabledensity sampling strategies we provide — which are independent of the ambient dimension up to logarithmic factors — allow for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by ℓ1minimization and by total variation minimization. 1
Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum
, 2013
"... The purpose of this paper is to report on recent approaches to reconstruction problems based on analog, or in other words, infinitedimensional, image and signal models. We describe three main contributions to this problem. First, linear reconstructions from sampled measurements via socalled genera ..."
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Cited by 8 (7 self)
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The purpose of this paper is to report on recent approaches to reconstruction problems based on analog, or in other words, infinitedimensional, image and signal models. We describe three main contributions to this problem. First, linear reconstructions from sampled measurements via socalled generalized sampling (GS). Second, the extension of generalized sampling to inverse and illposed problems. And third, the combination
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 loworder wavelets and loworder 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 socalled 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 powerlaw density, we can prove the restricted isometry property with nearoptimal embedding dimensions. Consequently, the variabledensity 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 ℓ1minimization 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
Generalized sampling: extension to frames and inverse and illposed problems. Inverse Problems
"... Generalized sampling is new framework for sampling and reconstruction in infinitedimensional Hilbert spaces. Given measurements (inner products) of an element with respect to one basis, it allows one to reconstruct in another, arbitrary basis, in a way that is both convergent and numerically stable ..."
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Cited by 7 (7 self)
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Generalized sampling is new framework for sampling and reconstruction in infinitedimensional Hilbert spaces. Given measurements (inner products) of an element with respect to one basis, it allows one to reconstruct in another, arbitrary basis, in a way that is both convergent and numerically stable. However, generalized sampling is thus far only valid for sampling and reconstruction in systems that comprise bases. Thus, in the first part of this paper we extend this framework from bases to frames, and provide fundamental sampling theorems for this more general case. The second part of the paper is concerned with extending the idea of generalized sampling to the solution of inverse and illposed problems. In particular, we introduce two generalized sampling frameworks for such problems, based on regularized and nonregularized approaches. We furnish evidence of the usefulness of the proposed theories by providing a number of numerical experiments. 1