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—We introduce a mathematical framework that bridges a substantial gap between compressed sensing theory and its current use in real-world applications. Although completely general, one of the principal applications for our framework is the Magnetic Resonance Imaging (MRI) problem. Our theory provides a comprehensive explanation for the abundance of(More)
This paper provides an extension of compressed sensing which bridges a substantial gap between existing theory and its current use in real-world applications. It introduces a mathematical framework that generalizes the three standard pillars of compressed sensing - namely, sparsity, incoherence and uniform random subsampling - to three new concepts:(More)
Generalized sampling is a recently developed linear framework for sampling and reconstruction in separable Hilbert spaces. It allows one to recover any element in any finite-dimensional subspace given finitely many of its samples with respect to an arbitrary frame. Unlike more common approaches for this problem, such as the consistent reconstruction(More)
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(More)
Many of the applications of compressed sensing have been based on variable density sampling, where certain sections of the sampling coefficients are sampled more densely. Furthermore, it has been observed that these sampling schemes are dependent not only on sparsity but also on the sparsity structure of the underlying signal. This paper extends the result(More)
—We introduce a mathematical framework that bridges a substantial gap between compressed sensing theory and its current use in applications. Although completely general, one of the principal applications for our framework is the Magnetic Resonance Imaging (MRI) problem. Our theory provides an explanation for the abundance of numerical evidence demonstrating(More)
—We consider 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 in the context of compactly supported orthonormal wavelet bases. Our first result demonstrates that using generalized sampling one obtains a stable and accurate(More)
This paper considers the use of the anisotropic total variation seminorm to recover a two dimensional vector x ∈ C N ×N from its partial Fourier coefficients, sampled along Cartesian lines. We prove that if (x k,j − x k−1,j) k,j has at most s1 nonzero coefficients in each column and (x k,j − x k,j−1) k,j has at most s2 nonzero coefficients in each row,(More)
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