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A signature result in compressed sensing is that Gaussian random sampling achieves stable and robust recovery of sparse vectors under optimal conditions on the number of measurements. However, in the context of image reconstruction, it has been extensively documented that sampling strategies based on Fourier measurements outperform this purportedly optimal approach. Motivated by this seeming paradox, we investigate the problem of optimal sampling for compressed sensing. Rigorously combining the theories of wavelet approximation and infinite-dimensional compressed sensing, our analysis leads to new error bounds in terms of the total number of measurements $m$ for the approximation of piecewise $alpha$-H{o}lder functions. Our theoretical findings suggest that Fourier sampling outperforms random Gaussian sampling when the Holder exponent $alpha$ is large enough. Moreover, we establish a provably optimal sampling strategy. This work is an important first step towards the resolution of the claimed paradox, and provides a clear theoretical justification for the practical success of compressed sensing techniques in imaging problems.
In this paper, based on a successively accuracy-increasing approximation of the $ell_0$ norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave
This letter investigates the joint recovery of a frequency-sparse signal ensemble sharing a common frequency-sparse component from the collection of their compressed measurements. Unlike conventional arts in compressed sensing, the frequencies follow
In this work, we analyze the failing sets of the interval-passing algorithm (IPA) for compressed sensing. The IPA is an efficient iterative algorithm for reconstructing a k-sparse nonnegative n-dimensional real signal x from a small number of linear
Evaluating the statistical dimension is a common tool to determine the asymptotic phase transition in compressed sensing problems with Gaussian ensemble. Unfortunately, the exact evaluation of the statistical dimension is very difficult and it has be
Snapshot compressed sensing (CS) refers to compressive imaging systems in which multiple frames are mapped into a single measurement frame. Each pixel in the acquired frame is a noisy linear mapping of the corresponding pixels in the frames that are