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- David L. Donoho
- IEEE Transactions on Information Theory
- 2006

Suppose x is an unknown vector in Ropf<sup>m</sup> (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the… (More)

With ideal spatial adaptation, an oracle furnishes information about how best to adapt a spatially variable estimator, whether piecewise constant, piecewise polynomial, variable knot spline, or variable bandwidth kernel, to the unknown function. Estimation with the aid of an oracle o ers dramatic advantages over traditional linear estimation by nonadaptive… (More)

- Scott Saobing Chen, David L. Donoho, Michael A. Saunders
- SIAM Review
- 1998

The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries — stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of… (More)

- David L. Donoho
- IEEE Trans. Information Theory
- 1995

Donoho and Johnstone (1992a) proposed a method for reconstructing an unknown function f on [0; 1] from noisy data di = f(ti) + zi, i = 0; : : : ; n 1, ti = i=n, zi iid N(0; 1). The reconstruction f̂ n is de ned in the wavelet domain by translating all the empirical wavelet coe cients of d towards 0 by an amount p 2 log(n) = p n. We prove two results about… (More)

We attempt to recover a function of unknown smoothness from noisy, sampled data. We introduce a procedure, SureShrink, which suppresses noise by thresholding the empirical wavelet coe cients. The thresholding is adaptive: a threshold level is assigned to each dyadic resolution level by the principle of minimizing the Stein Unbiased Estimate of Risk (Sure)… (More)

- Michael Lustig, David L. Donoho, John M. Pauly
- Magnetic resonance in medicine
- 2007

The sparsity which is implicit in MR images is exploited to significantly undersample k-space. Some MR images such as angiograms are already sparse in the pixel representation; other, more complicated images have a sparse representation in some transform domain-for example, in terms of spatial finite-differences or their wavelet coefficients. According to… (More)

- Emmanuel J. Candès, Laurent Demanet, David L. Donoho, Lexing Ying
- Multiscale Modeling & Simulation
- 2006

This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform [12, 10] in two and three dimensions. The first digital transformation is based on unequally-spaced fast Fourier transforms (USFFT) while the second is based on the wrapping of specially selected Fourier samples. The two… (More)

- David L. Donoho, Arian Maleki, Andrea Montanari
- Proceedings of the National Academy of Sciences…
- 2009

Compressed sensing aims to undersample certain high-dimensional signals yet accurately reconstruct them by exploiting signal characteristics. Accurate reconstruction is possible when the object to be recovered is sufficiently sparse in a known basis. Currently, the best known sparsity-undersampling tradeoff is achieved when reconstructing by convex… (More)

- David L. Donoho, Michael Elad, Vladimir N. Temlyakov
- IEEE Transactions on Information Theory
- 2006

Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of… (More)

- David L. Donoho, Michael Elad
- Proceedings of the National Academy of Sciences…
- 2003

Given a dictionary D = {d(k)} of vectors d(k), we seek to represent a signal S as a linear combination S = summation operator(k) gamma(k)d(k), with scalar coefficients gamma(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considered the special case where D is… (More)