David L. Donoho

Compressed sensing

Suppose x is an unknown vector in Ropf <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> (a digital image or signal); we pla...

Compressed sensing

Suppose x is an unknown vector in Ropfm (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 ...

Atomic Decomposition by Basis Pursuit

The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries --- stationary wavelets, wavelet packets, cosine packe...

Uncertainty principles and ideal atomic decomposition

Suppose a discrete-time signal S(t), 0/spl les/t<N, is a superposition of atoms taken from a combined time-frequency dictionary made of spike sequences 1/sub {t=/spl tau/}/ and ...

Uncertainty Principles and Signal Recovery

The uncertainty principle can easily be generalized to cases where the “sets of concentration” are not intervals. Such generalizations are presented for continuous and discrete-...

Rapid MR Imaging with "Compressed Sensing" and Randomly Under-Sampled 3DFT Trajectories

Recently a rapid imaging method was proposed [1] that exploits the fact that sparse or compressible signals, such as MR images, 3D randomly under-sampled Cartesian trajectory ca...

Atomic Decomposition by Basis Pursuit

The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries---stationary wavelets, wavelet packets, cosine packets...

For most large underdetermined systems of linear equations the minimal 𝓁₁‐norm solution is also the sparsest solution

Abstract We consider linear equations y = Φ x where y is a given vector in ℝ n and Φ is a given n × m matrix with n &lt; m ≤ τ n , and we wish to solve for x ∈ ℝ m . We suppose ...

De-noising by soft-thresholding

Donoho and Johnstone (1994) proposed a method for reconstructing an unknown function f on [0,1] from noisy data d/sub i/=f(t/sub i/)+/spl sigma/z/sub i/, i=0, ..., n-1,t/sub i/=...

Ideal Spatial Adaptation by Wavelet Shrinkage

Journal Article Ideal spatial adaptation by wavelet shrinkage Get access David L Donoho, David L Donoho Department of Statistics, Stanford University, Stanford, California, U.S....

Adapting to Unknown Smoothness via Wavelet Shrinkage

Abstract We attempt to recover a function of unknown smoothness from noisy sampled data. We introduce a procedure, SureShrink, that suppresses noise by thresholding the empirica...

Minimax estimation via wavelet shrinkage

We attempt to recover an unknown function from noisy, sampled data.\nUsing orthonormal bases of compactly supported wavelets, we develop a nonlinear\nmethod which works in the w...

Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition

Unconditional Bases Are Optimal Bases for Data Compression and for Statistical Estimation

Statistical Estimation and Optimal Recovery

New formulas are given for the minimax linear risk in estimating a linear functional of an unknown object from indirect data contaminated with random Gaussian noise. The formula...

Minimax Risk Over Hyperrectangles, and Implications

Consider estimating the mean of a standard Gaussian shift when that mean is known to lie in an orthosymmetric quadratically convex set in $l_2$. Such sets include ellipsoids, hy...

Asymptotic minimax risk for sup-norm loss: Solution via optimal recovery

Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ ¹ minimization

Given a dictionary D = { d k } of vectors d k , we seek to represent a signal S as a linear combination S = ∑ k γ( k ) d k , with scalar coefficients γ ( k ). In particular, we ...

Stable recovery of sparse overcomplete representations in the presence of noise

Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in ...

Image decomposition via the combination of sparse representations and a variational approach

The separation of image content into semantic parts plays a vital role in applications such as compression, enhancement, restoration, and more. In recent years, several pioneeri...

Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)

Ideal spatial adaptation by wavelet shrinkage

SUMMARY With ideal spatial adaptation, an oracle furnishes information about how best to adapt a spatially variable estimator, whether piecewise constant, piecewise polynomial, ...

Adapting to unknown sparsity by controlling the false discovery rate

We attempt to recover an n-dimensional vector observed in white noise, where n is large and the vector is known to be sparse, but the degree of sparsity is unknown. We consider ...

Minimax risk overl p -balls forl p -error