Results 1 
5 of
5
Compressive Phase Retrieval via Generalized Approximate Message Passing
"... Abstract—In this paper, we propose a novel approach to compressive phase retrieval based on loopy belief propagation and, in particular, on the generalized approximate message passing (GAMP) algorithm. Numerical results show that the proposed PRGAMP algorithm has excellent phasetransition behavior ..."
Abstract

Cited by 28 (8 self)
 Add to MetaCart
(Show Context)
Abstract—In this paper, we propose a novel approach to compressive phase retrieval based on loopy belief propagation and, in particular, on the generalized approximate message passing (GAMP) algorithm. Numerical results show that the proposed PRGAMP algorithm has excellent phasetransition behavior, noise robustness, and runtime. In particular, for successful recovery of synthetic BernoullicircularGaussian signals, PRGAMP requires ≈ 4 times the number of measurements as a phaseoracle version of GAMP and, at moderate to large SNR, the NMSE of PRGAMP is only ≈ 3 dB worse than that of phaseoracle GAMP. A comparison to the recently proposed convexrelation approach known as “CPRL ” reveals PRGAMP’s superior phase transition and ordersofmagnitude faster runtimes, especially as the problem dimensions increase. When applied to the recovery of a 65kpixel grayscale image from 32k randomly masked magnitude measurements, numerical results show a median PRGAMP runtime of only 13.4 seconds. A. Phase retrieval I.
Statistical Image Recovery: A MessagePassing Perspective
"... Abstract—We review MAP and MMSEbased approaches to image recovery and their implementation via generalized approximate messagepassing (GAMP), highlighting recent results on GAMP convergence for general measurement operators. We consider the recovery of image x ∈ CN from noisy outputs y ∈ CM of kno ..."
Abstract
 Add to MetaCart
Abstract—We review MAP and MMSEbased approaches to image recovery and their implementation via generalized approximate messagepassing (GAMP), highlighting recent results on GAMP convergence for general measurement operators. We consider the recovery of image x ∈ CN from noisy outputs y ∈ CM of known linear measurement operator Φ ∈ CM×N. The “statistical ” approach to image recovery models the image x as a realization of random x ∼ px and the measurements as a realization of random y whose statistics are governed by a likelihood function of the form pyz(yΦx̂). Here, pyz is the pdf of y conditioned on the (hidden) transform outputs z = Φx and x ̂ is a hypothesis of the image. For clarity, we denote random quantities in sanserif font. In the maximum a posteriori (MAP) approach to statistical image recovery, one computes the most probable estimate of x given y, i.e., x̂MAP = argmaxx ̂ pxy(x̂y) = argmaxx ̂ pyx(yx̂)px(x̂)/py(y) = argmin x̂ − log pyz(yΦx̂) − log px(x̂)
1SpikeandSlab Approximate MessagePassing for HighDimensional PiecewiseConstant Recovery
"... Abstract—One of the challenges in Big Data is efficient handling of highdimensional data or signals. This paper proposes a novel AMP algorithm for solving highdimensional linear systems Y = HX +W ∈ RM which has a piecewiseconstant solution X ∈ RN, under a compressed sensing framework (M ≤ N). We ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract—One of the challenges in Big Data is efficient handling of highdimensional data or signals. This paper proposes a novel AMP algorithm for solving highdimensional linear systems Y = HX +W ∈ RM which has a piecewiseconstant solution X ∈ RN, under a compressed sensing framework (M ≤ N). We refer to the proposed AMP as ssAMP. This ssAMP algorithm is derived from the classical messagepassing rule over a bipartite graph which includes spikeandslab potential functions to encourage the piecewiseconstant nature of X. The ssAMP iteration includes a novel scalarwise denoiser satisfying the Lipschitz continuity, generating an approximate MMSE estimate of the signal. The Lipschitz continuity of our denoiser enables the ssAMP to use the state evolution framework, given by the works [16],[19], for MSE prediction. In addition, we empirically show that ssAMP has better phase transition characteristic than TVAMP [22] and GrAMPA [26] which are the existing AMPs for piecewiseconstant recovery. We also discuss computational efficiency, empirically showing that ssAMP has computational advantage over the other recent algorithms under a highdimensional setting. Index Terms—Compressed sensing, piecewiseconstant signals, approximate messagepassing (AMP), TVAMP, total variation denoising I.
1 Compressive Phase Retrieval via Generalized Approximate Message Passing
"... ar ..."
(Show Context)
From Denoising to Compressed Sensing
, 2014
"... A denoising algorithm seeks to remove perturbations or errors from a signal. The last three decades have seen extensive research devoted to this arena, and as a result, today’s denoisers are highly optimized algorithms that effectively remove large amounts of additive white Gaussian noise. A compres ..."
Abstract
 Add to MetaCart
A denoising algorithm seeks to remove perturbations or errors from a signal. The last three decades have seen extensive research devoted to this arena, and as a result, today’s denoisers are highly optimized algorithms that effectively remove large amounts of additive white Gaussian noise. A compressive sensing (CS) reconstruction algorithm seeks to recover a structured signal acquired using a small number of randomized measurements. Typical CS reconstruction algorithms can be cast as iteratively estimating a signal from a perturbed observation. This paper answers a natural question: How can one effectively employ a generic denoiser in a CS reconstruction algorithm? In response, in this paper, we develop a denoisingbased approximate message passing (DAMP) algorithm that is capable of highperformance reconstruction. We demonstrate that, for an appropriate choice of denoiser, DAMP offers stateoftheart CS recovery performance for natural images. We explain the exceptional performance of DAMP by analyzing some of its theoretical features. A critical insight in our approach is the use of an appropriate Onsager correction term in the DAMP iterations, which coerces the signal perturbation at each iteration to be very close to the white Gaussian noise that denoisers are typically designed to remove.