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Compressive imaging via approximate message passing with image denoising,” arXiv:1405.4429
, 2014
"... We consider compressive imaging problems, where images are reconstructed from a reduced number of linear measurements. Our objective is to improve over existing compressive imaging algorithms in terms of both reconstruction error and runtime. To pursue our objective, we propose compressive imaging ..."
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We consider compressive imaging problems, where images are reconstructed from a reduced number of linear measurements. Our objective is to improve over existing compressive imaging algorithms in terms of both reconstruction error and runtime. To pursue our objective, we propose compressive imaging algorithms that employ the approximate message passing (AMP) algorithm. AMP is an iterative signal reconstruction algorithm that performs scalar denoising at each iteration; in order for AMP to reconstruct the original input signal well, a good scalar denoiser must be used. We apply two wavelet based image denoisers within AMP. The first denoiser is the “amplitudescaleinvariant Bayes estimator ” (ABE), and the second is an adaptive Wiener filter; we call our AMP based algorithms for compressive imaging AMPABE and AMPWiener. Numerical results show that both AMPABE and AMPWiener significantly improve over the state of the art in terms of runtime. In terms of reconstruction quality, AMPWiener offers lower mean square error (MSE) than existing compressive imaging algorithms. In contrast, AMPABE has higher MSE, because ABE does not denoise as well as the adaptive Wiener filter.
Sparse estimation with the swept approximated messagepassing algorithm,” Arxiv preprint arxiv:1406.4311
, 2014
"... Approximate Message Passing (AMP) has been shown to be a superior method for inference problems, such as the recovery of signals from sets of noisy, lowerdimensionality measurements, both in terms of reconstruction accuracy and in computational efficiency. However, AMP suffers from serious converge ..."
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Approximate Message Passing (AMP) has been shown to be a superior method for inference problems, such as the recovery of signals from sets of noisy, lowerdimensionality measurements, both in terms of reconstruction accuracy and in computational efficiency. However, AMP suffers from serious convergence issues in contexts that do not exactly match its assumptions. We propose a new approach to stabilizing AMP in these contexts by applying AMP updates to individual coefficients rather than in parallel. Our results show that this change to the AMP iteration can provide theoretically expected, but hitherto unobtainable, performance for problems on which the standard AMP iteration diverges. Additionally, we find that the computational costs of this swept coefficient update scheme is not unduly burdensome, allowing it to be applied efficiently to signals of large dimensionality. I.
Compressed Sensing via Universal Denoising and Approximate Message Passing
"... Abstract—We study compressed sensing (CS) signal reconstruction problems where an input signal is measured via matrix multiplication under additive white Gaussian noise. Our signals are assumed to be stationary and ergodic, but the input statistics are unknown; the goal is to provide reconstruction ..."
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Abstract—We study compressed sensing (CS) signal reconstruction problems where an input signal is measured via matrix multiplication under additive white Gaussian noise. Our signals are assumed to be stationary and ergodic, but the input statistics are unknown; the goal is to provide reconstruction algorithms that are universal to the input statistics. We present a novel algorithm that combines: (i) the approximate message passing (AMP) CS reconstruction framework, which converts the matrix channel recovery problem into scalar channel denoising; (ii) a universal denoising scheme based on context quantization, which partitions the stationary ergodic signal denoising into independent and identically distributed (i.i.d.) subsequence denoising; and (iii) a density estimation approach that approximates the probability distribution of an i.i.d. sequence by fitting a Gaussian mixture (GM) model. In addition to the algorithmic framework, we provide three contributions: (i) numerical results showing that state evolution holds for nonseparable Bayesian slidingwindow denoisers; (ii) a universal denoiser that does not require the input signal to be bounded; and (iii) we modify the GM learning algorithm, and extend it to an i.i.d. denoiser. Our universal CS recovery algorithm compares favorably with existing reconstruction algorithms in terms of both reconstruction quality and runtime, despite not knowing the input statistics of the stationary ergodic signal. Index Terms—approximate message passing, compressed sensing, Gaussian mixture model, universal denoising.
Approximate Message Passing with Universal Denoising
"... Abstract—We study compressed sensing (CS) signal reconstruction problems where an input signal is measured via matrix multiplication under additive white Gaussian noise. Our signals are assumed to be stationary and ergodic, but the input statistics are unknown; the goal is to provide reconstruction ..."
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Abstract—We study compressed sensing (CS) signal reconstruction problems where an input signal is measured via matrix multiplication under additive white Gaussian noise. Our signals are assumed to be stationary and ergodic, but the input statistics are unknown; the goal is to provide reconstruction algorithms that are universal to the input statistics. We present a novel algorithmic framework that combines: (i) the approximate message passing (AMP) CS reconstruction framework, which solves the matrix channel recovery problem by iterative scalar channel denoising; (ii) a universal denoising scheme based on context quantization, which partitions the stationary ergodic signal denoising into independent and identically distributed (i.i.d.) subsequence denoising; and (iii) a density estimation approach that approximates the probability distribution of an i.i.d. sequence by fitting a Gaussian mixture (GM) model. In addition to the algorithmic framework, we provide three contributions: (i) numerical results showing that state evolution holds for nonseparable Bayesian slidingwindow denoisers; (ii) an i.i.d. denoiser based on a modified GM learning algorithm; and (iii) a universal denoiser that does not require the input signal to be bounded. We provide two implementations of our universal CS recovery algorithm with one being faster and the other being more accurate. The two implementations compare favorably with existing reconstruction algorithms in terms of both reconstruction quality and runtime. Index Terms—approximate message passing, compressed sensing, Gaussian mixture model, universal denoising.
On the Performance of Turbo Signal Recovery with Partial DFT Sensing Matrices
"... Abstract—This letter is on the performance of the turbo signal recovery (TSR) algorithm for partial discrete Fourier transform (DFT) matrices based compressed sensing. Based on state evolution analysis, we prove that TSR with a partial DFT sensing matrix outperforms the wellknown approximate messa ..."
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Abstract—This letter is on the performance of the turbo signal recovery (TSR) algorithm for partial discrete Fourier transform (DFT) matrices based compressed sensing. Based on state evolution analysis, we prove that TSR with a partial DFT sensing matrix outperforms the wellknown approximate message passing (AMP) algorithm with an independent identically distributed (IID) sensing matrix. Index Terms—AMP, partial DFT, signal recovery, state evolution, turbo compressed sensing. I.
Compressive Hyperspectral Imaging via Approximate Message Passing
"... Abstract—We consider a compressive hyperspectral imaging reconstruction problem, where threedimensional spatiospectral information about a scene is sensed by a coded aperture snapshot spectral imager (CASSI). The CASSI imaging process can be modeled as suppressing threedimensional coded and shift ..."
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Abstract—We consider a compressive hyperspectral imaging reconstruction problem, where threedimensional spatiospectral information about a scene is sensed by a coded aperture snapshot spectral imager (CASSI). The CASSI imaging process can be modeled as suppressing threedimensional coded and shifted voxels and projecting these onto a twodimensional plane, such that the number of acquired measurements is greatly reduced. On the other hand, because the measurements are highly compressive, the reconstruction process becomes challenging. We previously proposed a compressive imaging reconstruction algorithm that is applied to twodimensional images based on the approximate message passing (AMP) framework. AMP is an iterative algorithm that can be used in signal and image reconstruction by performing denoising at each iteration. We employed an adaptive Wiener filter as the image denoiser, and called our algorithm “AMPWiener. ” In this paper, we extend AMPWiener to threedimensional hyperspectral image reconstruction, and call it “AMP3DWiener. ” Applying the AMP framework to the CASSI system is challenging, because the matrix that models the CASSI system is highly sparse, and such a matrix is not suitable to AMP and makes it difficult for AMP to converge. Therefore, we modify the adaptive Wiener filter and employ a technique called damping to solve for the divergence issue of AMP. Our approach is applied in nature, and the numerical experiments show that AMP3DWiener outperforms existing widelyused algorithms such as gradient projection for sparse reconstruction (GPSR) and twostep iterative shrinkage/thresholding (TwIST) given a similar amount of runtime. Moreover, in contrast to GPSR and TwIST, AMP3DWiener need not tune any parameters, which simplifies the reconstruction process. Index Terms—Approximate message passing, CASSI, compressive hyperspectral imaging, gradient projection for sparse reconstruction, image denoising, twostep iterative shrinkage/thresholdng, Wiener filtering.
Approximate Message Passing in Coded Aperture Snapshot Spectral Imaging
"... Abstract—We consider a compressive hyperspectral imaging reconstruction problem, where threedimensional spatiospectral information about a scene is sensed by a coded aperture snapshot spectral imager (CASSI). The approximate message passing (AMP) framework is utilized to reconstruct hyperspectral ..."
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Abstract—We consider a compressive hyperspectral imaging reconstruction problem, where threedimensional spatiospectral information about a scene is sensed by a coded aperture snapshot spectral imager (CASSI). The approximate message passing (AMP) framework is utilized to reconstruct hyperspectral images from CASSI measurements, and an adaptive Wiener filter is employed as a threedimensional image denoiser within AMP. We call our algorithm “AMP3DWiener.” The simulation results show that AMP3DWiener outperforms existing widelyused algorithms such as gradient projection for sparse reconstruction (GPSR) and twostep iterative shrinkage/thresholding (TwIST) given the same amount of runtime. Moreover, in contrast to GPSR and TwIST, AMP3DWiener need not tune any parameters, which simplifies the reconstruction process. Index Terms—Approximate message passing, CASSI, compressive hyperspectral imaging, Wiener filtering. I.
Scalable Inference for Neuronal Connectivity from Calcium Imaging
"... Fluorescent calcium imaging provides a potentially powerful tool for inferring connectivity in neural circuits with up to thousands of neurons. However, a key challenge in using calcium imaging for connectivity detection is that current systems often have a temporal response and frame rate that can ..."
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Fluorescent calcium imaging provides a potentially powerful tool for inferring connectivity in neural circuits with up to thousands of neurons. However, a key challenge in using calcium imaging for connectivity detection is that current systems often have a temporal response and frame rate that can be orders of magnitude slower than the underlying neural spiking process. Bayesian inference methods based on expectationmaximization (EM) have been proposed to overcome these limitations, but are often computationally demanding since the Estep in the EM procedure typically involves state estimation for a highdimensional nonlinear dynamical system. In this work, we propose a computationally fast method for the state estimation based on a hybrid of loopy belief propagation and approximate message passing (AMP). The key insight is that a neural system as viewed through calcium imaging can be factorized into simple scalar dynamical systems for each neuron with linear interconnections between the neurons. Using the structure, the updates in the proposed hybrid AMP methodology can be computed by a set of onedimensional state estimation procedures and linear transforms with the connectivity matrix. This yields a computationally scalable method for inferring connectivity of large neural circuits. Simulations of the method on realistic neural networks demonstrate good accuracy with computation times that are potentially significantly faster than current approaches based on Markov Chain Monte Carlo methods. 1
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 ..."
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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.
Approximate Message Passingbased Compressed Sensing Reconstruction with Generalized Elastic Net
"... In this paper, we study the compressed sensing reconstruction problem with generalized elastic net prior (GENP), where a sparse signal is sampled via a noisy underdetermined linear observation system, and an additional initial estimation of the signal (the GENP) is available during the reconstructi ..."
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In this paper, we study the compressed sensing reconstruction problem with generalized elastic net prior (GENP), where a sparse signal is sampled via a noisy underdetermined linear observation system, and an additional initial estimation of the signal (the GENP) is available during the reconstruction. We first incorporate the GENP into the LASSO and the approximate message passing (AMP) frameworks, denoted by GENPLASSO and GENPAMP respectively. We then focus on GENPAMP and investigate its parameter selection, state evolution, and noisesensitivity analysis. A practical parameterless version of the GENPAMP is also developed, which does not need to know the sparsity of the unknown signal and the variance of the GENP. Simulation results with 1D data and two different imaging applications are presented to demonstrate the efficiency of the proposed schemes. Keywords: Compressed sensing, approximate message passing, elastic net prior, state evolution, phase transition. 1.