Content-Aware Compressive Sensing Recovery Using Laplacian Scale Mixture Priors and Side Information

Nonlocal methods have shown great potential in many image restoration tasks including compressive sensing (CS) reconstruction through use of image self-similarity prior. However, they are still limited in recovering fine-scale details and sharp features, when rich repetitive patterns cannot be guaranteed; moreover the CS measurements are corrupted. In this paper, we propose a novel CS recovery algorithm that combines nonlocal sparsity with local and global prior, which soften and complement the self-similarity assumption for irregular structures. First, a Laplacian scale mixture (LSM) prior is utilized to model dependencies among similar patches. For achieving group sparsity, each singular value of similar packed patches is modeled as a Laplacian distribution with a variable scale parameter. Second, a global prior and a compensation-based sparsity prior of local patch are designed in order to maintain differences between packed patches. The former refers to a prediction which integrates the information at the independent processing stage and is used as side information, while the latter enforces a small (i.e., sparse) prediction error and is also modeled with the LSM model so as to obtain local sparsity. Afterward, we derive an efficient algorithm based on the expectation-maximization (EM) and approximate message passing (AMP) frame for the maximum a posteriori (MAP) estimation of the sparse coefficients. Numerical experiments show that the proposed method outperforms many CS recovery algorithms.

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An Entropy-Based Algorithm with Nonlocal Residual Learning for Image Compressive Sensing Recovery

Entropy ◽

10.3390/e21090900 ◽

2019 ◽

Vol 21 (9) ◽

pp. 900 ◽

Cited By ~ 2

Author(s):

Zhonghua Xie ◽

Lingjun Liu ◽

Cui Yang

Keyword(s):

Compressive Sensing ◽

Latent Variables ◽

Message Passing ◽

Signal Reconstruction ◽

Measurement Data ◽

Image Recovery ◽

Group Sparsity ◽

Approximate Message Passing ◽

Em Method ◽

Laplacian Distribution

Image recovery from compressive sensing (CS) measurement data, especially noisy data has always been challenging due to its implicit ill-posed nature, thus, to seek a domain where a signal can exhibit a high degree of sparsity and to design an effective algorithm have drawn increasingly more attention. Among various sparsity-based models, structured or group sparsity often leads to more powerful signal reconstruction techniques. In this paper, we propose a novel entropy-based algorithm for CS recovery to enhance image sparsity through learning the group sparsity of residual. To reduce the residual of similar packed patches, the group sparsity of residual is described by a Laplacian scale mixture (LSM) model, therefore, each singular value of the residual of similar packed patches is modeled as a Laplacian distribution with a variable scale parameter, to exploit the benefits of high-order dependency among sparse coefficients. Due to the latent variables, the maximum a posteriori (MAP) estimation of the sparse coefficients cannot be obtained, thus, we design a loss function for expectation–maximization (EM) method based on relative entropy. In the frame of EM iteration, the sparse coefficients can be estimated with the denoising-based approximate message passing (D-AMP) algorithm. Experimental results have shown that the proposed algorithm can significantly outperform existing CS techniques for image recovery.

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Near-optimal matrix recovery from random linear measurements

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1705490115 ◽

2018 ◽

Vol 115 (28) ◽

pp. 7200-7205 ◽

Cited By ~ 3

Author(s):

Elad Romanov ◽

Matan Gavish

Keyword(s):

Phase Transition ◽

Message Passing ◽

Iterative Solvers ◽

Transition Curve ◽

Linear Measurements ◽

Information Theoretic ◽

Matrix Estimation ◽

Recovery Algorithm ◽

Matrix Recovery ◽

Approximate Message Passing

In matrix recovery from random linear measurements, one is interested in recovering an unknown M-by-N matrixX0fromn<MNmeasurementsyi=Tr(Ai⊤X0), where eachAiis an M-by-N measurement matrix with i.i.d. random entries,i=1,…,n. We present a matrix recovery algorithm, based on approximate message passing, which iteratively applies an optimal singular-value shrinker—a nonconvex nonlinearity tailored specifically for matrix estimation. Our algorithm typically converges exponentially fast, offering a significant speedup over previously suggested matrix recovery algorithms, such as iterative solvers for nuclear norm minimization (NNM). It is well known that there is a recovery tradeoff between the information content of the objectX0to be recovered (specifically, its matrix rank r) and the number of linear measurements n from which recovery is to be attempted. The precise tradeoff between r and n, beyond which recovery by a given algorithm becomes possible, traces the so-called phase transition curve of that algorithm in the(r,n)plane. The phase transition curve of our algorithm is noticeably better than that of NNM. Interestingly, it is close to the information-theoretic lower bound for the minimal number of measurements needed for matrix recovery, making it not only state of the art in terms of convergence rate, but also near optimal in terms of the matrices it successfully recovers.

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