A novel multi-morphological representation model for solving the nonlocal similarity-based image reconstruction from compressed measurements is introduced in this paper. Under the probabilistic framework, the proposed approach provides the nonlocal similarity clustering for image patches by using the Gaussian mixture models, and endows a multi-morphological representation for image patches in each cluster by using the Gaussians that represent the different features to model the morphological components. Using the simple alternating iteration, the developed piecewise morphological diversity estimation (PMDE) algorithm can effectively estimate the MAP of morphological components, thus resulting in the nonlinear estimation for image patches. We extend the PMDE to a piecewise morphological diversity sparse estimation by using the constrained Gaussians with the low-rank covariance matrices, to gain the performance improvements. We report the experimental results on image compressed sensing in the case of sensing nonoverlapping patches with Gaussian random matrices. The results demonstrate that our algorithms can suppress undesirable block artifacts efficiently, and delivers reconstructed images with higher qualities than other state-of-the-art methods.

Compressive sensing (CS) is a theory that one may achieve an exact signal reconstruction from sufficient CS measurements taken from a sparse signal. However, in practical applications, the transform coefficients of SAR images usually have weak sparsity. Exactly reconstructing these images is very challenging. A new Bayesian evolutionary pursuit algorithm (BEPA) is proposed in this paper. A signal is represented as the sum of a main signal and some residual signals, and the generalized Gaussian distribution (GGD) is employed as the prior of the main signal and the residual signals. BEPA decomposes the residual iteratively and estimates the maximum a posteriori of the main signal and the residual signals by solving a sequence of subproblems to achieve the approximate CS reconstruction of the signal. Under the assumption of GGD with the parameter 0 < p < 1, the evolutionary algorithm (EA) is introduced to CS reconstruction for the first time. The better reconstruction performance can be achieved by searching the global optimal solutions of subproblems with EA. Numerical experiments demonstrate that the important features of SAR images (e.g., the point and line targets) can be well preserved by our algorithm, and the superior reconstruction performance can be obtained at the same time.

Most wavelet-based reconstruction methods of compressive sensing (CS) are developed under the independence assumption of the wavelet coefficients. However, the wavelet coefficients of images have significant statistical dependencies. Lots of multivariate prior models for the wavelet coefficients of images have been proposed and successfully applied to the image estimation problems. In this paper, the statistical structures of the wavelet coefficients are considered for CS reconstruction of images that are sparse or compressive in wavelet domain. A multivariate pursuit algorithm (MPA) based on the multivariate models is developed. Several multivariate scale mixture models are used as the prior distributions of MPA. Our method reconstructs the images by means of modeling the statistical dependencies of the wavelet coefficients in a neighborhood. The proposed algorithm based on these scale mixture models provides superior performance compared with many state-of-the-art compressive sensing reconstruction algorithms.