Compressive Sensing Reconstruction Based On Non-local Structured Sparse Models

Compressive sensing(CS)is a recently emerging technique in signal sampling and processing.By achieving reconstruction of a sparse signal from a small number of random measurements,the CS theory has the potential of significantly improving the energy efficiency of sensors in many applications,such as radar imaging,magnetic resonance imaging and wireless sensor networks.Designing the effective CS reconstruction models and algorithms is critical to the success of CS theory.Three difficult problems in the CS reconstruction of natural images are: 1)most natural images are insufficient sparse in their orthogonal transform domains which is in conflict with the precondition of CS accurate reconstruction;2)it is difficult to distinguish image content in the presence of noise,which makes the performance of image CS recovery algorithms degrades seriously;3)the amount of the pixel datas is very large,resulting in the high complexity of CS reconstruction algorithms.In this paper,three aspects are carried out: 1)constructing non-local sparse structured models of images with the tools of statistical distribution approximation,composite sparse representation and Graph structure simplification;2)designing signal enhancement strategies aided with auxiliary information in a noisy environment;3)studying fast algorithms for solving non-local sparse models based on the approximate message passing algorithm.The details are as follows:1)A novel structured approximate message passing algorithm using a Laplacian scale mixture(LSM)prior is proposed.The AMP algorithm exploits the sparsity of signal in wavelet or gradient domain wich is not suitable for non-sparse natural images.Thus,the LSM model is used to describe the non-local similarity of images and is used as the higher-order statistical constraint of the AMP algorithm.Its essence is that we use the LSM distribution to model the sparsity of the singular values of the matrices built by similar patches,which denotes the similarity of image patches,and thus utilize the LSM model to describe the nonlocal similarity of images;to obtain reliable prior information,the scale parameters of the LSM model are estimated using the expectation-maximization(EM)algorithm.Finally,the singular value minimization problem is solved by the AMP algorithm to achieve the accurate image reconstruction.2)A modified AMP method based on composite sparse constraint is proposed.The rich structures of natural images make each sparse structured constraint hard to apply widely.To represent natural images more comprehensive,the composite sparse constraint built with the low-rank regularization and bilateral filter is utilized to improve the CS reconstruction quality.The low-rank property of similar image patches is beneficial to recover self-repetitive structures however some local details are not clear.On the other hand,the bilateral filter is beneficial to maintain edges.This composite model is solved with forward-backward iterations alternate between a gradient descent step and a proximal denoising correction,where the composite proximal correction is realized by using the composite splitting algorithm based on the variable splitting technique.3)Signal enhancement strategies aided with side information are designed to enhance the robustness of the AMP algorithm to noise.Since the performance of image CS recovery algorithms degrades seriously in the presence of noise,to overcome this drawback,in the frame of the AMP algorithm with the low-rank regularization,the reconstructed image in the previous iteration that contains some identified components is taken as side information to identify the signal.First,we use factor graph to describe the statistical correlation among the original image,CS measurements and side information.Second,the objective function of this side information-aided problem is deduced with Bayes formula,then the forward-backward iterations of the AMP algorithm are applied to solve this function.At last,the parameters in this model are estimated by maximum likelihood estimate method.Through the use of side information,signal is enhanced and thus the recovery of image details is improved.4)A fast CS reconstruction algorithm based on the graph sparsity regularization is proposed.To solve the problem of the slow speed of reconstruction algorithms based on non-local sparse models,graph theory method is introduced for describing the non-local similarity of images,then a new sparse model is designed by simplifying the graph structure to improve the running speed of structured AMP algorithms.First,to achieve more efficient sparse representation,the nonlocal similarity of images is constrained to be graph structured sparse,meanwhile,the structure of sparse coefficients is simplified from the complete graph structure to a star graph one where the coefficients are only connected with the mean node.Second,for realizing the adaptive reconstruction,the weighted norm is utilized to reflect the different significances of sparsity coefficients.In the frame of the AMP algorithm,Weight parameters and sparse coefficients can be estimated easily by introducing auxiliary variables.