Bayesian Learning And Structure Prior Model Based Image Reconstruction For Compressive Sensing

The traditional sampling approach is based on Nyquist-Shanon sampling theorem,which states that the sampling rate must be at least twice the bandwidth of the signal.The ever growing demand for data, as well as advances in sening devices, havepromoted the use of high-bandwidth signals, for which the traditional smpling theoremhas to face the high cost, low efficiency and redundacy of data acquisition andprocessing, and the waste of resources of data storage and transmission. Compressivesensing (CS) theory brings an opportunity to solve the aforementioned problems of theconventional sampling. The data acquisition and compression are synchronouslyprocessed in CS. CS theory predicts that the signals or images that have a sparse orcompressible representation can be recovered from what was previously believed to behighly incomplete measurements, which are smpled at rates far lower than the Nyquistrate. CS enables a potentially large reduction in the sampling and computation costs forsensing signals.Designing the effective CS inversion algorithms is an important link ofsuccessfully extending and applying the theory for actual data models and acquisitionsystems. It is a goal of this thesis. After reviewing the recent development of CSreconstruction algorithms in detail, this thesis focuses on designing the reconstructionalgorithms for complicated images including the fast algorithms based onl1-minimization, lp-nonconvex minimization and structured CS. Using the methods ofmachine learning, Bayesian learning and intelligence computation as tools, we explorethe specific methods for improving the efficiency of algorithms, extracting andmodeling the structures of images, and designing the recovery algorithms, and evaluatethe performances of new schemes via experimental study. The main contributions ofthis thesis can be summarized as follows:(1) For the low computational efficiency of traditional CS algorithm based onl1-minimization, a fast stagewise LASSO algorithm is proposed. An l1-minimizationmodel based on the insensitive Huber Loss Function (IHLF), named insensitiveHuber-LASSO, is developed by introducing an IHLF to the objective function ofLASSO problem of CS reconstruction. A fast algorithm for solving the insensitiveHuber-LASSO is proposed by integrating a sequential learning scheme with a stagewiselearning strategy. The experimental results show that comparing with many traditionalCS algorithms, the proposed algorithm can provide the good reconstruction accuracy and remain high computational efficiency.(2) In Bayesian framework, lp-nonconvex minimization of the traditional CS forimage reconstruction is explored. The image data has an empirical distribution that ishighly peaked at zero with heavy tails. An lp-type sparse distribution, which well depictsthe above property of image, is used as the prior distribution for solving the sparselinear model. Under this prior assumption, estimating the maximum a posteriori (MAP)of image wavelet coefficients is an lp-nonconvex minimization problem due to thenonconvex of lp-type distribution. Motivated by the idea of MAP and sparse Bayesianlearning (SBL), lp-SBL algorithm based on learning the hyprparameters is proposed toestimate the MAP of image coefficients. A fast stagewise lp-SBL algorithm is developedby integrating a sequential learning scheme and a stagewise learning strategy withlp-SBL. Experimental results demonstrate that the proposed algorithm is a fast andeffective CS reconstruction algorithm.(3) As the further explore for lp-nonconvex optimization of CS reconstruction, theevolutionary algorithm is applied to solving the lp-problem of CS reconstruction for thefirst time. A new fast pursuit algorithm based on Bayesian framework and evolutionaycomputation is proposed in this thesis. In the proposed algorithm, a signal to berecovered is viewed as the sum of some signal components, the approximatereconstruction is achieved by iteratively estimating each signal component one by one.In Bayesian framework, the generalized Gaussian distributions with the differentparameters are employed as the prior of signal components. The high-dimensionalnonconvex optimization problem for CS reconstruction is decomposed into a series oflow-dimensinal nonconvex sub-optimization problems. The evolutionary algorithm isadopted to solve the sub-problems to estimate the MAP of signals. The proposedalgorithm has the lower computational complexity, and can improve the reconstructionqualities of those complicated images which have not strong sparsity.(4) The image reconstruction of structure-based CS is explored. To facilitate takingadvantage of the intra-statistical dependency model of the wavelet coefficients ofimages in CS recovery, a multivariate CS sampling scheme is proposed firstly totranslate the traditional CS problem to a multiple measurement vectors problem. A fastmultivariate pursuit algorithm is developed by modeling the intra-dependency of thewavelet coefficients during the provessing of CS recovery. In Bayesian framework, thedeveloped multivariate pursuit algorithms with several scale mixture models arecomputationally tractable and provide superior performance compared with manytraditional CS algorithms that do not employ the structure of the wavelet coefficients, and many state-of-the-art structured CS algorithms. In the thesis, we also develop aHMT based iterative reweighted multivariate pursuit algorithm by using the HMTstructure model of image wavelet coefficients to construct the weights. The non-zerosupports of the coefficients to be recovered can be automatically and efficientlydetrmined by using the HMT-based reweighted methods, thus enhancing the sparsity ofrecovered coefficients and impoving the reconstruction quality. A improved HMT-based weighted method is developed by means of the complete low frequencymeasurements, and the iterative reweighted multivariate pursuit algorithm based on theimproved HMT-based weights is derived. The experimental results show that theproposed algorithms perform better than many other structured CS algorithm in the caseof low CS measurement rates.(5) In the structure model based multivariate CS, edge information is introduced toCS reconstruction of images. In wavelet domain, an extraction scheme of image edgeinformation from the complete low frequency measurements is proposed, and theextracted edge information is used to determine the positions of non-zero coefficients tobe recovered. Then the position information is employed to guide the pursuit processingof multivariate algorithm based on joint recovery. The developed edge-basedmultivariate pursuit algorithm dramatically improves the reconstruction quality ofimages by means of the interaction of multivariate joint recovery and edge information,especially for those images with the obvious edges and high sparsity, such as syntheticimages and CT, MRI images.(6) The reconstruction problem of the structured CS in other transform domain ispreliminarily discussed in this thesis. Instead of orthogonal wavelet, wavelet-basedcontourlet transform (WBCT) is used as the sparse transform. The multivariatedependency of WBCT coefficients is modeled in CS recovery. The proposed algorithmprovides the superior reconstruction quality for those images with rich textures andcontours.