In a plethora of applications dealing with inverse problems, e.g. in image processing, social networks, compressive sensing, biological data processing etc., the signal of interest is known to be structured in several ways at the same time. This premise has recently guided the research to the innovative and meaningful idea of imposing multiple constraints on the unknown parameters involved in the problem under study. For instance, when dealing with problems whose unknown parameters form sparse and low-rank matrices, the adoption of suitably combined constraints imposing sparsity and lowrankness, is expected to yield substantially enhanced estimation results. In this paper, we address the spectral unmixing problem in hyperspectral images. Specifically, two novel unmixing algorithms are introduced, in an attempt to exploit both spatial correlation and sparse representation of pixels lying in homogeneous regions of hyperspectral images. To this end, a novel mixed penalty term is first defined consisting of the sum of the weighted 1 and the weighted nuclear norm of the abundance matrix corresponding to a small area of the image determined by a sliding square window. This penalty term is then used to regularize a conventional quadratic cost function and impose simultaneously sparsity and row-rankness on the abundance matrix. The resulting regularized cost function is minimized by a) an incremental proximal sparse and low-rank unmixing algorithm and b) an algorithm based on the alternating minimization method of multipliers (ADMM). The effectiveness of the proposed algorithms is illustrated in experiments conducted both on simulated and real data. Index TermsSemi-supervised spectral unmixing, hyperspectral images, simultaneously sparse and low-rank matrices, proximal methods, alternating direction method of multipliers (ADMM), abundance estimation

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Recently, a number of mostly -norm regularized least-squares-type deterministic algorithms have been proposed to address the problem of sparse adaptive signal estimation and system identification. From a Bayesian perspective, this task is equivalent to maximum a posteriori probability estimation under a sparsity promoting heavy-tailed prior for the parameters of interest. Following a different approach, this paper develops a unifying framework of sparse variational Bayes algorithms that employ heavy-tailed priors in conjugate hierarchical form to facilitate posterior inference. The resulting fully automated variational schemes are first presented in a batch iterative form. Then, it is shown that by properly exploiting the structure of the batch estimation task, new sparse adaptive variational Bayes algorithms can be derived, which have the ability to impose and track sparsity during real-time processing in a time-varying environment. The most important feature of the proposed algorithms is that they completely eliminate the need for computationally costly parameter fine-tuning, a necessary ingredient of sparse adaptive deterministic algorithms. Extensive simulation results are provided to demonstrate the effectiveness of the new sparse adaptive variational Bayes algorithms against state-of-the-art deterministic techniques for adaptive channel estimation. The results show that the proposed algorithms are numerically robust and exhibit in general superior estimation performance compared to their deterministic counterparts.Index Terms-Sparse adaptive estimation, online variational Bayes, Bayesian models, sparse Bayesian learning, Student-t distribution, Laplace distribution, generalized inverse Gaussian distribution, Bayesian inference. 1053-587X

This article presents a comparative study of three different types of estimators used for supervised linear unmixing of two MEx/OMEGA hyperspectral cubes. The algorithms take into account the constraints of the abundance fractions, in order to get physically interpretable results. Abundance maps show that the Bayesian maximum a posteriori probability (MAP) estimator proposed in Themelis and Rontogiannis (2008) outperforms the other two schemes, offering a compromise between complexity and estimation performance. Thus, the MAP estimator is a candidate algorithm to perform ice and minerals detection on large hyperspectral datasets