Compressed sensing imaging techniques for radio interferometry

Radio interferometry probes astrophysical signals through incomplete and noisy Fourier measurements. The theory of compressed sensing demonstrates that such measurements may actually suffice for accurate reconstruction of sparse or compressible signals. We propose new generic imaging techniques based on convex optimization for global minimization problems defined in this context. The versatility of the framework notably allows introduction of specific prior information on the signals, which offers the possibility of significant improvements of reconstruction relative to the standard local matching pursuit algorithm CLEAN used in radio astronomy. We illustrate the potential of the approach by studying reconstruction performances on simulations of two different kinds of signals observed with very generic interferometric configurations. The first kind is an intensity field of compact astrophysical objects. The second kind is the imprint of cosmic strings in the temperature field of the cosmic microwave background radiation, of particular interest for cosmology.Comment: 10 pages, 1 figure. Version 2 matches version accepted for publication in MNRAS. Changes includes: writing corrections, clarifications of arguments, figure update, and a new subsection 4.1 commenting on the exact compliance of radio interferometric measurements with compressed sensin

Non-convex regularization in remote sensing

In this paper, we study the effect of different regularizers and their implications in high dimensional image classification and sparse linear unmixing. Although kernelization or sparse methods are globally accepted solutions for processing data in high dimensions, we present here a study on the impact of the form of regularization used and its parametrization. We consider regularization via traditional squared (2) and sparsity-promoting (1) norms, as well as more unconventional nonconvex regularizers (p and Log Sum Penalty). We compare their properties and advantages on several classification and linear unmixing tasks and provide advices on the choice of the best regularizer for the problem at hand. Finally, we also provide a fully functional toolbox for the community.Comment: 11 pages, 11 figure

Composite Minimization: Proximity Algorithms and Their Applications

ABSTRACT Image and signal processing problems of practical importance, such as incomplete data recovery and compressed sensing, are often modeled as nonsmooth optimization problems whose objective functions are the sum of two terms, each of which is the composition of a prox-friendly function with a matrix. Therefore, there is a practical need to solve such optimization problems. Besides the nondifferentiability of the objective functions of the associated optimization problems and the larger dimension of the underlying images and signals, the sum of the objective functions is not, in general, prox-friendly, which makes solving the problems challenging. Many algorithms have been proposed in literature to attack these problems by making use of the prox-friendly functions in the problems. However, the efficiency of these algorithms relies heavily on the underlying structures of the matrices, particularly for large scale optimization problems. In this dissertation, we propose a novel algorithmic framework that exploits the availability of the prox-friendly functions, without requiring any structural information of the matrices. This makes our algorithms suitable for large scale optimization problems of interest. We also prove the convergence of the developed algorithms. This dissertation has three main parts. In part 1, we consider the minimization of functions that are the sum of the compositions of prox-friendly functions with matrices. We characterize the solutions to the associated optimization problems as the solutions of fixed point equations that are formulated in terms of the proximity operators of the dual of the prox-friendly functions. By making use of the flexibility provided by this characterization, we develop a block Gauss-Seidel iterative scheme for finding a solution to the optimization problem and prove its convergence. We discuss the connection of our developed algorithms with some existing ones and point out the advantages of our proposed scheme. In part 2, we give a comprehensive study on the computation of the proximity operator of the ℓp-norm with 0 ≤ p \u3c 1. Nonconvexity and non-smoothness have been recognized as important features of many optimization problems in image and signal processing. The nonconvex, nonsmooth ℓp-regularization has been recognized as an efficient tool to identify the sparsity of wavelet coefficients of an image or signal under investigation. To solve an ℓp-regularized optimization problem, the proximity operator of the ℓp-norm needs to be computed in an accurate and computationally efficient way. We first study the general properties of the proximity operator of the ℓp-norm. Then, we derive the explicit form of the proximity operators of the ℓp-norm for p ∈ {0, 1/2, 2/3, 1}. Using these explicit forms and the properties of the proximity operator of the ℓp-norm, we develop an efficient algorithm to compute the proximity operator of the ℓp-norm for any p between 0 and 1. In part 3, the usefulness of the research results developed in the previous two parts is demonstrated in two types of applications, namely, image restoration and compressed sensing. A comparison with the results from some existing algorithms is also presented. For image restoration, the results developed in part 1 are applied to solve the ℓ2-TV and ℓ1-TV models. The resulting restored images have higher peak signal-to-noise ratios and the developed algorithms require less CPU time than state-of-the-art algorithms. In addition, for compressed sensing applications, our algorithm has smaller ℓ2- and ℓ∞-errors and shorter computation times than state-ofthe- art algorithms. For compressed sensing with the ℓp-regularization, our numerical simulations show smaller ℓ2- and ℓ∞-errors than that from the ℓ0-regularization and ℓ1-regularization. In summary, our numerical simulations indicate that not only can our developed algorithms be applied to a wide variety of important optimization problems, but also they are more accurate and computationally efficient than stateof- the-art algorithms