PURIFY: a new algorithmic framework for next-generation radio-interferometric imaging

In recent works, compressed sensing (CS) and convex opti- mization techniques have been applied to radio-interferometric imaging showing the potential to outperform state-of-the-art imaging algorithms in the field. We review our latest contributions [1, 2, 3], which leverage the versatility of convex optimization to both handle realistic continuous visibilities and offer a highly parallelizable structure paving the way to significant acceleration of the reconstruction and high-dimensional data scalability. The new algorithmic structure promoted in a new software PURIFY (beta version) relies on the simultaneous-direction method of multipliers (SDMM). The performance of various sparsity priors is evaluated through simulations in the continuous visibility setting, confirming the superiority of our recent average sparsity approach SARA

Multiplicative Noise Removal Using Variable Splitting and Constrained Optimization

Multiplicative noise (also known as speckle noise) models are central to the study of coherent imaging systems, such as synthetic aperture radar and sonar, and ultrasound and laser imaging. These models introduce two additional layers of difficulties with respect to the standard Gaussian additive noise scenario: (1) the noise is multiplied by (rather than added to) the original image; (2) the noise is not Gaussian, with Rayleigh and Gamma being commonly used densities. These two features of multiplicative noise models preclude the direct application of most state-of-the-art algorithms, which are designed for solving unconstrained optimization problems where the objective has two terms: a quadratic data term (log-likelihood), reflecting the additive and Gaussian nature of the noise, plus a convex (possibly nonsmooth) regularizer (e.g., a total variation or wavelet-based regularizer/prior). In this paper, we address these difficulties by: (1) converting the multiplicative model into an additive one by taking logarithms, as proposed by some other authors; (2) using variable splitting to obtain an equivalent constrained problem; and (3) dealing with this optimization problem using the augmented Lagrangian framework. A set of experiments shows that the proposed method, which we name MIDAL (multiplicative image denoising by augmented Lagrangian), yields state-of-the-art results both in terms of speed and denoising performance.Comment: 11 pages, 7 figures, 2 tables. To appear in the IEEE Transactions on Image Processing

Efficient Output Kernel Learning for Multiple Tasks

The paradigm of multi-task learning is that one can achieve better generalization by learning tasks jointly and thus exploiting the similarity between the tasks rather than learning them independently of each other. While previously the relationship between tasks had to be user-defined in the form of an output kernel, recent approaches jointly learn the tasks and the output kernel. As the output kernel is a positive semidefinite matrix, the resulting optimization problems are not scalable in the number of tasks as an eigendecomposition is required in each step. \mbox{Using} the theory of positive semidefinite kernels we show in this paper that for a certain class of regularizers on the output kernel, the constraint of being positive semidefinite can be dropped as it is automatically satisfied for the relaxed problem. This leads to an unconstrained dual problem which can be solved efficiently. Experiments on several multi-task and multi-class data sets illustrate the efficacy of our approach in terms of computational efficiency as well as generalization performance

An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems

We propose a new fast algorithm for solving one of the standard approaches to ill-posed linear inverse problems (IPLIP), where a (possibly non-smooth) regularizer is minimized under the constraint that the solution explains the observations sufficiently well. Although the regularizer and constraint are usually convex, several particular features of these problems (huge dimensionality, non-smoothness) preclude the use of off-the-shelf optimization tools and have stimulated a considerable amount of research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (often known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the family of augmented Lagrangian methods, can be used to deal with a variety of imaging IPLIP, including deconvolution and reconstruction from compressive observations (such as MRI), using either total-variation or wavelet-based (or, more generally, frame-based) regularization. The proposed algorithm is an instance of the so-called "alternating direction method of multipliers", for which convergence sufficient conditions are known; we show that these conditions are satisfied by the proposed algorithm. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is a strong contender for the state-of-the-art.Comment: 13 pages, 8 figure, 8 tables. Submitted to the IEEE Transactions on Image Processin

Forward-backward truncated Newton methods for convex composite optimization

This paper proposes two proximal Newton-CG methods for convex nonsmooth optimization problems in composite form. The algorithms are based on a a reformulation of the original nonsmooth problem as the unconstrained minimization of a continuously differentiable function, namely the forward-backward envelope (FBE). The first algorithm is based on a standard line search strategy, whereas the second one combines the global efficiency estimates of the corresponding first-order methods, while achieving fast asymptotic convergence rates. Furthermore, they are computationally attractive since each Newton iteration requires the approximate solution of a linear system of usually small dimension

An optimal subgradient algorithm for large-scale convex optimization in simple domains

This paper shows that the optimal subgradient algorithm, OSGA, proposed in \cite{NeuO} can be used for solving structured large-scale convex constrained optimization problems. Only first-order information is required, and the optimal complexity bounds for both smooth and nonsmooth problems are attained. More specifically, we consider two classes of problems: (i) a convex objective with a simple closed convex domain, where the orthogonal projection on this feasible domain is efficiently available; (ii) a convex objective with a simple convex functional constraint. If we equip OSGA with an appropriate prox-function, the OSGA subproblem can be solved either in a closed form or by a simple iterative scheme, which is especially important for large-scale problems. We report numerical results for some applications to show the efficiency of the proposed scheme. A software package implementing OSGA for above domains is available