Abstract. We propose simple and extremely efficient methods for solving the basis pursuit problem min{‖u‖1: Au = f,u ∈ R n}, which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of 1 instances of the unconstrained problem minu∈Rn μ‖u‖1 + 2 ‖Au−fk ‖ 2 2 for given matrix A and vector f k. We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrix-vector operations involving A and A ⊤ can be computed by fast transforms. Utilizing a fast fixed-point continuation solver that is based solely on such operations for solving the above unconstrained subproblem, we were able to quickly solve huge instances of compressed sensing problems on a standard PC.

Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. Inspired by recent breakthroughs in the development of novel first-order methods in convex optimization, most notably Nesterov’s smoothing technique, this paper introduces a fast and accurate algorithm for solving common recovery problems in signal processing. In the spirit of Nesterov’s work, one of the key ideas of this algorithm is a subtle averaging of sequences of iterates, which has been shown to improve the convergence properties of standard gradient-descent algorithms. This paper demonstrates that this approach is ideally suited for solving large-scale compressed sensing reconstruction problems as 1) it is computationally efficient, 2) it is accurate and returns solutions with several correct digits, 3) it is flexible and amenable to many kinds of reconstruction problems, and 4) it is robust in the sense that its excellent performance across a wide range of problems does not depend on the fine tuning of several parameters. Comprehensive numerical experiments on realistic signals exhibiting a large dynamic range show that this algorithm compares favorably with recently proposed state-of-the-art methods. We also apply the algorithm to solve other problems for which there are fewer alternatives, such as total-variation minimization, and

We present a framework for solving large-scale ℓ1-regularized convex minimization problem: min �x�1 + µf(x). Our approach is based on two powerful algorithmic ideas: operator-splitting and continuation. Operator-splitting results in a fixed-point algorithm for any given scalar µ; continuation refers to approximately following the path traced by the optimal value of x as µ increases. In this paper, we study the structure of optimal solution sets; prove finite convergence for important quantities; and establish q-linear convergence rates for the fixed-point algorithm applied to problems with f(x) convex, but not necessarily strictly convex. The continuation framework, motivated by our convergence results, is demonstrated to facilitate the construction of practical algorithms.

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Abstract. Image inpainting is a fundamental problem in image processing and has many applications. Motivated by the recent tight frame based methods on image restoration in either the image or the transform domain, we propose an iterative tight frame algorithm for image inpainting. We consider the convergence of this framelet-based algorithm by interpreting it as an iteration for minimizing a special functional. The proof of the convergence is under the framework of convex analysis and optimization theory. We also discuss the relationship of our method with other wavelet-based methods. Numerical experiments are given to illustrate the performance of the proposed algorithm. Key words. Tight frame, inpainting, convex analysis 1. Introduction. The problem of inpainting [2] occurs when part of the pixel data in a picture is missing or over-written by other means. This arises for example in restoring ancient drawings, where a portion of the picture is missing or damaged due to aging or scratch; or when an image is transmitted through a noisy channel. The task of inpainting is to recover the missing region from the incomplete data observed. Ideally, the restored image should possess shapes and patterns consistent

We propose an image restoration technique exploiting regularized inversion and the recent block-matching and 3D filtering (BM3D) denoising filter. The BM3D employs a non-local modeling of images by collecting similar image patches in 3D arrays. The so-called collaborative filtering applied on such a 3D array is realized by transformdomain shrinkage. In this work, we propose an extension of the BM3D filter for colored noise, which we use in a two-step deblurring algorithm to improve the regularization after inversion in discrete Fourier domain. The first step of the algorithm is a regularized inversion using BM3D with collaborative hard-thresholding and the seconds step is a regularized Wiener inversion using BM3D with collaborative Wiener filtering. The experimental results show that the proposed technique is competitive with and in most cases outperforms the current best image restoration methods in terms of improvement in signal-to-noise ratio.

Shrinkage is a well known and appealing denoising technique. The use of shrinkage is known to be optimal for Gaussian white noise, provided that the sparsity on the signal’s representation is enforced using a unitary transform. Still, shrinkage is also practiced successfully with nonunitary, and even redundant representations. In this paper we shed some light on this behavior. We show that simple shrinkage could be interpreted as the first iteration of an algorithm that solves the basis pursuit denoising (BPDN) problem. Thus, this work leads to a novel iterative shrinkage algorithm that can be considered as an effective pursuit method. We demonstrate this algorithm, both on synthetic data, and for the image denoising problem, where we learn the image prior parameters directly from the given image. The results in both cases are superior to several popular alternatives. 1

image deblurring

Sparse and redundant representations – an emerging and powerful model for signals – suggests that a data source could be described as a linear combination of few atoms from a pre-specified and over-complete dictionary. This model has drawn a considerable attention in the past decade, due to its appealing theoretical foundations, and promising practical results it leads to. Many of the applications that use this model are formulated as a mixture of ℓ2-ℓp (p ≤ 1) optimization expressions. Iterated Shrinkage algorithms are a new family of highly effective numerical techniques for handling these optimization tasks, surpassing traditional optimization techniques. In this paper we aim to give a broad view of this group of methods, motivate their need, present their derivation, show their comparative performance, and most important of all, discuss their potential in various applications.

Abstract. A new iterative algorithm for the solution of minimization problems in infinitedimensional Hilbert spaces which involve sparsity constraints in form of ℓ p-penalties is proposed. In contrast to the well-known algorithm considered by Daubechies, Defrise and De Mol, it uses hard instead of soft shrinkage. It is shown that the hard shrinkage algorithm is a special case of the generalized conditional gradient method. Convergence properties of the generalized conditional gradient method with quadratic discrepancy term are analyzed. This leads to strong convergence of the iterates with convergence rates O(n −1/2) and O(λ n) for p = 1 and 1 < p ≤ 2 respectively. Numerical experiments on image deblurring, backwards heat conduction, and inverse integration are given. Key words. sparsity constraints, iterated hard shrinkage, generalized conditional gradient method, convergence analysis AMS subject classifications. 46N10, 49M05, 65K10 1. Introduction. This