Robust Principal Component Pursuit via Inexact Alternating Minimization on Matrix Manifolds

Robust principal component pursuit (RPCP) refers to a decomposition of a data matrix into a low-rank component and a sparse component. In this work, instead of invoking a convex-relaxation model based on the nuclear norm and the `1 -norm as is typically done in this context, RPCP is solved by considering a least-squares problem subject to rank and cardinality constraints. An inexact alternating minimization scheme, with guaranteed global convergence, is employed to solve the resulting constrained minimization problem. In particular, the low-rank matrix subproblem is resolved inexactly by a tailored Riemannian optimization technique, which favorably avoids singular value decompositions in full dimen- sion. For the overall method, a corresponding q-linear convergence theory is established. The numerical experiments show that the newly proposed method compares competitively with a popular convex-relaxation based approach.Peer Reviewe

Recent Progress in Image Deblurring

This paper comprehensively reviews the recent development of image deblurring, including non-blind/blind, spatially invariant/variant deblurring techniques. Indeed, these techniques share the same objective of inferring a latent sharp image from one or several corresponding blurry images, while the blind deblurring techniques are also required to derive an accurate blur kernel. Considering the critical role of image restoration in modern imaging systems to provide high-quality images under complex environments such as motion, undesirable lighting conditions, and imperfect system components, image deblurring has attracted growing attention in recent years. From the viewpoint of how to handle the ill-posedness which is a crucial issue in deblurring tasks, existing methods can be grouped into five categories: Bayesian inference framework, variational methods, sparse representation-based methods, homography-based modeling, and region-based methods. In spite of achieving a certain level of development, image deblurring, especially the blind case, is limited in its success by complex application conditions which make the blur kernel hard to obtain and be spatially variant. We provide a holistic understanding and deep insight into image deblurring in this review. An analysis of the empirical evidence for representative methods, practical issues, as well as a discussion of promising future directions are also presented.Comment: 53 pages, 17 figure

A Coordinate Descent Method for Total Variation Minimization

Total variation (TV) is a well-known image model with extensive applications in various images and vision tasks, for example, denoising, deblurring, superresolution, inpainting, and compressed sensing. In this paper, we systematically study the coordinate descent (CoD) method for solving general total variation (TV) minimization problems. Based on multidirectional gradients representation, the proposed CoD method provides a unified solution for both anisotropic and isotropic TV-based denoising (CoDenoise). With sequential sweeping and small random perturbations, CoDenoise is efficient in denoising and empirically converges to optimal solution. Moreover, CoDenoise also delivers new perspective on understanding recursive weighted median filtering. By incorporating with the Augmented Lagrangian Method (ALM), CoD was further extended to TV-based image deblurring (ALMCD). The results on denoising and deblurring validate the efficiency and effectiveness of the CoD-based methods

Generating structured non-smooth priors and associated primal-dual methods

The purpose of the present chapter is to bind together and extend some recent developments regarding data-driven non-smooth regularization techniques in image processing through the means of a bilevel minimization scheme. The scheme, considered in function space, takes advantage of a dualization framework and it is designed to produce spatially varying regularization parameters adapted to the data for well-known regularizers, e.g. Total Variation and Total Generalized variation, leading to automated (monolithic), image reconstruction workflows. An inclusion of the theory of bilevel optimization and the theoretical background of the dualization framework, as well as a brief review of the aforementioned regularizers and their parameterization, makes this chapter a self-contained one. Aspects of the numerical implementation of the scheme are discussed and numerical examples are provided

Convex Variational Approaches to Image Motion Estimation, Denoising and Segmentation

Energy minimization and variational methods are widely used in image processing and computer vision, where most energy functions and related constraints can be expressed as, or at least relaxed to, a convex formulation. In this regard, the central role is played by convexity, which not only provides an elegant analytical tool in mathematics but also facilitates the derivation of fast and tractable numerical solvers. In this thesis, four challenging topics of computer vision and image processing are studied by means of modern convex optimization techniques: non-rigid motion decomposition and estimation, TV-L1 image approximation, image segmentation, and multi-class image partition. Some of them are originally modelled in a convex formulation and can be directly solved by convex optimization methods, such as non-rigid flow estimation and non-smooth flow decomposition. The others are first stated as a non-convex model, then studied and solved in a convex relaxation manner, for which their dual models are employed to derive both novel analytical results and fast numerical solvers

Splitting Methods in Image Processing

It is often necessary to restore digital images which are affected by noise (denoising), blur (deblurring), or missing data (inpainting). We focus here on variational methods, i.e., the restored image is the minimizer of an energy functional. The first part of this thesis deals with the algorithmic framework of how to compute such a minimizer. It turns out that operator splitting methods are very useful in image processing to derive fast algorithms. The idea is that, in general, the functional we want to minimize has an additive structure and we treat its summands separately in each iteration of the algorithm which yields subproblems that are easier to solve. In our applications, these are typically projections onto simple sets, fast shrinkage operations, and linear systems of equations with a nice structure. The two operator splitting methods we focus on here are the forward-backward splitting algorithm and the Douglas-Rachford splitting algorithm. We show based on older results that the recently proposed alternating split Bregman algorithm is equivalent to the Douglas-Rachford splitting method applied to the dual problem, or, equivalently, to the alternating direction method of multipliers. Moreover, it is illustrated how this algorithm allows us to decouple functionals which are sums of more than two terms. In the second part, we apply the above techniques to existing and new image restoration models. For the Rudin-Osher-Fatemi model, which is well suited to remove Gaussian noise, the following topics are considered: we avoid the staircasing effect by using an additional gradient fitting term or by combining first- and second-order derivatives via an infimal-convolution functional. For a special setting based on Parseval frames, a strong connection between the forward-backward splitting algorithm, the alternating split Bregman method and iterated frame shrinkage is shown. Furthermore, the good performance of the alternating split Bregman algorithm compared to the popular multistep methods is illustrated. A special emphasis lies here on the choice of the step-length parameter. Turning to a corresponding model for removing Poisson noise, we show the advantages of the alternating split Bregman algorithm in the decoupling of more complicated functionals. For the inpainting problem, we improve an existing wavelet-based method by incorporating anisotropic regularization techniques to better restore boundaries in an image. The resulting algorithm is characterized as a forward-backward splitting method. Finally, we consider the denoising of a more general form of images, namely, tensor-valued images where a matrix is assigned to each pixel. This type of data arises in many important applications such as diffusion-tensor MRI