## Generalized Forward-Backward Splitting

This paper introduces the generalized forward-backward splitting algorithm for minimizing convex functions of the form $F + \sum_{i=1}^n G_i$, where $F$ has a Lipschitz-continuous gradient and the $G_i$’s are simple in the sense that their Moreau proximity operators are easy to compute. While the forward-backward algorithm cannot deal with more than $n = 1$ non-smooth … Read more

## Accelerated and Inexact forward-backward algorithms

We propose a convergence analysis of accelerated forward-backward splitting methods for minimizing composite functions, when the proximity operator is not available in closed form, and is thus computed up to a certain precision. We prove that the $1/k^2$ convergence rate for the function values can be achieved if the admissible errors are of a certain … Read more

## Convergence analysis of primal-dual algorithms for total variation image restoration

Recently, some attractive primal-dual algorithms have been proposed for solving a saddle-point problem, with particular applications in the area of total variation (TV) image restoration. This paper focuses on the convergence analysis of existing primal-dual algorithms and shows that the involved parameters of those primal-dual algorithms (including the step sizes) can be significantly enlarged if … Read more

## A splitting method for separate convex programming with linking linear constraints

We consider the separate convex programming problem with linking linear constraints, where the objective function is in the form of the sum of m individual functions without crossed variables. The special case with m=2 has been well studied in the literature and some algorithms are very influential, e.g. the alternating direction method. The research for … Read more

## A Fast Algorithm for Total Variation Image Reconstruction from Random Projections

Total variation (TV) regularization is popular in image restoration and reconstruction due to its ability to preserve image edges. To date, most research activities on TV models concentrate on image restoration from blurry and noisy observations, while discussions on image reconstruction from random projections are relatively fewer. In this paper, we propose, analyze, and test … Read more

## Alternating direction algorithms for total variation deconvolution in image reconstruction

Image restoration and reconstruction from blurry and noisy observation is known to be ill-posed. To stabilize the recovery, total variation (TV) regularization was introduced by Rudin, Osher and Fatemi in \cite{LIR92}, which has demonstrated superiority in preserving image edges. However, the nondifferentiability of TV makes the underlying optimization problems difficult to solve. In this paper, … Read more

## Solving Constrained Total-Variation Image Restoration and Reconstruction Problems via Alternating Direction Methods

In this paper, we study alternating direction methods for solving constrained total-variation image restoration and reconstruction problems. Alternating direction methods can be implementable variants of the classical augmented Lagrangian method for optimization problems with separable structures and linear constraints. The proposed framework allows us to solve problems of image restoration, impulse noise removal, inpainting and … Read more

## A Fast Algorithm For Image Deblurring with Total Variation Regularization

We propose and test a simple algorithmic framework for recovering images from blurry and noisy observations based on total variation (TV) regularization when a blurring point-spread function is given. Using a splitting technique, we construct an iterative procedure of alternately solving a pair of easy subproblems associated with an increasing sequence of penalty parameter values. … Read more

## Second-order Cone Programming Methods for Total Variation-based Image Restoration

In this paper we present optimization algorithms for image restoration based on the total variation (TV) minimization framework of L. Rudin, S. Osher and E. Fatemi (ROF). Our approach formulates TV minimization as a second-order cone program which is then solved by interior-point algorithms that are efficient both in practice (using nested dissection and domain … Read more