Inexact Alternating Direction Method Research Articles

An inexact alternating direction method with SQP regularization for the structured variational inequalities

In this paper, we propose an inexact alternating direction method with square quadratic proximal (SQP) regularization for the structured variational inequalities. The predictor is obtained via solving SQP system approximately under significantly relaxed accuracy criterion and the new iterate is computed directly by an explicit formula derived from the original SQP method. Under appropriate conditions, the global convergence of the proposed method is proved. We show the $O(1/t)$ convergence rate for the inexact SQP alternating direction method. We also reported some numerical results to illustrate the efficiency of the proposed method.

Inexact alternating direction method based on Newton descent algorithm with application to Poisson image deblurring

The recovery of images from the observations that are degraded by a linear operator and further corrupted by Poisson noise is an important task in modern imaging applications such as astronomical and biomedical ones. Gradient-based regularizers involve the popular total variation semi-norm have become standard techniques for Poisson image restoration due to its edge-preserving ability. Various efficient algorithms have been developed for solving the corresponding minimization problem with non-smooth regularization terms. In this paper, motivated by the idea of the alternating direction minimization algorithm and the Newton's method with upper convergent rate, we further propose inexact alternating direction methods utilizing the proximal Hessian matrix information of the objective function, in a way reminiscent of Newton descent methods. Besides, we also investigate the global convergence of the proposed algorithms under certain conditions. Finally, we illustrate that the proposed algorithms outperform the current state-of-the-art algorithms through numerical experiments on Poisson image deblurring.

Two-Stage Image Segmentation Scheme Based on Inexact Alternating Direction Method

AbstractImage segmentation is a fundamental problem in both image processing and computer vision with numerous applications. In this paper, we propose a two-stage image segmentation scheme based on inexact alternating direction method. Specifically, we first solve the convex variant of the Mumford-Shah model to get the smooth solution, the segmentation are then obtained by apply the K-means clustering method to the solution. Some numerical comparisons are arranged to show the effectiveness of our proposed schemes by segmenting many kinds of images such as artificial images, natural images, and brain MRI images.

An inexact alternating direction method of multipliers with relative error criteria

In this paper, we study an inexact version of the alternating direction method of multipliers (ADMM) for solving two-block separable linearly constrained convex optimization problems. Specifically, the two subproblems in the classic ADMM are allowed to be solved inexactly by certain relative error criteria, in the sense that only two parameters are needed to control the inexactness. Related convergence analysis are established under the assumption that the solution set to the KKT system of the problem is not empty. Numerical results on solving a class of sparse signal recovery problems are also provided to demonstrate the efficiency of the proposed algorithm.

Inexact alternating direction method based on proximity projection operator for image inpainting in wavelet domain

Wavelet domain inpainting refers to the process of recovering the missing coefficients during image compression or transmission stage. Recently, efficient algorithms including the augmented Lagrangian methods (ALMs) and the primal-dual hybrid gradient (PDHG) algorithm have been successfully applied to solve the classical variational model of wavelet inpainting. However, the standard ALMs and PDHG require the wavelet bases to be orthogonal. Very recently, an approximated PDHG algorithm was proposed to deal with the case of the non-orthogonal wavelet such as Daubechies 7–9 wavelet. In this paper, we further develop an inexact alternating direction method for solving the general non-orthogonal wavelet inpainting problem, where one sub-minimization problem is solved by the proximity projection algorithm based on the Newton iteration. The global convergence of the proposed algorithm is further investigated under mild conditions. Numerical experiments demonstrate that the proposed method overall outperforms the recently proposed approximated PDHG algorithm, especially in the computational time.

Column distribution reconstruction algorithm via the alternating direction method

With the development of compressive sensing theory, computed tomography (CT) image reconstruction from sparse-view projections has become a research hotspot. And total variation (TV)-based methods have been experimentally proved to be capable of achieving accurate reconstruction from sparse-view data. In this study, a column distributed reconstruction algorithm based on TV minimization and the alternating direction method (ADM) has been developed. The algorithm uses the inexact ADM, which involves linearization and proximal points techniques, hence it is relatively simple. Moreover, it distributes data to individual nodes and computes in parallel, so that can achieve outstanding acceleration factors. Experimental results demonstrate that the proposed method is well suited for distributed computing and can accelerate the alternating direction total variation minimization (ADTVM) algorithm with very little loss of accuracy.

Distributed CT image reconstruction algorithm based on the alternating direction method.

With the development of compressive sensing theory, image reconstruction from few-view projections has been paid considerable research attention in the field of computed tomography (CT). Total variation (TV)-based CT image reconstruction has been shown experimentally to be capable of producing accurate reconstructions from sparse-view data. Motivated by the need of solving few-view reconstruction problem with large scale data, a general block distribution reconstruction algorithm based on TV minimization and the alternating direction method (ADM) has been developed in this study. By utilizing the inexact ADM, which involves linearization and proximal point techniques, the algorithm is relatively simple and hence convenient for the derivation and distributed implementation. And because the data as well as the computation are distributed to individual nodes, an outstanding acceleration factor is achieved. Experimental results demonstrate that the proposed method can accelerate the alternating direction total variation minimization (ADTVM) algorithm with nearly no loss of accuracy, which means compared with ADTVM, the proposed algorithm has a better accuracy with same running time.

Linearized alternating directions method for [formula omitted]-norm inequality constrained [formula omitted]-norm minimization

The ℓ1-regularization is popular in compressive sensing due to its ability to promote sparsity property. In the past few years, intensive research activities have been attracted to the algorithms for ℓ1-regularized least squares or its multifarious variations. In this study, we consider the ℓ1-norm minimization problems simultaneously with ℓ1-norm inequality constraints. The formulation of this problem is preferable when the measurement of a large and sparse signal is corrupted by an impulsive noise, in the mean time the noise level is given. This study proposes and investigates an inexact alternating direction method. At each iteration, as the closed-form solution of the resulting subproblem is not clear, we apply a linearized technique such that the closed-form solutions of the linearized subproblem can be easily derived. Global convergence of the proposed method is established under some appropriate assumptions. Numerical results, including comparisons with another algorithm are reported which demonstrate the superiority of the proposed algorithm. Finally, we extend the algorithm to solve ℓ2-norm constrained ℓ1-norm minimization problem, and show that the linearized technique can be avoided.

Inexact Distributed Reconstruction via Alternating Direction Method

Total variation (TV)-based CT image reconstruction has been shown to be experimentally capable of producing accurate reconstructions from sparse-view data. In this study, an inexact distributed reconstruction algorithm based on TV minimization has been developed. The algorithm is relatively simple as it uses the inexact alternating direction method, which involves linearization and proximal points techniques. The outstanding acceleration factor is achieved as the algorithm distributes the data and computation to individual nodes. Experimental results demonstrate that the proposed method can accelerate the alternating direction total variation minimization (ADTVM) algorithm with very little accuracy loss.

GPU Based Cone Beam Computed Tomography Reconstruction by the Inexact Alternating Direction Method

GPU based sparse reconstruction shows great significance in cone beam computed tomography (CBCT). This paper proposes a GPU based efficient algorithm for sparse view CBCT reconstruction. The reconstruction problem is converted to a constrained optimization using total variation minimization. The alternating direction method is adopted to solve it efficiently. Furthermore, a linearized proximity and FFT techniques are used for improving computation efficiency. To tackle with the most time consumption of forward and backward projection operation, the GPU hardware acceleration is utilized. The simulation experiments indicate that the new method is able to realize high accuracy reconstruction for CBCT with high speed.

GPU Based Cone Beam Computed Tomography Reconstruction by the Inexact Alternating Direction Method

GPU Based Cone Beam Computed Tomography Reconstruction by the Inexact Alternating Direction Method

An inexact LQP alternating direction method for solving a class of structured variational inequalities

In this paper, we present an inexact alternating direction method. Each iteration of the proposed method contains a prediction and a correction, the predictor is obtained via solving the logarithmic-quadratic proximal (LQP) system approximately under significantly relaxed accuracy criterion and the new iterate is computed directly by an explicit formula derived from the original LQP method. Under certain conditions, the global convergence of the proposed method is proved.

An inexact alternating direction method for solving a class of structured variational inequalities

The alternating direction method is mainly adopted to solve large-scale variational inequality problems with separable structure. The method is effective because it solves the original high-dimensional variational inequality problem by solving a series of much easier low-dimensional subproblems. In this paper, we present an inexact alternating directions method. Compared with the quadratic proximal alternating direction methods, the proposed method solves a series of related systems of nonlinear equations instead of a series of sub-VIs. The inexact criteria are more relaxed than the ones used by He et al. [7]. The generated sequence is Fejér monotone with respect to the solution set and the convergence is proved under suitable conditions.

Image reconstruction algorithm based on inexact alternating direction total-variation minimization

Image reconstruction algorithms implemented in existing computed tomography (CT) scanners require that the projection data should be available in proportional-space. The image reconstruction from the projections viewed from few angles has already been one of the hot problems in the research of iterative reconstruction algorithms. Total variation (TV)-based CT image reconstruction has shown to be experimentally capable of producing accurate reconstructions from sparse-view data. Reconstruction algorithms based on alternating direction method (ADM) show higher performance among these TV-based algorithms. However, computing the pseudoinverse at each iteration is too costly to implement numerically in the exact ADM algorithm. For this problem, then inexact ADM is adopted, which uses linearization and proximal points techniques such that computing the pseudoinverse can be accomplished by fast Fourier transforms. Experimental results demonstrate that the proposed method can accelerate the exact ADM algorithm, with little accuracy loss, and the computing time is approximatively reduced by 30%.

A partial inexact alternating direction method for structured variational inequalities

In this article, a new method is proposed for solving a class of structured variational inequalities (SVIs). The proposed method is referred to as the partial inexact proximal alternating direction (piPAD) method. In the method, two subproblems are solved independently. One is handled by an inexact proximal point method and the other is solved directly. This feature is the major difference between the proposed method and some existing alternating direction-like methods. The convergence of the piPAD method is proved. Two examples of the modern convex optimization problem arising from engineering and information sciences, which can be reformulated into the encountered SVIs, are presented to demonstrate the applicability of the piPAD method. Also, some preliminary numerical results are reported to validate the feasibility and efficiency of the piPAD method.

Statistical Multiresolution Estimation for Variational Imaging: With an Application in Poisson-Biophotonics

In this paper we present a spatially-adaptive method for image reconstruction that is based on the concept of statistical multiresolution estimation as introduced in Frick et al. (Electron. J. Stat. 6:231---268, 2012). It constitutes a variational regularization technique that uses an ? ?-type distance measure as data-fidelity combined with a convex cost functional. The resulting convex optimization problem is approached by a combination of an inexact alternating direction method of multipliers and Dykstra's projection algorithm. We describe a novel method for balancing data-fit and regularity that is fully automatic and allows for a sound statistical interpretation. The performance of our estimation approach is studied for various problems in imaging. Among others, this includes deconvolution problems that arise in Poisson nanoscale fluorescence microscopy.

Fast minimization methods for solving constrained total-variation superresolution image reconstruction

In this paper, we study the problem of reconstructing a high-resolution image from several decimated, blurred and noisy low-resolution versions of the high-resolution image. The problem can be formulated as a combination of the total variation (TV) inpainting model and the superresolution image reconstruction model. The main purpose of this paper is to develop an inexact alternating direction method for solving such constrained TV image reconstruction problem. Experimental results are given to show that the proposed algorithm is effective and efficient.

The improvement with relative errors of he et al.'s inexact alternating direction method for monotone variational inequalities

The proximal alternating direction method (PADM) solves a kind of structured monotone variational inequalities (VI) via solving a series of lower-dimensional and strongly monotone sub-VIs. We present a new inexact PADM which solves the involved sub-VIs approximately. The new criteria for solving the involved sub-VIs approximately are relaxed than those presented by He et al. [1]. In addition, the new criteria make use of the relative errors of each iteration instead of the absolute errors to control the accuracy of the inexact solutions of the sub-VIs.