Xiao-Guang Lv

Kronecker product approximations for image restoration with whole-sample symmetric boundary conditions

Reflexive boundary conditions (BCs) assume that the array values outside the viewable region are given by a symmetry of the array values inside. The reflection guarantees the continuity of the image. In fact, there are usually two choices for the symmetry: symmetry around the meshpoint and symmetry around the midpoint. The first is called whole-sample symmetry in signal and image processing, the second is half-sample. Many researchers have developed some fast algorithms for the problems of image restoration with the half-sample symmetric BCs over the years. However, little attention has been given to the whole-sample symmetric BCs. In this paper, we consider the use of the whole-sample symmetric boundary conditions in image restoration. The blurring matrices constructed from the point spread functions (PSFs) for the BCs have block Toeplitz-plus-PseudoHankel with Toeplitz-plus-PseudoHankel blocks structures. Recently, regardless of symmetric properties of the PSFs, a technique of Kronecker product approximations was successfully applied to restore images with the zero BCs, half-sample symmetric BCs and anti-reflexive BCs, respectively. All these results extend quite naturally to the whole-sample symmetric BCs, since the resulting matrices have similar structures. It is interesting to note that when the size of the true PSF is small, the computational complexity of the algorithm obtained for the Kronecker product approximation of the resulting matrix in this paper is very small. It is clear that in this case all calculations in the algorithm are implemented only at the upper left corner submatrices of the big matrices. Finally, detailed experimental results reporting the performance of the proposed algorithm are presented.

Two soft-thresholding based iterative algorithms for image deblurring

Iterative regularization algorithms, such as the conjugate gradient algorithm for least squares problems (CGLS) and the modified residual norm steepest descent (MRNSD) algorithm, are popular tools for solving large-scale linear systems arising from image deblurring problems. These algorithms, however, are hindered by a semi-convergence behavior, in that the quality of the computed solution first increases and then decreases. In this paper, in order to overcome the semi-convergence behavior, we propose two iterative algorithms based on soft-thresholding for image deblurring problems. One of them combines CGLS with a denoising technique like soft-thresholding at each iteration and another combines MRNSD with soft-thresholding in a similar way. We prove the convergence of MRNSD and soft-thresholding based algorithm. Numerical results show that the proposed algorithms overcome the semi-convergence behavior and the restoration results are slightly better than those of CGLS and MRNSD with their optimal stopping iterations.

High-order total variation-based multiplicative noise removal with spatially adapted parameter selection

Multiplicative noise is one common type of noise in imaging science. For coherent image-acquisition systems, such as synthetic aperture radar, the observed images are often contaminated by multiplicative noise. Total variation (TV) regularization has been widely researched for multiplicative noise removal in the literature due to its edge-preserving feature. However, the TV-based solutions sometimes have an undesirable staircase artifact. In this paper, we propose a model to take advantage of the good nature of the TV norm and high-order TV norm to balance the edge and smoothness region. Besides, we adopt a spatially regularization parameter updating scheme. Numerical results illustrate the efficiency of our method in terms of the signal-to-noise ratio and structure similarity index.

A note on inversion of Toeplitz matrices

It is shown that the invertibility of a Toeplitz matrix can be determined through the solvability of two standard equations. The inverse matrix can be denoted as a sum of products of circulant matrices and upper triangular Toeplitz matrices. The stability of the inversion formula for a Toeplitz matrix is also considered.

Restoration of blurred color images with impulse noise

Restoration of images degraded by blurring and impulse noise has received considerable attention recently. Guo et?al. (2009) proposed a fast l 1 -total variation algorithm for grayscale image restoration with impulse noise. In this paper, we extend their idea for deblurring color images with impulse noise. An alternating iteration scheme is adopted for solving the corresponding problem. More importantly, we employ the five-point property to analyze the convergence of the proposed alternating algorithm. Numerical experiments demonstrate that the proposed method could deblur color images with good quality.

Split Bregman Iteration Algorithm for Image Deblurring Using Fourth-Order Total Bounded Variation Regularization Model

We propose a fourth-order total bounded variation regularization model which could reduce undesirable effects effectively. Based on this model, we introduce an improved split Bregman iteration algorithm to obtain the optimum solution. The convergence property of our algorithm is provided. Numerical experiments show the more excellent visual quality of the proposed model compared with the second-order total bounded variation model which is proposed by Liu and Huang (2010).

A note on solving nearly penta-diagonal linear systems

In this paper, a new efficient computational algorithm is presented for solving nearly penta-diagonal linear systems based on the use of any penta-diagonal linear solver. The implementation of the algorithm using computer algebra systems (CAS) such as MAPLE and MATLAB is straightforward. Numerical examples are given to illustrate the effectiveness of our method.

Alternating direction method for the high-order total variation-based Poisson noise removal problem

The restoration of blurred images corrupted by Poisson noise is an important task in various applications such as astronomical imaging, electronic microscopy, single particle emission computed tomography (SPECT) and positron emission tomography (PET). The problem has received significant attention in recent years. Total variation (TV) regularization, one of the standard regularization techniques in image restoration, is well known for recovering sharp edges of an image, but also for producing staircase artifacts. In order to remedy the shortcoming of TV in Poissonian image restoration, we consider a high-order total variation-based optimization model. The optimization model is converted to a constrained problem by variable splitting and then is addressed with the alternating direction method. Numerical results from the Poisson noise removal problem are given to illustrate the validity and efficiency of the proposed method.

Total Variation with Overlapping Group Sparsity for Image Deblurring under Impulse Noise

The total variation (TV) regularization method is an effective method for image deblurring in preserving edges. However, the TV based solutions usually have some staircase effects. In order to alleviate the staircase effects, we propose a new model for restoring blurred images under impulse noise. The model consists of an ℓ1-fidelity term and a TV with overlapping group sparsity (OGS) regularization term. Moreover, we impose a box constraint to the proposed model for getting more accurate solutions. The solving algorithm for our model is under the framework of the alternating direction method of multipliers (ADMM). We use an inner loop which is nested inside the majorization minimization (MM) iteration for the subproblem of the proposed method. Compared with other TV-based methods, numerical results illustrate that the proposed method can significantly improve the restoration quality, both in terms of peak signal-to-noise ratio (PSNR) and relative error (ReE).

An augmented Lagrangian algorithm for total bounded variation regularization based image deblurring

Abstract The augmented Lagrangian strategy has recently emerged as an important methodology for image processing problems. In this paper, based on this strategy, we propose a new projected gradient algorithm for image deblurring with total bounded variation regularization. The convergence property of our algorithm is discussed. Numerical experiments show that the proposed algorithm can yield better visual quality than the Rudin–Osher–Fatemi (ROF) method and the split Bregman iteration method.