A TRUST REGION METHOD FOR SOLVING DISTRIBUTED PARAMETER IDENTIFICATION PROBLEMS

This paper is concerned with the ill-posed problems of identifying a parameter in an elliptic equation which appears in many applications in science and industry. Its solution is obtained by applying trust region method to a nonlinear least squares error problem.Trust region method has long been a popular method for well-posed problems. This paper indicates that it is also suitable for ill-posed problems, Numerical experiment is given to compare the trust region method with the Tikhonov regularization method. It seems that the trust region method is more promising.     

A new trust region algorithm for image restoration

The image restoration problems play an important role in remote sensing and astronomical image analysis. One common method for the recovery of a true image from corrupted or blurred image is the least squares error (LSE) method. But the LSE method is unstable in practical applications. A popular way to overcome instability is the Tikhonov regularization. However, difficulties will encounter when adjusting the so-called regularization parameter a. Moreover, how to truncate the iteration at appropriate steps is also challenging. In this paper we use the trust region method to deal with the image restoration problem, meanwhile, the trust region subproblem is solved by the truncated Lanczos method and the preconditioned truncated Lanczos method. We also develop a fast algorithm for evaluating the Kronecker matrix-vector product when the matrix is banded. The trust region method is very stable and robust, and it has the nice property of updating the trust region automatically. This releases us from tedious fi     

A feasible direction method for image restoration

In this work, a feasible direction method is proposed for computing the regularized solution of image restoration problems by simply using an estimate of the noise present on the data. The problem is formulated as an optimization problem with one quadratic constraint. The proposed method computes a feasible search direction by inexactly solving a trust region subproblem with the truncated Conjugate Gradient method of Steihaug. The trust region radius is adjusted to maintain feasibility and a line-search globalization strategy is employed. The global convergence of the method is proved. The results of image denoising and deblurring are presented in order to illustrate the effectiveness and efficiency of the proposed method.     

A special Hermitian and skew-Hermitian splitting method for image restoration

In this paper, we consider an ill-posed image restoration problem with a noise contaminated observation, and a known convolution kernel. A special Hermitian and skew-Hermitian splitting (HSS) iterative method is established for solving the linear systems from image restoration. Our approach is based on an augmented system formulation. The convergence and operation cost of the special HSS iterative method for image restoration problems are discussed. The optimal parameter minimizing the spectral radius of the iteration matrix is derived. We present a detailed algorithm for image restoration problems. Numerical examples are given to demonstrate the performance of the presented method. Finally, the SOR acceleration scheme for the special HSS iterative method is discussed.     

Sylvester Tikhonov-regularization methods in image restoration

In this paper, we consider large-scale linear discrete ill-posed problems where the right-hand side contains noise. Regularization techniques such as Tikhonov regularization are needed to control the effect of the noise on the solution. In many applications such as in image restoration the coefficient matrix is given as a Kronecker product of two matrices and then Tikhonov regularization problem leads to the generalized Sylvester matrix equation. For large-scale problems, we use the global-GMRES method which is an orthogonal projection method onto a matrix Krylov subspace. We present some theoretical results and give numerical tests in image restoration.     

基于分数阶微积分正则化的图像处理

全变分正则化方法已被广泛地应用于图像处理,利用此方法可以较好地去除噪声,并保持图像的边缘特征,但得到的优化解会产生"阶梯"效应.为了克服这一缺点,本文通过分数阶微积分正则化方法,建立了一个新的图像处理模型.为了克服此模型中非光滑项对求解带来的困难,本文研究了基于不动点方程的迫近梯度算法.最后,本文利用提出的模型与算法进行了图像去噪、图像去模糊与图像超分辨率实验,实验结果表明分数阶微积分正则化方法能较好的保留图像纹理等细节信息.     

A GCV based Arnoldi-Tikhonov regularization method

For the solution of linear discrete ill-posed problems, in this paper we consider the Arnoldi-Tikhonov method coupled with the Generalized Cross Validation for the computation of the regularization parameter at each iteration. We study the convergence behavior of the Arnoldi method and its properties for the approximation of the (generalized) singular values, under the hypothesis that Picard condition is satisfied. Numerical experiments on classical test problems and on image restoration are presented.     

The Lagrange method for the regularization of discrete ill-posed problems

In many science and engineering applications, the discretization of linear ill-posed problems gives rise to large ill-conditioned linear systems with the right-hand side degraded by noise. The solution of such linear systems requires the solution of minimization problems with one quadratic constraint, depending on an estimate of the variance of the noise. This strategy is known as regularization. In this work, we propose a modification of the Lagrange method for the solution of the noise constrained regularization problem. We present the numerical results of test problems, image restoration and medical imaging denoising. Our results indicate that the proposed Lagrange method is effective and efficient in computing good regularized solutions of ill-conditioned linear systems and in computing the corresponding Lagrange multipliers. Moreover, our numerical experiments show that the Lagrange method is computationally convenient. Therefore, the Lagrange method is a promising approach for dealing with ill-posed problems. This work was supported by the Italian FIRB Project “Parallel algorithms and Nonlinear Numerical Optimization” RBAU01JYPN.     

On a class of ill-posed minimization problems in image processing

In this paper, we show that minimization problems involving sublinear regularizing terms are ill-posed, in general, although numerical experiments in image processing give very good results. The energies studied here are inspired by image restoration and image decomposition. Rewriting the nonconvex sublinear regularizing terms as weighted total variations, we give a new approach to perform minimization via the well-known Chambolle's algorithm. The approach developed here provides an alternative to the well-known half-quadratic minimization one.     

Regularization using a parameterized trust region subproblem

We present a new method for regularization of ill-conditioned problems, such as those that arise in image restoration or mathematical processing of medical data. The method extends the traditional trust-region subproblem, TRS, approach that makes use of the L-curve maximum curvature criterion, a strategy recently proposed to find a good regularization parameter. We apply a parameterized trust region approach to estimate the region of maximum curvature of the L-curve and find the regularized solution. This exploits the close connections between various parameters used to solve TRS. A MATLAB code for the algorithm is tested and a comparison to the conjugate gradient least squares, CGLS, approach is given and analysed.     

Existence,uniqueness and stability of minimizers of generalized Tikhonov–Phillips functionals

The Tikhonov–Phillips method is widely used for regularizing ill-posed inverse problems mainly due to the simplicity of its formulation as an optimization problem. The use of different penalizers in the functionals associated to the corresponding optimization problems has originated a variety of other methods which can be considered as “variants” of the traditional Tikhonov–Phillips method of order zero. Such is the case for instance of the Tikhonov–Phillips method of order one, the total variation regularization method, etc. In this article we find sufficient conditions on the penalizers in generalized Tikhonov–Phillips functionals which guarantee existence, uniqueness and stability of the minimizers. The particular cases in which the penalizers are given by the bounded variation norm, by powers of seminorms and by linear combinations of powers of seminorms associated to closed operators, are studied. Several examples are presented and a few results on image restoration are shown.     

Quasi-Newton projection methods and the discrepancy principle in image restoration

In this work, the problem of the restoration of images corrupted by space invariant blur and noise is considered. This problem is ill-posed and regularization is required. The image restoration problem is formulated as a nonnegatively constrained minimization problem whose objective function depends on the statistical properties of the noise corrupting the observed image. The cases of Gaussian and Poisson noise are both considered. A Newton-like projection method with early stopping of the iterates is proposed as an iterative regularization method in order to determine a nonnegative approximation to the original image. A suitable approximation of the Hessian of the objective function is proposed for a fast solution of the Newton system. The results of the numerical experiments show the effectiveness of the method in computing a good solution in few iterations, when compared with some methods recently proposed as best performing.     

Iterative exponential filtering for large discrete ill-posed problems

Summary. We describe a new iterative method for the solution of large, very ill-conditioned linear systems of equations that arise when discretizing linear ill-posed problems. The right-hand side vector represents the given data and is assumed to be contaminated by measurement errors. Our method applies a filter function of the form with the purpose of reducing the influence of the errors in the right-hand side vector on the computed approximate solution of the linear system. Here is a regularization parameter. The iterative method is derived by expanding in terms of Chebyshev polynomials. The method requires only little computer memory and is well suited for the solution of large-scale problems. We also show how a value of and an associated approximate solution that satisfies the Morozov discrepancy principle can be computed efficiently. An application to image restoration illustrates the performance of the method. Received January 25, 1997 / Revised version received February 9, 1998 / Published online July 28, 1999     

Regularized fast multiple-image deconvolution for LBT

In this paper we study a fast deconvolution technique for the image restoration problem of the Large Binocular Telescope (LBT) interferometer. Since LBT provides several blurred and noisy images of the same object, it requires the use of multiple-image deconvolution methods in order to produce a unique image with high resolution. Hence the restoration process is basically a linear ill-posed problem, with overdetermined system and data corrupted by several components of noise.     

Noise-reducing cascadic multilevel methods for linear discrete ill-posed problems

Cascadic multilevel methods for the solution of linear discrete ill-posed problems with noise-reducing restriction and prolongation operators recently have been developed for the restoration of blur- and noise-contaminated images. This is a particular ill-posed problem. The multilevel methods were found to determine accurate restorations with fairly little computational work. This paper describes noise-reducing multilevel methods for the solution of general linear discrete ill-posed problems.     

A fast subspace method for image deblurring

Image deblurring problems appear frequently in astronomical image analysis. For image deblurring problems, it is reasonable to add a non-negativity constraint because of the physical meaning of the image. Previous research works are mainly full-space methods, i.e., solving a regularized optimization problem in a full space. To solve the problem more efficiently, we propose a subspace method. We first formulate the problem from full space to subspace and then use an interior-point trust-region method to solve it. The numerical experiments show that this method is suitable for ill-posed image deblurring problems.     

A new modified nonmonotone adaptive trust region method for unconstrained optimization

In this paper, we present an adaptive trust region method for solving unconstrained optimization problems which combines nonmonotone technique with a new update rule for the trust region radius. At each iteration, our method can adjust the trust region radius of related subproblem. We construct a new ratio to adjust the next trust region radius which is different from the ratio in the traditional trust region methods. The global and superlinear convergence results of the method are established under reasonable assumptions. Numerical results show that the new method is efficient for unconstrained optimization problems.     

A new trust region method with adaptive radius

In this paper we develop a new trust region method with adaptive radius for unconstrained optimization problems. The new method can adjust the trust region radius automatically at each iteration and possibly reduces the number of solving subproblems. We investigate the global convergence and convergence rate of this new method under some mild conditions. Theoretical analysis and numerical results show that the new adaptive trust region radius is available and reasonable and the resultant trust region method is efficient in solving practical optimization problems. The work was supported in part by NSF grant CNS-0521142, USA.     

A new nonmonotone adaptive trust region method based on simple quadratic models

Based on simple quadratic models of the trust region subproblem, we combine the trust region method with the nonmonotone and adaptive techniques to propose a new nonmonotone adaptive trust region algorithm for unconstrained optimization. Unlike traditional trust region method, our trust region subproblem is very simple by using a new scale approximation of the minimizing function??s Hessian. The new method needs less memory capacitance and computational complexity. The convergence results of the method are proved under certain conditions. Numerical results show that the new method is effective and attractive for large scale unconstrained problems.     

The convergence of subspace trust region methods

The trust region method is an effective approach for solving optimization problems due to its robustness and strong convergence. However, the subproblem in the trust region method is difficult or time-consuming to solve in practical computation, especially in large-scale problems. In this paper we consider a new class of trust region methods, specifically subspace trust region methods. The subproblem in these methods has an adequate initial trust region radius and can be solved in a simple subspace. It is easier to solve than the original subproblem because the dimension of the subproblem in the subspace is reduced substantially. We investigate the global convergence and convergence rate of these methods.