Applications Of TSVD In Numerical Differential And Image Recovering

Inverse problem, which is a hot topic in the computational mathematics, applied mathematics and system science, has extensive background in many domains. Inverse problem is always ill-posed, and this property results in the instability of its solution. That means the solution of the equation (if exists) does not continuously depend on the right-handed data and the error of right-handed data will produce large error on the solution. Therefore we must use special method to obtain reasonable solution. The most popular and effective method for ill-posed solving is regularization method. The key issues of inverse problem study are creating effective regularization method, the selection of parameters and the realization of the algorithm.This paper first introduces the definitions of inverse problem and ill-posed problem by considering some instants and discusses the Moore-Penrose generalized solution and Moore-Penrose generalized inverse. On the base of spectrum analysis of linear compact self-operator and singular value decomposition (SVD) of compact operator, we present the expression of the singular system and conclude that the linear compact operator equation is ill-posed that give us the theory: the generalized solution of Moore-Penrose is unstable. The reason of the instability of compact operator equation is the property that the singular value of compact operator approaches zero. Therefore we introduce the regularization filter function to weaken or leach the impact of this property on stability of solution. By creating the regularization operator, we construct the theory of regularization method.The numerical calculation of inverse problem ordinarily needs discretization of problem and Truncated Singular Value Decomposition (TSVD) regularization method is then simple but effect method. This paper detailedly discusses the error estimation of TSVD regularization solution and the selection of parameters. By the comparison of the prior and posteriori selection of parameters, we prove that the error of TSVD regularization solution has optimum asymptotic convergence order. To apply this regularization method, we study two typical ill-posed problems from two different domains of numerical differentiation and image restoration.Problem of numerical differentiation is ill-posed. In order to get the stable approximate derivative of a given function which is also able to indicate the discontinuity of the derivative, in this paper, we discuss pp-TSVD method. The regularization solution of pp-TSVD can indicate the discontinuity without any priori information. We apply this method in numerical differentiation and the experiment shows that this method is very effective.We also study on the digital image restoration of point spread function which has the feature of translation invariance, under the Neumann boundary assumption. This problem of image restoration can be changed to a ill-posed deconvolution problem. We analyze the properties of convolution operator and then apply the TSVD method to deconvolution problem. We present the corresponding algorithm by use of fast cosine transform, and the experiment accounts for the effectivity.