Using generalized cross validation to select regularization parameter for total variation regularization problems

You-Wei Wen,Raymond Honfu Chan

doi:10.3934/ipi.2018046

Abstract

The regularization approach is used widely in image restoration problems. The visual quality of the restored image depends highly on the regularization parameter. In this paper, we develop an automatic way to choose a good regularization parameter for total variation (TV) image restoration problems. It is based on the generalized cross validation (GCV) approach and hence no knowledge of noise variance is required. Due to the lack of the closed-form solution of the TV regularization problem, difficulty arises in finding the minimizer of the GCV function directly. We reformulate the TV regularization problem as a minimax problem and then apply a first-order primal-dual method to solve it. The primal subproblem is rearranged so that it becomes a special Tikhonov regularization problem for which the minimizer of the GCV function is readily computable. Hence we can determine the best regularization parameter in each iteration of the primal-dual method. The regularization parameter for the original TV regularization problem is then obtained by an averaging scheme. In essence, our method needs only to solve the TV regulation problem twice: one to determine the regularization parameter and one to restore the image with that parameter. Numerical results show that our method gives near optimal parameter, and excellent performance when compared with other state-of-the-art adaptive image restoration algorithms.

Highlights

Image restoration is an important task in image processing
In [33], we have developed a primal-dual framework based on discrepancy principle (DP) to find α and recover the original image simultaneously
When the regularization term is non-quadratic, there is no closed-form solution of the regularization problem, and the minimizer of the generalized cross validation (GCV) function can not be derived explicitly

Summary

Introduction

Image restoration is an important task in image processing. Its aim is to recover the original image f from the degraded observed image g. Numerical results show that our method gives near optimal regularization parameter, and excellent performance when compared with other state-of-the-art adaptive image restoration algorithms. The optimal regularization parameter λ should be chosen to minimize the GCV function, which is defined as the weighted predicted error over all the observed data:. We can observe that the optimal regularization parameters vary with the image, the noise level, and the blur type It means that one should choose a difference α for different case in order to obtain an optimal performance and our method provides a robust and adaptive approach to find such an α. The experimental results demonstrate that our method is robust and provides excellent performance when compared to existing adaptive image restoration algorithms

Conclusions

Using GCV to Select Regularization Parameter