Some novel linear regularization methods for a deblurring problem

Jin Wen,Jinmei Li,Xiangtuan Xiong

doi:10.3934/ipi.2017019

Abstract

In this article, we consider a fractional backward heat conduction problem (BHCP) in the two-dimensional space which is associated with a deblurring problem. It is well-known that the classical Tikhonov method is the most important regularization method for linear ill-posed problems. However, the classical Tikhonov method over-smooths the solution. As a remedy, we propose two quasi-boundary regularization methods and their variants. We prove that these two methods are better than Tikhonov method in the absence of noise in the data. Deblurring experiment is conducted by comparing with some classical linear filtering methods for BHCP and the total variation method with the proposed methods.

Highlights

Since the physical imaging system usually is imperfect in the application where the image data are recorded, a recorded image presents a noisy and blurred version of an original scene in many practical situations
In order to overcome the over-smoothing shortcoming of Tikhonov method, in this paper we present two quasi-boundary regularization methods for the fractional backward heat conduction problem (BHCP) which is associated with a wide class of deblurring problems
Motivated by the fractional Tikhonov regularization method in the discrete setting [18], we present the fractional filtering methods in order to improve on the over-smoothing disadvantage of the classical Tikhonov method

Summary

Introduction

Since the physical imaging system usually is imperfect in the application where the image data are recorded, a recorded image presents a noisy and blurred version of an original scene in many practical situations. Suppose u(x, y, t) is the solution of the problem with the exact data g(x, y) and uαδ (x, y, t) is the regularization solution whose Fourier transform is given by (19) with the noisy data gδ(x, y), let (12) and the a-priori condition (13) hold. A modified quasi-boundary regularization method can be devised, i.e., let us consider the following problem vtα(x, y, t) = −γ(−∆)βvα(x, y, t),. Suppose u(x, y, t) is the solution of the problem with the exact data g(x, y) and vδα(x, y, t) is the regularization solution whose Fourier transform is given by (30) with the noisy data gδ(x, y), let (12) and the a-priori condition (13) hold. The fractional modified quasi-boundary method (FMQBM) is given by:

Pure deblurring

Conclusion

A deblurring test