# American Institute of Mathematical Sciences

April  2017, 11(2): 403-426. doi: 10.3934/ipi.2017019

## Some novel linear regularization methods for a deblurring problem

 Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China

* Corresponding author: xiongxt@gmail.com; xiongxt@fudan.edu.cn.

*The first author is supported by NSF grant of China 11661072

Received  November 2012 Revised  November 2016 Published  March 2017

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.

Citation: Xiangtuan Xiong, Jinmei Li, Jin Wen. Some novel linear regularization methods for a deblurring problem. Inverse Problems & Imaging, 2017, 11 (2) : 403-426. doi: 10.3934/ipi.2017019
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##### References:
(left): the original image $f(x,y)$, (right): the blurred image $g(x,y)$.
Comparison of the quasi-boundary methods and Tikhonov method.(A): quasi-boundary method. (B): modified quasi-boundary method. (C): Tikhonov method.
(A): the original image $f(x,y)$, (B): the blurred image $g(x,y)$.
Comparison of the quasi-boundary methods and Tikhonov method.(A): quasi-boundary method. (B): modified quasi-boundary method. (C): Tikhonov method.
. (A): quasi-boundary method.(B): modified quasi-boundary method. (C): Tikhonov method.">Figure 5.  Zooming on the partial region in Figure 4. (A): quasi-boundary method.(B): modified quasi-boundary method. (C): Tikhonov method.
Profiles for the convergence rates of quasi-boundary methods.
Comparison of different deblurring methods. (A) the original image. (B) the blurred and noisy image. (C)the true Wiener filtering method.
Comparison of different deblurring methods. (A) the SECB method. (B) the backward beam method. (C)the Tikhonov method.
Comparison of different deblurring methods. (A) the modified quasi-boundary method. (B) the Spectral method 2. (C)the convolution-type method.
Comparison of different deblurring methods. (A) the TV method. (B) the FQBM. (C)the FMQBM.
behavior of PSNR for the three methods.
 Method PSNR Quasi-boundary regularization 16.5 Modified quasi-boundary regularization 15.9 Tikhonov 14.7
 Method PSNR Quasi-boundary regularization 16.5 Modified quasi-boundary regularization 15.9 Tikhonov 14.7
behavior of PSNR for the three methods.
 Method PSNR Quasi-boundary regularization 30.0 Modified quasi-boundary regularization 29.1 Tikhonov 25.2
 Method PSNR Quasi-boundary regularization 30.0 Modified quasi-boundary regularization 29.1 Tikhonov 25.2
behavior in the Lena image in Fig. 7, Fig. 8, Fig. 9, Fig. 10.
 Deblurring method Parameter $(t=0.01,\, \, \mbox {if, necessary})$ PSNR Ture Wiener filtering 8.80% 28 SECB $s^*=0.0016,\,s=0.01,\,K=12.0$ 27 Backward beam $\rho=7.0$ 25 Tikhonov $\alpha=8*10^{-7}$ 26 Modified quasi-boundary $\alpha=0.001$ 26 Spectral $\alpha=0.01$ 24 Convolution-type $\epsilon=0.005$ 22 TV $k=1800,\epsilon=4*10^{-3}, \alpha=6*10^{-4}, \tau=0.86$ 25 FQBM $s=1.5, \alpha=1*10^{-4}$ 27 FMQBM $s=1.2, \alpha=1*10^{-4}$ 27
 Deblurring method Parameter $(t=0.01,\, \, \mbox {if, necessary})$ PSNR Ture Wiener filtering 8.80% 28 SECB $s^*=0.0016,\,s=0.01,\,K=12.0$ 27 Backward beam $\rho=7.0$ 25 Tikhonov $\alpha=8*10^{-7}$ 26 Modified quasi-boundary $\alpha=0.001$ 26 Spectral $\alpha=0.01$ 24 Convolution-type $\epsilon=0.005$ 22 TV $k=1800,\epsilon=4*10^{-3}, \alpha=6*10^{-4}, \tau=0.86$ 25 FQBM $s=1.5, \alpha=1*10^{-4}$ 27 FMQBM $s=1.2, \alpha=1*10^{-4}$ 27
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