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Some novel linear regularization methods for a deblurring problem

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

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  • 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.

    Mathematics Subject Classification: Primary: 35S10, 65M12; Secondary: 65M32.


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  • Figure 1.  (left): the original image $f(x,y)$, (right): the blurred image $g(x,y)$.

    Figure 2.  Comparison of the quasi-boundary methods and Tikhonov method.(A): quasi-boundary method. (B): modified quasi-boundary method. (C): Tikhonov method.

    Figure 3.  (A): the original image $f(x,y)$, (B): the blurred image $g(x,y)$.

    Figure 4.  Comparison of the quasi-boundary methods and 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.

    Figure 6.  Profiles for the convergence rates of quasi-boundary methods.

    Figure 7.  Comparison of different deblurring methods. (A) the original image. (B) the blurred and noisy image. (C)the true Wiener filtering method.

    Figure 8.  Comparison of different deblurring methods. (A) the SECB method. (B) the backward beam method. (C)the Tikhonov method.

    Figure 9.  Comparison of different deblurring methods. (A) the modified quasi-boundary method. (B) the Spectral method 2. (C)the convolution-type method.

    Figure 10.  Comparison of different deblurring methods. (A) the TV method. (B) the FQBM. (C)the FMQBM.

    Table 1.  behavior of PSNR for the three methods.

    Quasi-boundary regularization16.5
    Modified quasi-boundary regularization15.9
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    Table 2.  behavior of PSNR for the three methods.

    Quasi-boundary regularization30.0
    Modified quasi-boundary regularization29.1
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    Table 3.  behavior in the Lena image in Fig. 7, Fig. 8, Fig. 9, Fig. 10.

    Deblurring methodParameter $(t=0.01,\, \, \mbox {if, necessary})$PSNR
    Ture Wiener filtering8.80%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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