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Nonconvex regularization for blurred images with Cauchy noise

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  • In this paper, we propose a nonconvex regularization model for images damaged by Cauchy noise and blur. This model is based on the method of the total variational proposed by Federica, Dong and Zeng [SIAM J. Imaging Sci.(2015)], where a variational approach for restoring blurred images with Cauchy noise is used. Here we consider the nonconvex regularization, namely a weighted difference of $ l_1 $-norm and $ l_2 $-norm coupled with wavelet frame, the alternating direction method of multiplier is carried out for this minimization problem, we describe the details of the algorithm and prove its convergence. Numerical experiments are tested by adding different levels of noise and blur, results show that our method can denoise and deblur the image better.

    Mathematics Subject Classification: 58F15, 58F17.

    Citation:

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  • Figure 1.  Testing images

    Figure 2.  Restored images from blurred and noised images by different methods. (a):Images with Cauchy noise ($ \xi = 0.02 $) and Gaussian blur; (b): Results by median filter (MD); (c): Results by convex TV method; (d): Results by our method

    Figure 3.  Restored images from blurred and noised images by different methods. (a):Images with Cauchy noise ($ \xi = 0.04 $) and Gaussian blur; (b): Results by median filter (MD); (c): Results by convex TV method; (d): Results by our method

    Figure 4.  Restored images from blurred and noised images by different methods. (a):Images destroyed by Cauchy noise($ \xi = 0.02 $) and motion blur; (b): Results by median filter (MD); (c): Results by convex TV method; (d): Results by our method

    Figure 5.  Restored images from blurred and noised images by different methods. (a):Images destroyed by Cauchy noise($ \xi = 0.02 $) and average blur; (b): Results by median filter (MD); (c): Results by convex TV method; (d): Results by our method

    Table 1.  The SNR, PSNR and SSIM values for corrupted images and recovered images given by different methods ($ \xi = 0.02 $, Gaussian blur)

    Image Value Corrupted MD TV Ours
    Lena SNR 5.76 12.43 14.31 14.93
    PSNR 18.47 24.39 27.64 28.21
    SSIM 0.1951 0.7884 0.8168 0.8389
    Cameraman SNR 6.16 11.72 13.69 14.57
    PSNR 18.19 24.39 26.08 26.84
    SSIM 0.1566 0.7483 0.8061 0.8083
    Parrot SNR 6.65 12.15 14.85 15.72
    PSNR 18.27 24.42 26.72 27.50
    SSIM 0.1909 0.7833 0.8239 0.8394
    Boat SNR 6.80 13.85 15.40 16.52
    PSNR 18.02 25.70 27.02 28.07
    SSIM 0.1981 0.7935 0.8286 0.8559
    Kitten SNR 6.15 10.13 11.86 13.01
    PSNR 17.98 22.67 24.03 25.06
    SSIM 0.2496 0.7195 0.7696 0.8161
    House SNR 5.03 13.70 16.11 17.05
    PSNR 18.38 28.48 30.52 31.44
    SSIM 0.1425 0.7908 0.8304 0.8494
     | Show Table
    DownLoad: CSV

    Table 2.  The SNR, PSNR and SSIM values for corrupted images and recovered images given by different methods ($ \xi = 0.04 $, Gaussian blur)

    Image Value Corrupted MD TV Ours
    Lena SNR 3.8 11.24 12.94 13.35
    PSNR 15.87 23.53 26.31 26.65
    SSIM 0.0928 0.6944 0.7675 0.7928
    Cameraman SNR 4.16 10.88 12.39 13.09
    PSNR 15.67 23.53 24.94 25.44
    SSIM 0.0761 0.6241 0.7532 0.7602
    Parrot SNR 4.58 11.19 13.46 14.12
    PSNR 15.75 23.55 25.40 25.91
    SSIM 0.0964 0.6752 0.7763 0.7847
    Boat SNR 4.65 12.59 14.13 14.81
    PSNR 15.33 24.50 25.83 26.41
    SSIM 0.0932 0.6936 0.7843 0.8074
    Kitten SNR 4.24 9.52 10.48 11.37
    PSNR 15.46 22.02 22.76 23.48
    SSIM 0.1235 0.6611 0.6967 0.7413
    House SNR 3.19 11.83 14.61 15.29
    PSNR 15.53 26.49 29.16 29.70
    SSIM 0.0636 0.6628 0.8159 0.8216
     | Show Table
    DownLoad: CSV

    Table 3.  The SNR, PSNR and SSIM values for corrupted images and recovered images given by different methods($ \xi = 0.02 $, motion blur)

    Image Value Corrupted MD TV Ours
    Lena SNR 4.15 7.78 11.66 12.40
    PSNR 17.37 22.05 25.16 25.73
    SSIM 0.1294 0.6209 0.7328 0.7969
    Cameraman SNR 4.98 8.60 11.60 12.42
    PSNR 17.19 21.53 24.13 24.60
    SSIM 0.1149 0.6478 0.7777 0.7797
    Parrot SNR 5.73 9.61 12.16 13.28
    PSNR 17.46 22.00 24.15 24.73
    SSIM 0.1574 0.7011 0.7782 0.7848
    Boat SNR 5.93 10.80 13.02 13.68
    PSNR 17.23 22.77 24.77 25.27
    SSIM 0.1458 0.6731 0.7652 0.7675
    Kitten SNR 4.93 7.63 10.15 10.76
    PSNR 16.92 20.37 22.44 22.87
    SSIM 0.1802 0.5619 0.6901 0.7216
    House SNR 4.54 10.72 14.85 15.65
    PSNR 18.01 25.60 29.34 30.03
    SSIM 0.1272 0.7211 0.8192 0.8264
     | Show Table
    DownLoad: CSV

    Table 4.  The SNR, PSNR and SSIM values for corrupted images and recovered images given by different methods($ \xi = 0.02 $, average blur)

    Image Value Corrupted MD TV Ours
    Lena SNR 5.87 12.88 14.56 15.35
    PSNR 18.55 26.47 27.87 28.64
    SSIM 0.1991 0.8031 0.8203 0.8534
    Cameraman SNR 6.20 12.04 14.17 15.16
    PSNR 18.15 24.65 26.54 27.40
    SSIM 0.1571 0.7623 0.8077 0.8309
    Parrot SNR 6.78 12.64 15.35 16.11
    PSNR 18.34 24.83 27.20 27.88
    SSIM 0.1938 0.7974 0.8228 0.8542
    Boat SNR 6.86 14.24 15.79 16.90
    PSNR 18.00 26.04 27.38 28.45
    SSIM 0.2014 0.8075 0.8345 0.8701
    Kitten SNR 6.31 10.62 12.46 13.39
    PSNR 18.06 23.08 24.56 25.38
    SSIM 0.2598 0.7467 0.7956 0.8327
    House SNR 5.09 14.08 16.54 17.41
    PSNR 18.39 28.70 30.98 31.81
    SSIM 0.1436 0.8013 0.8448 0.8581
     | Show Table
    DownLoad: CSV
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