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A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising

  • * Corresponding author: Tianling Gao

    * Corresponding author: Tianling Gao
The first author is supported by Young Foundation of Three Gorges University(19QN09). The last two authors are supported by NSF of Guandong grant 2018A030310454, 2020A1515010554.
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  • In this paper, we mainly show a novel fast fractional order anisotropic diffusion algorithm for noise removal based on the recent numerical scheme called the Fast Explicit Diffusion. To balance the efficiency and accuracy of the algorithm, the truncated matrix method is used to deal with the iterative matrix in the model and its error is also estimated. In particular, we obtain the stability condition of the iteration by the spectrum analysis method. Through implementing the fast explicit format iteration algorithm with periodic change of time step size, the efficiency of the algorithm is greatly improved. At last, we show some numerical results on denoising tasks. Many experimental results confirm that the algorithm can more quickly achieve satisfactory denoising results.

    Mathematics Subject Classification: Primary: 68U10; Secondary: 35R11.

    Citation:

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  • Figure 1.  The truncation error of the truncated polynomial $ {{\rho _N}\left( t \right)} $ at $ z = 1 $ with truncated lengths $ t $ from 10 to 50 and fractional-orders $ \alpha $ from 1.1 to 1.9

    Figure 2.  First row to last row: comparing the convergence rate of three algorithms for the camera image, the Lena image, the butterfly image and the house image with additive Gauss noise levels of 10, 20 and 40

    Figure 3.  First column to last column: Origin images, noisy images with $ \sigma {\rm{ = }}10 $, and the restored images by the FOFED algorithm, FOFED$ _f $ algorithm and FOFFT algorithm in [6], respectively

    Figure 4.  First column to last column: Origin images, noisy images with $ \sigma {\rm{ = }}20 $, and the restored images by the FOFED algorithm, FOFED$ _f $ algorithm and FOFFT algorithm in [6], respectively

    Figure 5.  First column to last column: Origin images, noisy images with $ \sigma {\rm{ = }}40 $, and the restored images by the FOFED algorithm, FOFED$ _f $ algorithm and FOFFT algorithm in [6], respectively

    Table 1.  The truncation error of the truncated polynomial $ {{\rho _N}\left( t \right)} $ at $ z = 1 $ with truncated lengths $ t $ from 10 to 50 (rows) and fractional-orders $ \alpha $ from 1.1 to 1.9 (columns)

    t $ \alpha = 1.1 $ $ \alpha = 1.2 $ $ \alpha = 1.3 $ $ \alpha = 1.4 $ $ \alpha = 1.5 $ $ \alpha = 1.6 $ $ \alpha = 1.7 $ $ \alpha = 1.8 $ $ \alpha = 1.9 $
    10 0.0084 0.0125 0.0136 0.0128 0.0109 0.0085 0.0060 0.0036 0.0016
    20 0.0037 0.0051 0.0051 0.0044 0.0035 0.0025 0.0016 0.0009 0.0004
    30 0.0023 0.0030 0.0029 0.0024 0.0018 0.0013 0.0008 0.0004 0.0002
    40 0.0017 0.0021 0.0020 0.0016 0.0012 0.0008 0.0005 0.0002 0.00009
    50 0.0013 0.0016 0.0015 0.0012 0.0008 0.0005 0.0003 0.00016 0.00005
     | Show Table
    DownLoad: CSV

    Table 2.  PSNR, MAE, and SSIM results for the camera image, the Lena image, the butterfly image and the house image with additive Gauss noise levels of 10, 20 and 40. The PSNR, MAE and SSIM values are similar

    PSNR MAE SSIM
    $ \sigma $ 10 20 40 10 20 40 10 20 40
    Cameraman image ($ 1024 \times 1024 $)
    FOFED 39.4084 35.4073 33.7938 2.0434 3.2504 5.1143 0.5978 0.4621 0.3333
    FOFED$ _{f} $ 39.6005 35.6151 33.9866 1.9789 3.1111 4.8185 0.6030 0.4671 0.3374
    FOFFT 39.4881 35.5515 33.8737 1.9958 3.1240 4.8238 0.6021 0.4665 0.3370
    Lena image ($ 512 \times 512 $)
    FOFED 34.2268 30.9968 28.5389 3.7184 5.2972 7.3706 0.6056 0.4904 0.3856
    FOFED$ _{f} $ 34.2882 31.0802 28.6008 3.6588 5.0912 7.1460 0.6082 0.4957 0.3908
    FOFFT 34.2816 31.0847 28.5937 3.6564 5.0856 7.1361 0.6095 0.4962 0.3910
    Butterfly image ($ 256 \times 256 $)
    FOFED 32.8641 28.5347 26.6302 4.2297 6.8386 10.5351 0.7961 0.6960 0.5777
    FOFED$ _{f} $ 32.8944 28.5595 26.6618 4.1910 6.7488 10.3155 0.8126 0.7198 0.5960
    FOFFT 32.8459 28.5339 26.6119 4.2067 6.7536 10.2976 0.8134 0.7209 0.5973
    House image ($ 128 \times 128 $)
    FOFED 31.9328 27.8935 27.0384 4.7282 7.3484 11.1483 0.5607 0.4463 0.3439
    FOFED$ _{f} $ 31.9543 27.9203 27.0600 4.6964 7.2673 10.6816 0.5610 0.4470 0.3446
    FOFFT 31.9849 27.9562 27.0898 4.6672 7.2156 10.6107 0.5624 0.4476 0.3453
     | Show Table
    DownLoad: CSV

    Table 3.  Number of Iteration and CPU time for the camera image, the Lena image, the butterfly image and the house image with additive Gauss noise levels of 10, 20 and 40. The best CPU times are in black

    Number of Iteration CPU time(s)
    $ \sigma $ 10 20 40 10 20 40
    Cameraman image ($ 1024 \times 1024 $)
    FOFED 41 71 116 $ 12.0397 $ $ 15.5452 $ $ 25.8294 $
    FOFED$ _{f} $ 100 230 553 32.8190 77.5531 187.0590
    FOFFT 114 258 612 98.8958 227.8830 518.8290
    Lena image ($ 512 \times 512 $)
    FOFED 30 56 129 $ 1.9445 $ $ 3.5040 $ $ 8.1629 $
    FOFED$ _{f} $ 58 157 420 4.6834 12.4698 32.5470
    FOFFT 69 181 473 14.8598 39.0244 101.7675
    Butterfly image ($ 256 \times 256 $)
    FOFED 25 55 88 $ 0.3193 $ $ 0.6546 $ $ 1.0031 $
    FOFED$ _{f} $ 47 112 289 0.6994 1.5716 4.0302
    FOFFT 56 134 336 2.8438 6.7368 16.9259
    House image ($ 128 \times 128 $)
    FOFED 40 54 27.0384 $ 0.3604 $ $ 0.4095 $ $ 0.7060 $
    FOFED$ _{f} $ 41 104 27.0600 0.3820 0.8661 2.2998
    FOFFT 50 125 27.0898 0.7193 1.6759 3.9392
     | Show Table
    DownLoad: CSV
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