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Image retinex based on the nonconvex TV-type regularization

  • * Corresponding author: Zhi-Feng Pang

    * Corresponding author: Zhi-Feng Pang 
Dr. Z.-F. Pang was partially supported by National Basic Research Program of China (973 Program No.2015CB856003), and also gratefully acknowledges financial support from China Scholarship Council(CSC) as a research scholar to visit the University of Liverpool from August 2017 to August 2018
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  • Retinex theory is introduced to show how the human visual system perceives the color and the illumination effect such as Retinex illusions, medical image intensity inhomogeneity and color shadow effect etc.. Many researchers have studied this ill-posed problem based on the framework of the variation energy functional for decades. However, to the best of our knowledge, the existing models via the sparsity of the image based on the nonconvex $ \ell^p $-quasinorm were limited. To deal with this problem, this paper considers a TV$ _p $-HOTV$ _q $-based retinex model with $ p, q\in(0, 1) $. Specially, the TV$ _p $ term based on the total variation(TV) regularization can describe the reflectance efficiently, which has the piecewise constant structure. The HOTV$ _q $ term based on the high order total variation(HOTV) regularization can penalize the smooth structure called the illumination. Since the proposed model is non-convex, non-smooth and non-Lipschitz, we employ the iteratively reweighed $ \ell_1 $ (IRL1) algorithm to solve it. We also discuss some properties of our proposed model and algorithm. Experimental experiments on the simulated and real images illustrate the effectiveness and the robustness of our proposed model both visually and quantitatively by compared with some related state-of-the-art variational models.

    Mathematics Subject Classification: 80M30, 80M50, 68U10.


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  • Figure 1.  Two synthetic testing images

    Figure 2.  Reconstruction images based on the different model. The first and third rows: recovered reflectance $ r $ by five methods. The second and fourth rows: illumination $ l $ by five methods

    Figure 3.  Comparison of five methods on $ T_1 $-weighted brain MRIs with different levels of noises

    Figure 4.  Comparison among five models in terms of the PSNR and the MSSIM. Here the x-axis denotes the the serial number of the image

    Figure 5.  Comparison of the performance in terms of CV(%)

    Figure 6.  $ T_1 $-weighted brain images with different intensity inhomogeneities and noises

    Figure 7.  Performances of five methods on $ T_1 $-weighted brain images with different intensity inhomogeneities for the first, third and fifth rows; Colorbar to different between the clean images and restored images for the second, fourth and sixth rows

    Figure 8.  The relative errors of the original images and the restored images in Figure 6.(a) First image, (b) Second image and (c) Third image

    Figure 9.  The relative errors of $ r $ and $ l $ and numerical energy of our model for the third image in Figure 6

    Figure 10.  MRIs with noises and bias field

    Figure 11.  Bias field correction for the different MRIs

    Figure 12.  The gray values of underlined part with five methods

    Figure 13.  Original corrupted images and the corresponding contour images

    Figure 14.  Comparisons of restored images with five methods. The head 1 for the first row, the head 2 for the second row and the head 3 for the third row

    Figure 15.  FIGURE 14 corresponding contour images. The head 1 for the first row, the head 2 for the second row and the head 3 for the third row

    Figure 16.  Two test images for the illusion problem. (a) Adelson's checkerboard shadow image. (b) Logvinenko's cube shadow image

    Figure 17.  Decomposition comparison of the checkerboard image and the cube image. The first and third rows: recovered reflectance $ r $ by five methods. The second and fourth rows: illumination $ l $ by five methods

    Table 1.  PSNR and MSSIM of the reconstructed synthetic images

    TVH1 30.9323 0.9812
    Shape1 L0MS 31.7962 0.9837
    HOTVL 18.9389 0.9374
    ETV 36.2657 0.9967
    OUR 36.8019 0.9973
    TVH1 31.5509 0.9514
    Shape2 L0MS 31.6024 0.9463
    HoTVL1 15.6438 0.8137
    ETV 36.6105 0.9972
    OUR 37.6838 0.9975
     | Show Table
    DownLoad: CSV

    Table 2.  PSNR and MSSIM of $ T_1 $-weighted brain MRIs with different levels of Gaussian white noises

    $ 0.03 $ $ 0.05 $ $ 0.07 $ $ 0.09 $
    TVH1 26.8509 0.9436 25.3513 0.9176 24.5925 0.8939 23.4850 0.8642
    Test HoTVL1 30.2055 0.9515 27.7438 0.9288 26.4877 0.9069 24.7440 0.8774
    Image1 L0MS 29.7718 0.9227 27.8711 0.9088 26.1986 0.8936 24.3809 0.8649
    ETV 32.6699 0.9904 31.1178 0.9844 29.1294 0.9767 28.4238 0.9686
    OUR 33.0513 0.9909 31.2246 0.9846 30.1037 0.9781 28.4666 0.9686
    TVH1 27.4061 0.9230 26.5258 0.8935 25.5879 0.8661 24.0629 0.8348
    Test HoTVL1 30.0791 0.9317 29.0495 0.9065 27.0113 0.8795 25.1211 0.8481
    Image2 L0MS 32.0987 0.9246 29.0807 0.9016 27.1344 0.8760 24.9108 0.8382
    ETV 33.4363 0.9911 31.4108 0.9842 29.8199 0.9764 28.6496 0.9698
    OUR 33.9694 0.9916 31.4609 0.9841 29.8766 0.9760 29.0947 0.9703
     | Show Table
    DownLoad: CSV

    Table 3.  PSNR and MSSIM of Retinex illusion images

    TVH1 31.7514 0.8887
    Checkboard L0MS 33.1312 0.9602
    HOTVL1 29.5090 0.8725
    ETV 38.7053 0.9936
    OUR 39.2105 0.9934
    TVH1 30.7779 0.9799
    L0MS 30.0935 0.9774
    Cube HOTVL1 27.9950 0.9692
    ETV 29.6898 0.9866
    OUR 33.3660 0.9896
     | Show Table
    DownLoad: CSV
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