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A variational saturation-value model for image decomposition: Illumination and reflectance

  • *Corresponding author: Wei Wang

    *Corresponding author: Wei Wang 

Wei Wang is supported by Natural Science Foundation of Shanghai (18ZR1441800)

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  • In this paper, we study to decompose a color image into the illumination and reflectance components in saturation-value color space. By considering the spatial smoothness of the illumination component, the total variation regularization of the reflectance component, and the data-fitting in saturation-value color space, we develop a novel variational saturation-value model for image decomposition. The main aim of the proposed model is to formulate the decomposition of a color image such that the illumination component is uniform with only brightness information, and the reflectance component contains the color information. We establish the theoretical result about the existence of the solution of the proposed minimization problem. We employ a primal-dual algorithm to solve the proposed minimization problem. Experimental results are shown to illustrate the effectiveness of the proposed decomposition model in saturation-value color space, and demonstrate the performance of the proposed method is better than the other testing methods.

    Mathematics Subject Classification: 68U10, 65K10, 65J22, 90C25.

    Citation:

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  • Figure 1.  Results by using different pairs of $ \mu $ and $ \lambda $. The first row is the original image; the second and fourth rows are the reflectance components; the third and the fifth rows are the illumination components

    Figure 2.  The first column: the original image; the second column: the reflectance components; the third column: the illumination components. From top to bottom: the results by using the proposed model, L1-Ma model, TV-Ma model, TV-Ng-HSV model, TV-Ng-RGB model, Dictionary model, Multigrid model respectively

    Figure 3.  The first column: the original image; the second column: the reflectance components; the third column: the illumination components. From top to bottom: the results by using the proposed model, L1-Ma model, TV-Ma model, TV-Ng-HSV model, TV-Ng-RGB model, Dictionary model, Multigrid model respectively

    Figure 4.  The first column: the original image; the second column: the reflectance components; the third column: the illumination components. From top to bottom: the results by using the proposed model, L1-Ma model, TV-Ma model, TV-Ng-HSV model, TV-Ng-RGB model, Dictionary model, Multigrid model respectively

    Figure 5.  The first column: the original image; the second column: the reflectance components; the third column: the illumination components. From top to bottom: the results by using the proposed model, L1-Ma model, TV-Ma model, TV-Ng-HSV model, TV-Ng-RGB model, Dictionary model, Multigrid model respectively

    Figure 6.  Adelson's checker shadow illusion example. From top to bottom: the results by using the proposed SV model, L1-Ma model, TV-Ma model, TV-Ng-HSV model, TV-Ng-RGB model, Dictionary method, Multigrid method respectively

    Figure 7.  Deer example. (a): the original image; (b): the bias image; (c) - (i): the reflectance results by using the proposed SV model, L1-Ma model, TV-Ma model, TV-Ng-HSV model, TV-Ng-RGB model, Dictionary method, Multigrid method respectively

    Figure 8.  Parthenon example. (a): the original image; (b): the bias image; (c) - (i): the reflectance results by using the proposed SV model, L1-Ma model, TV-Ma model, TV-Ng-HSV model, TV-Ng-RGB model, Dictionary method, Multigrid method respectively

    Figure 9.  Clay figure example. (a): the original image; (b): the bias image; (c) - (i): the reflectance results by using the proposed SV model, L1-Ma model, TV-Ma model, TV-Ng-HSV model, TV-Ng-RGB model, Dictionary method, Multigrid method respectively

    Figure 10.  Bridge example. (a): the original image; (b): the bias image; (c) - (i): the reflectance results by using the proposed SV model, L1-Ma model, TV-Ma model, TV-Ng-HSV model, TV-Ng-RGB model, Dictionary method, Multigrid method respectively

    Figure 11.  Soldiers example. (a): the original image; (b): the bias image; (c) - (i): the reflectance results by using the proposed SV model, L1-Ma model, TV-Ma model, TV-Ng-HSV model, TV-Ng-RGB model, Dictionary method, Multigrid method respectively

    Algorithm A:
       Step1: Choose stopping criteria $ \epsilon $ and initialization $ {\bf{u}}^{0} $.
       Step2: For fixed $ {\bf{u}}^{n} $, update $ {\bf{\bar{u}}} $ by solving:
    $ \begin{equation} (8) \;\;\;\;\;\;\;\; \max\limits_{{\bf{\bar{u}}}} \left\lbrace -<{\bf{u}}^{n}, \text{div}{\bf{\bar{u}}}> - f^{*}({\bf{\bar{u}}}) - \frac{1}{2\sigma}\left\| {\bf{\bar{u}}} - {\bf{\bar{u}}}^{n}\right\| _{2}^{2}\right\rbrace. \end{equation}$
       Step3: For fixed $ {\bf{\bar{u}}}^{n+1} $, update $ {\bf{u}} $ by solving:
    $ \begin{equation} (9)\;\;\;\;\;\;\;\;\;\; \min\limits_{{\bf{u}}} \left\lbrace -<{\bf{u}},\text{div}{\bf{\bar{u}}}^{n+1}> + g({\bf{u}}) - \frac{1}{2\tau}\left\| {\bf{u}} - {\bf{u}}^{n}\right\|_{2}^{2} \right\rbrace. \end{equation} $
       Step4: Iterate until $ \frac{\parallel{\bf{u}}^{n+1}-{\bf{u}}^{n}\parallel^2}{\parallel{\bf{u}}^{n}\parallel^2}\leq \epsilon $.
     | Show Table
    DownLoad: CSV
    Algorithm B:
       Step1: Choose step size $ \sigma > 0 $ and $ \tau >0 $. Initialize $ {\bf{u}}^{0} $ and $ {\bf{\bar{u}}}^{0} $.
       Step2: For each iteration:
    $\begin{equation} (14)\;\;\;\;\;\;\;\; \left\lbrace \begin{aligned} {\bf{\bar{u}}}^{n+1} & = {\bf{prox}}_{f^{*}}\left( {\bf{\bar{u}}}^{n} + \sigma\bigtriangledown{\bf{u}}^{n}\right),\\ {\bf{u}}^{n+1} & = {\bf{prox}}_{g}\left( {\bf{u}}^{n} - \tau \text{div}{\bf{\bar{u}}}^{n+1}\right). \end{aligned} \right. \end{equation} $
       Step3: Repeat $ {\bf{Step 2}} $ until $ \frac{\parallel{\bf{u}}^{n+1}-{\bf{u}}^{n}\parallel^2}{\parallel{\bf{u}}^{n}\parallel^2}\leq \epsilon $.
     | Show Table
    DownLoad: CSV

    Table 1.  SSIM, PSNR and S-CIELAB color error between original images and reflectance images

    SSIM original image
    SV TV-Ma L1-Ma TV-Ng-HSV TV-Ng-RGB Dictionary Multigrid
    deer 0.8804 0.4212 0.6019 0.9098 0.6017 0.3684 0.8634
    Parthenon 0.8476 0.6347 0.6036 0.7735 0.6996 0.7733 0.7932
    clay figure 0.8703 0.1671 0.8666 0.8569 0.6425 0.4150 0.9051
    bridge 0.8646 0.6285 0.6645 0.7508 0.5885 0.8166 0.8349
    soldiers 0.8564 0.2343 0.8289 0.7625 0.5295 0.6100 0.8041
    PSNR original image
    SV TV-Ma L1-Ma TV-Ng-HSV TV-Ng-RGB Dictionary Multigrid
    deer 19.5951 17.2721 18.5804 15.5689 8.5341 9.5946 13.2474
    Parthenon 17.5351 17.0368 13.1229 10.9352 8.5158 13.0687 11.5762
    clay figure 18.5353 15.0454 15.5129 13.2862 11.1804 8.9712 15.6808
    bridge 17.9966 18.1970 18.6187 9.9868 8.4870 14.0895 12.1009
    soldiers 17.3405 15.3468 18.9972 10.7686 7.8472 11.8415 11.7106
    S-CIELAB error original image
    SV TV-Ma L1-Ma TV-Ng-HSV TV-Ng-RGB Dictionary Multigrid
    deer 4.87% 25.46% 10.47% 7.24% 91.33% 82.33% 24.13%
    Parthenon 3.03% 8.54% 36.89% 44.82% 62.12% 37.75% 43.87%
    clay figure 10.07% 77.20% 15.42% 31.47% 67.22% 92.31% 15.37%
    bridge 2.63% 16.08% 3.89% 71.92% 81.82% 29.05% 31.93%
    soldiers 11.00% 78.01% 3.89% 63.79% 94.14% 75.67% 55.22%
     | Show Table
    DownLoad: CSV
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