# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2021061
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## A variational saturation-value model for image decomposition: Illumination and reflectance

 School of Mathematical Sciences, Tongji University, Shanghai, China

*Corresponding author: Wei Wang

Received  January 2021 Revised  July 2021 Early access October 2021

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

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.

Citation: Wei Wang, Caifei Li. A variational saturation-value model for image decomposition: Illumination and reflectance. Inverse Problems & Imaging, doi: 10.3934/ipi.2021061
##### References:
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Woodell, A multiscale Retinex for bridging the gap between color image and the human observation of scenes, IEEE Transactions on Image Processing, 6 (1997), 965-976.   Google Scholar [19] R. Kimmel, M. Elad, D. Shaked, R. Keshet and I. Sobel, A variational framework for Retinex, International Journal of Computer Vision, 52 (2003), 7-23.   Google Scholar [20] E. H. Land and J. J. McCann, Lightness and retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11.   Google Scholar [21] E. H. Land, The retinex theory of color vision, Scientific American, 237 (1977), 108-128.   Google Scholar [22] E. H. Land, Recent advances in the retinex theory and some implications for cortical computations: Color vision and natural image, Proceedings of the National Academy of Sciences of the United States of America, 80 (1983), 5163-5169.   Google Scholar [23] N. Limare, A.-B. Petro, C. Sbert and J.-M. Morel, Retinex poisson equation: A model for color perception, Image Processing On Line, 1 (2011), 39-50.   Google Scholar [24] W. Ma and S. Osher, A TV Bregman iterative model of retinex theory, Inverse Probl. Imaging, 6 (2012), 697-708.  doi: 10.3934/ipi.2012.6.697.  Google Scholar [25] W. Ma, J.-M. Morel, S. Osher and A. Chien, An L1-based variational model for Retinex theory and its application to medical images, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2011), 153–160. Google Scholar [26] J.-M. Morel, A. B. Petro and C. Sbert, Fast implementation of color constancy algorithms, Proceedings of the SPIE, 7241 (2009). Google Scholar [27] J. M. Morel, A. B. Petro and C. Sbert, A PDE formalization of retinex theory, IEEE Transactions on Image Processing, 19 (2010), 2825-2837.  doi: 10.1109/TIP.2010.2049239.  Google Scholar [28] M. K. Ng and W. Wang, A total variation model for retinex, SIAM J. Imaging Sci., 4 (2011), 345-365.  doi: 10.1137/100806588.  Google Scholar [29] Z.-F. Pang, Y.-M. Zhou, T. Wu and D.-J. Li, Image denoising via a new anisotropic total-variation-based model, Signal Processing: Image Communication, 74 2019,140–152. Google Scholar [30] L. Sun and Y.-M. Huang, A modulus-based multigrid method for image retinex, Appl. Numer. Math., 164 (2021), 199-210.  doi: 10.1016/j.apnum.2020.11.011.  Google Scholar [31] D. Terzopoulos, Image analysis using multigrid relaxation method, IEEE Transactions on Pattern Analysis and Machine Intelligence, 8 (1986), 129-139.   Google Scholar [32] T. Wu, X. Gu, Y. Wang and T. Zeng, Adaptive total variation based image segmentation with semi-proximal alternating minimization, Signal Processing, 183 (2021). Google Scholar [33] X. Zhang and B. A. Wandell, A spatial extension of CIELAB for digital color image reproduction, Journal of the Society for Information Display, 5 (1997), 61-63.   Google Scholar

show all references

##### References:
 [1] P. Arbelaez, M. Maire, C. Fowlkes and J. Malik, Contour detection and hierarchical image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 898-916.   Google Scholar [2] T. Batard, Heat equations on vector bundles - application to color image regularization, J. Math. Imaging Vision, 41 (2011), 59-85.  doi: 10.1007/s10851-011-0265-3.  Google Scholar [3] M. Bertalmio and J. D. Cowan, Implementing the retinex algorithm with wilson-cowan equations, Journal of Physiology Paris, 103 (2009), 69-72.   Google Scholar [4] A. Blakea, Boundary conditions for lightness computation in mondrian world, Computer Vision, Graphics and Image Processing, 32 (1985), 314-327.   Google Scholar [5] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar [6] H. Chang, M. K. Ng, W. Wang and T. Zeng, Retinex image enhancement via a learned dictionary, Optical Engineering, 54 (2015), 013107.   Google Scholar [7] P. Denis, P. Carre and C. Fernandez-Maloigne, Spatial and spectral quaternionic approaches for colour images, Computer Vision and Image Understanding, 107 (2007), 74-87.   Google Scholar [8] M. Elad, Retinex by two bilateral filters, Lecture Notes in Computer Science, 3459 (2005), 217-229.   Google Scholar [9] E. Esser, X. Zhang and T. F. Chan, A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, SIAM J. Imaging Sci., 3 (2010), 1015-1046.  doi: 10.1137/09076934X.  Google Scholar [10] J. Frankle and J. McCann, Method and apparatus for lightness imaging, US Patent, 1983. Google Scholar [11] B. Funt, F. Ciuera and J. McCann, Retinex in matlab, Journal of Electronic Imaging, 13 (2004), 48-57.   Google Scholar [12] B. Funt, M. Drew and M. Brockington, Recovering shading from color images, Lecture Notes in Computer Science, 588 (1992), 124-132.   Google Scholar [13] R. C. Gonzales and R. E. Woods, Digital Image Processing, 3$^{nd}$ edition, Pearson International Edition, Prentice Hall, 2008. Google Scholar [14] W. R. Hamilton, Elements of Quaternions, Longmans, Green and Co., London, 1866. Google Scholar [15] B. K. P. Horn, Determining lightness from an image, Computer Graphics and Image Processing, 3 (1974), 277-299.   Google Scholar [16] Z. Jia, M. K. Ng and W. Wang, Color image restoration by saturation-value total variation, SIAM J. Imaging Sci, 12 (2019), 972-1000.  doi: 10.1137/18M1230451.  Google Scholar [17] D. J. Jobson, Z. Rahman and G. A. Woodell, Properties and performance of the center/surround retinex, IEEE Transactions on Image Processing, 6 (1997), 451-462.   Google Scholar [18] D. J. Jobson, Z. Rahman and G. A. Woodell, A multiscale Retinex for bridging the gap between color image and the human observation of scenes, IEEE Transactions on Image Processing, 6 (1997), 965-976.   Google Scholar [19] R. Kimmel, M. Elad, D. Shaked, R. Keshet and I. Sobel, A variational framework for Retinex, International Journal of Computer Vision, 52 (2003), 7-23.   Google Scholar [20] E. H. Land and J. J. McCann, Lightness and retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11.   Google Scholar [21] E. H. Land, The retinex theory of color vision, Scientific American, 237 (1977), 108-128.   Google Scholar [22] E. H. Land, Recent advances in the retinex theory and some implications for cortical computations: Color vision and natural image, Proceedings of the National Academy of Sciences of the United States of America, 80 (1983), 5163-5169.   Google Scholar [23] N. Limare, A.-B. Petro, C. Sbert and J.-M. Morel, Retinex poisson equation: A model for color perception, Image Processing On Line, 1 (2011), 39-50.   Google Scholar [24] W. Ma and S. Osher, A TV Bregman iterative model of retinex theory, Inverse Probl. Imaging, 6 (2012), 697-708.  doi: 10.3934/ipi.2012.6.697.  Google Scholar [25] W. Ma, J.-M. Morel, S. Osher and A. Chien, An L1-based variational model for Retinex theory and its application to medical images, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2011), 153–160. Google Scholar [26] J.-M. Morel, A. B. Petro and C. Sbert, Fast implementation of color constancy algorithms, Proceedings of the SPIE, 7241 (2009). Google Scholar [27] J. M. Morel, A. B. Petro and C. Sbert, A PDE formalization of retinex theory, IEEE Transactions on Image Processing, 19 (2010), 2825-2837.  doi: 10.1109/TIP.2010.2049239.  Google Scholar [28] M. K. Ng and W. Wang, A total variation model for retinex, SIAM J. Imaging Sci., 4 (2011), 345-365.  doi: 10.1137/100806588.  Google Scholar [29] Z.-F. Pang, Y.-M. Zhou, T. Wu and D.-J. Li, Image denoising via a new anisotropic total-variation-based model, Signal Processing: Image Communication, 74 2019,140–152. Google Scholar [30] L. Sun and Y.-M. Huang, A modulus-based multigrid method for image retinex, Appl. Numer. Math., 164 (2021), 199-210.  doi: 10.1016/j.apnum.2020.11.011.  Google Scholar [31] D. Terzopoulos, Image analysis using multigrid relaxation method, IEEE Transactions on Pattern Analysis and Machine Intelligence, 8 (1986), 129-139.   Google Scholar [32] T. Wu, X. Gu, Y. Wang and T. Zeng, Adaptive total variation based image segmentation with semi-proximal alternating minimization, Signal Processing, 183 (2021). Google Scholar [33] X. Zhang and B. A. Wandell, A spatial extension of CIELAB for digital color image reproduction, Journal of the Society for Information Display, 5 (1997), 61-63.   Google Scholar
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
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
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
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
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
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
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
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
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
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
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: $$$(8) \;\;\;\;\;\;\;\; \max\limits_{{\bf{\bar{u}}}} \left\lbrace -<{\bf{u}}^{n}, \text{div}{\bf{\bar{u}}}> - f^{*}({\bf{\bar{u}}}) - \frac{1}{2\sigma}\left\| {\bf{\bar{u}}} - {\bf{\bar{u}}}^{n}\right\| _{2}^{2}\right\rbrace.$$$    Step3: For fixed ${\bf{\bar{u}}}^{n+1}$, update ${\bf{u}}$ by solving: $$$(9)\;\;\;\;\;\;\;\;\;\; \min\limits_{{\bf{u}}} \left\lbrace -<{\bf{u}},\text{div}{\bf{\bar{u}}}^{n+1}> + g({\bf{u}}) - \frac{1}{2\tau}\left\| {\bf{u}} - {\bf{u}}^{n}\right\|_{2}^{2} \right\rbrace.$$$    Step4: Iterate until $\frac{\parallel{\bf{u}}^{n+1}-{\bf{u}}^{n}\parallel^2}{\parallel{\bf{u}}^{n}\parallel^2}\leq \epsilon$.
 Algorithm A:    Step1: Choose stopping criteria $\epsilon$ and initialization ${\bf{u}}^{0}$.    Step2: For fixed ${\bf{u}}^{n}$, update ${\bf{\bar{u}}}$ by solving: $$$(8) \;\;\;\;\;\;\;\; \max\limits_{{\bf{\bar{u}}}} \left\lbrace -<{\bf{u}}^{n}, \text{div}{\bf{\bar{u}}}> - f^{*}({\bf{\bar{u}}}) - \frac{1}{2\sigma}\left\| {\bf{\bar{u}}} - {\bf{\bar{u}}}^{n}\right\| _{2}^{2}\right\rbrace.$$$    Step3: For fixed ${\bf{\bar{u}}}^{n+1}$, update ${\bf{u}}$ by solving: $$$(9)\;\;\;\;\;\;\;\;\;\; \min\limits_{{\bf{u}}} \left\lbrace -<{\bf{u}},\text{div}{\bf{\bar{u}}}^{n+1}> + g({\bf{u}}) - \frac{1}{2\tau}\left\| {\bf{u}} - {\bf{u}}^{n}\right\|_{2}^{2} \right\rbrace.$$$    Step4: Iterate until $\frac{\parallel{\bf{u}}^{n+1}-{\bf{u}}^{n}\parallel^2}{\parallel{\bf{u}}^{n}\parallel^2}\leq \epsilon$.
 Algorithm B:    Step1: Choose step size $\sigma > 0$ and $\tau >0$. Initialize ${\bf{u}}^{0}$ and ${\bf{\bar{u}}}^{0}$.    Step2: For each iteration: (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.    Step3: Repeat ${\bf{Step 2}}$ until $\frac{\parallel{\bf{u}}^{n+1}-{\bf{u}}^{n}\parallel^2}{\parallel{\bf{u}}^{n}\parallel^2}\leq \epsilon$.
 Algorithm B:    Step1: Choose step size $\sigma > 0$ and $\tau >0$. Initialize ${\bf{u}}^{0}$ and ${\bf{\bar{u}}}^{0}$.    Step2: For each iteration: (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.    Step3: Repeat ${\bf{Step 2}}$ until $\frac{\parallel{\bf{u}}^{n+1}-{\bf{u}}^{n}\parallel^2}{\parallel{\bf{u}}^{n}\parallel^2}\leq \epsilon$.
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%
 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%
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