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Generative imaging and image processing via generative encoder
A variational saturation-value model for image decomposition: Illumination and reflectance
School of Mathematical Sciences, Tongji University, Shanghai, China |
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.
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.
|
[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. |
[3] |
M. Bertalmio and J. D. Cowan,
Implementing the retinex algorithm with wilson-cowan equations, Journal of Physiology Paris, 103 (2009), 69-72.
|
[4] |
A. Blakea,
Boundary conditions for lightness computation in mondrian world, Computer Vision, Graphics and Image Processing, 32 (1985), 314-327.
|
[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. |
[6] |
H. Chang, M. K. Ng, W. Wang and T. Zeng,
Retinex image enhancement via a learned dictionary, Optical Engineering, 54 (2015), 013107.
|
[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.
|
[8] |
M. Elad,
Retinex by two bilateral filters, Lecture Notes in Computer Science, 3459 (2005), 217-229.
|
[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. |
[10] |
J. Frankle and J. McCann, Method and apparatus for lightness imaging, US Patent, 1983. |
[11] |
B. Funt, F. Ciuera and J. McCann,
Retinex in matlab, Journal of Electronic Imaging, 13 (2004), 48-57.
|
[12] |
B. Funt, M. Drew and M. Brockington,
Recovering shading from color images, Lecture Notes in Computer Science, 588 (1992), 124-132.
|
[13] |
R. C. Gonzales and R. E. Woods, Digital Image Processing, 3$^{nd}$ edition, Pearson International Edition, Prentice Hall, 2008. |
[14] |
W. R. Hamilton, Elements of Quaternions, Longmans, Green and Co., London, 1866. |
[15] |
B. K. P. Horn,
Determining lightness from an image, Computer Graphics and Image Processing, 3 (1974), 277-299.
|
[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. |
[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.
|
[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.
|
[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.
|
[20] |
E. H. Land and J. J. McCann,
Lightness and retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11.
|
[21] |
E. H. Land,
The retinex theory of color vision, Scientific American, 237 (1977), 108-128.
|
[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.
|
[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.
|
[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. |
[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. |
[26] |
J.-M. Morel, A. B. Petro and C. Sbert, Fast implementation of color constancy algorithms, Proceedings of the SPIE, 7241 (2009). |
[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. |
[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. |
[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. |
[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. |
[31] |
D. Terzopoulos,
Image analysis using multigrid relaxation method, IEEE Transactions on Pattern Analysis and Machine Intelligence, 8 (1986), 129-139.
|
[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). |
[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.
|
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.
|
[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. |
[3] |
M. Bertalmio and J. D. Cowan,
Implementing the retinex algorithm with wilson-cowan equations, Journal of Physiology Paris, 103 (2009), 69-72.
|
[4] |
A. Blakea,
Boundary conditions for lightness computation in mondrian world, Computer Vision, Graphics and Image Processing, 32 (1985), 314-327.
|
[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. |
[6] |
H. Chang, M. K. Ng, W. Wang and T. Zeng,
Retinex image enhancement via a learned dictionary, Optical Engineering, 54 (2015), 013107.
|
[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.
|
[8] |
M. Elad,
Retinex by two bilateral filters, Lecture Notes in Computer Science, 3459 (2005), 217-229.
|
[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. |
[10] |
J. Frankle and J. McCann, Method and apparatus for lightness imaging, US Patent, 1983. |
[11] |
B. Funt, F. Ciuera and J. McCann,
Retinex in matlab, Journal of Electronic Imaging, 13 (2004), 48-57.
|
[12] |
B. Funt, M. Drew and M. Brockington,
Recovering shading from color images, Lecture Notes in Computer Science, 588 (1992), 124-132.
|
[13] |
R. C. Gonzales and R. E. Woods, Digital Image Processing, 3$^{nd}$ edition, Pearson International Edition, Prentice Hall, 2008. |
[14] |
W. R. Hamilton, Elements of Quaternions, Longmans, Green and Co., London, 1866. |
[15] |
B. K. P. Horn,
Determining lightness from an image, Computer Graphics and Image Processing, 3 (1974), 277-299.
|
[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. |
[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.
|
[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.
|
[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.
|
[20] |
E. H. Land and J. J. McCann,
Lightness and retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11.
|
[21] |
E. H. Land,
The retinex theory of color vision, Scientific American, 237 (1977), 108-128.
|
[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.
|
[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.
|
[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. |
[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. |
[26] |
J.-M. Morel, A. B. Petro and C. Sbert, Fast implementation of color constancy algorithms, Proceedings of the SPIE, 7241 (2009). |
[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. |
[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. |
[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. |
[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. |
[31] |
D. Terzopoulos,
Image analysis using multigrid relaxation method, IEEE Transactions on Pattern Analysis and Machine Intelligence, 8 (1986), 129-139.
|
[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). |
[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.
|











Algorithm A:
Step1: Choose stopping criteria Step2: For fixed Step3: For fixed Step4: Iterate until |
Algorithm A:
Step1: Choose stopping criteria Step2: For fixed Step3: For fixed Step4: Iterate until |
Algorithm B: Step1: Choose step size Step2: For each iteration: Step3: Repeat |
Algorithm B: Step1: Choose step size Step2: For each iteration: Step3: Repeat |
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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