American Institute of Mathematical Sciences

Advanced
doi: 10.3934/ipi.2021061
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

Wei Wang , and  Caifei Li

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.

Keywords: Image decomposition, illumination, reflectance, saturation-value, primal-dual algorithm.
Mathematics Subject Classification: 68U10, 65K10, 65J22, 90C25.
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:
[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

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

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

Table0 
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 $.
Table Options

Download as excel

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 $.
Table00 
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 $.
Table Options

Download as excel

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 $.
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%
Table Options

Download as excel

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%
[1]

Gianni Di Pillo, Giampaolo Liuzzi, Stefano Lucidi. A primal-dual algorithm for nonlinear programming exploiting negative curvature directions. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 509-528. doi: 10.3934/naco.2011.1.509
[2]

Kai Wang, Deren Han. On the linear convergence of the general first order primal-dual algorithm. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021134
[3]

Xiaojing Ye, Haomin Zhou. Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm. Inverse Problems & Imaging, 2013, 7 (3) : 1031-1050. doi: 10.3934/ipi.2013.7.1031
[4]

Yu-Hong Dai, Zhouhong Wang, Fengmin Xu. A Primal-dual algorithm for unfolding neutron energy spectrum from multiple activation foils. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2367-2387. doi: 10.3934/jimo.2020073
[5]

Wei Wang, Ling Pi, Michael K. Ng. Saturation-Value Total Variation model for chromatic aberration correction. Inverse Problems & Imaging, 2020, 14 (4) : 733-755. doi: 10.3934/ipi.2020034
[6]

Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial & Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723
[7]

Fengmin Wang, Dachuan Xu, Donglei Du, Chenchen Wu. Primal-dual approximation algorithms for submodular cost set cover problems with linear/submodular penalties. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 91-100. doi: 10.3934/naco.2015.5.91
[8]

Yu-Hong Dai, Xin-Wei Liu, Jie Sun. A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs. Journal of Industrial & Management Optimization, 2020, 16 (2) : 1009-1035. doi: 10.3934/jimo.2018190
[9]

Yanqin Bai, Xuerui Gao, Guoqiang Wang. Primal-dual interior-point algorithms for convex quadratic circular cone optimization. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 211-231. doi: 10.3934/naco.2015.5.211
[10]

Siqi Li, Weiyi Qian. Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 37-46. doi: 10.3934/naco.2015.5.37
[11]

Yixuan Yang, Yuchao Tang, Meng Wen, Tieyong Zeng. Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications. Inverse Problems & Imaging, 2021, 15 (4) : 787-825. doi: 10.3934/ipi.2021014
[12]

Guoqiang Wang, Zhongchen Wu, Zhongtuan Zheng, Xinzhong Cai. Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 101-113. doi: 10.3934/naco.2015.5.101
[13]

Rongliang Chen, Jizu Huang, Xiao-Chuan Cai. A parallel domain decomposition algorithm for large scale image denoising. Inverse Problems & Imaging, 2019, 13 (6) : 1259-1282. doi: 10.3934/ipi.2019055
[14]

Shi Yan, Jun Liu, Haiyang Huang, Xue-Cheng Tai. A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation. Inverse Problems & Imaging, 2019, 13 (3) : 653-677. doi: 10.3934/ipi.2019030
[15]

Nadia Hazzam, Zakia Kebbiche. A primal-dual interior point method for $ P_{\ast }\left( \kappa \right) $-HLCP based on a class of parametric kernel functions. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 513-531. doi: 10.3934/naco.2020053
[16]

Ying Zhang, Xuhua Ren, Bryan Alexander Clifford, Qian Wang, Xiaoqun Zhang. Image fusion network for dual-modal restoration. Inverse Problems & Imaging, 2021, 15 (6) : 1409-1419. doi: 10.3934/ipi.2021067
[17]

Chunrong Chen, Shengji Li. Upper Hölder estimates of solutions to parametric primal and dual vector quasi-equilibria. Journal of Industrial & Management Optimization, 2012, 8 (3) : 691-703. doi: 10.3934/jimo.2012.8.691
[18]

Wenzhong Zhu, Huanlong Jiang, Erli Wang, Yani Hou, Lidong Xian, Joyati Debnath. X-ray image global enhancement algorithm in medical image classification. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1297-1309. doi: 10.3934/dcdss.2019089
[19]

Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems & Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523
[20]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4963-4998. doi: 10.3934/dcdsb.2020321

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (64)
  • HTML views (79)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy