A game theory approach for joint blind deconvolution and inpainting

: pp. 674–681
Received: February 15, 2023
Revised: July 08, 2023
Accepted: July 09, 2023

Mathematical Modeling and Computing, Vol. 10, No. 3, pp. 674–681 (2023)

LMCSA, FSTM, Hassan II University of Casablanca
LMCSA, FSTM, Hassan II University of Casablanca
LMCSA, FSTM, Hassan II University of Casablanca

In this paper we propose a new mathematical model for joint Blind Deconvolution and Inpainting.  The main objective is the treatment of blurred images with missing parts, through the game theory framework, in particular, a Nash game, we define two players: Player 1 handles the image intensity while Player 2, operates on the blur kernel.  The two engage in a game until the equilibrium is reached.  Finally, we provide some numerical examples: we compare the efficiency of our proposed approach to other existing methods in the literature that deals with Blind Deconvolution and Inpainting separately.

  1. Xu L., Zheng S., Jia J.  Unnatural L0 Sparse Representation for Natural Image Deblurring.  2013 IEEE Conference on Computer Vision and Pattern Recognition. 1107–1114 (2013).
  2. Zhang G., Kingsbury N.  Fast l0-based image deconvolution with variational Bayesian inference and majorization-minimization.  IEEE Global Conference on Signal and Information Processing. 1081–1084 (2013).
  3. Xu L., Jia J.  Two-phase kernel estimation for robust motion deblurring.  European Conference on Computer Vision (ECCV 2010). 157–170 (2010).
  4. Lin Y., Kandel Y., Zotta M., Lifshin E.  SEM Resolution improvement using semi-blind restoration with hybrid l1–l2 regularization.  IEEE Southwest Symposium on Image Analysis and Interpretation (SSIAI). 33–36 (2016).
  5. Huang Y., Ng M. K., Wen Y. W.  A fast total variation minimization method for image restoration.  Multiscale Modeling & Simulation.  7 (2), 774–795 (2008).
  6. Krishnan D., Tay T., Fergus R.  Blind deconvolution using a normalized sparsity measure.  CVPR 2011. 233–240 (2011).
  7. Li Z.-M., Zheng Y., Jing W.-F., Zhao R.-S., Jing K.-L.  Hyper-Laplacian non-blind deblurring model based on regional division.  2015 International Conference on Network and Information Systems for Computers. 223–226 (2015).
  8. You Y.-L., Kaveh M.  Blind image restoration by anisotropic regularization.  IEEE Transactions on Image Processing.  8 (3), 396–407 (1999).
  9. Perona P., Malik J.  Scale-space and edge detection using anisotropic diffusion.  IEEE Transactions on Pattern Analysis and Machine Intelligence.  12 (7), 629–639 (1990).
  10. Chan T. F., Wong C.-K.  Total variation blind deconvolution.  IEEE Transactions on Image Processing.  7 (3), 370–375 (1998).
  11. Liu H., Gu M., Meng M. Q.-H., Lu W.-S.  Fast weighted total variation regularization algorithm for blur identification and image restoration.  IEEE Access.  4, 6792–6801 (2016).
  12. Bertalmio M., Sapiro G., Caselles V., Ballester C.  Image Inpainting.  Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques. 417–424 (2000).
  13. Criminisi A., Perez P., Toyama K.  Region filling and object removal by exemplar-based image inpaintin.  IEEE Transactions on Image Processing.  13 (9), 1200–1212 (2004).
  14. Getreuer P.  Total Variation Inpainting using Split Bregman.  Image Processing on Line.  2, 147–157 (2012).
  15. Esedoglu S., Shen J.  Digital inpainting based on the Mumford–Shah–Euler image model.  European Journal of Applied Mathematics.  13 (4), 353–370 (2002).
  16. Boujena S., Bellaj K., Gouasnouane O., El Guarmah E.  An improved nonlinear model for image inpainting.  Applied Mathematical Sciences.  9 (124), 6189–6205 (2015).
  17. Gouasnouane O., Moussaid N., Boujena S., Kabli K.  A nonlinear fractional partial differential equation for image inpainting.  Mathematical Modeling and Computing.  9 (3), 536–546 (2022).
  18. Caselles V., Morel J.-M., Sbert C.  An axiomatic approach to image interpolation.  IEEE Transactions on Image Processing Journal of Applied Mathematics.  7 (3), 376–386 (1998).
  19. Elmoumen S., Moussaid N., Aboulaich R.  Image retrieval using Nash equilibrium and Kalai–Smorodinsky solution.  Mathematical Modeling and Computing.  8 (4), 646–657 (2021).
  20. Meskine D., Moussaid N., Berhich S.  Blind image deblurring by game theory.  NISS19: Proceedings of the 2nd International Conference on Networking, Information Systems & Security. 1–7 (2019).
  21. Nasr N., Moussaid N., Gouasnouane O.  A Nash-game approach to Blind Image Deblurring.  2021 Third International Conference on Transportation and Smart Technologies (TST). 36–41 (2021).
  22. Nasr N., Moussaid N., Gouasnouane O.  The Kalai Smorodinsky solution for blind deconvolution.  Computational and Applied Mathematics.  41 (5), 222 (2022).
  23. Chan T. F., Yip A. M., Park F. E.  Simultaneous total variation image inpainting and blind deconvolution.  International Journal of Imaging Systems and Technology.  15 (1), 92–102 (2005).
  24. Lagendijk R. L., Biemond J.  Iterative Identification and Restoration of Images.  Springer, New York (1991).
  25. Chen Y., Wunderli T.  Adaptive total variation for image restoration in BV space.  Journal of Mathematical Analysis and Applications.  272 (1), 117–137 (2002).
  26. Wang Z., Bovik A. C., Sheikh H. R., Simoncelli E. P.  Image quality assessment: from error visibility to structural similarity.  IEEE Transactions on Image Processing.  13 (4), 600–612 (2004).
  27. Yin M., Gao J., Tien D., Cai S.  Blind image deblurring via coupled sparse representation.  Journal of Visual Communication and Image Representation.  25 (5), 814–821 (2014).