An adaptive image inpainting method based on the modified mumford-shah model and multiscale parameter estimation
Thanh D.N.H., Prasath V.B.S., Son N.V., Hieu L.M.


Department of Information Technology, Hue College of Industry, Hue 530000 VN,
Division of Biomedical Informatics, Cincinnati Children’s Hospital Medical Center, Cincinnati, OH 45229 USA,
Department of Biomedical Informatics, College of Medicine, University of Cincinnati, OH 45267 USA
Department of Electrical Engineering and Computer Science, University of Cincinnati, OH 45221 USA,
Department of Robotics and Production Adaptation, Tula State University, Tula 300012, Russia,
Ballistic Research Laboratory, Military Weapon Institute, Hanoi 100000, Vietnam,
Department of Economics, University of Economics, The University of Danang, Danang 550000, Vietnam

Image inpainting is a process of filling missing and damaged parts of image. By using the Mumford-Shah image model, the image inpainting can be formulated as a constrained optimization problem. The Mumford-Shah model is a famous and effective model to solve the image inpainting problem. In this paper, we propose an adaptive image inpainting method based on multiscale parameter estimation for the modified Mumford-Shah model. In the experiments, we will handle the comparison with other similar inpainting methods to prove that the combination of classic model such the modified Mumford-Shah model and the multiscale parameter estimation is an effective method to solve the inpainting problem.

image inpainting, Mumford-Shah model, modified Mumford-Shah model, regularization, Euler-Lagrange equation, inverse gradient, multiscale.

Thanh DNH, Prasath VBS, Son NV, Son NV, Hieu LM. An adaptive image inpainting method based on the modified Mumford-Shah model and multiscale parameter estimation. Computer Optics 2019; 43(2): 251-257. DOI: 10.18287/2412-6179-2019-43-2-251-257.


  1. Chan T, Shen, J. Image processing and analysis: variational, PDE, wavelet, and stochastic methods. Philadelphia: Society for Industrial and Applied Mathematics Philadelphia; 2005.
  2. Grossauer H. Digital image inpainting: Completion of images with missing data regions. Innsbruck: Simon & Schuster; 2008.
  3. Esedoglu S, Shen J. Digital inpainting based on the Mumford-Shah-Euler image model. European Journal of Applied Mathematics 2002; 13(4): 353-370.
  4. Tauber Z, Li ZN, Drew MS. Review and preview: Disocclusion by inpainting for image-based rendering. IEEE Transactions on Systems, Man, and Cybernetics 2007; 37(4): 527-540.
  5. Zayed A. Advances in Shannon's sampling theory. CRC Press, 2018.
  6. Prasath VBS, Thanh DNH, Cuong NX, Hai NH. Image restoration with total variation and iterative regularization parameter estimation. ACM The Eighth International Symposium on Information and Communication Technology (SoICT 2017) 2017: 378-384.
  7. Thanh DNH, Dvoenko S. A method of total variation to remove the mixed Poisson-Gaussian noise. Pattern Recognition and Image Analysis 2016; 26(2): 285-293.
  8. Thanh DNH, Dvoenko S. Image noise removal based on total variation. Computer Optics 2015; 39(4): 564-571. DOI: 10.18287/0134-2452-2015-39-4-564-571.
  9. Rogers CA. Hausdorff measures. Cambridge: Cambridge University Press; 1998.
  10. Torben P, Robert MK, Tobias P. Ambrosio-Tortorelli segmentation of stochastic images: Model extensions, theoretical investigations and numerical methods. Inter­national Journal of Computer Vision 2013; 103(2): 190-212.
  11. Ambrosio L, Tortorelli M. Approximation of functional depending on jumps by elliptic functional via gamma convergence. Communications on Pure and Applied Mathematics 1990; 43(8): 999-1036.
  12. Prasath VBS. Quantum noise removal in X-Ray images with adaptive total variation regularization. Informatica 2017; 28(3): 505-515.
  13. Shen J, Chan TF. Mathematical models for local nontex­ture inpaintings. SIAM Journal on Applied Mathematics 2002; 62(3): 1019-1043.
  14. Schönlieb CB. Partial differential equation methods for image inpainting. Cambridge: Cambridge University Press; 2015.
  15. Dahl J, Hansen PC, Jensen SH, Jensen TL. Algorithms and software for total variation image reconstruction via first-order methods. Numer Algo 2010; 52: 67-91.
  16. Rudin LI, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D 1990; 60: 259-268.
  17. Mumford D, Shah, J. Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics 1989;42(5): 577-685.
  18. Matus PP, Hieu LM. Difference schemes on nonuniform grids for the two-dimensional convection–diffusion equation. Computational Mathematics and Mathematical Physics 2017; 57(12): 1994-2004.
  19. Prasath VBS, Vorotnikov D, Pelapur R, Jose S, Seetharaman G, Palaniappan K. Multiscale Tikhonov-total variation image restoration using spatially varying edge coherence exponent. IEEE Transactions on Image Processing 2015; 24(12): 5220-5235.
  20. Thanh DNH, Prasath VBS, Hieu LM. A review on CT and X-ray images denoising methods. Informatica 2019; 43: (forthcoming).

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