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An image inpainting model based on the mixture of Perona–Malik equation and Cahn–Hilliard equation

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Abstract

In this paper, a fourth-order PDE model is proposed for image inpainting. This model is based on the mixture of Perona–Malik equation and Cahn–Hilliard equation. Using the idea of energy splitting, a numerical scheme is introduced for solving the proposed PDE model. Numerical experiments for testing the proposed model are provided at the end of the paper. The numerical results indicate that the proposed model can provide better inpainting performance with less computational time comparing to the classical PDE based inpainting models.

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References

  1. Barbu, T.: Detail-preserving fourth-order nonlinear pde-based image restoration framework. J. Image Graphics 8(1), 5–8 (2020)

    Google Scholar 

  2. Barbu, T., Munteanu, I.: A nonlinear fourth-order diffusion-based model for image denoising and restoration. Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 18(2), 108–115 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, pp. 417–424 (2000)

  4. Bertozzi, A., Schönlieb, C.-B.: Unconditionally stable schemes for higher order inpainting. Commun. Math. Sci. 9(2), 413–457 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Bertozzi, A.L., Esedoglu, S., Gillette, A.: Inpainting of binary images using the Cahn–Hilliard equation. IEEE Trans. Image Process. 16(1), 285–291 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Bosch, J., Stoll, M.: A fractional inpainting model based on the vector-valued Cahn–Hilliard equation. SIAM J. Imaging Sci. 8(4), 2352–2382 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Brkić, A.L., Mitrović, D., Novak, A.: On the image inpainting problem from the viewpoint of a nonlocal Cahn-Hilliard type equation. J. Adv. Res. (2020). https://doi.org/10.1016/j.jare.2020.04.015

    Article  Google Scholar 

  8. Burger, M., He, L., Schönlieb, C.-B.: Cahn-hilliard inpainting and a generalization for grayvalue images. SIAM J. Imaging Sci. 2(4), 1129–1167 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Chan, T.F., Shen, J.: Nontexture inpainting by curvature-driven diffusions. J. Vis. Commun. Image Represent. 12(4), 436–449 (2001)

    Google Scholar 

  10. Chan, T.F., Shen, J., Zhou, H.-M.: Total variation wavelet inpainting. J. Math. Imaging Vis. 25(1), 107–125 (2006)

    MathSciNet  Google Scholar 

  11. Criminisi, A., Perez, P., Toyama, K.: Object removal by exemplar-based inpainting. In: 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003 Proceedings, volume 2, pp. II–II. IEEE (2003)

  12. Dobrosotskaya, J.A., Bertozzi, A.L.: A wavelet-laplace variational technique for image deconvolution and inpainting. IEEE Trans. Image Process. 17(5), 657–663 (2008)

    MathSciNet  Google Scholar 

  13. Gelfand, I .M., Silverman, R .A.: Calculus of Variations. Courier Corporation, North Chelmsford (2000)

    Google Scholar 

  14. Gillette, A.: Image Inpainting Using a Modified Cahn–Hilliard Equation. PhD thesis, University of California Los Angeles (2006)

  15. Kim, J., Lee, C.-O.: Three-dimensional volume reconstruction using two-dimensional parallel slices. SIAM J. Imaging Sci. 12(1), 1–27 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Latecki, L. J., Lakamper, R., Eckhardt, T.: Shape descriptors for non-rigid shapes with a single closed contour. In: Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No. PR00662), volume 1, pp. 424–429. IEEE (2000)

  17. Li, X.: Image recovery via hybrid sparse representations: a deterministic annealing approach. IEEE J. Sel. Topics Signal Process. 5(5), 953–962 (2011)

    Google Scholar 

  18. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    Google Scholar 

  19. Shen, J., Chan, T.: Variational restoration of nonflat image features: models and algorithms. SIAM J. Appl. Math. 61(4), 1338–1361 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Shen, J., Chan, T.F.: Mathematical models for local nontexture inpaintings. SIAM J. Appl. Math. 62(3), 1019–1043 (2002)

    MathSciNet  MATH  Google Scholar 

  21. Tsai, A., Yezzi, A., Willsky, A.S.: Curve evolution implementation of the Mumford–Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Trans. Image Process. 10(8), 1169–1186 (2001)

    MATH  Google Scholar 

  22. Vijayakrishna, R.: A unified model of Cahn–Hilliard greyscale inpainting and multiphase classification. PhD thesis, PhD thesis, Indian Institute of Technology Kanpur, Kanpur (2015)

  23. Vollmayr-Lee, B.P., Rutenberg, A.D.: Fast and accurate coarsening simulation with an unconditionally stable time step. Phys. Rev. E 68(6), 066703 (2003)

    Google Scholar 

  24. Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)

    Google Scholar 

  25. Zhang, K., Crooks, E., Orlando, A.: Compensated convexity methods for approximations and interpolations of sampled functions in euclidean spaces: applications to contour lines, sparse data, and inpainting. SIAM J. Imaging Sci. 11(4), 2368–2428 (2018)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Applied Mathematics and Computational Sciences, The University of Iowa, Iowa City, IA, 52242, USA

    Qing Zou

  2. Computational Biomedical Imaging Group, The University of Iowa, Iowa City, IA, 52242, USA

    Qing Zou

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  1. Qing Zou
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Correspondence to Qing Zou.

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Zou, Q. An image inpainting model based on the mixture of Perona–Malik equation and Cahn–Hilliard equation. J. Appl. Math. Comput. 66, 21–38 (2021). https://doi.org/10.1007/s12190-020-01422-8

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  • DOI: https://doi.org/10.1007/s12190-020-01422-8

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