Multidimensional Systems and Signal Processing

, Volume 29, Issue 1, pp 299–320 | Cite as

Wavelet inpainting by fractional order total variation

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Abstract

Image inpainting in the wavelet domain refers to the recovery of an image from incomplete wavelet coefficients. In this paper, we propose a wavelet inpainting model by using fractional order total variation regularization approach. Moreover, we use a simple but very efficient primal–dual algorithm to calculate the optimal solution. In the light of saddle-point theory, the convergence of new algorithm is guaranteed. Experimental results are presented to show performance of the proposed algorithm.

Keywords

Fractional order derivative Total variation Primal–dual Wavelet inpainting 

Notes

Acknowledgements

The authors would like to thank the associate editor and reviewers for helpful comments that greatly improved the paper. This work was supported by the National Natural Science Foundation of China (Nos. 61301243, 61201455), Natural Science Foundation of Shandong Province of China (Nos. ZR2013FQ007, ZR2014AQ014), and the Fundamental Research Funds for the Central Universities (No. 15CX02060A) the China Scholarship Council.

References

  1. Arrow, K., Hurwicz, L., & Uzawa, H. (1958). Studies in linear and nonlinear programming. In K. J. Arrow (Ed.), Mathematical studies in the social sciences. Palo Alto, CA: Stanford University Press.Google Scholar
  2. Aujol, J.-F., Ladjal, S., & Masnou, S. (2011). Exemplar-based inpainting from a variational point of view. International Journal of Computer Vision, 93(3), 319–347.MathSciNetMATHCrossRefGoogle Scholar
  3. Bai, J., & Feng, X. (2007). Fractional-order anisotropic diffusion for image denoising. IEEE Transactions on Image Processing, 16(10), 2492–2502.MathSciNetCrossRefGoogle Scholar
  4. Bertalmo, M., Sapiro, G., Caselles, V., & Ballester, C. (2000). Image inpainting. In Proceedings of SIGGRAPH (pp. 417–424).Google Scholar
  5. Bonettini, S., & Ruggiero, V. (2012). On the convergence of primal–dual hybrid gradient algorithms for total variation image restoration. Journal of mathematical Imaging and Vision, 44(3), 236–253.MathSciNetMATHCrossRefGoogle Scholar
  6. Cai, J. F., Chan, R. H., & Shen, Z. (2008). A framelet-based image inpainting algorithm. Applied and Computational Harmonic Analysis, 24(2), 131–149.MathSciNetMATHCrossRefGoogle Scholar
  7. Chambolle, A. (2004). An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 20(1–2), 89–97.MathSciNetMATHGoogle Scholar
  8. Chambolle, A., & Pock, T. (2010). A first-order primal–dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision, 40(1), 120–145.MathSciNetMATHCrossRefGoogle Scholar
  9. Chan, T., & Shen, J. (2001a). Mathematical models for local nontexture inpaintings. SIAM Journal on Applied Mathematics, 62(3), 1019–1043.MathSciNetMATHGoogle Scholar
  10. Chan, T., & Shen, J. (2001b). Non-texture inpainting by curvature-driven diffusions. Journal of Visual Communication and Image Representation, 4(12), 436–449.CrossRefGoogle Scholar
  11. Chan, T., & Shen, J. (2002). Euler’s elastica and curvature-based inpainting. SIAM Journal on Applied Mathematics, 63(2), 564–592.MathSciNetMATHGoogle Scholar
  12. Chan, T., Shen, J., & Zhou, H. (2006). Total variation wavelet inpainting. Journal of Mathematical Imaging and Vision, 25(1), 107–125.MathSciNetCrossRefGoogle Scholar
  13. Chen, D., Chen, Y., & Xue, D. (2011). Digital fractional order Savitzky–Golay differentiator. IEEE Transactions on Circuits and Systems II, 58(11), 758–762.CrossRefGoogle Scholar
  14. Chen, D., Sheng, H., Chen, Y., & Xue, D. (1990). Fractional-order variational optical flow model for motion estimation. Philosophical Transactions of the Royal Society A, 2013, 371.Google Scholar
  15. Chen, D., Sun, S., Zhang, C., Chen, Y., & Xue, D. (2013). Fractional order TV-L2 model for image denoising. Berlin: Central European Journal of Physics.Google Scholar
  16. Chen, D. Q., & Cheng, L. Z. (2010). Alternative minimisation algorithm for non-local total variational image deblurring. IET Image Processing, 4(5), 353–364.CrossRefGoogle Scholar
  17. Cohen, A., Daubeches, I., & Feauveau, J. C. (1992). Biorthogonal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics, 45(5), 485–560.MathSciNetMATHCrossRefGoogle Scholar
  18. Criminisi, A., Perez, P., & Toyama, K. (2004). Region filling and object removal by exemplar-based image inpainting. IEEE Transactions on Image Processing, 13(9), 200–1212.CrossRefGoogle Scholar
  19. Durand, S., & Froment, J. (2003). Reconstruction of wavelet coefficients using total variation minimization. SIAM Journal on Scientific Computing, 24, 1754–1767.MathSciNetMATHCrossRefGoogle Scholar
  20. Elad, M., Starck, J., Querre, P., & Donoho, D. (2005). Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA). Applied and Computational Harmonic Analysis, 19, 340–358.MathSciNetMATHCrossRefGoogle Scholar
  21. Esser, E., Zhang, X., & Chan, T. F. (2010). A general framework for a class of first order primal–dual algorithms for convex optimization in imaging science. SIAM Journal on Imaging Sciences, 3(4), 1015–1046.MathSciNetMATHCrossRefGoogle Scholar
  22. Gilboa, G., & Osher, S. (2008). Nonlocal operators with applications to image processing. SIAM Multiscale Modeling and Simulation, 7(3), 1005–1028.MathSciNetMATHCrossRefGoogle Scholar
  23. He, B., & Yuan, X. (2012). Convergence analysis of primal–dual algorithms for a saddle-point problem: From contraction perspective. SIAM Journal on Imaging Sciences, 5(1), 119–149.MathSciNetMATHCrossRefGoogle Scholar
  24. He, L., & Wang, Y. (2014). Iterative support detection-based split Bregman method for wavelet frame-based image inpainting. IEEE Transactions on Image Processing, 23(12), 5470–5485.MathSciNetMATHCrossRefGoogle Scholar
  25. Idczak, D., & Kamocki, R. (2015). Fractional differential repetitive processes with Riemann–Liouville and Caputo derivatives. Multidimensional Systems and Signal Processing, 26, 193–206.MathSciNetMATHCrossRefGoogle Scholar
  26. Jin, K. H., & Ye, J. C. (2015). Annihilating filter-based low-rank Hankel matrix approach for image inpainting. IEEE Transactions on Image Processing, 24(11), 3498–3511.MathSciNetCrossRefGoogle Scholar
  27. Kingsbury, N. (2001). Complex wavelets for shift invariant analysis and filtering of signals. Applied and Computational Harmonic Analysis, 10(3), 234–253.MathSciNetMATHCrossRefGoogle Scholar
  28. Liu, S.-C., & Chang, S. (1997). Dimension estimation of discrete-time fractional Brownian motion with applications to image texture classification. IEEE Transactions on Image Processing, 6(8), 1176–1184.CrossRefGoogle Scholar
  29. Peyre, G., Bougleux, S., & Cohen, L. (2011). Non-local regularization of inverse problems. Inverse Problems and Imaging, 5(2), 511–530.MathSciNetMATHCrossRefGoogle Scholar
  30. Pu, Y. F., Zhou, J. L., & Yuan, X. (2010). Fractional differential mask: A fractional differential-based approach for multiscale texture enhancement. IEEE Transactions on Image Processing, 19(2), 491–511.MathSciNetMATHCrossRefGoogle Scholar
  31. Ron, A., & Shen, Z. (1997). Affine systems in \(l^{2}(r^{d})\): The analysis of the analysis operator. Journal of Functional Analysis, 148, 408–447.MathSciNetMATHCrossRefGoogle Scholar
  32. Zhang, J., & Chen, K. (2015). A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution. SIAM Journal on Imaging Sciences, 8(4), 2487–2518.MathSciNetMATHCrossRefGoogle Scholar
  33. Zhang, J., Wei, Z., & Xiao, L. (2012). Adaptive fractional-order multiscale method for image denoising. Journal of Mathematical Imaging and Vision, 43(1), 39–49.MathSciNetMATHCrossRefGoogle Scholar
  34. Zhang, X., Burger, M., & Osher, S. (2011). A unified primal–dual algorithm framework based on Bregman iteration. Journal of Scientific Computing, 46(1), 20–46.MathSciNetMATHCrossRefGoogle Scholar
  35. Zhang, X. Q., Burger, M., Bresson, X., & Osher, S. (2010). Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM Journal on Imaging Sciences, 3(3), 253–276.MathSciNetMATHCrossRefGoogle Scholar
  36. Zhang, X. Q., & Chan, T. F. (2010). Wavelet inpainting by nonlocal total variation. Inverse Problems and Imaging, 4(1), 191–210.MathSciNetMATHCrossRefGoogle Scholar
  37. Zhu, M., & Chan, T. (2008). An efficient primal–dual hybrid gradient algorithm for total variation image restoration. CAM Report 08-34, UCLA, Los Angeles, CAGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.College of ScienceChina University of PetroleumQingdaoChina

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