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Wavelet Frame Based Image Restoration via Combined Sparsity and Nonlocal Prior of Coefficients

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Abstract

Owing to the good ability of sparsely approximating piece-wise smooth functions like images, the (tight) wavelet frame has been widely investigated and applied for image restoration and other image processing problems over the past few years. Most of the variational models based on wavelet frame proposed in the past utilize the \(l_{1}\) norm of frame coefficients as a sparsity prior. Very recently, the variational model which penalizes the \(l_{0}\) “norm” of frame coefficients was proposed for image restoration, and proved to outperform the commonly used \(l_{1}\) minimization methods in the quality of restored images. Though the \(l_{0}\) “norm” has the ability of preserving sharp edges and smooth regions, textures and small details may be mistakenly removed at the same time. Therefore, we introduce a \(l_0-l_2\) regularization model which contains a nonlocal prior of frame coefficients to avoid this issue in this paper. Meanwhile, a narrow-band technique is introduced to further improve the computational efficiency of the proposed algorithm. Numerical experiments demonstrate that the propose algorithm is superior to the recently proposed algorithm for \(l_{0}\) “norm” minimization in both iterative time and recovery quality.

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References

  1. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1), 259–268 (1992)

    Article  MATH  Google Scholar 

  2. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Springer, Berlin (2006)

    Google Scholar 

  3. Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chan, R.H., Chan, T.F., Shen, L., et al.: Wavelet algorithms for high-resolution image reconstruction. SIAM J. Sci. Comput. 24(4), 1408–1432 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cai, J.F., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8(2), 337–369 (2009)

    Article  MathSciNet  Google Scholar 

  6. Dong, B., Shen, Z.: MRA based wavelet frames and applications. IAS Lecture Notes Series, Summer Program on The Mathematics of Image Processing, Park City Mathematics Institute (2010)

  7. Shen, Z., Toh, K.C., Yun, S.: An accelerated proximal gradient algorithm for frame-based image restoration via the balanced approach. SIAM J. Imaging Sci. 4(2), 573–596 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, D.Q.: Regularized generalized inverse accelerating linearized alternating minimization algorithm for frame-based poissonian image deblurring. SIAM J. Imaging Sci. 7(2), 716–739 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Durand, S., Nikolova, M.: Denoising of frame coefficients using \(l^1\) data-fidelity term and edge-preserving regularization. Multiscale Model. Simul. 6(2), 547–576 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, D.Q., Cheng, L.Z., Su, F.: A new TV-stokes model with augmented Lagrangian method for image denoising and deconvolution. J. Sci. Comput. 51(3), 505–526 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Benning, M., Brune, C., Burger, M., Müller, J.: Higher-order TV methods-enhancement via Bregman iteration. J. Sci. Comput. 54(2–3), 269–310 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lefkimmiatis, S., Ward, J.P., Unser, M.: Hessian Schatten-norm regularization for linear inverse problems. IEEE Trans. Image Process. 22(5), 1873 (2013)

    Article  MathSciNet  Google Scholar 

  15. Chan, T.F., Shen, J.J.: Image processing and analysis: variational, PDE, wavelet, and stochastic methods. SIAM (2005)

  16. Cai, J.F., Osher, S., Shen, Z.: Linearized Bregman iterations for frame-based image deblurring. SIAM J. Imaging Sci. 2(1), 226–252 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cai, J.F., Dong, B., Osher, S., Shen, Z.: Image restoration: total variation, wavelet frames, and beyond. J. Am. Math. Soc. 25(4), 1033–1089 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. CaI, J.F., Dong, B., Shen, Z.: Image restoration: a wavelet frame based model for piecewise smooth functions and beyond. UCLA CAM Report, pp. 14–28 (2014)

  19. Dong, B., Jiang, Q., Shen, Z.: Image restoration: wavelet frame shrinkage, nonlinear evolution pdes, and beyond. UCLA CAM Report, pp. 13–78 (2013)

  20. Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Candès, E.J., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)

    Article  MATH  Google Scholar 

  22. Candès, E.J., Wakin, M.B.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25(2), 21–30 (2008)

    Article  Google Scholar 

  23. Chartrand, R.: Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data. In: IEEE International Symposium on Biomedical Imaging (ISBI), pp. 262–265 (2009)

  24. Chartrand, R., Yin, W.: Iteratively reweighted algorithms for compressive sensing. In: 33rd International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 3869–3872 (2008)

  25. Zhang, Y., Dong, B., Lu, Z.: \(l_0\) Minimization for wavelet frame based image restoration. Math. Comput. 82(282), 995–1015 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lu, Z., Zhang, Y.: Penalty decomposition methods for l0-norm minimization. Preprint (2010)

  27. Dong, B., Zhang, Y.: An efficient algorithm for \(l_0\) minimization in wavelet frame based image restoration. J. Sci. Comput. 54(2–3), 350–368 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kindermann, S., Osher, S., Jones, P.W.: Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul. 4(4), 1091–1115 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang, X., Burger, M., Bresson, X., et al.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM J. Imaging Sci. 3(3), 253–276 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Deledalle, C.A., Denis, L., Tupin, F.: Iterative weighted maximum likelihood denoising with probabilistic patch-based weights. IEEE Trans. Image Process. 18(12), 2661–2672 (2009)

    Article  MathSciNet  Google Scholar 

  32. Dabov, K., Foi, A., Katkovnik, V., et al.: Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007)

    Article  MathSciNet  Google Scholar 

  33. Dong, W., Li, X., Zhang, D., et al.: Sparsity-based image denoising via dictionary learning and structural clustering. IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 457–464 (2011)

  34. Dong, W., Zhang, L., Shi, G., et al.: Nonlocally centralized sparse representation for image restoration. IEEE Trans. Image Process. 22(4), 1620–1630 (2013)

    Article  MathSciNet  Google Scholar 

  35. Mairal, J., Bach, F., Ponce, J., et al.: Non-local sparse models for image restoration. IEEE International Conference on Computer Vision (ICCV), pp. 2272–2279 (2009)

  36. Dong, W., Shi, G., Li, X.: Nonlocal image restoration with bilateral variance estimation: a low-rank approach. Image Process. IEEE Trans. 22(2), 700–711 (2013)

    Article  MathSciNet  Google Scholar 

  37. Cai, J.F., Ji, H., Shen, Z., et al.: Data-driven tight frame construction and image denoising. Appl. Comput. Harmonic Anal. 37(1), 89–105 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Quan, Y., Ji, H., Shen, Z.: Data-Driven Multi-scale Non-local Wavelet Frame Construction and Image Recovery. J. Sci. Comput. (2014). doi:10.1007/s10915-014-9893-2

  39. Daubechies, I., Han, B., Ron, A., et al.: Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmonic Anal. 14(1), 1–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Dong, B., Shen, Z.: Pseudo-splines, wavelets and framelets. Appl. Comput. Harmonic Anal. 22(1), 78–104 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1(2), 97–116 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, X., Burger, M., Osher, S.: A unified primal-dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46(1), 20–46 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tao, M., Yuan, X.: On the O(1/t) convergence rate of alternating direction method with logarithmic-quadratic proximal regularization. SIAM J. Optim. 22(4), 1431–1448 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  45. Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image restoration by sparse 3D transform-domain collaborative filtering. International Society for Optics and Photonics, 681207-681207-12 (2008)

  46. Danielyan, A., Katkovnik, V., Egiazarian, K.: BM3D frames and variational image deblurring. IEEE Trans. Image Process. 21(4), 1715–1728 (2012)

    Article  MathSciNet  Google Scholar 

  47. Lou, Y., Zhang, X., Osher, S., et al.: Image recovery via nonlocal operators. J. Sci. Comput. 42(2), 185–197 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

Download references

Acknowledgments

The work was supported in part by the National Natural Science Foundation of China under Grant 61401473 and 61271014. We appreciate the constructive comments of the anonymous reviewers, which led to great improvements in this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Third Military Medical University, Chongqing, 400038, People’s Republic of China

    Dai-Qiang Chen & Yan Zhou

  2. School of Automation, Chongqing University, Chongqing, 401331, People’s Republic of China

    Yan Zhou

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  1. Dai-Qiang Chen
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  2. Yan Zhou
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Correspondence to Dai-Qiang Chen.

Additional information

The work was supported in part by the National Natural Science Foundation of China under Grant 61271014 and 61401473.

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Chen, DQ., Zhou, Y. Wavelet Frame Based Image Restoration via Combined Sparsity and Nonlocal Prior of Coefficients. J Sci Comput 66, 196–224 (2016). https://doi.org/10.1007/s10915-015-0018-3

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