Recursive SURE for image recovery via total variation minimization

  • Feng XueEmail author
  • Jiaqi Liu
  • Xia Ai
Original Paper


Recently, total variation regularization has become a standard technique, and even a basic tool for image denoising and deconvolution. Generally, the recovery quality strongly depends on the regularization parameter. In this work, we develop a recursive evaluation of Stein’s unbiased risk estimate (SURE) for the parameter selection, based on specific reconstruction algorithms. It enables us to monitor the evolution of mean squared error (MSE) during the iterations. In particular, to deal with large-scale data, we propose a Monte Carlo simulation for the practical computation of SURE, which is free of any explicit matrix operation. Experimental results show that the proposed recursive SURE could lead to highly accurate estimate of regularization parameter and nearly optimal restoration performance in terms of MSE.


Total variation Denoising Deconvolution Stein’s unbiased risk estimate (SURE) Jacobian recursion 



  1. 1.
    Blu, T., Luisier, F.: The SURE-LET approach to image denoising. IEEE Trans. Image Process. 16(11), 2778–2786 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, D.: Inext alternating direction method based on Newton descent algorithm with application to Poisson image deblurring. Signal Image Video Process. 11, 89–96 (2017)CrossRefGoogle Scholar
  4. 4.
    Dell’Acqua, P.: \(\nu \) acceleration of statistical iterative methods for image restoration. Signal Image Video Process. 10, 927–934 (2016)CrossRefGoogle Scholar
  5. 5.
    Golub, G., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21(2), 215–223 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Guo, C., Li, Q., Wei, C., Tan, J., Liu, S., Liu, Z.: Axial multi-image phase retrieval under tilt illumination. Sci. Rep. 7, 7562 (2017)CrossRefGoogle Scholar
  7. 7.
    Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34(4), 561–580 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Morozov, V.: Methods for Solving Incorrectly Posed Problems. Springer, New York (1984)CrossRefGoogle Scholar
  9. 9.
    Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. SIAM J. Multiscale Model. Simul. 4(2), 460–489 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pan, H., Blu, T.: An iterative linear expansion of thresholds for \(\ell _1\)-based image restoration. IEEE Trans. Image Process. 22(9), 3715–3728 (2013)CrossRefGoogle Scholar
  11. 11.
    Ramirez, C., Argaez, M.: An \(\ell _1\) minimization algorithm for non-smooth regularization in image processing. Signal Image Video Process. 9, 373–386 (2015)CrossRefGoogle Scholar
  12. 12.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Selesnick, I., Parekh, A., Bayram, I.: Convex 1-D total variation denoising with non-convex regularization. IEEE Signal Process. Lett. 22(2), 141–144 (2015)CrossRefGoogle Scholar
  14. 14.
    Shen, C., Bao, X., Tan, J., Liu, S., Liu, Z.: Two noise-robust axial scanning multi-image phase retrieval algorithms based on Pauta criterion and smoothness constraint. Opt. Express 25(14), 16235–16249 (2017)CrossRefGoogle Scholar
  15. 15.
    Siadat, M., Aghazadeh, N., Öktem, O.: Reordering for improving global Arnoldi–Tikhonov method in image restoration problems. Signal Image Video Process. 12, 497–504 (2018)CrossRefGoogle Scholar
  16. 16.
    Stein, C.M.: Estimation of the mean of a multivariate normal distribution. In: Stein, C.M. (ed.) The Annals of Statistics, vol. 9, No. 6, pp. 1135–1151 (1981)Google Scholar
  17. 17.
    Tao, M., Yang, J.: Alternating Direction Algorithms for Total Variation Deconvolution in Image Reconstruction. Optimization Online, TR0918, Department of Mathmatics, Nanjing University (2009)Google Scholar
  18. 18.
    Vonesch, C., Ramani, S., Unser, M.: Recursive risk estimation for non-linear image deconvolution with a wavelet-domain sparsity constraint. In: IEEE International conference on Image processing, pp. 665–668 (2008)Google Scholar
  19. 19.
    Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1(3), 248–272 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Xue, F., Blu, T.: A novel SURE-based criterion for parametric PSF estimation. IEEE Trans. Image Process. 24(2), 595–607 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Xue, F., Du, R., Liu, J.: A recursive predictive risk estimate for proximal algorithms. In: Proceedings of the 41st IEEE International Conference on Acoustics, Speech and Signal Processing, Shanghai, China, March 20–25, pp. 4498–4502 (2016)Google Scholar
  22. 22.
    Xue, F., Luisier, F., Blu, T.: Multi-Wiener SURE-LET deconvolution. IEEE Trans. Image Process. 22(5), 1954–1968 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Xue, F., Yagola, A.G., Liu, J., Meng, G.: Recursive SURE for iterative reweighted least square algorithms. Inverse Probl. Sci. Eng. 24(4), 625–646 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.National Key Laboratory of Science and Technology on Test Physics and Numerical MathematicsBeijingChina

Personalised recommendations