Journal of Mathematical Imaging and Vision

, Volume 59, Issue 3, pp 415–431 | Cite as

Dual Block-Coordinate Forward-Backward Algorithm with Application to Deconvolution and Deinterlacing of Video Sequences

  • Feriel Abboud
  • Emilie Chouzenoux
  • Jean-Christophe Pesquet
  • Jean-Hugues Chenot
  • Louis Laborelli
Article

Abstract

Optimization methods play a central role in the solution of a wide array of problems encountered in various application fields, such as signal and image processing. Especially when the problems are highly dimensional, proximal methods have shown their efficiency through their capability to deal with composite, possibly nonsmooth objective functions. The cornerstone of these approaches is the proximity operator, which has become a quite popular tool in optimization. In this work, we propose new dual forward-backward formulations for computing the proximity operator of a sum of convex functions involving linear operators. The proposed algorithms are accelerated thanks to the introduction of a block-coordinate strategy combined with a preconditioning technique. Numerical simulations emphasize the good performance of our approach for the problem of jointly deconvoluting and deinterlacing video sequences.

Keywords

Proximity operator Duality Block-coordinate approach Video processing Deconvolution Deinterlacing 

References

  1. 1.
    Abboud, F., Chouzenoux, E., Pesquet, J.C., Chenot, J.H., Laborelli, L.: A hybrid alternating proximal method for blind video restoration. In: 22nd IEEE 22nd European Signal Processing Conference (EUSIPCO 2014), pp. 1811–1815. Lisbon, Portugal (2014)Google Scholar
  2. 2.
    Bauschke, H.H., Combettes, P.L.: A Dykstra-like algorithm for two monotone operators. Pac. J. Optim. 4(3), 383–391 (2008)MathSciNetMATHGoogle Scholar
  3. 3.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefMATHGoogle Scholar
  4. 4.
    Bolte, J., Nguyen, T.P., Peypouquet, J., Suter, B.: From error bounds to the complexity of first-order descent methods for convex functions. Tech. rep. (2015). http://www.optimization-online.org/DB_HTML/2015/10/5176.html
  5. 5.
    Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146(1–2), 459–494 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 8(1), 1–122 (2011)MATHGoogle Scholar
  7. 7.
    Chambolle, A., Pock, T.: A remark on accelerated block coordinate descent for computing the proximity operators of a sum of convex functions. SMAI J. Comput. Math. 1, 29–54 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chouzenoux, E., Pesquet, J.C., Repetti, A.: Variable metric forward-backward algorithm for minimizing the sum of a differentiable function and a convex function. J. Optim. Theory App. 162(1), 107–132 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chouzenoux, E., Pesquet, J.C., Repetti, A.: A block coordinate variable metric forward-backward algorithm. J. Global Optim. 1–29 (2016)Google Scholar
  10. 10.
    Combettes, P.L., Dũng, D., Vũ, B.C.: Proximity for sums of composite functions. J. Math. Anal. Appl. 380(2), 680–688 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Combettes, P.L., Dung, D., Vũ, B.C.: Dualization of signal recovery problems. Set-Valued Var. Anal. 18(3), 373–404 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., Burachik, R., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer-Verlag, New York (2010)Google Scholar
  13. 13.
    Combettes, P.L., Pesquet, J.C.: Stochastic quasi-Fejér block-coordinate fixed point iterations with random sweeping. SIAM J. Optim. 25(2), 1221–1248 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Combettes, P.L., Vũ, B.C.: Variable metric forward-backward splitting with applications to monotone inclusions in duality. Optimization 63(9), 1289–1318 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Condat, L.: Semi-local total variation for regularization of inverse problems. In: 22nd IEEE European Signal Processing Conference (EUSIPCO 2014), pp. 1806–1810. Lisbon, Portugal (2014)Google Scholar
  17. 17.
    Csiba, D., Qu, Z., Richtárik, P.: Stochastic dual coordinate ascent with adaptive probabilities. In: 32nd International Conference on Machine Learning (ICML 2015), pp. 674–683. Lille, France (2015)Google Scholar
  18. 18.
    Fu, X., He, B., Wang, X., Yuan, X.: Block wise alternating direction method of multipliers with gaussian back substitution for multiple block convex programming (2014). http://www.optimization-online.org/DB_FILE/2014/09/4544.pdf
  19. 19.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)MATHGoogle Scholar
  20. 20.
    Jaggi, M., Smith, V., Takác, M., Terhorst, J., Krishnan, S., Hofmann, T., Jordan, M.I.: Communication-efficient distributed dual coordinate ascent. In: Advances in and Neural Information Processing Systems (NIPS 2014), pp. 3068–3076. Montréal, Canada (2014)Google Scholar
  21. 21.
    Keller, S.H.: Video upscaling using variational methods. Ph.D. thesis, The Image Group, Department of Computer Science Faculty of Science, University of Copenhagen (2007)Google Scholar
  22. 22.
    Keller, S.H., Lauze, F., Nielsen, M.: A total variation motion adaptive deinterlacing scheme. In: 5th International Conference Scale-Space 2005, pp. 408–418. Hofgeismar, Germany (2005)Google Scholar
  23. 23.
    Krishnan, D., Tay, T., Fergus, R.: Blind deconvolution using a normalized sparsity measure. In: IEEE Conference Computer Vision Pattern Recognition (CVPR 2011), pp. 233–240. Colorado Springs, CO (2011)Google Scholar
  24. 24.
    Liu, C., Freeman, W.T., Adelson, E.H., Weiss, Y.: Human-assisted motion annotation. In: IEEE Conference Computer Vision Pattern Recognition (CVPR 2008), pp. 1–8. Anchorage, Alaska (2008)Google Scholar
  25. 25.
    Lojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels. Editions du centre National de la Recherche Scientifique pp. 87–89 (1963)Google Scholar
  26. 26.
    Lorenz, D.A., Wenger, S., Schöpfer, F., Magnor, M.A.: A sparse Kaczmarz solver and a linearized Bregman method for online compressed sensing. In: IEEE International Conference on Image Processing (ICIP 2014), pp. 1347–1351. Paris, France (2014)Google Scholar
  27. 27.
    Mallat, S.: Super resolution bandlet upconversion for HD tv. Tech. rep. (2006). http://www.di.ens.fr/~mallat/papiers/whitepaper.pdf
  28. 28.
    Pesquet, J.C., Repetti, A.: A class of randomized primal-dual algorithms for distributed optimization. J. Nonlinear Convex Anal. 16(12), 2453–2490 (2015)MathSciNetMATHGoogle Scholar
  29. 29.
    Qu, Z., Richtárik, P., Zhang, T.: Randomized dual coordinate ascent with arbitrary sampling. In: Advances in Neural Information Processing System (NIPS 2015), pp. 865–873. Montréal, Canada (2015)Google Scholar
  30. 30.
    Richtárik, P., Takác, M.: Efficient serial and parallel coordinate descent methods for huge-scale truss topology design. In: International Conference on Operations Research (OR 2011), pp. 27–32 (2011)Google Scholar
  31. 31.
    Saha, A., Tewari, A.: On the finite time convergence of cyclic coordinate descent methods. SIAM J. Optim. 23(1), 576–601 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Seshadrinathan, k, Bovik, A.C.: Motion tuned spatio-temporal quality assessment of natural videos. IEEE Trans. Image Process. 19(2), 335–350 (2010)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Shalev-Shwartz, S., Zhang, T.: Stochastic dual coordinate ascent methods for regularized loss minimization. J. Mach. Learn. Res. 14(1), 567–599 (2013)Google Scholar
  34. 34.
    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)CrossRefGoogle Scholar
  35. 35.
    Xu, Z., Gan, Z., Zhu, X.: Compressed video super-resolution reconstruction based on regularized algorithm. In: 6th IEEE International Conference Signal Processing (ICSP 2006). Beijing, China (2006)Google Scholar
  36. 36.
    Zhang, X., Xiong, R., Ma, S., Gao, W.: A robust video super-resolution algorithm. In: 28th IEEE Picture Coding Symposium (PCS 2010), pp. 574–577. Nagoya, Japan (2010)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Feriel Abboud
    • 1
    • 3
  • Emilie Chouzenoux
    • 1
    • 2
  • Jean-Christophe Pesquet
    • 2
  • Jean-Hugues Chenot
    • 3
  • Louis Laborelli
    • 3
  1. 1.Université Paris-Est, LIGMChamps Sur MarneFrance
  2. 2.Center for Visual ComputingCentraleSupelec, Université Paris-SaclayChâtenay-malabryFrance
  3. 3.INA, Institut National de L’AudiovisuelBry Sur MarneFrance

Personalised recommendations