Advertisement

Journal of Mathematical Imaging and Vision

, Volume 47, Issue 3, pp 167–178 | Cite as

A Parallel Proximal Splitting Method for Disparity Estimation from Multicomponent Images Under Illumination Variation

  • C. Chaux
  • M. El-Gheche
  • J. Farah
  • J.-C. Pesquet
  • B. Pesquet-Popescu
Article

Abstract

Proximal splitting algorithms play a central role in finding the numerical solution of convex optimization problems. This paper addresses the problem of stereo matching of multi-component images by jointly estimating the disparity and the illumination variation. The global formulation being non-convex, the problem is addressed by solving a sequence of convex relaxations. Each convex relaxation is non trivial and involves many constraints aiming at imposing some regularity on the solution. Experiments demonstrate that the method is efficient and provides better results compared with other approaches.

Keywords

Stereo Disparity Color images Illumination variation Proximity operator Total variation Tight frame Convex optimization Parallel proximal algorithms 

Notes

Acknowledgements

We would like to thank Prof. Wided Miled for providing us with her codes. We would also like to thank Dr. Raffaele Gaetano and Giovanni Chierchia for their implementation of our approach on GPU architectures.

References

  1. 1.
    Kolmogorov, V., Zabih, R.: Computing visual correspondence with occlusions using graph cuts. In: Proc. IEEE Int. Conf. Comput. Vis, Vancouver, BC, Canada, Jul. 9–12, vol. 2, pp. 508–515 (2001) Google Scholar
  2. 2.
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 23(11), 1222–1239 (2011) CrossRefGoogle Scholar
  3. 3.
    Klaus, A., Sormann, M., Karner, K.: Segment-based stereo matching using belief propagation and a self-adapting dissimilarity measure. In: Proc. Int. Conf. Pattern Recognition, Hong Kong, Aug. 20–24, vol. 3, pp. 15–18 (2006) Google Scholar
  4. 4.
    Yang, Q., Wang, L., Yang, R., Wang, S., Liao, M., Nistér, D.: Realtime global stereo matching using hierarchical belief propagation. In: Proc. British Machine Vision Conference, Edinburgh, UK, Sept. 4–7, pp. 989–998 (2006). Google Scholar
  5. 5.
    Deriche, R., Kornprobst, P., Aubert, G.: Optical-flow estimation while preserving its discontinuities: A variational approach. In: Li, S., Mital, D., Teoh, E., Wang, H. (eds.) Recent Developments in Computer Vision. Lecture Notes in Computer Science, vol. 1035, pp. 69–80. Springer, Berlin (1996) CrossRefGoogle Scholar
  6. 6.
    Miled, W., Pesquet, J.-C., Parent, M.: Disparity map estimation using a total variation bound. In: The 3rd Canadian Conf. on Computer and Robot Vision, Quebec, Canada, June, pp. 48–55 (2006) Google Scholar
  7. 7.
    Miled, W., Pesquet, J.-C., Parent, M.: A convex optimisation approach for depth estimation under illumination variation. IEEE Trans. Image Process. 18(4), 813–830 (2009) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cox, I.J., Roy, S., Hingorani, S.L.: Dynamic histogram warping of image pairs for constant image brightness. In: Proc. Int. Conf. Image Process, Washington, DC, Oct. 23–26, vol. 2, pp. 366–369 (1995) CrossRefGoogle Scholar
  9. 9.
    Zabih, R., Woodfill, J.: Non-parametric local transforms for computing visual correspondence. In: Proc. Eur. Conf. Comput. Vis, Stockholm, Sweden, May 2–6, pp. 15–158 (1994) Google Scholar
  10. 10.
    Davis, J.E., Yang, R., Wang, L.: BRDF invariant stereo using light transport constancy. In: Proc. IEEE Int. Conf. Comput. Vis, Beijing, China, Oct. 15–21, vol. 1, pp. 436–443 (2005) Google Scholar
  11. 11.
    Gennert, M.A.: Brightness-based stereo matching. In: Proc. IEEE Int. Conf. Comput. Vis, Tampa, FL, Dec. 5–8, pp. 139–143 (1988) Google Scholar
  12. 12.
    Pesquet, J.-C., Pustelnik, N.: A parallel inertial proximal optimization method. Pac. J. Optim. 8(2), 273–305 (2012) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fusiello, A., Trucco, E., Verri, A.: A compact algorithm for rectification of stereo pairs. Mach. Vis. Appl. 12(1), 16–22 (2000) CrossRefGoogle Scholar
  14. 14.
    Tsai, D., Lin, C., Chen, J.: The evaluation of normalized cross correlations for defect detection. Pattern Recognit. Lett. 24(15), 2525–2535 (2003) zbMATHCrossRefGoogle Scholar
  15. 15.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992) zbMATHCrossRefGoogle Scholar
  16. 16.
    Combettes, P.L., Pesquet, J.-C.: Image restoration subject to a total variation constraint. IEEE Trans. Image Process. 13(9), 1213–1222 (2004) CrossRefGoogle Scholar
  17. 17.
    Combettes, P.L., Pesquet, J.-C.: A proximal decomposition method for solving convex variational inverse problems. Inverse Probl. 24(6), 27 (2008) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pustelnik, N., Chaux, C., Pesquet, J.-C.: Parallel ProXimal algorithm for image restoration using hybrid regularization. IEEE Trans. Image Process. 20(9), 2450–2462 (2011) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Aujol, J.F., Aubert, G., Blanc-Féraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vis. 22(1), 71–88 (2005) CrossRefGoogle Scholar
  20. 20.
    Chambolle, A., DeVore, R.A., Lee, N.-Y., Lucier, B.J.: Nonlinear wavelet image processing: variational problems, compression and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7, 319–335 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Han, D., Larson, D.R.: Frames, Bases, and Group Representations. Mem. Amer. Math. Soc., vol. 147, p. 94. AMS, Providence (2000) Google Scholar
  22. 22.
    Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, San Diego (1997) Google Scholar
  23. 23.
    Lefkimmiatis, S., Bourquard, A., Unser, M.: Hessian-based norm regularization for image restoration with biomedical applications. IEEE Trans. Image Process. 21(3), 983–995 (2012) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Moreau, J.J.: Fonctions convexes duales et points proximaux dans un espace hilbertien. C. R. Acad. Sci. 255, 2897–2899 (1962) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Combettes, P.L.: The foundations of set theoretic estimation. Proc. IEEE 81(2), 182–208 (1993) CrossRefGoogle Scholar
  26. 26.
    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, New York (2011) CrossRefGoogle Scholar
  27. 27.
    Youla, D.C., Webb, H.: Image restoration by the method of convex projections: Part I - theory. IEEE Trans. Med. Imaging 1(2), 81–94 (1982) CrossRefGoogle Scholar
  28. 28.
    van den Berg, E., Friedlander, M.P.: Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2), 890–912 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Eldar, Y.C., Mishali, M.: Robust recovery of signals from a structured union of subspaces. IEEE Trans. Inf. Theory 55(11), 5302–5316 (2009) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Afonso, M., Bioucas-Dias, J., Figueiredo, M.A.T.: An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Trans. Image Process. 20(3), 681–695 (2011) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Setzer, S., Steidl, G., Teuber, T.: Deblurring Poissonian images by split Bregman techniques. J. Vis. Commun. Image Represent. 21, 193–199 (2010) CrossRefGoogle Scholar
  32. 32.
    Attouch, H., Soueycatt, M.: Augmented Lagrangian and proximal alternating direction methods of multipliers in Hilbert spaces—applications to games, PDE’s and control. Pac. J. Optim. 5(1), 17–37 (2009) MathSciNetzbMATHGoogle Scholar
  33. 33.
    Chen, G., Teboulle, M.: A proximal-based decomposition method for convex minimization problems. Math. Program., Ser. A 64(1), 81–101 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Esser, E., Zhang, X., Chan, T.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci. 3(4), 1015–1046 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Combettes, P.L., Dũng, Ð., Vũ, B.C.: Proximity for sums of composite functions. J. Math. Anal. Appl. 320(2), 680–688 (2011) CrossRefGoogle Scholar
  37. 37.
    Briceño-Arias, L.M., Combettes, P.L.: A monotone + skew splitting model for composite monotone inclusions in duality. SIAM J. Optim. 21(4), 1230–1250 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Combettes, P.L., Pesquet, J.-C.: Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators. Set-Valued Var. Anal. 20(2), 307–330 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Pustelnik, N., Pesquet, J.-C., Chaux, C.: Relaxing tight frame condition in parallel proximal methods for signal restoration. IEEE Trans. Signal Process. 60(2), 968–973 (2012) MathSciNetCrossRefGoogle Scholar
  40. 40.
    Yoon, K.-J., Kweon, I.-S.: Adaptive support-weight approach for correspondence search. IEEE Trans. Pattern Anal. Mach. Intell. 28(4), 650–656 (2006) CrossRefGoogle Scholar
  41. 41.
    Yuille, A.L., Poggio, T.: A generalized ordering constraint for stereo correspondence. Technical report aim-777 (1984) Google Scholar
  42. 42.
    Pock, T., Schoenemann, T., Graber, G., Bischof, H., Cremers, D.: A convex formulation of continuous multilabel problems. In: Proc. Eur. Conf. Comput. Vis, Marseille, France, Oct., pp. 792–805 (2008) Google Scholar
  43. 43.
    Wang, L., Yang, R.: Global stereo matching leveraged by sparse ground control points. In: Proc. IEEE Conf. Comput. Vis. and Patt. Rec, Colorado Springs, USA, Jun. 20–25, pp. 3033–3040 (2011) Google Scholar
  44. 44.
    Min, D., Lu, J., Do, M.: A revisit to cost aggregation in stereo matching: how far can we reduce its computational redundancy? In: Proc. IEEE Int. Conf. Comput. Vis, Barcelona, Spain, Nov., pp. 1567–1574 (2011) Google Scholar
  45. 45.
    Mukherjee, D., Wang, G., Wu, J.: Stereo matching algorithm based on curvelet decomposition and modified support weights. In: Proc. Int. Conf. Acoust., Speech Signal Process, Dallas, Texas, USA, Mar. 14–19, pp. 758–761 (2010) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • C. Chaux
    • 1
  • M. El-Gheche
    • 1
  • J. Farah
    • 2
  • J.-C. Pesquet
    • 1
  • B. Pesquet-Popescu
    • 3
  1. 1.Laboratoire d’Informatique Gaspard Monge—CNRS UMR 8049Université Paris-EstMarne la Vallée Cedex 2France
  2. 2.Department of Telecommunications, Faculty of EngineeringHoly-Spirit University of KaslikJouniehLebanon
  3. 3.Signal and Image Proc. Dept.Telecom ParisTechParisFrance

Personalised recommendations