Advertisement

Feynman-Kac Formula and Restoration of High ISO Images

  • Dariusz Borkowski
  • Adam Jakubowski
  • Katarzyna Jańczak-Borkowska
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8671)

Abstract

In this paper we explore the problem of reconstruction of RGB images with additive Gaussian noise. In order to solve this problem we use Feynman-Kac formula and non local means algorithm. Expressing the problem in stochastic terms allows us to adapt to anisotropic diffusion the concept of similarity patches used in non local means. This novel look on the reconstruction is fruitful, gives encouraging results and can be successfully applied to denoising of high ISO images.

Keywords

Image Restoration Noisy Image Image Denoising Grey Level Image Additive Gaussian Noise 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aharon, M., Elad, M., Bruckstein, A.: K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans. Image Process. 54(11), 4311–4322 (2006)CrossRefGoogle Scholar
  2. 2.
    Aubert, G., Kornprobst, P.: Mathematical problems in image processing. Springer, New York (2002)Google Scholar
  3. 3.
    Borkowski, D.: Stochastic approximation to reconstruction of vector-valued images. In: Burduk, R., Jackowski, K., Kurzynski, M., Wozniak, M., Zolnierek, A. (eds.) CORES 2013. AISC, vol. 226, pp. 395–404. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Borkowski, D., Jańczak-Borkowska, K.: Image restoration using anisotropic stochastic diffusion collaborated with non local means. In: Saeed, K., Chaki, R., Cortesi, A., Wierzchoń, S. (eds.) CISIM 2013. LNCS, vol. 8104, pp. 177–189. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    Buades, A., Coll, B., Morel, J.M.: A non local algorithm for image denoising. IEEE Computer Vision and Pattern Recognition 2, 60–65 (2005)Google Scholar
  6. 6.
    Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2006)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Buades, A., Coll, B., Morel, J.M.: Non-local means denoising. Image Processing on Line (2011)Google Scholar
  8. 8.
    Catte, F., Lions, P.L., Morel, J.M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29(1), 182–193 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3D transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Danielyan, A., Katkovnik, V., Egiazarian, K.: Bm3d frames and variational image deblurring. IEEE Trans. Image Process. 21(4), 1715–1728 (2012)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Donoho, D.L., Johnstone, I.M.: Ideal spatial adaptation via wavelet shrinkage. Biometrika 81(3), 425–455 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Geman, S., Geman, D.: Stochastic relaxation, gibbs distributions and the bayesian restoration of images. IEEE Pat. Anal. Mach. Intell. 6, 721–741 (1984)CrossRefzbMATHGoogle Scholar
  13. 13.
    Katkovnik, V., Danielyan, A., Egiazarian, K.: Decoupled inverse and denoising for image deblurring: variational BM3D-frame technique. In: 2011 18th IEEE International Conference on Image Processing (ICIP), pp. 3453–3456 (2011)Google Scholar
  14. 14.
    Lebrun, M., Buades, A., Morel, J.M.: Implementation of the Non-local Bayes image denoising. Image Processing on Line (2011)Google Scholar
  15. 15.
    Mairal, J., Elad, M., Sapiro, G.: Sparse representation for color image restoration. IEEE Trans. Image Process. 17(1), 53–69 (2008)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Oksendal, B.: Stochastic differential equations: an introduction with applications, 3rd edn. Springer (1992)Google Scholar
  17. 17.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)CrossRefGoogle Scholar
  18. 18.
    Richardson, W.H.: Bayesian-based iterative method of image restoration. JOSA 62(1), 55–59 (1972)CrossRefGoogle Scholar
  19. 19.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1992)CrossRefzbMATHGoogle Scholar
  20. 20.
    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
  21. 21.
    Weickert, J.: Theoretical foundations of anisotropic diffusion in image processing. Computing Suppement 11, 221–236 (1996)CrossRefGoogle Scholar
  22. 22.
    Weickert, J.: Coherence-enhancing diffusion filtering. Int. J. Comput. Vision 31(2/3), 111–127 (1999)CrossRefGoogle Scholar
  23. 23.
    Yaroslavsky, L.P.: Local adaptive image restoration and enhancement with the use of DFT and DCT in a running window. In: Proceedings of SPIE 2825, pp. 2–13 (1996)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Dariusz Borkowski
    • 1
  • Adam Jakubowski
    • 1
  • Katarzyna Jańczak-Borkowska
    • 2
  1. 1.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland
  2. 2.Institute of Mathematics and PhysicsUniversity of Technology and Life SciencesBydgoszczPoland

Personalised recommendations