Acta Applicandae Mathematicae

, Volume 153, Issue 1, pp 163–169 | Cite as

Image Processing and ‘Noise Removal Algorithms’—The Pdes and Their Invariance Properties & Conservation Laws

  • B. Al Qurashi
  • A. H. Kara
  • H. Akca


We analyse the symmetry, invariance properties and conservation laws of the partial differential equations (pdes) and minimization problems (variational functionals) that arise in the analyses of some noise removal algorithms.


Noise removal algorithms Invariance Conservation laws 

Mathematics Subject Classification

58D19 58J70 70H33 


  1. 1.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D, Nonlinear Phenom. 60, 259–268 (1992) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Esedolu, S., Osher, S.J.: Decomposition of images by the anisotropic Rudin-Osher-Fatemi model. Commun. Pure Appl. Math. 57, 1609–1626 (2004) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Osher, S., Solé, A., Vese, L.: Image decomposition and restoration using total variation minimization and the \(H^{-1}\)-norm. Multiscale Model. Simul. 1, 349–370 (2003) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Vese, L.A., Osher, S.J.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19, 553–572 (2003) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Shi, J., Osher, S.: A nonlinear inverse scale space method for a convex multiplicative noise model. SIAM J. Imaging Sci. 1, 294–321 (2008) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chang, S.G., Yu, B., Vetterli, M.: Adaptive wavelet thresholding for image denoising and compression. IEEE Trans. Image Process. 9, 1532–1546 (2000) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Elad, M., Aharon, M.: Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15, 3736–3745 (2006) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Mahmoudi, M., Sapiro, G.: Fast image and video denoising via nonlocal means of similar neighborhoods. IEEE Signal Process. Lett. 12, 839–842 (2005) CrossRefGoogle Scholar
  9. 9.
    Portilla, J., Strela, V., Wainwright, M.J., Simoncelli, E.P.: Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Trans. Image Process. 12, 1338–1351 (2003) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Noether, E.: Nachr. Akad. Wiss. Gött. Math.-Phys. Kl. 2, 235 (1918), English translation in Transp. Theory Stat. Phys. 1, 186 (1971) Google Scholar
  11. 11.
    Anderson, I.M., Pohjanpelt, o.J.: The cohomology of invariant variational bicomplexes. In: Geometric and Algebraic Structures in Differential Equations, pp. 3–19. Springer, Amsterdam (1995) CrossRefGoogle Scholar
  12. 12.
    Olver, P.: Application of Lie Groups to Differential Equations. Springer, New York (1986) CrossRefMATHGoogle Scholar
  13. 13.
    Ibragimov, N.H. (ed.): CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1. CRC Press, Boca Raton (1996) MATHGoogle Scholar
  14. 14.
    Stephani, H.: Differential Equations: Their Solution Using Symmetries. Cambridge Univ. Press, Cambridge (1989) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.School of MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa
  2. 2.College of Arts and SciencesAbu Dhabi UniversityAbu DhabiUnited Arab Emirates

Personalised recommendations