Nonlinear Multiscale Analysis Models for Filtering of 3D + Time Biomedical Images

  • A. Sarti
  • K. Mikula
  • F. Sgallari
  • C. Lamberti
Part of the Mathematics and Visualization book series (MATHVISUAL)


We review nonlinear partial differential equations (PDEs) in the processing of 2D and 3D images. At the same time we present recent models introduced for processing of space-time image sequences and apply them to 3D echocardiography. The nonlinear (degenerate) diffusion equations filter the sequence with a keeping of space-time coherent structures. They have been developed using ideas of regularized Perona-Malik anisotropic diffusion and geometrical diffusion of mean curvature flow type, combined with Galilean invariant movie multiscale analysis of Alvarez, Guichard, Lions and Morel. A discretization of space-time filtering equations is discussed. Computational results in processing of 3D echocardiographic sequences obtained by rotational acquisition technique and by Real-Time 3D Echo Volumetrics aquisition technique are presented.


Image Sequence Neumann Boundary Condition Multiscale Analysis Nonlinear Filter Motion Coherence 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M.: Axioms and Fundamental Equations of Image Processing. Arch. Rat. Mech. Anal. 123 (1993) 200–257MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alvarez, L., Lions, P.L., Morel, J.M.: Image selective smoothing and edge detection by nonlinear diffusion II. SIAM J. Numer. Anal. 29 (1992) 845–866MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bänsch, E., Mikula, K.: Adaptivity in 3D image processing. Computing and Visualization in Science (2001)Google Scholar
  4. 4.
    Catte, F., Lions, P.L., Morel, J.M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29 (1992) 182–193MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Evans, L.C., Spruck, J.: Motion of level sets by mean curvature I. J. Diff. Geom. 33 (1991) 635–681MathSciNetzbMATHGoogle Scholar
  6. 6.
    Guichard, F.: Axiomatisation des analyses multi-echelles d’images et de films. PhD Thesis Univerity Paris IX Dauphine (1994)Google Scholar
  7. 7.
    Handlovičová, A., Mikula, K., Sarti, A.: Numerical solution of parabolic equations related to level set formulation of mean curvature flow. Computing and Visualization in Science 1, No. 2 (1999) 179–182Google Scholar
  8. 8.
    Handlovičová, A., Mikula, K., Sgallari, F.: Variational numerical methods for solving nonlinear diffusion equations arising in image processing, Journal of Visual Communication and Image Representation (2001)Google Scholar
  9. 9.
    Kačur, J., Mikula, K.: Solution of nonlinear diffusion appearing in image smoothing and edge detection. Applied Numerical Mathematics 17 (1995) 47–59MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kichenassamy, R.: The Perona-Malik paradox. SIAM J. Appl. Math. 57, No.5 (1997) 1328–1342MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Malladi, R., Sethian, J.A., Vemuri, B.: Shape modeling with front propagation: a level set approach. IEEE Trans. Pattern Analysis Machine Intelligence 17, No.2 (1995) 158–174CrossRefGoogle Scholar
  12. 12.
    Mikula, K., Ramarosy, N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing. Numerische Mathematik (2001)Google Scholar
  13. 13.
    Mikula, K., Sarti, A., Lamberti, C: Geometrical diffusion in 3D-echocardiography. Proceedings of ALGORITMY’97 — Conference on Sci. Comp., Zuberec, Slovakia (1997) 167–181Google Scholar
  14. 14.
    Perona, P., Malik, J., Scale space and edge detection using anisotropic diffusion. In Proc. IEEE Computer Society Workshop on Computer Vision (1987)Google Scholar
  15. 15.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60 (1992) 259–268zbMATHCrossRefGoogle Scholar
  16. 16.
    Saad, Y.: Iterative methods for sparse linear systems. PWS Publ. Comp. (1996)Google Scholar
  17. 17.
    Sarti, A., Mikula, K., Sgallari, F.: Nonlinear multiscale analysis of 3D echocardiographic sequences. IEEE Transactions on Medical Imaging 18, No. 6 (1999) 453–466CrossRefGoogle Scholar
  18. 18.
    Sarti, A., Ortiz de Solorzano, C., Lockett, S. and Malladi, R.: A Geometric Model for 3-D Confocal Image Analysis. IEEE Transactions on Biomedical Engineering 45, No. 12, (2000), 1600–1610Google Scholar
  19. 19.
    Sarti, A., Malladi, R., Sethian, J.A.: Subjective Surfaces: A Method for Completing Missing Boundaries. Proceedings of the National Academy of Sciences of the United States of America, Vol 12, N.97, pag. 6258–6263, 2000.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Material Science. Cambridge University Press (1999)Google Scholar
  21. 21.
    Weickert, J., Romeny, B.M.t.H., Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. on Image Processing 7 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • A. Sarti
    • 1
  • K. Mikula
    • 2
  • F. Sgallari
    • 3
  • C. Lamberti
    • 1
  1. 1.DEISUniversity of BolognaItaly
  2. 2.Department of MathematicsSlovak University of TechnologyBratislavaSlovakia
  3. 3.Department of MathematicsUniversity of BolognaItaly

Personalised recommendations