Skip to main content

Covariant fractional extension of the modified Laplace-operator used in 3D-shape recovery


Extending the Liouville-Caputo definition of a fractional derivative to a nonlocal covariant generalization of arbitrary bound operators acting on multidimensional Riemannian spaces an appropriate approach for the 3D shape recovery of aperture afflicted 2D slide sequences is proposed. We demonstrate, that the step from a local to a nonlocal algorithm yields an order of magnitude in accuracy and by using the specific fractional approach an additional factor 2 in accuracy of the derived results.

This is a preview of subscription content, access via your institution.


  1. F. Falzon, and G. Giraudon, Singularity analysis and derivative scalespace. In: Proceedings CVPR’ 94, IEEE Computer Society Conference (1994), 245–250.

  2. W. Feller, On a generalization of Marcel Riesz’ potentials and the semigroups generated by them. Comm. Sem. Mathem. Universite de Lund (1952), 72–81.

  3. R. Herrmann, Fractional Calculus — An Introduction for Physicists. World Scientific Publishing, Singapore (2011);

    Book  Google Scholar 

  4. Z. Jiao and Y.Q. Chen, Impulse response of a generalized fractional second order filter, Fract. Calc. Appl. Anal. 15, No 1 (2012), 97–116; DOI: 10.2478/s13540-012-0007-2; at

    Google Scholar 

  5. J. Liouville, Sur le calcul des differentielles à indices quelconques. J. École Polytechnique 13 (1832), 1–162.

    Google Scholar 

  6. B. Mathieu, P. Melchior, A. Oustaloup and Ch. Ceyral, Fractional differentiation for edge detection. Signal Processing 83 (2003), 2421–2432.

    MATH  Article  Google Scholar 

  7. K. B. Oldham and J. Spanier, The Fractional Calculus. Academic Press, New York (1974).

    Google Scholar 

  8. M. D. Ortigueira and J. A. Tenreiro Machado, Fractional signal processing and applications. Signal Processing 83 (2003), 2285–2286.

    Article  Google Scholar 

  9. I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).

    Google Scholar 

  10. H. Rafeiro and S. Samko, On multidimensional analogue of Marchaud formula for fractional Riesz-type derivatives in domains in R n. Fract. Calc. Appl. Anal. 8, No 4 (2005), 393–402; at

    MathSciNet  MATH  Google Scholar 

  11. M. Riesz, L’integrale de Riemann-Liouville et le probléme de Cauchy. Acta Math. 81 (1949), 1–223.

    MathSciNet  MATH  Article  Google Scholar 

  12. B. Rubin, On some inversion formulas for the Riesz potentials and k-plane transforms. Fract. Calc. Appl. Anal. 15, No 1 (2012), 34–44; DOI: 10.2478/s13540-012-0004-5; at

    Google Scholar 

  13. POV-Ray: Persistence of Vision Raytracer (2011);

  14. C. E. Shannon, Communication in the presence of noise, Proc. Inst. of Radio Engineers 37, No 1 (1949), 10–21.

    MathSciNet  Google Scholar 

  15. A. C. Sparavigna, Using fractional differentiation in astronomy, (2009).

  16. M. Spencer, Fundamentals of Light Microscopy. Cambridge University Press, Cambridge (1982).

    Google Scholar 

  17. M. Zernike, Das Phasenkontrastverfaren bei der mikroskopischen Beobachtung, Z. Tech. Phys. 16 (1935), 454–457.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Richard Herrmann.

About this article

Cite this article

Herrmann, R. Covariant fractional extension of the modified Laplace-operator used in 3D-shape recovery. fcaa 15, 332–343 (2012).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:

MSC 2010

  • 26A33
  • 35R11
  • 62H35
  • 65D18
  • 68U10
  • 94A08
  • 35J05

Key Words and Phrases

  • fractional calculus
  • computer graphics
  • image processing
  • shape recovery
  • confocal microscopy
  • modified Laplacian