Advertisement

Inverse Problems for Granulometries by Erosion

  • Juliette Mattioli
  • Michel Schmitt

Abstract

Let us associate to any binary planar shape X the erosion curve Ψ X defined by Ψ X : r ∈ IR+A(XrB), where A(X) stands for the surface area of X and XrB for the eroded set of X with respect to the ball rB of size r. Note the analogy to shape quantification by granulometry. This paper describes the relationship between sets X and Y verifyingΨ X =Ψ Y . Under some regularity conditions on X, Ψ X is expressed as an integral on its skeleton of the quench function qx (distance to the boundary of X). We first prove that a bending of arcs of the skeleton of X does not affectΨ X : Ψ X quantifies soft shapes. We then prove, in the generic case, that the five possible cases of behavior of the second derivative Ψ” X characterize five different situations on the skeleton Sk(X) and on the quench function qx: simple points of Sk(X) where qx is a local minimum, a local maximum, or neither, multiple points of Sk(X) where qx is a local maximum or not. Finally, we give infinitesimal generators of the reconstruction process for the entire family of shapes Y verifyingΨ X =Ψ Y for a given X.

Key words

mathematical morphology erosion curve skeleton quench function granulometry 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. Matheron, Random Sets and Integral Geometry, John Wiley: New York, 1975.zbMATHGoogle Scholar
  2. 2.
    M. Coster and J.L. Chermant, “Précis d’analyse d’images,” CNRS Etud., 1985.Google Scholar
  3. 3.
    D. Stoyan, W.S. Kendall, and J. Mecke, Stochastic Geometry and Its Applications, John Wiley: New York, 1987.zbMATHGoogle Scholar
  4. 4.
    L. Calabi and J.A. Riley, “The skeletons of stable plane sets,” Parke Math. Lab., Inc., Carlisle, MA, Tech. Rep. AF 19, 1967.Google Scholar
  5. 5.
    J. Serra, Image Analysis and Mathematical Morphology, Academic Press: London, 1982.zbMATHGoogle Scholar
  6. 6.
    G. Matheron, “Examples of topological properties of skeletons,” in Image Analysis and Mathematical Morphology, Vol. 2: Theoretical Advances, J. Serra, ed., Academic Press: London, 1988.Google Scholar
  7. 7.
    G. Matheron, “On the negligibility of the skeleton and the absolute continuity of erosions,” in Image Analysis and Mathematical Morphology, Vol. 2: Theoretical Advances, J. Serra, ed., Academic Press: London, 1988.Google Scholar
  8. 8.
    J. Riley and L. Calabi, “Certain properties of circles inscribed in simple closed curves,” Park Math. Lab., Inc., Carlisle, MA, Tech. Rep. 59281, 1964.Google Scholar
  9. 9.
    J. Riley, “Plane graphs and their skeletons,” Park Math. Lab., Inc., Carlisle, MA, Tech. Rep. 60429, 1965.Google Scholar
  10. 10.
    J. Dieudonné, Eléments d’analyse,vol. I, Gauthier-Villars: Paris, 1969.Google Scholar
  11. 11.
    J. Mattioli, “Squelette, érosion et fonction spectrale par érosion d’une forme binaire planaire,” Rapport Interne, ASRF-91–8, 1991.Google Scholar
  12. 12.
    G. Matheron, La formule de Steiner pour les érosions, Centre de Géostatistique, Ecole des Mines, Paris, Tech. Rep. 496, 1977.Google Scholar
  13. 13.
    M. Schmitt, “On two inverse problems in mathematical morphology,” in Mathematical Morphology in Image Processing, E.R. Dougherty, ed., Marcel Dekker: New York, 1991.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Juliette Mattioli
    • 1
  • Michel Schmitt
    • 1
  1. 1.Thomson-CSF, Laboratoire Central de RecherchesDomaine de CorbevilleOrsay CedexFrance

Personalised recommendations