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Minkowski operations and vector spaces

Abstract

Mathematical morphology started as a set of tools for analysing images by the use of transformations based on set-theoretical operations which are the Minkowski sum and subtraction. It was first developed for the analysis of binary images. Its extension to grey-level images was a later development with the extension of the Minkowski operations to real-valued functions in terms of sup-convolution and inf-convolution. The purpose of this paper is to define a type of convolution between set-valued maps, to study its properties, and to establish some associated differential relations. This set-convolution map allows us to extend the Minkowski sum and substraction to multivalued functions and to functions with vectorial values.

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References

  1. Aubin, J.-P.:Morphological and Mutational Analysis, Tools for Shape Regulation and Optimization, Commett Matari Programme, CEREMADE, University of Paris-Dauphine, France, 1993.

    Google Scholar 

  2. Aubin, J.-P. and Cellina, A.:Differential Inclusions (Set-Valued Maps and Viability Theory), Springer, Berlin, 1984.

    Google Scholar 

  3. Aubin, J.-P. and Frankowska, H.:Set-Valued Analysis, Systems and Control: Foundations and Applications, Birkhäuser, Basle, 1990.

    Google Scholar 

  4. Beucher, S. Watersheds of functions and picture segmentations, in:Proc. IEEE Internat. Conf. on Acoustics, Speech and Signal Processing, Paris, May 1982, 1928–1931.

  5. Heijmans, H.J.A.M.: Theoretical aspects of gray-level morphology,IEEE Transactions on Pattern Analysis and Machine Intelligence 13 (1991), 568–582.

    Google Scholar 

  6. Matheron, G.:Random Sets and Integral Geometry, Wiley, New York, 1975.

    Google Scholar 

  7. Matheron, G.: Filters and lattices. Technical Report 851, CGMM, École des Mines, Sep. 1983.

  8. Matheron, G.: Filters and lattices, in: J. Serra (ed.),Image Analysis and Mathematical Morphology, Vol. 2, Theoretical Advances, Academic Press, London, 1988.

    Google Scholar 

  9. Matheron, G.: Les treillis compacts. Technical Report N-23/90/G, Centre de Géostatistique, École des Mines de Paris, Nov. 1990.

  10. Mattioli, J.: Differential Relations of Morphological Operators, in:Mathematical Morphology and Its Applications to Signal Processing, Barcelona, Spain, 12–14, May 1993, pp. 162–167.

  11. Mattioli, J.: Relations différentielles d'opérations de la morphologie mathématique,C.R. Acad. Sci. Paris 316 (1993), 879–884.

    Google Scholar 

  12. Meyer, F.: Contrast feature extraction, in: J.-L. Chermant (ed.),Quantitative Analysis of Microstructures in Material Sciences, Biology and Medicine, Stuttgart, FRG, 1978, Riederer-Verlag, Special issue ofPractical Metallography.

  13. Moreau, J.-J.: Fonctionnelles convexes,Séminaire du Collège de France, 1966.

  14. Rockafellar, R.T.:Convex Analysis, Princeton University Press, Princeton, NJ, 1970.

    Google Scholar 

  15. Serra, J.:Image Analysis and Mathematical Morphology, Academic Press, London, 1982.

    Google Scholar 

  16. Serra, J.: Eléments de théorie pour l'optique morphologique, Thèse d'Etat, Université de Paris VI, 1986.

  17. Serra, J.: ed.,Image Analysis and Mathematical Morphology, Vol. 2, Theoretical Advances, Academic Press, London, 1988.

    Google Scholar 

  18. Sternberg, S.R.: Grayscale Morphology,Computer Vision, Graphics and Image Processing 35 (1986), 333–355.

    Google Scholar 

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Mattioli, J. Minkowski operations and vector spaces. Set-Valued Anal 3, 33–50 (1995). https://doi.org/10.1007/BF01033640

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  • DOI: https://doi.org/10.1007/BF01033640

Mathematics Subject Classifications (1991)

  • 47H04
  • 68U10

Key words

  • mathematical morphology
  • Minkowski sum
  • Minkowski subtraction
  • set-convolution
  • internal set-convolution
  • Steiner selection