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Annali dell’Università di Ferrara

, Volume 37, Issue 1, pp 185–203 | Cite as

Existence result for a new variational problem in one-dimensional segmentation theory

  • Alessandra Coscia
Article
  • 25 Downloads

Abstract

In order to study some segmentation problems in dimension one, we propose a new functional, whose leading term includes the second order derivative of the unknown function. We prove an existence result for the associated minimization problem, by relying on the compactness and the lower semicontinuity of the functional with respect to theL 1-convergence.

Keywords

Minimization Problem Variational Problem Existence Result Limit Function Minimum Point 
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.

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References

  1. [1]
    R. Adams,Sobolev Spaces, Academic Press, New York (1975).MATHGoogle Scholar
  2. [2]
    L. Ambrosio,A compactness theorem for a special class of functions of bounded variation, Boll. Un. Mat. It.,3 B (1989), pp. 857–881.MathSciNetGoogle Scholar
  3. [3]
    L. Ambrosio,Variational problems in SBV, Acta Appl. Math.,17 (1989), pp. 1–40.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    L. Ambrosio,Existence theory for a new class of variational problems, Arch. Rational Mech. Anal., (4)111 (1990), pp. 291–322.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    L. AmbrosioV. M. Tortorelli,Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence, Comm. Pure Appl. Math.,XLIII (1990), pp. 999–1036.CrossRefMathSciNetGoogle Scholar
  6. [6]
    A. BlakeA. Zisserman,Visual Reconstruction, The MIT Press, Cambridge, Massachusetts (1987).Google Scholar
  7. [7]
    H. Brezis,Analyse fonctionelle, Masson, Paris (1983).Google Scholar
  8. [8]
    G. Dal MasoJ. M. MorelS. Solimini,A variational method in image segmentation: existence and approximation results, Acta Math.,168 (1992), pp. 89–151.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    E. De Giorgi,Free discontinuity problems in calculus of variations, Analyse Mathématique et Applications (Paris, 1988), Gauthier-Villars, Paris (1988).Google Scholar
  10. [10]
    E. De GiorgiL. Ambrosio,Un nuovo tipo di funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (8)LXXXII (1988), pp. 199–210.Google Scholar
  11. [11]
    E. De GiorgiM. CarrieroA. Leaci,Existence theorem for a minimum problem with free discontinuity set, Arch. Rational Mech. Anal., (3)108 (1989), pp. 195–218.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    H. Federer,Geometric Measure Theory, Springer-Verlag, New York (1969).MATHGoogle Scholar
  13. [13]
    J. M. MorelS. Solimini,Segmentation of images by variational methods: a constructive approach, Revista Matematica Universidad Complutense de Madrid,1 (1988), pp. 169–182.MATHMathSciNetGoogle Scholar
  14. [14]
    J. M. MorelS. Solimini,Segmentation d’images par méthode variationelle: une preuve constructive d’existence, C. R. Acad. Sci. Paris, t. 308, série I (1989), pp. 465–470.MathSciNetGoogle Scholar
  15. [15]
    D. Mumford—J. Shah,Boundary detection by minimizing functionals, Proc. IEEE Conf. on Computer Vision and Pattern Recognition (San Francisco, 1985).Google Scholar
  16. [16]
    D. MumfordJ. Shah,Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math.,XLII (1989), pp. 577–685.CrossRefMathSciNetGoogle Scholar
  17. [17]
    A. RosenfeldA. C. Kak,Digital Picture Processing, Academic Press, New York (1982).Google Scholar
  18. [18]
    J. Shah,Properties of energy-minimizing segmentations, Manuscript Northeastern Univ., Boston (1988).Google Scholar

Copyright information

© Università degli Studi di Ferrara 1991

Authors and Affiliations

  • Alessandra Coscia
    • 1
  1. 1.S.I.S.S.A.TriesteItaly

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