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Generic uniqueness of minimizer for Blake & Zisserman functional

Abstract

Blake-Zisserman functional \(F_{\alpha ,\beta}^{g} \) achieves a finite minimum for any pair of real numbers α, β such that 0<βα≤2β and any gL 2(0,1). Uniqueness of minimizer does not hold in general. Nevertheless, in the 1D case uniqueness of minimizer is a generic property for \(F_{\alpha ,\beta }^{g}\) in the sense that it holds true for almost all gray levels data g and parameters α, β: we prove that, whenever α/β∉ℚ, the minimizer is unique for any g belonging to a dense G δ set of L 2(0,1) dependent on α and β.

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Fig. 1

Notes

  1. 1.

    With the same arrangement we would have uniqueness by Remark 3, Theorem 3 and Theorem 7.

References

  1. 1.

    Ambrosio, L., Faina, L., March, R.: Variational approximation of a second order free discontinuity problem in computer vision. SIAM J. Math. Anal. 32, 1171–1197 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)

    MATH  Google Scholar 

  3. 3.

    Amar, M., De Cicco, V.: The uniqueness as a generic property for some one dimensional segmentation problems. Rend. Semin. Mat. Univ. Padova 88, 151–173 (1992)

    MATH  Google Scholar 

  4. 4.

    Blake, A., Zisserman, A.: Visual Reconstruction. MIT Press, Cambridge (1987)

    Google Scholar 

  5. 5.

    Boccellari, T., Tomarelli, F.: About well-posedness of optimal segmentation for Blake & Zisserman functional. Istit. Lomb. (Rend. Sci.) 142, 237–266 (2008)

    MathSciNet  Google Scholar 

  6. 6.

    Braides, A., Defranceschi, A., Vitali, E.: A compactness result for a second-order variational discrete model. ESAIM Math. Model. Numer. Anal. 46(2), 389–410 (2012)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Carriero, M., Leaci, A., Tomarelli, F.: A second order model in image segmentation: Blake& Zisserman functional. In: Variational Methods for Discontinuous Structures, Como, 1994. Progr. Nonlinear Differential Equations Appl., vol. 25, pp. 57–72. Birkäuser, Basel (1996)

    Chapter  Google Scholar 

  8. 8.

    Carriero, M., Leaci, A., Tomarelli, F.: Strong minimizers of Blake & Zisserman functional. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 25(1–2), 257–285 (1997)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Carriero, M., Leaci, A., Tomarelli, F.: Necessary conditions for extremals of Blake & Zisserman functional. C. R. Math. Acad. Sci. Paris 334, 343–348 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Carriero, M., Leaci, A., Tomarelli, F.: Calculus of variations and image segmentation. J. Physiol., Paris 97(2–3), 343–353 (2003)

    Article  Google Scholar 

  11. 11.

    Carriero, M., Leaci, A., Tomarelli, F.: Second order variational problems with free discontinuity and free gradient discontinuity. In: Calculus of Variations: Topics from the Mathematical Heritage of Ennio De Giorgi. Quad. Mat., vol. 14, pp. 135–186 (2004). Dept. Math., Seconda Univ. Napoli, Caserta

    Google Scholar 

  12. 12.

    Carriero, M., Leaci, A., Tomarelli, F.: Euler equations for Blake & Zisserman functional. Calc. Var. Partial Differ. Equ. 32(1), 81–110 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Carriero, M., Leaci, A., Tomarelli, F.: A Dirichlet problem with free gradient discontinuity. Adv. Math. Sci. Appl. 20(1), 107–141 (2010)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Carriero, M., Leaci, A., Tomarelli, F.: Candidate local minimizer of Blake & Zisserman functional. J. Math. Pures Appl. 96, 58–87 (2011)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Carriero, M., Leaci, A., Tomarelli, F.: Variational approach to image segmentation. Pure Math. Appl. 20, 141–156 (2009)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Carriero, M., Leaci, A., Tomarelli, F.: Uniform density estimates for Blake & Zisserman functional. Discrete Contin. Dyn. Syst., Ser. A 31(4), 1129–1150 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Carriero, M., Leaci, A., Tomarelli, F.: About Poincaré inequalities for functions lacking summability. Note Mat. 31(1), 67–85 (2011)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Carriero, M., Leaci, A., Tomarelli, F.: Free gradient discontinuity and image inpainting. Zap. Nauč. Semin. POMI 390, 92–116 (2011)

    Google Scholar 

  19. 19.

    Carriero, M., Leaci, A., Tomarelli, F.: Free gradient discontinuity and image inpainting. J. Math. Sci. (N.Y.) 181(6), 805–819 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Coscia, A.: Existence result for a new variational problem in one-dimensional segmentation theory. Ann. Univ. Ferrara - Sez. VII - Sc. Mat. XXXVII, 185–203 (1991)

    MathSciNet  Google Scholar 

  21. 21.

    Dal Maso, G., Morel, J.M., Solimini, S.: A variational method in image segmentation: existence and approximation results. Acta Math. 168(1–2), 89–151 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    De Giorgi, E., Ambrosio, L.: Un nuovo tipo di funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 82, 199–210 (1988)

    MATH  Google Scholar 

  23. 23.

    De Giorgi, E., Carriero, M., Leaci, A.: Existence theorem for a minimum problem with free discontinuity set. Arch. Ration. Mech. Anal. (3) 108, 195–218 (1989)

    MATH  Article  Google Scholar 

  24. 24.

    Lojasiewicz, S.: Triangulation of semi-analytic sets. Ann. Sc. Norm. Super. Pisa (3) 18, 449–474 (1964)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Morel, J.M., Solimini, S.: Variational Methods in Image Segmentation, PNLDE, vol. 14. Birkhäuser, Berlin (1995)

    Book  Google Scholar 

  26. 26.

    Mumford, D., Shah, J.: Boundary detection by minimizing functionals. In: Proc. IEEE Conf. on Computer Vision and Pattern Recognition, San Francisco (1985)

    Google Scholar 

  27. 27.

    Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. XLII, 577–685 (1989)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Switzer, R.M.: Algebraic Topology—Homotopy and Homology. Springer, Berlin (1975)

    MATH  Book  Google Scholar 

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Acknowledgements

Authors want to thank M.I.U.R. (Ministero Istruzione Università e Ricerca) which supported this research in 2008 as part of the P.R.I.N. (Progetti di Ricerca di Interesse Nazionale) project.

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Correspondence to Tommaso Boccellari.

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Boccellari, T., Tomarelli, F. Generic uniqueness of minimizer for Blake & Zisserman functional. Rev Mat Complut 26, 361–408 (2013). https://doi.org/10.1007/s13163-012-0103-1

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Keywords

  • Calculus of variations
  • Generic uniqueness
  • Euler equations
  • Image segmentation

Mathematics Subject Classification

  • 35A02
  • 35Q31
  • 49K05
  • 46N60