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Euler equations for Blake and Zisserman functional

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Abstract

We derive many necessary conditions for minimizers of a functional depending on free discontinuities, free gradient discontinuities and second derivatives, which is related to image segmentation.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Ambrosio L., Fusco N. and Pallara D. (2000). Functions of bounded variations and free discontinuity problems. Oxford Mathematical Monographs, Oxford University Press, Oxford

    Google Scholar 

  3. Boccellari, T., Tomarelli, F.: Generic uniqueness of minimizer for 1D Blake and Zisserman functional (to appear)

  4. Blake A. and Zisserman A. (1987). Visual Reconstruction. MIT, Cambridge

    Google Scholar 

  5. Carriero, M., Farina, A., Sgura, I.: Image segmentation in the framework of free discontinuity problems. In: Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi, Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta, pp. 85–133, (2004)

  6. Carriero, M., Leaci, A., Pallara, D., Pascali, E.: Euler Conditions for a Minimum Problem with Free Discontinuity Surfaces. Preprint Dip. di Matematica n.8, Lecce (1988)

  7. Carriero M., Leaci A. and Tomarelli F. (1992). Special Bounded Hessian and elastic–plastic plate. Rend. Accad. Naz. Sci. XL Mem. Math. 16(5): 223–258

    MATH  MathSciNet  Google Scholar 

  8. Carriero, M., Leaci, A., Tomarelli, F.: Free gradient discontinuities. In: Calculus of Variations, Homogeneization and Continuum Mechanics (Marseille 1993), Ser. Adv. Math. Appl. Sci., vol. 18, pp. 131–147. World Sci. Publishing, River Edge (1994)

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

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

    MATH  MathSciNet  Google Scholar 

  11. Carriero, M., Leaci, A., Tomarelli, F.: Density estimates and further properties of Blake and Zisserman functional. In: Panagiotopoulos, P.D., Gilbert, R., Pardalos (eds.) From Convexity to Nonconvexity, Nonconvex Optim. Appl., vol. 55, pp. 381–392. Kluwer, Dordrecht (2001)

  12. Carriero, M., Leaci, A., Tomarelli, F.: Second order functionals for image segmentation. In: Advanced Mathematical Methods in Measurement and Instrumentation (Como 1998), Esculapio, pp. 169–179 (2000)

  13. Carriero M., Leaci A. and Tomarelli F. (2002). Necessary conditions for extremals of Blake and Zisserman functional. C. R. Math. Acad. Sci. Paris 334(4): 343–348

    MATH  MathSciNet  Google Scholar 

  14. Carriero, M., Leaci, A., Tomarelli, F.: Local minimizers for a free gradient discontinuity problem in image segmentation. In: Variational Methods for Discontinuous Structures, Progr. Nonlinear Differential Equations Appl. vol. 51, pp. 67–80. Birkhäuser, Basel (2002)

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

    Article  Google Scholar 

  16. 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 E. De Giorgi, Quad. Mat., 14, pp. 135–186. Dept. Math., Seconda Univ. Napoli, Caserta (2004)

  17. Carriero, M., Leaci, A., Tomarelli, F.: Candidate local minimizer of Blake and Zisserman functional. Arch. Rational. Mech. Anal. (submitted)

  18. De Giorgi, E.: Free discontinuity problems in calculus of variations. In: Dautray, R. (ed.) Frontiers in Pure and Appl. Math., pp. 55–61. North-Holland, Amsterdam (1991)

  19. De Giorgi E. and Ambrosio L. (1988). Un nuovo tipo di funzionale del Calcolo delle Variazioni. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur. 82: 199–210

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Federer H. (1969). Geometric Measure Theory. Springer, Berlin

    MATH  Google Scholar 

  22. Fonseca I., Leoni G. and Paroni R. (2005). On hessian matrices in the space BH. Commun. Contemp. Math. 7: 401–420

    Article  MATH  MathSciNet  Google Scholar 

  23. Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Ann. Math. Stud. Princeton UP (1983)

  24. Giusti E. (1984). Minimal Surfaces and Functions of Bounded Variations. Birkhäuser Monographs in Mathematics, Boston

    Google Scholar 

  25. Lions J.L. (1955). Problèmes aux limites en théorie des distributions. Acta Math. 94: 13–153

    Article  MATH  MathSciNet  Google Scholar 

  26. March R. (1992). Visual reconstruction with discontinuities using variational methods. Image Vis. Comput. 10: 30–38

    Article  Google Scholar 

  27. Morel J.M. and Solimini S. (1995). Variational Models in Image Segmentation, PNLDE, N. 14. Birkhäuser, Basel

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Pallara D. (1990). Some new results on functions of bounded variation. Rend. Accad. Naz. Sci. dei XL, Mem. Mat. XIV: 295–321

    MathSciNet  Google Scholar 

  30. Simon L. (1983). Lectures on Geometric Measure Theory. Proc. of the Center for Mathematical Analysis, N.3. Australian National University, Canberra

    Google Scholar 

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Correspondence to Franco Tomarelli.

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Carriero, M., Leaci, A. & Tomarelli, F. Euler equations for Blake and Zisserman functional. Calc. Var. 32, 81–110 (2008). https://doi.org/10.1007/s00526-007-0129-2

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  • DOI: https://doi.org/10.1007/s00526-007-0129-2

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