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Variational approximation for detectingpoint-like target problems*

Published online by Cambridge University Press:  06 August 2010

Gilles Aubert  and
Daniele Graziani
Gilles Aubert
Affiliation:
Laboratoire J.A. Dieudonné, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 2, France. [email protected]
Daniele Graziani
Affiliation:
Ariana CNRS/INRIA/UNSA Sophia Antipolis, Inria, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France. [email protected]
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Abstract

The aim of this paper is to provide a rigorous variational formulation forthe detection of points in 2-d biological images. To this purposewe introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals forwhich we prove the Γ-convergence to the initial one.

Keywords

Points detectionbiological imagesdivergence-measure fieldsp-capacityΓ-convergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 909 - 930
Copyright
© EDP Sciences, SMAI, 2010

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References

L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford (2000).
Anzellotti, G., Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. 135 (1983) 293–318. CrossRef
Aubert, G., Aujol, J. and Blanc-Feraud, L., Detecting codimension – Two objects in an image with Ginzburg-Landau models. Int. J. Comput. Vis. 65 (2005) 29–42. CrossRef
Bellettini, G., Variational approximation of functionals with curvatures and related properties. J. Conv. Anal. 4 (1997) 91–108.
G. Bellettini and M. Paolini, Approssimazione variazionale di funzionali con curvatura. Seminario di Analisi Matematica, Dipartimento di Matematica dell'Università di Bologna (1993).
F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices. Birkäuser, Boston (1994).
A. Braides, Γ-convergence for beginners. Oxford University Press, New York (2000).
Braides, A. and Malchiodi, A., Curvature theory of boundary phases: the two dimensional case. Interfaces Free Bound. 4 (2002) 345–370. CrossRef
Braides, A. and March, R., Approximation by Γ-convergence of a curvature-depending functional in Visual Reconstruction. Comm. Pure Appl. Math. 59 (2006) 71–121. CrossRef
Chambolle, A. and Doveri, F., Continuity of Neumann linear elliptic problems on varying two-dimensionals bounded open sets. Comm. Partial Diff. Eq. 22 (1997) 811–840. CrossRef
Chen, G.Q. and Fried, H., Divergence-measure fields and conservation laws. Arch. Rational Mech. Anal. 147 (1999) 35–51. CrossRef
Chen, G.Q. and Fried, H., On the theory of divergence-measure fields and its applications. Bol. Soc. Bras. Math. 32 (2001) 1–33. CrossRef
G. Dal Maso, Introduction to Γ-convergence. Birkhäuser, Boston (1993).
G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999) 741–808.
E. De Giorgi, Some remarks on Γ-convergence and least square methods, in Composite Media and Homogenization Theory, G. Dal Maso and G.F. Dell'Antonio Eds., Birkhäuser, Boston (1991) 135–142.
De Giorgi, E. and Franzoni, T., Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat. Natur. 58 (1975) 842–850.
De Giorgi, E. and Franzoni, T., Su un tipo di convergenza variazionale. Rend. Sem. Mat. Brescia 3 (1979) 63–101.
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press (1992).
D. Graziani, L. Blanc-Feraud and G. Aubert, A formal Γ-convergence approach for the detection of points in 2-D images. SIAM J. Imaging Sci. (to appear).
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993).
Modica, L., The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123–142. CrossRef
L. Modica and S. Mortola, Un esempio di Γ-convergenza. Boll. Un. Mat. Ital. 14-B (1977) 285–299.
Röger, M. and Shätzle, R., On a modified conjecture of De Giorgi. Math. Zeitschrift 254 (2006) 675–714. CrossRef
Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 180–258. CrossRef
W. Ziemer, Weakly Differentiable Functions. Springer-Verlag, New York (1989).