no. 4
Variational approximation for detecting point-like target problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 909-930

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

DOI : 10.1051/cocv/2010029
Classification : 49J45, 49Q20
Keywords: points detection, biological images, divergence-measure fields, p-capacity, Γ-convergence 
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     title = {Variational approximation for detecting point-like target problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {909--930},
     year = {2011},
     publisher = {EDP Sciences},
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     mrnumber = {2859858},
     zbl = {1238.49024},
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Aubert, Gilles; Graziani, Daniele. Variational approximation for detecting point-like target problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 909-930. doi: 10.1051/cocv/2010029

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford (2000). | Zbl | MR

[2] G. Anzellotti, Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. 135 (1983) 293-318. | Zbl | MR

[3] G. Aubert, J. Aujol and L. Blanc-Feraud, Detecting codimension - Two objects in an image with Ginzburg-Landau models. Int. J. Comput. Vis. 65 (2005) 29-42. | Zbl

[4] G. Bellettini, Variational approximation of functionals with curvatures and related properties. J. Conv. Anal. 4 (1997) 91-108. | Zbl | MR

[5] G. Bellettini and M. Paolini, Approssimazione variazionale di funzionali con curvatura. Seminario di Analisi Matematica, Dipartimento di Matematica dell'Università di Bologna (1993).

[6] F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices. Birkäuser, Boston (1994). | Zbl | MR

[7] A. Braides, Γ-convergence for beginners. Oxford University Press, New York (2000). | Zbl | MR

[8] A. Braides and A. Malchiodi, Curvature theory of boundary phases: the two dimensional case. Interfaces Free Bound. 4 (2002) 345-370. | Zbl | MR

[9] A. Braides and R. March, Approximation by Γ-convergence of a curvature-depending functional in Visual Reconstruction. Comm. Pure Appl. Math. 59 (2006) 71-121. | Zbl | MR

[10] A. Chambolle and F. Doveri, Continuity of Neumann linear elliptic problems on varying two-dimensionals bounded open sets. Comm. Partial Diff. Eq. 22 (1997) 811-840. | Zbl | MR

[11] G.Q. Chen and H. Fried, Divergence-measure fields and conservation laws. Arch. Rational Mech. Anal. 147 (1999) 35-51. | Zbl

[12] G.Q. Chen and H. Fried, On the theory of divergence-measure fields and its applications. Bol. Soc. Bras. Math. 32 (2001) 1-33. | Zbl | MR

[13] G. Dal Maso, Introduction to Γ-convergence. Birkhäuser, Boston (1993). | Zbl | MR

[14] 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. | Zbl | MR | Numdam

[15] 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. | Zbl

[16] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat. Natur. 58 (1975) 842-850. | Zbl | MR

[17] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Rend. Sem. Mat. Brescia 3 (1979) 63-101. | Zbl

[18] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press (1992). | Zbl | MR

[19] 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). | Zbl | MR

[20] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993). | Zbl | MR

[21] L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123-142. | Zbl | MR

[22] L. Modica and S. Mortola, Un esempio di Γ-convergenza. Boll. Un. Mat. Ital. 14-B (1977) 285-299. | Zbl | MR

[23] M. Röger and R. Shätzle, On a modified conjecture of De Giorgi. Math. Zeitschrift 254 (2006) 675-714. | Zbl | MR

[24] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 180-258. | Zbl | MR | Numdam

[25] W. Ziemer, Weakly Differentiable Functions. Springer-Verlag, New York (1989). | Zbl | MR

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