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Variational approximation for detecting point-like target problems

Gilles Aubert; Daniele Graziani

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 4, page 909-930
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Aubert, Gilles, and Graziani, Daniele. "Variational approximation for detecting point-like target problems." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 909-930. <http://eudml.org/doc/272870>.

@article{Aubert2011,
abstract = {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.},
author = {Aubert, Gilles, Graziani, Daniele},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {points detection; biological images; divergence-measure fields; p-capacity; Γ-convergence; point detection; 2-d biological images; -capacity; -convergence},
language = {eng},
number = {4},
pages = {909-930},
publisher = {EDP-Sciences},
title = {Variational approximation for detecting point-like target problems},
url = {http://eudml.org/doc/272870},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Aubert, Gilles
AU - Graziani, Daniele
TI - Variational approximation for detecting point-like target problems
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 4
SP - 909
EP - 930
AB - 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.
LA - eng
KW - points detection; biological images; divergence-measure fields; p-capacity; Γ-convergence; point detection; 2-d biological images; -capacity; -convergence
UR - http://eudml.org/doc/272870
ER -

References

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  1. [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford (2000). Zbl0957.49001MR1857292
  2. [2] G. Anzellotti, Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl.135 (1983) 293–318. Zbl0572.46023MR750538
  3. [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. Zbl1287.94008
  4. [4] G. Bellettini, Variational approximation of functionals with curvatures and related properties. J. Conv. Anal.4 (1997) 91–108. Zbl0882.49013MR1459883
  5. [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. [6] F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices. Birkäuser, Boston (1994). Zbl0802.35142MR1269538
  7. [7] A. Braides, Γ-convergence for beginners. Oxford University Press, New York (2000). Zbl1198.49001MR1968440
  8. [8] A. Braides and A. Malchiodi, Curvature theory of boundary phases: the two dimensional case. Interfaces Free Bound.4 (2002) 345–370. Zbl1029.49039MR1935643
  9. [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. Zbl1098.49012MR2180084
  10. [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. Zbl0901.35019MR1452169
  11. [11] G.Q. Chen and H. Fried, Divergence-measure fields and conservation laws. Arch. Rational Mech. Anal.147 (1999) 35–51. Zbl0942.35111
  12. [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. Zbl1024.28009MR1894566
  13. [13] G. Dal Maso, Introduction to Γ-convergence. Birkhäuser, Boston (1993). Zbl0816.49001MR1201152
  14. [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. Zbl0958.35045MR1760541
  15. [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. Zbl0747.49008
  16. [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. Zbl0339.49005MR448194
  17. [17] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Rend. Sem. Mat. Brescia3 (1979) 63–101. Zbl0339.49005
  18. [18] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press (1992). Zbl0804.28001MR1158660
  19. [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). Zbl1254.92054MR2736021
  20. [20] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993). Zbl0780.31001MR1207810
  21. [21] L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal.98 (1987) 123–142. Zbl0616.76004MR866718
  22. [22] L. Modica and S. Mortola, Un esempio di Γ-convergenza. Boll. Un. Mat. Ital. 14-B (1977) 285–299. Zbl0356.49008MR445362
  23. [23] M. Röger and R. Shätzle, On a modified conjecture of De Giorgi. Math. Zeitschrift254 (2006) 675–714. Zbl1126.49010MR2253464
  24. [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. Zbl0151.15401MR192177
  25. [25] W. Ziemer, Weakly Differentiable Functions. Springer-Verlag, New York (1989). Zbl0692.46022MR1014685

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