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

top
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

top

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/221923>.

@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},
month = {11},
number = {4},
pages = {909-930},
publisher = {EDP Sciences},
title = {Variational approximation for detecting point-like target problems*},
url = {http://eudml.org/doc/221923},
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
DA - 2011/11//
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/221923
ER -

References

top
  1. L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford (2000).  Zbl0957.49001
  2. G. Anzellotti, Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl.135 (1983) 293–318.  Zbl0572.46023
  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. G. Bellettini, Variational approximation of functionals with curvatures and related properties. J. Conv. Anal.4 (1997) 91–108.  Zbl0882.49013
  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).  
  7. A. Braides, Γ-convergence for beginners. Oxford University Press, New York (2000).  
  8. A. Braides and A. Malchiodi, Curvature theory of boundary phases: the two dimensional case. Interfaces Free Bound.4 (2002) 345–370.  Zbl1029.49039
  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.  Zbl1098.49012
  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.  Zbl0901.35019
  11. G.Q. Chen and H. Fried, Divergence-measure fields and conservation laws. Arch. Rational Mech. Anal.147 (1999) 35–51.  
  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.  
  13. G. Dal Maso, Introduction to Γ-convergence. Birkhäuser, Boston (1993).  
  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.  Zbl0958.35045
  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.  
  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.  
  17. E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Rend. Sem. Mat. Brescia3 (1979) 63–101.  
  18. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press (1992).  Zbl0804.28001
  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).  Zbl1195.92046
  20. J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993).  Zbl0780.31001
  21. L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal.98 (1987) 123–142.  Zbl0616.76004
  22. L. Modica and S. Mortola, Un esempio di Γ-convergenza. Boll. Un. Mat. Ital.14-B (1977) 285–299.  
  23. M. Röger and R. Shätzle, On a modified conjecture of De Giorgi. Math. Zeitschrift254 (2006) 675–714.  Zbl1126.49010
  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.  Zbl0151.15401
  25. W. Ziemer, Weakly Differentiable Functions. Springer-Verlag, New York (1989).  Zbl0692.46022

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.