A variational model in image processing with focal points

Andrea Braides; Giuseppe Riey

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 5, page 729-748
  • ISSN: 0764-583X

Abstract

top
We propose a model for segmentation problems involving an energy concentrated on the vertices of an unknown polyhedral set, where the contours of the images to be recovered have preferred directions and focal points. We prove that such an energy is obtained as a Γ-limit of functionals defined on sets with smooth boundary that involve curvature terms of the boundary. The minimizers of the limit functional are polygons with edges either parallel to some prescribed directions or pointing to some fixed points, that can also be taken as unknown of the problem.

How to cite

top

Braides, Andrea, and Riey, Giuseppe. "A variational model in image processing with focal points." ESAIM: Mathematical Modelling and Numerical Analysis 42.5 (2008): 729-748. <http://eudml.org/doc/250363>.

@article{Braides2008,
abstract = { We propose a model for segmentation problems involving an energy concentrated on the vertices of an unknown polyhedral set, where the contours of the images to be recovered have preferred directions and focal points. We prove that such an energy is obtained as a Γ-limit of functionals defined on sets with smooth boundary that involve curvature terms of the boundary. The minimizers of the limit functional are polygons with edges either parallel to some prescribed directions or pointing to some fixed points, that can also be taken as unknown of the problem. },
author = {Braides, Andrea, Riey, Giuseppe},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Γ-convergence; curvature functionals; segmentation problems; image processing.; -convergence; segmentation problems; image processing},
language = {eng},
month = {7},
number = {5},
pages = {729-748},
publisher = {EDP Sciences},
title = {A variational model in image processing with focal points},
url = {http://eudml.org/doc/250363},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Braides, Andrea
AU - Riey, Giuseppe
TI - A variational model in image processing with focal points
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/7//
PB - EDP Sciences
VL - 42
IS - 5
SP - 729
EP - 748
AB - We propose a model for segmentation problems involving an energy concentrated on the vertices of an unknown polyhedral set, where the contours of the images to be recovered have preferred directions and focal points. We prove that such an energy is obtained as a Γ-limit of functionals defined on sets with smooth boundary that involve curvature terms of the boundary. The minimizers of the limit functional are polygons with edges either parallel to some prescribed directions or pointing to some fixed points, that can also be taken as unknown of the problem.
LA - eng
KW - Γ-convergence; curvature functionals; segmentation problems; image processing.; -convergence; segmentation problems; image processing
UR - http://eudml.org/doc/250363
ER -

References

top
  1. L. Ambrosio and A Braides, Functionals defined on partitions of sets of finite perimeter, I and II. J. Math. Pures. Appl.69 (1990) 285–305 and 307–333.  Zbl0676.49028
  2. L. Ambrosio and V.M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math.43 (1990) 999–1036.  Zbl0722.49020
  3. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000).  Zbl0957.49001
  4. G. Aubert and P. Kornprobst, Mathematical problems in image processing. Partial differential equations and the calculus of variations. Springer, New York (2006).  Zbl1110.35001
  5. G. Bellettini and R. March, An image segmentation variational model with free discontinuities and contour curvature. Math. Mod. Meth. Appl. Sci.14 (2004) 1–45.  Zbl1044.49009
  6. G. Bellettini and L. Mugnai, Characterization and representation of the lower semicontinuous envelope of the elastica functional. Ann. Inst. Henri Poincaré, Anal. Non Linéaire21 (2004) 839–880.  Zbl1110.49014
  7. G. Bellettini, G. Dal Maso and M. Paolini, Semicontinuity and relaxation properties of a curvature depending functional in 2D. Ann. Scuola Norm. Sup. Pisa (4)20 (1993) 247–297.  Zbl0797.49013
  8. A. Blake and A. Zisserman, Visual Reconstruction. MIT Press, Cambridge, MA (1987).  
  9. A. Braides, Approximation of Free-Discontinuity Problems, Lecture Notes in Mathematics. Springer-Verlag, Berlin (1998).  Zbl0909.49001
  10. A. Braides, Γ-Convergence for Beginners. Oxford University Press, Oxford (2002).  Zbl1198.49001
  11. A. Braides and A. Malchiodi, Curvature theory of boundary phases: the two-dimensional case. Interfaces Free Bound.4 (2002) 345–370.  Zbl1029.49039
  12. A. Braides and R. March, Approximation by -convergence of a curvature-depending functional in visual reconstruction. Comm. Pure Appl. Math. 59 (2006) 71–121.  Zbl1098.49012
  13. A. Braides, A. Chambolle and M. Solci, A relaxation result for energies defined on pairs set-function and applications. ESAIM: COCV13 (2007) 717–734.  Zbl1149.49017
  14. A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. SIAM J. Appl. Math.55 (1995) 827–863.  Zbl0830.49015
  15. A. Chambolle, Finite-differences discretizations of the Mumford-Shah functional. ESAIM: M2AN33 (1999) 261–288.  Zbl0947.65076
  16. A. Chambolle and G. Dal Maso, Discrete approximation of the Mumford-Shah functional in dimension two. ESAIM: M2AN33 (1999) 651–672.  Zbl0943.49011
  17. A. Coscia, On curvature sensitive image segmentation. Nonlin. Anal.39 (2000) 711–730.  Zbl0942.68135
  18. G. Dal Maso, An Introduction to -Convergence. Birkhäuser, Boston (1993).  Zbl0816.49001
  19. G. Dal Maso, J.M. Morel and S. Solimini, A variational method in image segmentation: existence and approximation results. Acta Math.168 (1992) 89–151.  Zbl0772.49006
  20. C. Mantegazza, Curvature varifolds with boundary. J. Diff. Geom.43 (1996) 807–843.  Zbl0865.49030
  21. R. March, Visual reconstruction with discontinuities using variational methods. Image Vis. Comput.10 (1992) 30–38.  
  22. L. Modica and S. Mortola, Il limite nella -convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5)3 (1977) 526–529.  Zbl0364.49006
  23. J.M. Morel and S. Solimimi, Variational Methods in Image Segmentation, Progress in Nonlinear Differential Equations and Their Applications14. Birkhäuser, Basel (1995).  
  24. D. Mumford, Elastica and computer vision, in Algebraic Geometry and its Applications (West Lafayette, IN 1990), Springer, New York (1994) 491–506.  Zbl0798.53003
  25. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math.42 (1989) 577–685.  Zbl0691.49036
  26. M. Nitzberg, D. Mumford and T. Shiota, Filtering, Segmentation and Depth, in Lecture Notes in Computer Science662, Springer-Verlag, Berlin (1993).  Zbl0801.68171
  27. M. Röger and R. Schätzle, On a modified conjecture of De Giorgi. Math. Z.254 (2006) 675–714.  Zbl1126.49010
  28. J. Shah, Uses of elliptic approximations in computer vision, in Variational Methods for Discontinuous Structures, Birkhäuser, Basel (1996) 19–34.  Zbl0871.65120
  29. J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion, in IEEE Conference on Computer Vision and Pattern Recognition, June (1996).  

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.