Hostname: page-component-cb9f654ff-plnhv Total loading time: 0.001 Render date: 2025-08-23T07:03:38.404Z Has data issue: false hasContentIssue false
  • English
  • Français

A variational model in image processing with focal points

Published online by Cambridge University Press:  04 July 2008

Andrea Braides  and
Giuseppe Riey
Andrea Braides
Affiliation:
Dipartimento di Matematica, Università di Roma `Tor Vergata', Via della Ricerca scientifica 1, 00133 Roma, Italy. riey@mat.unical.it
Giuseppe Riey
Affiliation:
Dipartimento di Matematica, Università della Calabria, Via P. Bucci, 87036 Arcavacata di Rende (CS), Italy.
Get access

Abstract

We propose a model for segmentation problemsinvolving an energy concentrated on the vertices of an unknownpolyhedral set, where the contours of the images to be recoveredhave preferred directions and focal points.We prove that such an energy is obtained as a Γ-limit offunctionals defined on sets with smooth boundary thatinvolve curvature terms of the boundary.The minimizers of the limit functional are polygons withedges either parallel to some prescribed directions or pointing to somefixed points, that can also be taken as unknown of the problem.

Keywords

Γ-convergencecurvature functionalssegmentation problemsimage processing.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 5 , September 2008 , pp. 729 - 748
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ambrosio, L. and Braides, A, Functionals defined on partitions of sets of finite perimeter, I and II. J. Math. Pures. Appl. 69 (1990) 285–305 and 307–333.
Ambrosio, L. and Tortorelli, V.M., Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43 (1990) 999–1036. CrossRef
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000).
G. Aubert and P. Kornprobst, Mathematical problems in image processing. Partial differential equations and the calculus of variations. Springer, New York (2006).
Bellettini, G. and March, R., An image segmentation variational model with free discontinuities and contour curvature. Math. Mod. Meth. Appl. Sci. 14 (2004) 1–45. CrossRef
G. Bellettini and L. Mugnai, Characterization and representation of the lower semicontinuous envelope of the elastica functional. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21 (2004) 839–880.
Bellettini, G., Dal Maso, G. and Paolini, M., Semicontinuity and relaxation properties of a curvature depending functional in 2D. Ann. Scuola Norm. Sup. Pisa (4) 20 (1993) 247–297.
A. Blake and A. Zisserman, Visual Reconstruction. MIT Press, Cambridge, MA (1987).
A. Braides, Approximation of Free-Discontinuity Problems, Lecture Notes in Mathematics. Springer-Verlag, Berlin (1998).
A. Braides, Γ-Convergence for Beginners. Oxford University Press, Oxford (2002).
Braides, A. and Malchiodi, A., Curvature theory of boundary phases: the two-dimensional case. Interfaces Free Bound. 4 (2002) 345–370. CrossRef
A. Braides and R. March, Approximation by -convergence of a curvature-depending functional in visual reconstruction. Comm. Pure Appl. Math. 59 (2006) 71–121.
Braides, A., Chambolle, A. and Solci, M., A relaxation result for energies defined on pairs set-function and applications. ESAIM: COCV 13 (2007) 717–734. CrossRef
Chambolle, A., Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. SIAM J. Appl. Math. 55 (1995) 827–863. CrossRef
Chambolle, A., Finite-differences discretizations of the Mumford-Shah functional. ESAIM: M2AN 33 (1999) 261–288. CrossRef
Chambolle, A. and Dal Maso, G., Discrete approximation of the Mumford-Shah functional in dimension two. ESAIM: M2AN 33 (1999) 651–672. CrossRef
Coscia, A., On curvature sensitive image segmentation. Nonlin. Anal. 39 (2000) 711–730. CrossRef
G. Dal Maso, An Introduction to -Convergence. Birkhäuser, Boston (1993).
Dal Maso, G., Morel, J.M. and Solimini, S., A variational method in image segmentation: existence and approximation results. Acta Math. 168 (1992) 89–151. CrossRef
Mantegazza, C., Curvature varifolds with boundary. J. Diff. Geom. 43 (1996) 807–843. CrossRef
March, R., Visual reconstruction with discontinuities using variational methods. Image Vis. Comput. 10 (1992) 30–38. CrossRef
Modica, L. and Mortola, S., Il limite nella -convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5) 3 (1977) 526–529.
J.M. Morel and S. Solimimi, Variational Methods in Image Segmentation, Progress in Nonlinear Differential Equations and Their Applications 14. Birkhäuser, Basel (1995).
D. Mumford, Elastica and computer vision, in Algebraic Geometry and its Applications (West Lafayette, IN 1990), Springer, New York (1994) 491–506.
Mumford, D. and Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577–685. CrossRef
M. Nitzberg, D. Mumford and T. Shiota, Filtering, Segmentation and Depth, in Lecture Notes in Computer Science 662, Springer-Verlag, Berlin (1993).
Röger, M. and Schätzle, R., On a modified conjecture of De Giorgi. Math. Z. 254 (2006) 675–714. CrossRef
J. Shah, Uses of elliptic approximations in computer vision, in Variational Methods for Discontinuous Structures, Birkhäuser, Basel (1996) 19–34.
J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion, in IEEE Conference on Computer Vision and Pattern Recognition, June (1996).