A variational model in image processing with focal points

Braides, Andrea; Riey, Giuseppe

ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 5, pp. 729-748.

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 $\Gamma$-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.

Classification : 68U10, 94A08, 49J45
Mots clés : $\Gamma $-convergence, curvature functionals, segmentation problems, image processing

[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. | Zbl

[2] L. Ambrosio and V.M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence. Comm. Pure Appl. Math. 43 (1990) 999-1036. | MR | Zbl

[3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000). | MR | Zbl

[4] G. Aubert and P. Kornprobst, Mathematical problems in image processing. Partial differential equations and the calculus of variations. Springer, New York (2006). | MR | Zbl

[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. | MR | Zbl

[6] 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. | Numdam | MR | Zbl

[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. | Numdam | MR | Zbl

[8] A. Blake and A. Zisserman, Visual Reconstruction. MIT Press, Cambridge, MA (1987). | MR

[9] A. Braides, Approximation of Free-Discontinuity Problems, Lecture Notes in Mathematics. Springer-Verlag, Berlin (1998). | MR | Zbl

[10] A. Braides, $\Gamma$-Convergence for Beginners. Oxford University Press, Oxford (2002). | MR

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

[12] A. Braides and R. March, Approximation by $\Gamma$-convergence of a curvature-depending functional in visual reconstruction. Comm. Pure Appl. Math. 59 (2006) 71-121. | MR | Zbl

[13] A. Braides, A. Chambolle and M. Solci, A relaxation result for energies defined on pairs set-function and applications. ESAIM: COCV 13 (2007) 717-734. | Numdam | MR | Zbl

[14] A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. SIAM J. Appl. Math. 55 (1995) 827-863. | MR | Zbl

[15] A. Chambolle, Finite-differences discretizations of the Mumford-Shah functional. ESAIM: M2AN 33 (1999) 261-288. | Numdam | MR | Zbl

[16] A. Chambolle and G. Dal Maso, Discrete approximation of the Mumford-Shah functional in dimension two. ESAIM: M2AN 33 (1999) 651-672. | Numdam | MR | Zbl

[17] A. Coscia, On curvature sensitive image segmentation. Nonlin. Anal. 39 (2000) 711-730. | MR | Zbl

[18] G. Dal Maso, An Introduction to $\Gamma$-Convergence. Birkhäuser, Boston (1993). | MR | Zbl

[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. | MR | Zbl

[20] C. Mantegazza, Curvature varifolds with boundary. J. Diff. Geom. 43 (1996) 807-843. | MR | Zbl

[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 $\Gamma$-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5) 3 (1977) 526-529. | MR | Zbl

[23] J.M. Morel and S. Solimimi, Variational Methods in Image Segmentation, Progress in Nonlinear Differential Equations and Their Applications 14. Birkhäuser, Basel (1995). | MR

[24] D. Mumford, Elastica and computer vision, in Algebraic Geometry and its Applications (West Lafayette, IN 1990), Springer, New York (1994) 491-506. | MR | Zbl

[25] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577-685. | MR | Zbl

[26] M. Nitzberg, D. Mumford and T. Shiota, Filtering, Segmentation and Depth, in Lecture Notes in Computer Science 662, Springer-Verlag, Berlin (1993). | MR | Zbl

[27] M. Röger and R. Schätzle, On a modified conjecture of De Giorgi. Math. Z. 254 (2006) 675-714. | Zbl

[28] J. Shah, Uses of elliptic approximations in computer vision, in Variational Methods for Discontinuous Structures, Birkhäuser, Basel (1996) 19-34. | MR | Zbl

[29] J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion, in IEEE Conference on Computer Vision and Pattern Recognition, June (1996).