Abstract
We consider the Nitzberg-Mumford variational formulation of the segmentation with depth problem. This is an image segmentation model that allows regions to overlap in order to take into account occlusions between different objects. The model gives rise to a variational problem with free boundaries. We discuss some qualitative properties of the Nitzberg-Mumford functional within the framework of the relaxation methods of the Calculus of Variations. We try to characterize minimizing segmentations of images made up of smooth overlapping regions, when the weight of the fidelity term in the functional becomes large. This should give some theoretical information about the capability of the model to reconstruct both occluded boundaries and depth order.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer, Applied Mathematical Sciences 147, 2002.
G. Bellettini, G. Dal Maso and M. Paolini, Semi-continuity and relaxation properties of a curvature depending functional in 2D. Ann. Scuola Norm. Sup. Pisa (4) 20 (1993), 247–299.
G. Bellettini and M. Paolini, Variational properties of an image segmentation functional depending on contours curvature. Adv. Math. Sci. Appl. 5 (1995), 681–715.
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.
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.
S. Esedoglu and R. March, Segmentation with depth but without detecting junctions. J. Math. Imaging Vision 18 (2003), 7–15.
J.M. Morel and S. Solimini, Variational Methods in Image Segmentation, Vol. 14 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Basel, 1995.
D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989), 577–685.
D. Mumford, Elastica and computer vision. Algebraic Geometry and Applications, C. Bajaj ed., Springer-Verlag, Heidelberg, 1992.
M. Nitzberg and D. Mumford, The 2.1-D sketch. Proceedings of the Third International Conference on Computer Vision, Osaka, 1990.
M. Nitzberg, D. Mumford and T. Shiota, Filtering, Segmentation and Depth, Springer, Lecture Notes in Computer Science 662, 1993.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Bellettini, G., March, R. (2006). Asymptotic Properties of the Nitzberg-Mumford Variational Model for Segmentation with Depth. In: Figueiredo, I.N., Rodrigues, J.F., Santos, L. (eds) Free Boundary Problems. International Series of Numerical Mathematics, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7719-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7719-9_8
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-7718-2
Online ISBN: 978-3-7643-7719-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)