Abstract
The variational methods of image segmentation discussed in this book minimize functionals. In this chapter, we review some formulas we use repeatedly in the definition and minimization of these functionals: Euler-Lagrange equations, gradient descent minimization, level set representation. We also review optical flow basic expressions used in motion based segmentation.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
R. Weinstock, Calculus of variations. Dover, 1974.
J. E. Marsden and A. J. Tromba, Vector calculus. W. H. Freeman and Company, 1976.
M. P. Do Carmo, Differential geometry of curves and surfaces. Prentice Hall, 1976.
A. Mitiche, R. Feghali, and A. Mansouri, “Motion tracking as spatio-temporal motion boundary detection,” Journal of Robotics and Autonomous Systems, vol. 43, pp. 39–50, 2003.
M. C. Delfour and J. P. Zolesio, Shapes and Geometries: Analysis, Differential Calculus and Optimization. SIAM series on Advances in Design and Control, 2001.
S. Jehan-Besson, M. Barlaud, G. Aubert, and O. Faugeras, “Shape gradients for histogram segmentation using active contours,” in International Conference on Computer Vision (ICCV), 2003, pp. 408–415.
G. Aubert, M. Barlaud, O. Faugeras, and S. Jehan-Besson, “Image segmentation using active contours: Calculus of variations or shape gradients?” SIAM Journal of Applied Mathematics, vol. 63, no. 6, pp. 2128–2154, 2003.
M. Minoux, Programmation mathématique. Dunod, Vol. 1, 1983.
J. A. Sethian, Level set Methods and Fast Marching Methods. Cambridge University Press, 1999.
A. Mansouri and J. Konrad, “Multiple motion segmentation with level sets,” IEEE Transactions on Image Processing, vol. 12, no. 2, pp. 201–220, 2003.
B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artificial Intelligence, vol. 17, no. 17, pp. 185–203, 1981.
J. J. Gibson, The perception of the visual world. Houghton Mifflin, 1950.
A. Mitiche and A. Mansouri, “On convergence of the Horn and Schunck optical flow estimation method,” IEEE Transactions on Image Processing, vol. 13, no. 6, pp. 848–852, 2004.
G. Aubert, R. Deriche, and P. Kornprobst, “Computing optical flow via variational techniques,” SIAM Journal of Applied Mathematics, vol. 60, no. 1, pp. 156–182, 1999.
R. Deriche, P. Kornprobst, and G. Aubert, “Optical-flow estimation while preserving its discontinuities: A variational approach,” in Asian Conference on Computer Vision (ACCV), 1995, pp. 71–80.
G. Aubert and P. Kornpbrost, Mathematical problems in image processing: Partial differential equations and the calculus of variations. Springer Verlag, 2006.
H. Longuet-Higgins and K. Prazdny, “The interpretation of a moving retinal image,” Proceedings of the Royal Society of London, B, vol. 208, pp. 385–397, 1981.
X. Zhuang and R. Haralick, “Rigid body motion and the optical flow image,” in International Conference on Artificial Intelligence Applications, 1984, pp. 366–375.
A. Mitiche, Computational Analysis of Visual Motion. Plenum Press, New York, 1994.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Mitiche, A., Ayed, I.B. (2010). Introductory Background. In: Variational and Level Set Methods in Image Segmentation. Springer Topics in Signal Processing, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15352-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-15352-5_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15351-8
Online ISBN: 978-3-642-15352-5
eBook Packages: EngineeringEngineering (R0)