Skip to main content

Part of the book series: Springer Topics in Signal Processing ((STSP,volume 5))

  • 1551 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Weinstock, Calculus of variations. Dover, 1974.

    Google Scholar 

  2. J. E. Marsden and A. J. Tromba, Vector calculus. W. H. Freeman and Company, 1976.

    Google Scholar 

  3. M. P. Do Carmo, Differential geometry of curves and surfaces. Prentice Hall, 1976.

    Google Scholar 

  4. 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.

    Article  Google Scholar 

  5. M. C. Delfour and J. P. Zolesio, Shapes and Geometries: Analysis, Differential Calculus and Optimization. SIAM series on Advances in Design and Control, 2001.

    Google Scholar 

  6. 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.

    Google Scholar 

  7. 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.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Minoux, Programmation mathématique. Dunod, Vol. 1, 1983.

    Google Scholar 

  9. J. A. Sethian, Level set Methods and Fast Marching Methods. Cambridge University Press, 1999.

    Google Scholar 

  10. A. Mansouri and J. Konrad, “Multiple motion segmentation with level sets,” IEEE Transactions on Image Processing, vol. 12, no. 2, pp. 201–220, 2003.

    Article  Google Scholar 

  11. B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artificial Intelligence, vol. 17, no. 17, pp. 185–203, 1981.

    Article  Google Scholar 

  12. J. J. Gibson, The perception of the visual world. Houghton Mifflin, 1950.

    Google Scholar 

  13. 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.

    Article  Google Scholar 

  14. 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.

    Article  MathSciNet  MATH  Google Scholar 

  15. 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.

    Google Scholar 

  16. G. Aubert and P. Kornpbrost, Mathematical problems in image processing: Partial differential equations and the calculus of variations. Springer Verlag, 2006.

    Google Scholar 

  17. 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.

    Google Scholar 

  18. X. Zhuang and R. Haralick, “Rigid body motion and the optical flow image,” in International Conference on Artificial Intelligence Applications, 1984, pp. 366–375.

    Google Scholar 

  19. A. Mitiche, Computational Analysis of Visual Motion. Plenum Press, New York, 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Amar Mitiche .

Rights and permissions

Reprints 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)

Publish with us

Policies and ethics