Skip to main content
Springer Nature Link
Log in

Visual tracking, active vision, and gradient flows

  • Conference paper
  • First Online:
The confluence of vision and control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 237))

Abstract

In this note, we discuss the minimization of certain functionals and the resulting gradient flows for problems in active vision. In particular, we consider how these techniques may be applied to deformable contours, and L 1-based methods for optical flow. Such techniques are becoming essential tools in the emerging field of controlled active vision.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Alvarez, F. Guichard, P. L. Lions, and J. M. Morel, “Axioms and fundamental equations of image processing,” Arch. Rational Mechanics 123 (1993), pp. 200–257.

    MathSciNet  Google Scholar 

  2. L. Alvarez, P. L. Lions, and J. M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29 (1992), pp. 845–866.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. L. Barron, D. J. Fleet, and S. S. Beauchemin, “Performance of optical flow techniques,” International Journal of Computer Vision, 12:43–77, 1994.

    Article  Google Scholar 

  4. A. D. Bimbo, P. Nesi, and J. L. C. Sanz, “Optical flow computation using extended constraints,” Technical report, Dept. of Systems and Informatics, University of Florence, 1992.

    Google Scholar 

  5. A. Blake, R. Curwen, and A. Zisserman, “A framework for spatio-temporal control in the tracking of visual contours,” to appear in Int. J. Computer Vision.

    Google Scholar 

  6. V. Casselles, F. Catte, T. Coll, and F. Dibos, “A geomtric model for active contours in image processing,” Numerische Mathematik 66 (1993), pp. 1–31.

    Article  MathSciNet  Google Scholar 

  7. V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic snakes,” to appear in Int. J. Computer Vision.

    Google Scholar 

  8. C. Foias, H. Ozbay, and A. Tannenbaum, Robust Control of Distributed Parameter Systems, Lecture Notes in Control and Information Sciences 209, Springer-Verlag, New York, 1995.

    Google Scholar 

  9. M. Grayson, “The heat equation shrinks embedded plane curves to round points,” J. Differential Geometry 26 (1987), pp. 285–314.

    MATH  MathSciNet  Google Scholar 

  10. E. C. Hildreth, “Computations underlying the measurement of visual motion,” Artificial Intelligence, 23:309–354, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  11. B. K. P. Horn, Robot Vision, MIT Press, Cambridge, Mass., 1986.

    Google Scholar 

  12. B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artificial Intelligence, 23:185–203, 1981.

    Article  Google Scholar 

  13. S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi, “Conformal curvature flows: from phase transitions to active vision,” Archive of Rational Mechanics and Analysis 134 (1996), pp. 275–301.

    Article  MATH  MathSciNet  Google Scholar 

  14. B. B. Kimia, A. Tannenbaum, and S. W. Zucker, “Shapes, shocks, and deformations, I,” Int. J. Computer Vision 15 (1995), pp. 189–224.

    Article  Google Scholar 

  15. B. B. Kimia, A. Tannenbaum, and S. W. Zucker, “On the evolution of curves via a function of curvature, I: the classical case,” J. of Math. Analysis and Applications 163 (1992), pp. 438–458.

    Article  MATH  MathSciNet  Google Scholar 

  16. B. B. Kimia, A. Tannenbaum, and S. W. Zucker, “Optimal control methods in computer vision and image processing,” in Geometry Driven Diffusion in Computer Vision, edited by Bart ter Haar Romeny, Kluwer, 1994.

    Google Scholar 

  17. A. Kumar, A. Tannenbaum, and G. Balas, “Optical flow: a curve evolution approach,” IEEE Transactions on Image Processing 5 (1996), pp. 598–611.

    Article  Google Scholar 

  18. A. Kumar, S. Haker, C. Vogel, A. Tannenbaum, and S. Zucker, “Stereo disparity and L 1 minimization,” to appear in Proceedings of CDC, December 1997.

    Google Scholar 

  19. R. Malladi, J. Sethian, and B. Vermuri, “Shape modelling with front propagation: a level set approach,” IEEE PAMI 17 (1995), pp. 158–175.

    Google Scholar 

  20. D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems,” Comm. on Pure and Applied Math. 42 (1989).

    Google Scholar 

  21. H.-H. Nagel and W. Enkelmann, “An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences,” IEEE Trans. Pattern Analysis and Machine Intelligence PAMI-8 (1986), pp. 565–593.

    Article  Google Scholar 

  22. B. K. P. Horn, Robot Vision, MIT Press, Cambridge, Mass., 1986.

    Google Scholar 

  23. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, pp. 259–268, 1992.

    Article  MATH  Google Scholar 

  24. S. J. Osher and J. A. Sethian, “Fronts propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations,” Journal of Computational Physics 79 (1988), pp. 12–49.

    Article  MATH  MathSciNet  Google Scholar 

  25. P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell. 12 (1990), pp. 629–639.

    Article  Google Scholar 

  26. T. Poggio, V. Torre, and C. Koch, “Computational vision and regularization theory,” Nature 317 (1985), pp. 314–319.

    Article  Google Scholar 

  27. B. ter Haar Romeny (editor), Geometry-Driven Diffusion in Computer Vision, Kluwer, Holland, 1994.

    MATH  Google Scholar 

  28. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60 (1993), 259–268.

    Article  Google Scholar 

  29. G. Sapiro and A. Tannenbaum, “On affine plane curve evolution,” Journal of Functional Analysis 119 (1994), pp. 79–120.

    Article  MATH  MathSciNet  Google Scholar 

  30. G. Sapiro and A. Tannenbaum, “Invariant curve evolution and image analysis,” Indiana University J. of Mathematics 42 (1993), pp. 985–1009.

    Article  MATH  MathSciNet  Google Scholar 

  31. B. G. Schunck, “The motion constraints equation for optical flow,” Proceedings of the Seventh IEEE International Conference on Pattern Recognition, pages 20–22, 1984.

    Google Scholar 

  32. J. A. Sethian, “Curvature and the evolution of fronts,” Commun. Math. Phys. 101 (1985), pp. 487–499.

    Article  MATH  MathSciNet  Google Scholar 

  33. J. A. Sethian, “A review of recent numerical algorithms for hypersurfaces moving with curvature dependent speed,” J. Differential Geometry 31 (1989), pp., 131–161.

    MathSciNet  Google Scholar 

  34. J. Shah, “Recovery of shapes by evolution of zero-crossings,” Technical Report, Math. Dept. Northeastern Univ, Boston MA, 1995.

    Google Scholar 

  35. K. Siddiqi, Y. Lauziere, A. Tannenbaum, S. Zucker, “Area and length minimizing flows for segmentation,” to appear in IEEE Transactions Image Processing, 1998.

    Google Scholar 

  36. C. Vogel, “Total variation regularization for ill-posed problems,” Technical Report, Department of Mathematics, Montana State University, April 1993.

    Google Scholar 

  37. A. Yezzi, S. Kichenesamy, A. Kumar, P. Olver, and A. Tannenbaum, “Geometric active contours for segmentation of medical imagery,” IEEE Trans. Medical Imaging 16 (1997), pp. 199–209.

    Article  Google Scholar 

  38. A. Yezzi, “Modified mean curvature motion for image smoothing and enhancement, to appear in IEEE Trans. Image Processing, 1998.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Minnesota, 55455, Minneapolis, MN, USA

    Allen Tannenbaum & Anthony Yezzi

Authors
  1. Allen Tannenbaum
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anthony Yezzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

David J. Kriegman PhD Gregory D. Hager PhD A. Stephen Morse PhD

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag

About this paper

Cite this paper

Tannenbaum, A., Yezzi, A. (1998). Visual tracking, active vision, and gradient flows. In: Kriegman, D.J., Hager, G.D., Morse, A.S. (eds) The confluence of vision and control. Lecture Notes in Control and Information Sciences, vol 237. Springer, London. https://doi.org/10.1007/BFb0109672

Download citation

  • DOI: https://doi.org/10.1007/BFb0109672

  • Published:

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-025-5

  • Online ISBN: 978-1-84628-528-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics