Abstract
In this paper, we discuss the use of differential invariants, and curvature driven flows for active vision. We concentrate on three problem areas: invariant flows, active contours, and L 1-based methods for optical flow and stereo. The solutions to these key problems will all be based on curvature based evolutions which are obtained in a completely natural manner from geometric and physical principles.
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
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.
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.
S. Angenent, G. Sapiro, and A. Tannenbaum, “On the affine heat equation for non-convex curves,” submitted for publication, 1996.
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.
A. Blake, R. Curwen, and A. Zisserman, “A framework for spatio-temporal control in the tracking of visual contours,” to appear in Int. J. Compter Vision.
W. Blaschke, Vorlesungen über Differentialgeometrie II, Verlag Von Julius Springer, Berlin, 1923.
E. Calabi, P. Olver, and A. Tannenbaum, “Affine geometry, curve flows, and invariant numerical approximations,” Advances in Mathematics 124 (1996), pp. 154–196.
E. Calabi, P. Olver, C. Shakiban, and A. Tannenbaum, “Differential and numerically invariant signature curves applied to object recognition,” to appear in Int. J. Computer Vision, 1996.
E. Cartan, La Méthode du Repére Mobile, la Théorie des Groupes Continus, et les Espaces Généralisés; Exposés de Géométrie, Hermann, Paris, 1935.
V. Casselles, F. Catte, T. Coll, and F. Dibos, “A geomteric model for active contours in image processing,” Numerische Mathematik 66 (1993), pp. 1–31.
V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic snakes,” to appear in Int. J. Computer Vision.
O. Faugeras, “On the evolution of simple curves of the real projective plane,” Comptes rendus de l’Acad. des Sciences de Paris 317, pp. 565–570, September 1993.
O. Faugeras and R. Keriven, “Scale-spaces and affine curvature,” Proc. Europe-China Workshop on Geometric Moddeling and Invariants for Computer Vision, edited by R. Mohr and C. Wu, 1995, pp. 17–24.
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.
M. Grayson, “The heat equation shrinks embedded plane curves to round points,” J. Differential Geometry 26 (1987), pp. 285–314.
B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artificial Intelligence, 23: 185–203, 1981.
G. R. Jensen, Higher order contact of submanifolds of homogeneous spaces, Lecture Notes in Math. 610, New York, Springer-Verlag, 1977.
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.
A. Kumar, A. Tannenbaum, and G. Balas, “Optical flow: a curve evolution approach,” IEEE Transactions on Image Processing 5 (1996), pp. 598–611.
A. Kumar, S. Haker, C. Vogel, A. Tannenbaum, and S. Zucker, “Stereo disparity and L’ minimization,” to appear in Proceedings of CDC, December 1997.
S. Lie, “Theorie der Transformationsgruppen I,” Math. Ann. 16 (1880), pp. 441–528.
R. Malladi, J. Sethian, B. and Vermuri, “Shape modelling with front propagation: a level set approach,” IEEE PAMI 17 (1995), pp. 158–175.
D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems,” Comm. on Pure and Applied Math. 42 (1989).
P. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, 1995.
P. Olver, G. Sapiro, and A. Tannenbaum, “Affine invariant edge maps and active contours,” submitted for publication, 1997.
P. Olver, G. Sapiro, and A. Tannenbaum, “Differential invariant signatures and flows in computer vision: a symmetry group approach,” in Geometry Driven Diffusion in Computer Vision, edited by Bart ter Haar Romeny, Kluwer, 1994.
P. Olver, G. Sapiro, and A. Tannenbaum, “Invariant geometric evolutions of surfaces and volumetric smoothing,” SIAM J. on Analysis, February 1997.
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.
P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell. 12 (1990), pp. 629–639.
T. Poggio, V. Torre, and C. Koch, “Computational vision and regularization theory,” Nature 317 (1985), pp. 314–319.
B. ter Haar Romeny (editor), Geometry-Driven Diffusion in Computer Vision, Kluwer, Holland, 1994.
L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60 (1993), 259–268.
G. Sapiro and A. Tannenbaum, “On affine plane curve evolution,” Journal of Functional Analysis 119 (1994), pp. 79–120.
G. Sapiro and A. Tannenbaum, “Affine invariant scale-space,” International Journal of Computer Vision 11 (1993), pp. 25–44.
G. Sapiro and A. Tannenbaum, “Invariant curve evolution and image analysis,” Indiana University J. of Mathematics 42 (1993), pp. 985–1009.
J. A. Sethian, “A review of recent numerical algorithms for hypersurfaces moving with curvature dependent speed,” J. Differential Geometry 31 (1989), pp. 131–161.
J. Shah, “Recovery of shapes by evolution of zero-crossings,” Technical Report, Math. Dept. Northeastern Univ, Boston MA, 1995.
C. Vogel, “Total variation regularization for ill-posed problems,” Technical Report, Department of Mathematics, Montana State University, April 1993.
A. P. Witkin, “Scale-space filtering,” Int. Joint. Conf. Artificial Intelligence, pp. 1019–1021, 1983.
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.
A. Yezzi, “Modified mean curvature motion for image smoothing and enhancement, to appear in IEEE Trans. Image Processing, 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
This paper is dedicated with much admiration and affection to Professor Paul Fuhrmann on the occasion of his 60th birthday.
Rights and permissions
Copyright information
© 1997 Springer Fachmedien Wiesbaden
About this chapter
Cite this chapter
Tannenbaum, A., Yezzi, A. (1997). Differential Invariants and Curvature Flows in Active Vision. In: Helmke, U., Prätzel-Wolters, D., Zerz, E. (eds) Operators, Systems and Linear Algebra. European Consortium for Mathematics in Industry. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-09823-2_16
Download citation
DOI: https://doi.org/10.1007/978-3-663-09823-2_16
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-663-09824-9
Online ISBN: 978-3-663-09823-2
eBook Packages: Springer Book Archive