Abstract
The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shapefromshading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front, and is typically based on level set methods introduced by Osher and Sethian. However, there are more classical approaches rooted in Hamiltonian physics, which have received little consideration in the computer vision literature. Here the front is interpreted as minimizing a particular action functional. In this context, we introduce a new algorithm for simulating the eikonal equation, which offers a number of computational advantages over the earlier methods. In particular, the locus of shocks is computed in a robust and efficient manner. We illustrate the approach with several numerical examples.
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References
V. Arnold. Mathematical Methods of Classical Mechanics, Second Edition. Springer-Verlag, 1989.
H. Blum. Biological shape and visual science. J. Theor. Biol., 38:205–287, 1973.
R. Brockett and P. Maragos. Evolution equations for continuous-scale morphology. In Proceedings of the IEEE Conference on Acoustics, Speech and Signal Processing, San Francisco, CA, March 1992.
A. R. Bruss. The eikonal equation: Some results applicable to computer vision. In B. K. P. Horn and M. J. Brooks, editors, Shape From Shading, pages 69–87, Cambridge, MA, 1989. MIT Press.
V. Caselles, J.-M. Morel, G. Sapiro, and A. Tannenbaum, editors. IEEE Transactions on Image Processing, Special Issue on PDEs and and Geometry-Driven Diffusion in Image Processing and Analysis, 1998.
M. G. Crandall, H. Ishii, and P.-L. Lions. User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society, 27(1):1–67, 1992.
A. D. J. Cross and E. R. Hancock. Scale-space vector fields for feature analysis. In Conference on Computer Vision and Pattern Recognition, pages 738–743, June 1997.
O. Faugeras and R. Keriven. Complete dense stereovision using level set methods. In Fifth European Conference on Computer Vision, volume 1, pages 379–393, 1998.
B. B. Kimia, A. Tannenbaum, and S. W. Zucker. Shape, shocks, and deformations I: The components of two-dimensional shape and the reaction-diffusion space. International Journal of Computer Vision, 15:189–224, 1995.
R. K. Siddiqi, B. B. Kimia, and A. Bruckstein. Shape from shading: Level set propagation and viscosity solutions. International Journal of Computer Vision, 16(2):107–133, 1995.
C. Lanczos. The Variational Principles of Mechanics. Dover, 1986.
R. J. LeVeque. Numerical Methods for Conservation Laws. Birkhauser Verlag, 1992.
S. Osher and C.-W. Shu. High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations. SIAM Journal of Numerical Analysis, 28:907–922, 1991.
S. J. Osher and J. A. Sethian. Fronts propagating with curvature dependent speed: Algorithms based on hamilton-jacobi formulations. Journal of Computational Physics, 79:12–49, 1988.
E. Rouy and A. Tourin. A viscosity solutions approach to shape-from-shading. SIAM. J. Numer. Analy., 29(3):867–884, June 1992.
G. Sapiro, B. B. Kimia, R. Kimmel, D. Shaked, and A. Bruckstein. Implementing continuous-scale morphology. Pattern Recognition, 26(9), 1992.
J. Sethian. Level Set Methods: evolving interfaces in geometry, fluid mechanics, computer vision, and materials science. Cambridge University Press, Cambridge, 1996.
J. A. Sethian. A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. USA, 93:1591–1595, February 1996.
J. Shah. A common framework for curve evolution, segmentation and anisotropic diffusion. In Conference on Computer Vision and Pattern Recognition, pages 136–142, June 1996.
K. Siddiqi, B. B. Kimia, and C. Shu. Geometric shock-capturing eno schemes for subpixel interpolation, computation and curve evolution. Graphical Models and Image Processing, 59(5):278–301, September 1997.
K. Siddiqi, A. Shokoufandeh, S. J. Dickinson, and S. W. Zucker. Shock graphs and shape matching. International Journal of Computer Vision, to appear, 1999.
Z. S. G. Tari, J. Shah, and H. Pien. Extraction of shape skeletons from grayscale images. Computer Vision and Image Understanding, 66:133–146, May 1997.
H. Tek and B. B. Kimia. Curve evolution, wave propagation and mathematical morphology. In Fourth International Symposium on Mathematical Morphology, June 1998.
R. van den Boomgaard. Mathematical morphology: extensions towards computer vision. Ph.D. dissertation, University of Amsterdam, March 1992.
R. van den Boomgaard and A. Smeulders. The morphological structure of images: The differential equations of morphological scale-space. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(11):1101–1113, November 1994.
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Siddiqi, K., Tannenbaum, A., Zucker, S.W. (1999). A Hamiltonian Approach to the Eikonal Equation. In: Hancock, E.R., Pelillo, M. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 1999. Lecture Notes in Computer Science, vol 1654. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48432-9_1
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DOI: https://doi.org/10.1007/3-540-48432-9_1
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