This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Finding Shortest Paths on Surfaces Using Level Sets Propagation
June 1995 (vol. 17 no. 6)
pp. 635-640

Abstract—We present a new algorithm for determining minimal length paths between two regions on a three dimensional surface. The numerical implementation is based on finding equal geodesic distance contours from a given area. These contours are calculated as zero sets of a bivariate function designed to evolve so as to track the equal distance curves on the given surface. The algorithm produces all paths of minimal length between the source and destination areas on the surface given as height values on a rectangular grid.

[1] D.L. Chopp,“Flow under geodesic curvature,” Dept. of Mathematics Report 92-23, UCLA, 1992
[2] M.P. Do Carmo,Differential Geometry of Curves and Surfaces.New Jersey: Prentice-Hall Inc., 1976.
[3] C.L. Epstein and M. Gage,“The curve shortening flow,” Wave Motion: Theory, Modeling, and Computation. A. Chorin and A. Majda, eds. New York: Springer-Verlag, 1987.
[4] R. Kimmel,A. Amir,, and A.M. Bruckstein,“Finding shortest paths on graph surfaces,” CIS Report #9301, Technion, Israel, Jan. 1993.
[5] R. Kimmel and A.M. Bruckstein,“Shape from shading via level sets,” CIS Report #9209, Technion, Israel, June 1992.
[6] R. Kimmel and A.M. Bruckstein,“Shape offsets via level sets,” CAD, vol. 25, no. 5, pp. 154-162, Mar. 1993.
[7] R. Kimmel,N. Kiryati,, and A.M. Bruckstein,“Distance maps and weighted distance transforms,” J. Mathematical Imaging and Vision, Special Issue on Topology and Geometry in Computer Vision, to be published 1994.
[8] R. Kimmel and G. Sapiro,“Shortening three dimensional curves via two dimensional flows,” Int’l J. Comp. Math. with App., vol. 29, no. 3, pp. 49-62, 1995.
[9] N. Kiryati and G. Székely,“Estimating shortest paths and minimal distances on digitized three dimensional surfaces,” Pattern Recognition, vol. 26, no. 11, pp. 1,623-1,637, 1993.
[10] J.S.B. Mitchell,D. Payton,, and D. Keirsey,“Planning and reasoning for autonomous vehicle control,” Int’l J. Intelligent Systems, vol. 2, pp. 129-198, 1987.
[11] W. Mulder,S. Osher,, and J. A. Sethian,“Computing interface motion in compressible gas dynamics,” J. of Computational Physics, vol. 100, no. 2, pp. 209-228, 1992.
[12] S. Osher and C.W. Shu,“High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations,” SIAM J. Numerical Analysis, vol. 28, no. 4, pp. 907-922, Aug. 1991.
[13] S.J. Osher and L.I. Rudin,“Feature-oriented image enhancement using shock filters,” SIAM J. Numerical Analysis, vol. 27, no. 4, pp. 919-940, Aug. 1990.
[14] S. Osher and J.A. Sethian, “Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations,” J. Computing in Physics, vol. 79, pp. 12-49, 1988.
[15] G. Sapiro,R. Kimmel,D. Shaked,B. Kimia,, and A.M. Bruckstein,“Implementing continuous-scale morphology via curve evolution,” Pattern Recognition, vol. 26, no. 9, pp. 1,363-1,372, 1993.
[16] J.A. Sethian,“Curvature and the evolution of fronts,” Comm. in Math. Phys., vol. 101, pp. 487-499, 1985.
[17] J.A. Sethian,“A review of recent numerical algorithms for hypersurfaces moving with curvature dependent speed,” J. Differential Geometry, vol. 33, pp. 131-161, 1989.

Index Terms:
Curve evolution, equal distance contours, geodesic path, numerical algorithms, minimal geodesics.
Citation:
Ron Kimmel, Arnon Amir, Alfred M. Bruckstein, "Finding Shortest Paths on Surfaces Using Level Sets Propagation," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 6, pp. 635-640, June 1995, doi:10.1109/34.387512
Usage of this product signifies your acceptance of the Terms of Use.