Abstract
The existing Fast Marching methods which are used to solve the Eikonal equation use a locally continuous model to estimate the accumulated cost, but a discontinuous (discretized) model for the traveling cost around each grid point. Because the accumulated cost and the traveling (local) cost are treated differently, the estimate of the accumulated cost at any point will vary based on the direction of the arriving front. Instead we propose to estimate the traveling cost at each grid point based on a locally continuous model, where we will interpolate the traveling cost along the direction of the propagating front. We further choose an interpolation scheme that is not biased by the direction of the front. Thus making the fast marching process truly isotropic. We show the significance of removing the directional bias in the computation of the cost in certain applications of fast marching method. We also compare the accuracy and computation times of our proposed methods with the existing state of the art fast marching techniques to demonstrate the superiority of our method.
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References
Adalsteinsson, D., Sethian, J.A.: A fast level set method for propagating interfaces. Journal of Computational Physics 118, 269–277 (1994)
Bronstein, A.M., Bronstein, M.M., Devir, Y.S., Kimmel, R., Weber, O.: Parallel algorithms for approximation of distance maps on parametric surfaces (2007)
Chan, T., Vese, L.: An active contour model without edges. In: Nielsen, M., Johansen, P., Fogh Olsen, O., Weickert, J. (eds.) Scale-Space 1999. LNCS, vol. 1682, pp. 141–151. Springer, Heidelberg (1999)
Cohen, L., Kimmel, R.: Global minimum for active contour models: A minimal path approach. International Journal of Computer Vision 24, 57–78 (1997)
Cohen, L.D., Kimmel, R.: Global minimum for active contour models: A minimal path approach. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, p. 666 (1996)
Danielsson, P.E., Lin, Q.: A modified fast marching method. In: Bigun, J., Gustavsson, T. (eds.) SCIA 2003. LNCS, vol. 2749, pp. 1154–1161. Springer, Heidelberg (2003)
Hassouna, M.S., Farag, A.A.: Multistencils fast marching methods: A highly accurate solution to the eikonal equation on cartesian domains. IEEE Transactions on Pattern Analysis and Machine Intelligence 29(9), 1563–1574 (2007)
Kim, S., Folie, D.: The group marching method: An o(n) level set eikonal solver
Polymenakos, L.C., Bertsekas, D.P., Tsitsiklis, J.N.: Implementation of efficient algorithms for globally optimal trajectories. IEEE Transactions on Automatic Control 43, 278–283 (1998)
Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)
Tsitsiklis, J.N.: Efficient algorithms for globally optimal trajectories. IEEE Transactions On Automatic Control 40(9), 1528–1538 (1995)
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Appia, V., Yezzi, A. (2010). Fully Isotropic Fast Marching Methods on Cartesian Grids. In: Daniilidis, K., Maragos, P., Paragios, N. (eds) Computer Vision – ECCV 2010. ECCV 2010. Lecture Notes in Computer Science, vol 6311. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15549-9_6
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DOI: https://doi.org/10.1007/978-3-642-15549-9_6
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