The Community for Technology Leaders
RSS Icon
Issue No.09 - Sept. (2013 vol.35)
pp: 2298-2304
V. Appia , Imaging Technol. Lab., Texas Instrum., Dallas, TX, USA
A. Yezzi , Sch. of Electr. & Comput. Eng., Georgia Inst. of Technol., Atlanta, GA, USA
Existing fast marching methods solve the Eikonal equation using a continuous (first-order) model to estimate the accumulated cost, but a discontinuous (zero-order) model for the traveling cost at each grid point. As a result the estimate of the accumulated cost (calculated numerically) at a given point will vary based on the direction of the arriving front, introducing an anisotropy into the discrete algorithm even though the continuous partial differential equation (PDE) is itself isotropic. To remove this anisotropy, we propose two very different schemes. In the first model, we utilize a continuous interpolation of the traveling cost, which is not biased by the direction of the propagating front. In the second model, we upsample the traveling cost on a higher resolution grid to overcome the directional bias. We show the significance of removing the directional bias in the computation of the cost in some applications of the fast marching method, demonstrating that both methods make the discrete implementation more isotropic, in accordance with the underlying continuous PDE.
Mathematical model, Equations, Interpolation, Cost function, Anisotropic magnetoresistance, Numerical models, Accuracy,global minimal path, Fast marching methods, isotropic fast marching, segmentation, FMM, Eikonal equation
V. Appia, A. Yezzi, "Symmetric Fast Marching Schemes for Better Numerical Isotropy", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.35, no. 9, pp. 2298-2304, Sept. 2013, doi:10.1109/TPAMI.2013.52
[1] D. Adalsteinsson and J.A. Sethian, "A Fast Level Set Method for Propagating Interfaces," J. Computational Physics, vol. 118, pp. 269-277, 1994.
[2] J.A. Sethian, Level Set Methods and Fast Marching Methods. Cambridge Univ. Press, 1999.
[3] J.N. Tsitsiklis, "Efficient Algorithms for Globally Optimal Trajectories," IEEE Trans. Automatic Control, vol. 40, no. 9, pp. 1528-1538, Sept. 1995.
[4] L. Cohen and R. Kimmel, "Global Minimum for Active Contour Models: A Minimal Path Approach," Int'l J. Computer Vision, vol. 24, pp. 57-78, 1997.
[5] L.D. Cohen and R. Kimmel, "Global Minimum for Active Contour Models: A Minimal Path Approach," Proc. IEEE Conf. Computer Vision and Pattern Recognition, p. 666, 1996.
[6] B. Appleton and H. Talbot, "Globally Optimal Geodesic Active Contours," J. Math. Imaging and Vision, vol. 23, pp. 67-86, 2005.
[7] V. Appia and A. Yezzi, "Active Geodesics: Region-Based Active Contour Segmentation with a Global Edge Based Constraint," Proc. IEEE Int'l Conf. Computer Vision, 2011.
[8] V. Appia, U. Patil, and B. Das, "Lung Fissure Detection in CT Images Using Global Minimal Paths," Proc. SPIE, vol. 7623, 2010.
[9] T. Deschamps and L.D. Cohen, "Fast Extraction of Minimal Paths in 3D Images and Applications to Virtual Endoscopy," Medical Image Analysis, vol. 5, pp. 281-299, 2000.
[10] H. Li and A. Yezzi, "Vessels as 4-D Curves: Global Minimal 4-D Paths to Extract 3-D Tubular Surfaces and Centerlines," IEEE Trans. Medical Imaging, vol. 26, no. 9, pp. 1213-1223, Sept. 2007.
[11] V. Kaul, Y. Tsai, and A.J. Yezzi, "Detection of Curves with Unknown Endpoints Using Minimal Path Techniques," Proc. British Machine Vision Conf., pp. 1-12, 2010.
[12] P.-E. Danielsson and Q. Lin, "A Modified Fast Marching Method," Image Analysis, pp. 631-644, 2003.
[13] M.S. Hassouna and A.A. Farag, "Multistencils Fast Marching Methods: A Highly-Accurate Solution to the Eikonal Equation on Cartesian Domains," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 29, no. 9, pp. 1563-1574, Sept. 2007.
[14] D.L. Chopp, "Some Improvements of the Fast Marching Method," SIAM J. Scientific Computing, vol. 23, pp. 230-244, Jan. 2001.
[15] S. Kim, "An o(n) Level Set Method for Eikonal Equations," SIAM J. Scientific Computing, vol. 22, pp. 2178-2193, 2000.
[16] L.C. Polymenakos, D.P. Bertsekas, and J.N. Tsitsiklis, "Implementation of Efficient Algorithms for Globally Optimal Trajectories," IEEE Trans. Automatic Control, vol. 43, no. 2, pp. 278-283, Feb. 1998.
[17] J.A. Sethian, Level Set Methods and Fast Marching Methods, second ed. Cambridge Univ. Press, June 1999.
[18] V. Appia and A. Yezzi, "Fully Isotropic Fast Marching Methods on Cartesian Grids," Proc. 11th European Conf. Computer Vision: Part VI, 2010.
[19] G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers. Courier Dover Publications, 2000.
[20] H. Zhao, "A Fast Sweeping Method for Eikonal Equations," Math. of Computation, vol. 74, pp. 603-627, 2005.
[21] L. Yatziv, A. Bartesaghi, and G. Sapiro, "O(N) Implementation of the Fast Marching Algorithm," J. Computational Physics, vol. 212, pp. 393-399, 2005.
[22] O. Weber, Y.S. Devir, A.M. Bronstein, M.M. Bronstein, and R. Kimmel, "Parallel Algorithms for Approximation of Distance Maps on Parametric Surfaces," ACM Trans. Graphics, vol. 27, article 104, 2008.
[23] Q. Lin, "Enhancement, Extraction, and Visualization of 3D Volume Data," PhD dissertation, Linköpings Universitet, Sweden, 2003.
[24] T. Chan and L. Vese, "An Active Contour Model without Edges," Proc. Int'l Conf. Scale-Space Theories in Computer Vision, pp. 141-151, 1999.
53 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool