A parallel fast sweeping method for the Eikonal equation

We present an algorithm for solving in parallel the Eikonal equation. The efficiency of our approach is rooted in the ordering and distribution of the grid points on the available processors; we utilize a Cuthill-McKee ordering. The advantages of our approach is that (1) the efficiency does not plateau for a large number of threads; we compare our approach to the current state-of-the-art parallel implementation of Zhao (2007) [14] and (2) the total number of iterations needed for convergence is the same as that of a sequential implementation, i.e. our parallel implementation does not increase the complexity of the underlying sequential algorithm. Numerical examples are used to illustrate the efficiency of our approach.

[1]  Anna R. Bruss.  The Eikonal equation: some results applicable to computer vision , 1982 .

[2]  E. Rouy,et al.  A viscosity solutions approach to shape-from-shading , 1992 .

[3]  Hongkai Zhao,et al.  A fast sweeping method for Eikonal equations , 2004, Math. Comput..

[4]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[5]  Hongkai Zhao,et al.  The Fast Sweeping Method , 2007 .

[6]  David L. Chopp,et al.  Some Improvements of the Fast Marching Method , 2001, SIAM J. Sci. Comput..

[7]  Ross T. Whitaker,et al.  A Fast Iterative Method for Eikonal Equations , 2008, SIAM J. Sci. Comput..

[8]  Ron Kimmel,et al.  Optimal Algorithm for Shape from Shading and Path Planning , 2001, Journal of Mathematical Imaging and Vision.

[9]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[10]  Ross T. Whitaker.  A FAST EIKONAL EQUATION SOLVER FOR PARALLEL SYSTEMS , 2007 .

[11]  James A. Sethian,et al.  A unified approach to noise removal, image enhancement, and shape recovery , 1996, IEEE Trans. Image Process..

[12]  Ian M. Mitchell,et al.  Optimal path planning under defferent norms in continuous state spaces , 2006, Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006..

[13]  Zhao,et al.  PARALLEL IMPLEMENTATIONS OF THE FAST SWEEPING METHOD , 2007 .

[14]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[15]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[16]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[17]  J. A. Sethian,et al.  Fast Marching Methods , 1999, SIAM Rev..