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Olivier Roy

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01 Dec 2010 Accurate Fast Marching Multistencils second order Fast Marching 2D and 3D including rk4 shortest path and skeletonize Author: Dirk-Jan Kroon

Great submission thanks. The tree structure you use is very efficient.

I compared, as you did, the accuracy of your implementation with the results reported in M. Sabry Hassouna et al. "Multistencils Fast Marching Methods: A Highly Accurate Solution to the Eikonal Equation on Cartesian Domains". Not sure exactly what they do but I found that the accuracy depends a lot on how the results of the two stencils are combined (which is somewhat arbitrary since we do not know a priori which stencil provides the most accurate result).

To improve the accuracy here is the trick: instead of computing the distance with the first and second stencils separately, simply sum the corresponding second order polynomials and solve the resulting second order equation. In other words, simply replace

if(usecross) {
Coeff[0]=0; Coeff[1]=0; Coeff[2]=-1/(max(pow2(Fij),eps));
...

by

if(usecross) {
Coeff[0]+=0; Coeff[1]+=0; Coeff[2]+=-1/(max(pow2(Fij),eps));

in the CalculateDistance function.

With this little modification, I could obtain better results that the aforementioned paper for their Experiment 1.

Note that a weighted sum of the two polynomials can also be done.

Hope this helps,
Olivier

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