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Symmetric Fast Marching Schemes for Better Numerical Isotropy
Sept. 2013 (vol. 35 no. 9)
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.
Index Terms:
partial differential equations,interpolation,directional bias,symmetric fast marching scheme,numerical isotropy,Eikonal equation,continuous model,first-order model,discontinuous model,zero-order model,discrete algorithm,continuous partial differential equation,PDE,continuous interpolation,traveling cost,Mathematical model,Equations,Interpolation,Cost function,Anisotropic magnetoresistance,Numerical models,Accuracy,global minimal path,Fast marching methods,isotropic fast marching,segmentation,FMM,Eikonal equation
Citation:
V. Appia, A. Yezzi, "Symmetric Fast Marching Schemes for Better Numerical Isotropy," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 9, pp. 2298-2304, Sept. 2013, doi:10.1109/TPAMI.2013.52
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