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Geometrical image segmentation by the Allen-Cahn equation
Applied Numerical Mathematics
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Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
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Proceedings of the 14th IEEE Visualization 2003 (VIS'03)
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Computer Aided Geometric Design
Duplex fitting of zero-level and offset surfaces
Computer-Aided Design
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Computer-Aided Design
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Computer-Aided Design
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In this paper, we present a novel fast and accurate numerical method for the surface embedding narrow volume reconstruction from unorganized points in R^3. Though the level set method prevails in the image processing, it requires a redistancing procedure to maintain a desired shape of the level set function. On the other hand, our method is based on the Allen-Cahn equation, which has been applied in image segmentation due to its motion by mean curvature property. We modify the original Allen-Cahn equation by multiplying a control function to restrict the evolution within a narrow band around the given surface data set. To improve the numerical stability of our proposed model, we split the governing equation into linear and nonlinear terms and use an operator splitting technique. The linear equation is solved by the multigrid method which is a fast solver and the nonlinear equation is solved analytically. The unconditional stability of the proposed scheme is also proved. Various numerical results are presented to demonstrate the robustness and accuracy of the proposed method.