Level Set Methods Based on Distance Function

WANG De-jun; TANG Yun; YU Hong-chuan; TANG Ze-sheng

Volume 24 Issue 8

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Article Navigation > Applied Mathematics and Mechanics > 2003 > 24(8): 839-848

WANG De-jun, TANG Yun, YU Hong-chuan, TANG Ze-sheng. Level Set Methods Based on Distance Function[J]. Applied Mathematics and Mechanics, 2003, 24(8): 839-848.

Citation:

WANG De-jun, TANG Yun, YU Hong-chuan, TANG Ze-sheng. Level Set Methods Based on Distance Function[J]. Applied Mathematics and Mechanics, 2003, 24(8): 839-848.

Citation:

WANG De-jun, TANG Yun, YU Hong-chuan, TANG Ze-sheng. Level Set Methods Based on Distance Function[J]. Applied Mathematics and Mechanics, 2003, 24(8): 839-848.

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Level Set Methods Based on Distance Function

1.
Institute of Software, Department of Computer Sciences, Tsinghua University, Beijing 100084, P. R. China;
2.
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China

Received Date: 2001-11-27
Rev Recd Date: 2003-05-06
Publish Date: 2003-08-15

Abstract

Abstract

Some basic problems on the levelset methods were discussed,such as the method used to preserve the distance function,the existence and uniqueness of solution for the level set equations.The main contribution is to prove that in a neighborhood of the initial zero level set,the level set equations with the restriction of the distance function have a unique solution,which must be the signed distance function with respect to the evolving surface.Some skillful approaches were used: Noticing that any solution for the original equation was a distance function,the original level set equations were transformedinto a simpler alternative form.Moreover,since the new system was not a classical one,the system was transformed into an ordinary one,for which the implicit function method was adopted.
- levelset methods,
- distance function,
- existence and uniqueness of the solution

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References

[1]	Osher S,Sethian J A.Fronts propagating with curvature dependent speed:algorithms based on the Hamilton-Jacobi formulation[J].Journal of Computational Physics,1988,79(1):12-49.
[2]	Kass M,Witkin A,Terzopoulos D.SNADES:Active contour models[J].Int'l Journal of Computer Vision,1988,1(3):321-332.
[3]	Cohen L,Cohen I.Finite element methods for active contour models and balloons for 2D and 3D images[J].IEEE Transaction on Pattern Analysis and Machine Intelligence,1993,15(11):1131-1147.
[4]	McInerney T,Terzopoulos D.Topologically adaptable snakes[A].In:IEEE Proc 5th Int'l Conf Computer Vision[C].Boston,M A:IEEE Computer Society Press,1995,694-699.
[5]	Sethian J A,Strain J.Crystal growth and dendritic solidification[J].Journal of Computational Physics,1992,98(2):231-253.
[6]	Zeng X,Staib L,Schultz R T,et al.Volumetric layer segmentation using coupled surfaces propagation[A].In:IEEE Proc Int'l Conf Computer Vision & Pattern Recognition[C].Santa Barbara,C A:IEEE Computer Society Press,1998,708-715.
[7]	Gomes J,Faugeras O.Level sets and distance functions[A].In:David Vernon Ed.Proc 6th European Conference Computer Vision[C].LNCS 1842,Berlin:Springer-Verlag,2000,588-602.
[8]	Sethian J A.Level Set Methods and Fast Matching Methods[M].Cambridge,U K:Cambridge University Press,1999.
[9]	张芷芬,于同仁,董金柱,等.常微分方程定性理论[M].北京:科学出版社,1985.

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