Reconciling Distance Functions and Level Sets

This paper is concerned with the simulation of the partial differential equation driven evolution of a closed surface by means of an implicit representation. In most applications, the natural choice for the implicit representation is the signed distance function to the closed surface. Osher and Sethian have proposed to evolve the distance function with a Hamilton-Jacobi equation. Unfortunately the solution to this equation is not a distance function. As a consequence, the practical application of the level set method is plagued with such questions as when do we have to reinitialize the distance function? how do we reinitialize the distance function?, which reveal a disagreement between the theory and its implementation. This paper proposes an alternative to the use of Hamilton-Jacobi equations which eliminates this contradiction: in our method the implicit representation always remains a distance function by construction, and the implementation does not differ from the theory any more. This is achieved through the introduction of a new equation. Besides its theoretical advantages, the proposed method also has several practical advantages which we demonstrate in three applications: (i) the segmentation of the human cortex surfaces from MRI images using two coupled surfaces (X. Zeng, et al., in Proceedings of the International Conference on Computer Vision and Pattern Recognition, June 1998), (ii) the construction of a hierarchy of Euclidean skeletons of a 3D surface, (iii) the reconstruction of the surface of 3D objects through stereo (O. Faugeras and R. Keriven, Lecture Notes in Computer Science, Vol. 1252, pp. 272-283).

[1]  G. Barles,et al.  Front propagation and phase field theory , 1993 .

[2]  J. Sethian,et al.  Crystal growth and dendritic solidification , 1992 .

[3]  M. Gage,et al.  The heat equation shrinking convex plane curves , 1986 .

[4]  Baba C. Vemuri,et al.  Evolutionary Fronts for Topology-Independent Shape Modeling and Recoveery , 1994, ECCV.

[5]  M. Grayson.  The heat equation shrinks embedded plane curves to round points , 1987 .

[6]  Yun-Gang Chen,et al.  Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations , 1989 .

[7]  L. Evans,et al.  Motion of level sets by mean curvature IV , 1995 .

[8]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[9]  D. Chopp.  Computing Minimal Surfaces via Level Set Curvature Flow , 1993 .

[10]  James A. Sethian,et al.  The Fast Construction of Extension Velocities in Level Set Methods , 1999 .

[11]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[12]  Luigi Ambrosio,et al.  Curvature and distance function from a manifold , 1998 .

[13]  Frederic Devernay.  Vision stéréoscopique et propriétés différentielles des surfaces , 1997 .

[14]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

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

[16]  J. Sethian,et al.  A Fast Level Set Method for Propagating Interfaces , 1995 .

[17]  Robert T. Schultz,et al.  Volumetric layer segmentation using coupled surfaces propagation , 1998, Proceedings. 1998 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No.98CB36231).

[18]  HARRY BLUM,et al.  Shape description using weighted symmetric axis features , 1978, Pattern Recognit..

[19]  T. Chan,et al.  A Variational Level Set Approach to Multiphase Motion , 1996 .

[20]  Grégoire Malandain,et al.  Euclidean skeletons , 1998, Image Vis. Comput..

[21]  Baba C. Vemuri,et al.  Shape Modeling with Front Propagation: A Level Set Approach , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[22]  Olivier D. Faugeras,et al.  Level Set Methods and the Stereo Problem , 1997, Scale-Space.

[23]  Anthony J. Yezzi,et al.  Gradient flows and geometric active contour models , 1995, Proceedings of IEEE International Conference on Computer Vision.