Reconciling distance functions and level sets

This paper is concerned with the simulation of the partial differential equ ation driven evolution of a closed surface by means of an implicit represen tation. In most applications, the natural choice for the implicit represent ation is the signed distance function to the closed surface. Osher and Seth ian have proposed to evolve the distance function with a Hamilton-Jacobi eq uation. Unfortunately the solution to this equation is not a distance funct ion. As a consequence, the practical application of the level set method is plagued with such questions as When do we have to reinitialize the distanc e function? How do we reinitialize the distance function?, which reveal a d isagreement between the theory and its implementation. This paper proposes an alternative to the use of Hamilton-Jacobi equations which eliminates thi s contradiction: in our method the implicit representation always remains a distance function by construction, and the implementation does not differ from the theory anymore. 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 Conf erence on Computer Vision and Pattern Recognition, June 1998), (ii) the con struction 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). (C ) 2000 Academic Press.