Reconciling Distance Functions and Level Sets

This paper is concerned with the simulation of the Par- tial Differential Equation (PDE) driven evolution of a closed surface by means of an implicit representation. In most applications, the natu- ral choice for the implicit representation is the signed distance function to the closed surface. Osher and Sethian propose 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 practi- cal 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? Etc... which reveal a disagreement between the theory and its implementation. This paper proposes an al- ternative 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 anymore. This is achieved through the introduc- tion of a new equation. Besides its theoretical advantages, the proposed method also has several practical advantages which we demonstrate in two applications: (i) the segmentation of the human cortex surfaces from MRI images using two coupled surfaces [26], (ii) the construction of a hierarchy of Euclidean skeletons of a 3D surface.