Rapid and Accurate Computation of the Distance Function Using Grids

We present two fast and simple algorithms for approximating the distance function for given isolated points on uniform grids. The algorithms are then generalized to compute the distance to piecewise linear objects. By incorporating the geometry of Huygensʹ principle in the reverse order with the classical viscosity solution theory for the eikonal equation |∇u|=1, the algorithms become almost purely algebraic and yield very accurate approximations. The generalized closest point formulation used in the second algorithm provides a framework for further extension to compute the distance accurately to smooth geometric objects on different grid geometries, without the construction of the Voronoi diagrams. This method provides a fast and simple translator of data commonly given in computational geometry to the volumetric representation used in level set methods.