This paper presents a general approach to modeling surface deformations of solid objects using level sets. Level-set models encode the shapes of 3D objects as the iso-surfaces of dense scalar fields. The embedding is stored as a discretely-sampled, rectilinear grid, i.e. a volume. Thus, level-set models can be described as a class of implicit models with basis functions that are fixed in number and location (grid points) and have a finite extent. The advantage of this representation is its ability to model incremental deformations in shape. We have shown that level-set models can mimic parametric surface models that seek to minimize various kinds of energy functionals. At the same time level-set models offer a number of advantages, including flexible topology, no need for reparameterization, concise descriptions of differential structure, and a natural scale space for hierarchical representations. These methods have a wide range of applications in 3D surface modeling.
This paper presents the mathematics for embedding deformable surfaces as level sets within volumes. It also describes our work in developing numerical algorithms for computing the resulting evolution equations. These algorithms allow level-set models to be manipulated and positioned to sub-voxel accuracy. This paper also presents some of our work in the application of level-set models to several different problems in 3D graphics: 3D surface morphing, filleting and blending, and 3D reconstruction from multiple range maps.