On the soundness of surface voxelizations

A voxelization of a mathematically defined continuous object in Euclidean 3-space is a set of voxels that may be used as a discrete representation of the continuous object. The generation of voxelizations is an important part of volume graphics. This chapter deals with the voxelization of surfaces. In this case the voxelization should be a thin connected set of voxels with appropriate local separation properties. The voxelization should be free of small holes that would allow a discrete ray in its complement to pass from one side of the continuous surface to the other. No part of the continuous surface should be far from the voxelization, and no part of the voxelization should be far from the surface. We state several mathematical conditions which express these requirements in a precise way. Some of the conditions were proposed a number of years ago (in a rather different context) by Morgenthaler and Rosenfeld. Two voxelizations of planes in 3-space are described, and shown to satisfy all of the conditions.