This paper describes a method of automatically performing the registration of two range images that have significant overlap. We first find points of interest in the intensity data that comes with each range image. Then we perform a triangulation of the 3D range points associated with these 2D interest points. All possible pairs of triangles between the two 3D triangulations are then matched. The fact that we have 3D data available makes it possible to efficiently prune matches. We do this pruning by using a simple and effective set of compatibility tests between potentially matching triangles and vertices. The best match is the one that aligns the largest number of interest points between the two range images. The algorithms are demonstrated experimentally on a number of different range image pairs.