Surface registration brings multiple scans into a common coordinate system by aligning their overlapping components. This can be achieved by finding a few pairs of matched points on different scans using local shape descriptors and employing the matches to compute transformations to produce the alignment. By defining a unique local reference frame (LRF) and attaching an LRF to shape descriptors, the transformation can be computed using only one match based on aligning the LRFs. This paper proposes a local voxelizer descriptor, and the key ideas are to define a unique LRF using the support around a basis point, to perform voxelization for the local shape within a cubical volume aligned with the LRF, and to concatenate local features extracted from each voxel to construct the descriptor. An automatic rigid registration approach is given based on the local voxelizer and an expanding strategy that merges descriptor representations of aligned scans. Experiments show that our registration approach allows the acquisition of 3D models of various objects, and that the local voxelizer is robust to mesh noise and varying mesh resolution, in comparison to two state-of-the-art shape descriptors.