Combinatorial methods for approximate pattern matching under rotations and translations in 3D arrays

We consider the problem of defining and evaluating the distance between a three-dimensional pattern P[1..m, 1..m, 1..m] of voxels and a three-dimensional volume V[1..n, 1..n, 1..n] of voxels when rotations of P are also allowed. In particular we are interested in finding the orientation and location of P with respect of V that gives the minimum distance. We consider several distance measures. Our basic method works for all distance measures such that the voxels affect the distance between P and V only locally, that is, the distance between two voxels can be completed in unit time. The number of different orientations that P can have is analyzed. We give incremental algorithms to compute the distance, and several filtering algorithms to compute the upper and lower bounds for the distance. We conclude with experimental results on real data (three dimensional reconstruction of a biological virus).