Algorithms For 2–D Hamming Distance Under Rotations

We consider the problem of defining and evaluating the Hamming distance between two– dimensional pattern P [1..m, 1..m] of pixels and two–dimensional text T [1..n, 1..n] of pixels when also rotations of P are allowed. In particular, we are interested in the orientation and location of P that gives the minimum Hamming distance. The number of different orientations for P is O(m3) when the center of rotation is fixed. We give two incremental algorithms. The first one computes the Hamming distance in all orientations of P for all text positions in time O(m3n2). The second algorithm works in time O(k3/2n2) where k denotes the maximum allowed Hamming distance. The preprocessing time is in both cases O(m3 log m). We show that the number of different locations (rotation and translation) for P becomes O(m7) if the center of rotation of P is not fixed. We also develop a variant of the same algorithms for computing a lower bound of the Hamming distance in this general setting also. We give experimental results for both algorithms.