On the ICP algorithm

We present upper and lower bounds for the number of iterations performed by the Iterative Closest Point (ICP) algorithm. This algorithm has been proposed by Besl and McKay 4 as a successful heuristics for pattern matching under translation, where the input consists of two point sets in d-space, for d≥1, but so far it seems not to have been rigorously analyzed. We consider two standard measures of resemblance that the algorithm attempts to optimize: The RMS (root mean squared distance) and the (one-directional) Hausdorff distance. We show that in both cases the number of iterations performed by the algorithm is polynomial in the number of input points. In particular, this bound is quadratic in the one-dimensional problem, for which we present a lower bound construction of Ω(n logn) iterations under the RMS measure, where n is the overall size of the input. Under the Hausdorff measure, this bound is only O(n) for input point sets whose spread is polynomial in n, and this is tight in the worst case.We also present several structural geometric properties of the algorithm under both measures. For the RMS measure, we show that at each iteration of the algorithm the cost function monotonically and strictly decreases along the vector Δt of the relative translation. As a result, we conclude that the polygonal path π, obtained by concatenating all the relative translations that are computed during the execution of the algorithm, does not intersect itself. In particular, in the one-dimensional problem all the relative translations of the ICP algorithm are in the same (left or right) direction. For the Hausdorff measure, some of these properties continue to hold (such as monotonicity in one dimension), whereas others do not.