## Abstract

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 as a successful heuristic for matching of point sets in d-space under translation, 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-sided) 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, under the RMS measure, for which we present a lower bound construction of Ω(nlogn) iterations, 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.

Original language | English |
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Pages (from-to) | 77-93 |

Number of pages | 17 |

Journal | Computational Geometry: Theory and Applications |

Volume | 41 |

Issue number | 1-2 |

DOIs | |

State | Published - Oct 2008 |

Externally published | Yes |

### Bibliographical note

Funding Information:✩ Work on this paper by the first two authors has been supported by NSF Grants CCR-00-98246 and CCF-05-14079, by a grant from the US– Israeli Binational Science Foundation, work by the second author was also supported by Grant 155/05 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work on this paper by the last author has been partially supported by an NSF CAREER award (CCR-03-48000) and an ITR/Collaborative Research grant (03-12443). A preliminary version of this paper has been presented in Proc. 22nd Annu. ACM Sympos. Comput. Geom. 2006, and in Proc. 22nd European Workshop on Computational Geometry, 2006. * Corresponding author. E-mail addresses: estere@tau.ac.il (E. Ezra), michas@tau.ac.il (M. Sharir), alon@cs.arizona.edu (A. Efrat).

## Keywords

- Hausdorff distance
- ICP algorithm
- Nearest neighbors
- Pattern matching
- RMS
- Voronoi diagrams