Partial-Matching and Hausdorff RMS Distance Under Translation: Combinatorics and Algorithms

TL;DR

This paper studies the RMS distance under translation between two point sets, providing structural insights, improved algorithms for partial matching and Hausdorff cases, and enhancing existing heuristics with more efficient solutions.

Contribution

It introduces new structural properties and algorithms for minimizing RMS distance under translation in partial matching and Hausdorff scenarios, with improved complexity bounds.

Findings

Best known algorithm for partial-matching RMS distance minimization.

Polynomial-time computation of local minima for partial matching.

Nearly linear and quadratic time algorithms for Hausdorff RMS distance in 1D and 2D.

Abstract

We consider the RMS distance (sum of squared distances between pairs of points) under translation between two point sets in the plane, in two different setups. In the partial-matching setup, each point in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time. In the Hausdorff setup, each point is paired to its nearest neighbor in the other set. We develop algorithms for finding a local minimum of the Hausdorff RMS distance in nearly linear time on the line, and in nearly quadratic time in the plane. These improve substantially the worst-case behavior of the popular ICP heuristics for solving this problem.