next up previous
Next: Solving the Correspondence (Matching) Up: Problem Definition Previous: Analogy with Object Recognition

Superposition - best least squares (RMSD) rigid alignment

Given two set of 3D points: \( P=\{p_{i}\},Q=\{q_{i}\},i=1...n \)
find a 3D rotation R0 and translation a0 such that:

\( min_{R,a}\sum _{i}\vert Rp_{i}+a-q_{i}\vert^{2}=\sum _{i}\vert R_{0}p_{i}+a_{0}-q_{i}\vert^{2} \)
A closed form solution exists for this task. It can be computed in O(n) time (Umeyama, 1991 [12]).
The problem is related to the well known Procrustees problems in statistics and involves eigenvalue analysis of a correlation matrix of the points.


Peer Itsik
2001-03-04