be a set of 
points 
and 
be the model.
Let 
be the Euclidean distance
between point 
and
.
Let CP(
) be the closest point in to
the scene point 
.
Robert B. Fisher
The ICP algorithm [1] is an iterative alignment algorithm that works in three phases: 1) establish correspondence between pairs of features in the two structures that are to be aligned based on proximity, 2) estimate the rigid transformation that best maps the first member of the pair onto the second and then 3) apply that transformation to all features in the first structure. These three steps are then reapplied until convergence is concluded. Although simple, the algorithm works quite effectively when given a good initial estimate.
More precisely, the point matching algorithm [6] is:
Rigid Iterative Closest Point Algorithm
Let be a set of points
and be the model.
Let be the Euclidean distance
between point and
.
Let CP() be the closest point in to
the scene point .
be an initial estimate of the rigid transformation.
or until convergence:
.
that minimizes
the mean square error between point pairs in 
.
The basic algorithm has been previously extended in a number of ways: 1) correspondence between a point and a tangent plane to overcome the lack of an exact correspondence between the two sets [3], 2) robustifying the algorithm to the influence of outliers and features lacking correspondences [7] [5], 3) using a weighted least-square error metric [4], and 4) matching between features using a metric trading off distance and feature similarity (based local shape invariances) [6]. All of these approaches assume a rigid Euclidean transformation between the corresponding features, whereas the method presented here uses projective correspondence.