The registration procedure allows the determination of the linear
relationship between two coordinate systems. Let 
and 
represent the same vector expressed in two different coordinate
systems, and related by the following expression:
where 
is a rotation matrix, 
is a vector
giving the translation between the coordinate systems and s is a
scale factor. Together, 
and 
unambiguously describe
any arbitrary rigid body transformation. In a real registration,
measurement errors will prevent one from finding the exact
transformation. In fact the derived transformed coordinate system will
in general be slightly rotated, translated and scaled with respect to
the exact one. This can be expressed by:
where 
is a rotation matrix, 
is a translation vector and 
is the scaling error. approximate, written as
with 
, 
and 
, clearly demonstrates the
fact that one can be mislead by merely comparing the rotation matrices
and the translation vectors to evaluate the registration error. Since
depends on 
, a small rotational error can
make 
and 
significantly different if the modulus of
is large (which is often the case in ).
We define instead the registration error 
as follows:
From exact and approximate:
