In this section we compute the covariance of the homography H
estimated from n image-world point correspondences.
We consider all the computation points to be measured with error
modelled as an homogeneous, isotropic
Gaussian noise process.
For the image computation points we define 
, and
for the world ones 
.
It is not strictly necessary to have such idealised distributions
but this has not been found to be a restriction in practice.
From section 3.1 we seek the eigenvector 
with
smallest eigenvalue 
of 
.
If the measured points are noise-free, or n = 4, then
and in general we can assume that for the
residual error 
.
We now use matrix perturbation theory [7] to compute the
covariance 
of based on this zero
approximation. In a similar manner to [16], it can be shown
that the 
covariance matrix is
where
, with
the 
eigenvector of the matrix and
the corresponding eigenvalue. S is the matrix:
with 
row vector of the A matrix and
The above theory has a double advantage over other methods such
as [2, ] which require the inverse of in order to compute . These methods are
poorly conditioned if only four correspondences are used, or if n >
4 and the correspondences are (almost) noise-free. In both cases
matrix is singular and thus is not invertible. Because
the derivation of expression (7) has not
involved the inverse, it is well conditioned in both these cases.