There may be errors in the world and image points used to compute the homography, and there may be errors in the image points back projected to make world measurements. All of these uncertainties must be taken into account in order to compute a cumulative uncertainty for the world measurement.
In this section we list the formulas used to compute the uncertainty for measurements under various error situations. The first order analysis is assumed sufficient. The uncertainty in the homography is computed as described in the previous section.
We first introduce new notation that will simplify the formulas.
Equation (
1
) can be written as
, where
B
is a
matrix in the form:
The formulas predict the
covariance matrix
of the homogeneous world point
where
. The conversion to a
covariance matrix
for inhomogeneous coordinates is given by:
where
and
. The opposite conversion for a point
is simply given by:
.
, given an uncertain
H
and exact
.
.
, given an exact
H
and uncertain
.
.
, given an uncertain
H
and and uncertain
.
This is the sum of the previous two terms:
.
Suppose
and
are back-projected to
and
with covariances
and
, computed as above. Then, the uncertainty on the Euclidean world
distance
L
between the two world points is
where
.
Antonio Criminisi