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A Plane Measuring Device
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2 The Camera Model
From equation (
1
) each image to world point correspondence provides two equations linear
in the
H
matrix elements. For
n
correspondences we obtain a system of 2
n
equation in 8 unknowns. If
n
= 4 then an exact solution is obtained. Otherwise, if
n
> 4, the matrix is over determined, and
H
is estimated by a suitable minimisation scheme.
The covariance of the estimated
H
matrix depends both on the errors in the position of the points used for
its computation
and
the estimation method. There are three standard methods for estimating
H
:
One of the 9 matrix elements is given a fixed value, usually unity,
and the resulting simultaneous equations for the other 8 elements are
then solved using a pseudo-inverse. This is the most commonly used
method. It has the disadvantage that poor estimates are obtained if the
chosen element should actually have the value zero.
The solution is obtained using SVD. This is the method used here and
is explained in more detail below. It does not have the disadvantage of
the non-homogeneous method.
The summed Euclidean distances between the measured and a mapped point
is minimised. This method has the advantage, over the above two
algebraic methods, that the quantity minimised is meaningful and
corresponds to the error involved in the measurement (similar
minimisations are used in estimate the fundamental matrix and trifocal
tensor [
15
,
17
]). There is no closed form solution.
Writing the
H
matrix in vector form as
the homogeneous equations (
1
) for
n
points become
with
A
the
matrix:
It is a standard result of linear algebra that the vector
that minimises the algebraic residuals
, subject to
, is given by the eigenvector of least eigenvalue of
. This eigenvector can be obtained directly from the SVD of
A
. In the case of
n
= 4,
is the null-vector of
A
and the residuals are zero.
Next:
4 First and Second
Up:
A Plane Measuring Device
Previous:
2 The Camera Model
Antonio Criminisi
Sun Jul 13 11:42:29 GMT 1997