Clearly, not every matrix can be written in the form
of equation (3) above. Indeed, this matrix depends upon ten parameters,
and the three independent
degrees of freedom associated with the rotation matrix
R =
A general
projective matrix has eleven degrees of freedom: it has 12 entries,
but an arbitrary scale factor is involved, so one of the entries can be
set to 1 without loss of generality.
Let us write C in the following form:
To see why this is true, we will consider the following proof, and in the process compute explicitly the ten unknown parameters:
If C is in the form of equation (3), thenBecause of the choices of, and since
is a row of a rotation matrix, its norm is 1. Moreover,
So in the case when C is a matrix in the form of equation (3), both these conditions clearly hold.
In the other direction, we need to show that if both these conditions hold, then C is actually in the form of equation (3). Understanding why this is so will help us develop algorithms for estimating C, and subsequently for extracting the intrinsic and extrinsic parameters of the camera.
We know that C is known up to a scale factor s, and s must be
because
. So if we have
then
(4) Taking the inner products ofwith
and
yields uc and vc:
Computing the squared magnitudes ofand
yields
Substituting the previous values for uc and vc and remembering that
has norm 1, we can simplify even further to get
Havingand vc, we can now compute
and ty: