The idea of a projective plane can be applied to $n$-dimensional space in order to define projective n-space. Of particular interest is projective 3-space. Transformations within and between projective spaces are called projectivities and are the fundamental concern of projective geometry. Certain properties and measurements remain invariant under the action of a projectivity - invariant properties include collinearity, concurrency, tangency and incidence; invariant measurements, which are referred to as projective invariants.
There is no emphasis on projective spaces of any particular dimension in a purely mathematical study of projective geometry, but in computer vision some cases are of more interest than others. We consider two projectivities in the rest of the section - perspective projection to the image plane, and plane-to-plane projection - and in doing so cover some general points.
(a) Perspective projection to the image plane.
Perspective projection is a projectivity from projective 3-space to the projective plane. It has the form
where 
are the homogeneous coordinates
of a point on the image plane, 
is a 3-by-4 matrix, and 
are
the homogeneous coordinates of a point in the world.
Perspective projection is a particular type of projectivity called a perspectivity, in which all rays of projection pass through a single point - this puts constraints on the form of the matrix P as described in [Mundy 1992].
(b) Plane-to-plane projection.
A projectivity from a projective plane to a projective plane is called a plane-to-plane projectivity, although it is often referred to by simply using the more general term of projectivity. It acts on, and generates, a homogeneous 3-vector and is therefore a 3-by-3 matrix.
To see how such a projectivity arises, consider two images taken from
different viewpoints of a plane 
in a scene, Figure
1. The mapping of points on 
to the corresponding
points in image 1 is described by a projectivity 
.
Similarly, the mapping of points on 
to the corresponding
points in image 2 is described by a projectivity 
.
An important property of
projectivities is that they form a group. It follows that there is a
projectivity 
which describes the mapping of the image of
in image 1 to the image of 
in image 2, where
Figure:
Two images are taken of the plane 
in the scene.
There is a projectivity between each image and the plane, and between
the image of 
in image 1 and the image of 
in image 2.
Chapter 2 contains an example of a projectivity computed between one image and another, and Chapter 8 an example of a projectivity computed between an image and a plane in the scene.
Computation of a projectivity such as 
requires four points
on 
together with their mapped points in the image.
The following method of computation is given in [Rothwell 1992].
Let the projectivity map point 
to point 
.
These points 
and 
are normalised by scaling them so that their third components are
equal to one.
The scale of 
is arbitrary and is
set by making 
equal to one. Then the
mapping of 
to 
is described by
where 
is a scale factor.
The matrix equation (19) yields three linear
equations, and eliminating the unknown scale factor 
leaves
two linear equations
in the eight unknowns 
,
Thus four pairs of points 
,
give eight linear equations in the eight unknowns and it is possible to
solve for 
. The condition for linear independence of the
equations is that no three of the four points are collinear. Given more than four pairs of points, the system
can be solved using a least-squares method such as the pseudo-inverse.
Since there is a duality between points and lines on the projective
plane, it is natural to ask if 
also describes the mapping of
lines. In fact, it is simple to show [Mundy 1992] that if 
is a point projectivity describing the mapping of points
from projective plane 1 to projective plane 2,
then the line projectivity describing the
mapping of lines from projective plane 1 to projective plane 2
is 
(the inverse transpose of 
). Also, the inverse point
projectivity for the mapping of points from projective plane 2
to projective plane 1 is 
, and
the inverse line projectivity is 
(existence of the
inverse is guaranteed because projectivities form a group).
(c) Extra discussion on the plane-to-plane projectivity.
There are some problems with the method in (b) for computing a plane-to-plane
projectivity. The normalisation stage, where the homogeneous vectors
are scaled so
that their third component is one, and 
is set to one, is carried out to make the
equations linear. This
is inappropriate if the elements which are being normalised are
actually of value zero. (It is possible to detect when the
third component of a homogeneous vector is zero and remove it from the
processing, but there is no way to trap the case of 
being zero because this is one of the unknowns in the
system). Although these failure conditions did not cause difficulty
in practice in the work in this thesis, an alternative method for
computing the projectivity would be preferable. A possible approach
is to find the projectivity 
which minimises the expression
where 
is the desired position to which a point 
is to
be mapped. (There will always be noise in a real system, so there is
unlikely to be a 
which maps all 
exactly to the
corresponding 
. Equation (22)
identifies the 
which maps the 
``as closely as
possible'' to the corresponding 
.)
The scale of 
is arbitrary, so Equation (22)
is used in conjunction with a normalisation
condition such as,
This approach overcomes the problems associated with the method in (b) at the expense of being a non-linear system.
