matrices of dimension (m+1)×(n+1). The word projectivity will also be used to denote
an invertible projective mapping.
If A is a 3 × 3 non-singular matrix representing a projective transformation of P
2
,then
A
∗
is the corresponding line map. In other words, if u
1
and u
2
line on a line λ,then
Au
1
and Au
2
line on the line A
∗
λ :insymbols
A
∗
(u
1
× u
2
)=(Au
1
) × (Au
2
) .
This formula is easily derived from Proposition 1.1.
Most of the vectors and matrices used in this paper will be defined only up to multi-
plication by a nonzero factor. Usually, we will ignore multiplicative factors and use the
equality sign (=) to denote equality up to a constant factor. Exceptions to this rule will
be noted specifically.
Camera Model. The camera model considered in this paper is that of central pro-
jection, otherwise known as the pinhole or perspective camera model. Such a camera
maps a region of R
3
lying in front of the camera into a region of the image plane R
2
.
For mathematical convenience we extend this mapping to a mapping between projective
spaces P
3
(the object space) and P
2
(image space). The map is defined everywhere in
P
3
except at the centre of projection of the camera (or camera centre).
Points in object space will therefore be denoted by homogeneous 4-vectors x =(x, y, z, t)
T
,
or more usually as (x, y, z, 1)
T
. Image space points will be represented by u =(u, v, w)
T
.
The projection from object to image space is a projective mapping represented by a 3×4
matrix P of rank 3, known as the camera matrix. The camera matrix transforms points
in 3-dimensional object space to points in 2-dimensional image space according to the
equation u = P x. The camera matrix P is defined up to a scale factor only, and hence
has 11 independent entries. This model allows for the modeling of several parameters,
in particular : the location and orientation of the camera; the principal point offsets in
the image space; and unequal scale factors in two orthogonal directions not necessarily
parallel to the axes in image space.
Suppose the camera centre is not at infinity, and let its Euclidean coordinates be t =
(t
x
,t
y
,t
z
)
T
. The camera mapping is undefined at t in that P (t
x
,t
y
,t
z
, 1)
T
=0. IfP is
writteninblockformasP =(M | v), then it follows that M t + v =0,andsov = −Mt.
Thus, the camera matrix may be written in the form
P =(M |−Mt)
where t is the camera centre. Since P has rank 3, it follows that M is non-singular.
2EpipolarGeometry
Suppose that we have two images of a common scene and let u be a point in the first
image. The locus of all points in P
3
that map to u consists of a straight line through
the centre of the first camera. As seen from the second camera this straight line maps to
a straight line in the image known as a epipolar line.Anypointu
in the second image
matching point u must lie on this epipolar line. The epipolar lines in the second image
corresponding to points u in the first image all meet in a point p
, called the epipole.
4
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