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Image formation involves the projection of points in
(the world) to points in
(the image plane). The perspective
projection equations with which we are familiar,
where the point (X,Y,Z) in the world is projected to the point (x,y)on the image plane, are inherently nonlinear. Converting to
homogeneous coordinates, however, makes them linear:
where
,
,
and
the perspective projection matrix T is given by:
The entire image formation process includes perspective projection,
along with matrices for internal and external calibration:
(4)
where
and
are the scale factors of the image plane
(in units of the focal length f),
is the skew (
for most real cameras),
the point (u_{0},v_{0}) is the principal point,
R is the
rotation matrix,
and
is the
translation vector. The matrix A contains
the internal parameters and perspective projection, while D contains the
external parameters.
It is sometimes convenient to decompose the
projection matrix
into a
matrix P and a
vector p
so that

(5) 
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Stanley Birchfield
19980423