The camera is modelled by its optical centre and its
retinal plane (or image plane)
. In each camera,
a 3-D point
in world coordinates (where the
world coordinate frame is fixed arbitrarily) is projected into an
image point
in camera coordinates, where
is the intersection of
with the line containing
and
. In projective (or homogeneous) coordinates, the
transformation from
to
is modelled by the linear
transformation
where
The points for which S=0 define the focal plane and
are projected to infinity.
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Each pinhole camera is therefore modelled by its perspective
projection matrix (PPM) , which can be decomposed into
the product
The matrix gathers the intrinsic parameters of the
camera, and has the following form:
where are the focal lengths in vertical and
horizontal pixels, respectively, and
are the coordinates
of the principal point. The matrix
is composed by a
rotation matrix
and a vector
, encoding
the camera position and orientation (extrinsic parameters) in
the world reference frame, respectively:
Let us write the PPM as
The plane (S=0) is
the focal plane, and the two planes
and
intersect
the retinal plane in the vertical (U=0) and horizontal (V=0) axis
of the retinal coordinates, respectively.
The optical centre is the intersection of the three
planes introduced in the previous paragraph; therefore
and
The optical ray associated to an image point is the line
, i.e. the set of points
. The equation of this ray can
be written in parametric form as
with an arbitrary real number.