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.

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.

Tue Feb 3 17:18:41 MET 1998