Real Cameras

Perspective (*i.e.* pinhole) projection is an idealized mathematical
model of the behaviour of real cameras. How good is this model? --
There are two aspects to this: extrinsic and intrinsic.

Light entering a camera has to pass through a complex lens
system. However, lenses are designed to mimic point-like elements
(pinholes), and in any case the camera and lens is usually negligibly
small compared to the viewed region. Hence, in most practical
situations the camera is ``effectively point-like'' and rather
accurately satisfies the *extrinsic perspective* assumptions: *(i)* for each pixel, the set of 3D points projecting to the pixel
(*i.e.* whose possibly-blurred images are centered on the pixel)
is a straight line in 3D space; and *(ii)* all of the lines meet
at a single 3D point (the *optical center*).

On the other hand, practical lens systems are nonlinear and can easily
introduce significant distortions in the *intrinsic perspective*
mapping from external optical rays to internal pixel coordinates.
This sort of distortion can be corrected by a nonlinear deformation of
the image-plane coordinates.

There are several ways to do this. One method, well known in the
photogrammetry and vision communities, is to explicitly model the
radial and decentering distortion (see [24]): if the
center of the image is
(*u*_{0}, *v*_{0}), the new coordinates (*x*',
*y*') of the corrected point are given by

This linearizes the image geometry to an accuracy that can reach of the image size [1]. The first order radial distortion correction

A more general method that does not require knowledge of the principal point and makes no assumptions about the symmetry of the distortion is based on a fundamental result in projective geometry:

Hence, to correct for distortion, all we need to do is to observe
straight lines in the world and deform the image to make their images
straight. Experiments described in [2] show accuracies of
up to
of the image for standard off-the-shelf CCD
cameras. Figure 1.4 illustrates the process: line
intersections are accurately detected in the image, four of them are
selected to define a projective basis for the plane, and the others
are re-expressed in this frame and perturbed so that they are
accurately aligned. The resulting distortion corrections are then
interpolated across the whole image.

Careful experiments showed that an off-the-shelf ( ) CCD camera with a standard frame-grabber could be stably rectified to a fully projective camera model to an accuracy of 1/20 of a pixel.