On the absolute quadratic complex and its application to autocalibration

This article introduces the absolute quadratic complex formed by all lines that intersect the absolute conic. If ω denotes the 3 × 3 symmetric matrix representing the image of that conic under the action of a camera with projection matrix P, it is shown that ω ≈ P^~Ω_{_}P^~T where V is the 3 × 6 line projection matrix associated with P and Ω_{_} is a 6 × 6 symmetric matrix of rank 3 representing the absolute quadratic complex. This simple relation between a camera´s intrinsic parameters, its projection matrix expressed in a projective coordinate frame, and the metric upgrade separating this frame from a metric one - as respectively captured by the matrices ω, P^~ and Ω_{_} - provides a new framework for autocalibration, particularly well suited to typical digital cameras with rectangular or square pixels since the skew and aspect ratio are decoupled from the other intrinsic parameters in ω.