Camera Calibration Problem

The result of camera calibration is an explicit transformation that maps a 3D world point M=(X,Y,Z,1)^T into a 2D pixel m=(u,v,1)^T. This mapping can be represented by a 3 x 4  projection matrix, P, that encompasses 11 physical parameters: rotation angles R_x, R_y and R_z, translations t_x, t_y and t_z, the coordinates of the principal point (u₀,v₀), two scale factors $\alpha _u$ and $\alpha_v$, and the skewness c between the image axes. This camera model thus ignores lens distortion which is often accounted for in the camera model by adding some distortion parameters [12]. However, these parameters can be estimated in the captured images by a pre-calibration process [9],[6]. Then the images (or image features) can be undistorted before calibration proceeds. The decoupling between distortion parameters from the others will allow us to maintain the simple relation of the distortion-free model thus making subsequent vision tasks (e.g., stereo reconstruction) easier. Moreover, the decoupling would reduce the effect of the correlation between lens distortion coefficients and other camera model parameters [8] on parameter estimation.

Given a sufficient number, N, of reference world points, M_i, as well as their corresponding pixel positions, m_i, the camera calibration problem is to estimate the 11 camera parameters or the projection matrix P, that minimize

$$E = \sum_{i=1}^N \parallel \textbf{P} ~\textbf{M}_i - \textbf{m}_i \parallel^2.$$ (1)

However, since the camera calibration parameters may vary as the lens setting is changed, the calibration problem of a zoom-lens camera system becomes finding the intrinsic and extrinsic camera parameters, expressed as functions of the controllable camera settings, which can be composed for any fixed camera setting in order to obtain the projection matrix.