Geometric calibration of a camera or a stereoscopic vision sensor

As indicated on Figure 2, represents a transformation between the world reference frame \(R_{w}\) (arbitrarily chosen) and the camera reference frame \(R_{c}\) (which origin is located in the optical center of the camera). This rigid transformation consists of a rotation \([R]\) and a translation \([t]\). The parameters of this transformation are called extrinsic parameters of the camera.

\(\left[ \begin{array}{c} X_{c} \\ Y_{c} \\ Z_{c} \\ 1 \end{array} \right] = \left[ \mathbf{R} \right] \left[ \begin{array}{c} X \\ Y \\ Z \\ 1 \end{array} \right] + \mathbf{t} = \left[ \begin{array}{cc} \mathbf{R} & t \\ \mathbf{0^{t}} & 1 \end{array} \right] \left[ \begin{array}{c} X \\ Y \\ Z \\ 1 \end{array} \right] = \left[ \mathbf{T} \right] \left[ \begin{array}{c} X \\ Y \\ Z \\ 1 \end{array} \right]\)

with:

\(\mathbf{t} = \left[ \begin{array}{c} t_{x} \\ t_{y} \\ t_{z}  \end{array} \right] ; \left[ \mathbf{R} \right] = \left[ \begin{array}{ccc} r_{11} & r_{12} &r_{13} \\ r_{21} & r_{22} & r_{23} \\ r_{31} & r_{32} & r_{33}\end{array} \right]\)

\(T\) is a \(4 \times 4\) matrix.