Geometric calibration of a camera or a stereoscopic vision sensor

The third and last transformation, referred as on Figure 2, describes the process that transforms image coordinates \((x,y)\) (in metric units) into discrete image coordinates \((u,v)\) (pixels).

\(\left[ \begin{array}{c} u \\ v \\ 1 \end{array} \right] = \left[ \begin{array}{ccc} k_{x} & k_{x} \cot(\theta) & c_{x} + c_{y} \cot(\theta) \\ 0 & \frac{k_{y}}{\sin(\theta)} & \frac{c_{y}}{\sin(\theta)} \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{c} x \\ y \\ 1 \end{array} \right] = \left[ \mathbf{A} \right] \left[ \begin{array}{c} x \\ y\\ 1 \end{array} \right]\)

where:

• \(c_{x}\) and \(c_{y}\) (in pixels) represent the coordinates of the optical axis intersection with the image plane (situated, in theory, at the center of the image)

• \(k_{x}\) and \(k_{y}\) represent the number of pixels per unit of length along directions \(x\) and \(y\) of the sensor respectively (\(k_{x} = k_{y}\) in the case of square pixels)

• \(\theta\) takes into account the possible non-orthogonality of the lines and the columns in the image. In practice, \(\theta\) is very close from to \(\pi /2\). This parameter is referred as “skew factor”.

It is often considered that the “skew factor” is negligible \(\theta = \frac{\pi}{2}\)\(\) and equation (3) is then simplified as below:

\(\left[ \begin{array}{c} u \\ v \\ 1 \end{array} \right] = \left[ \begin{array}{ccc} k_{x} & 0 & c_{x} \\ 0 & k_{y} & c_{y} \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{c} x \\ y \\ 1 \end{array} \right] = \left[ \mathbf{A_{simplifi\acute{e}e}} \right] \left[ \begin{array}{c} x \\ y \\ 1 \end{array} \right]\)