Geometric calibration of a camera or a stereoscopic vision sensor

The composition of transformations , and may be summarized by the equation shown in Figure 3.

\(\left( \begin{array}{ccc} X & Y & Z \end{array} \right) \overset{\mathbf{T}}{\longrightarrow} \left( \begin{array}{ccc} X_{c} & Y_{c} & Z_{c} \end{array} \right) \overset{\mathbf{P}}{\longrightarrow} \left( \begin{array}{cc} x & y \end{array} \right) \overset{\mathbf{A}}{\longrightarrow} \left( \begin{array}{cc} u & v \end{array} \right)\)

That leads to the equation of the pinhole camera model:

\(\tilde{m} = \underbrace{\mathbf{AP}}_{\mathbf{K}} \mathit{\mathbf{T}} \tilde{M}\)

with:

\(\mathbf{K} = \mathbf{AP} = \left[ \begin{array}{ccc} k_{x} & k_{x} \cot(\theta) & c_{x} + c_{y} \cot(\theta) \\ 0 & \frac{k_{y}}{\sin(\theta)} & \frac{c_{y}}{\sin(\theta)} \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{cccc} f & 0 & 0 & 0 \\ 0 & f & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array} \right] = \left[ \begin{array}{cccc} f_{x} & f_{x} \cot(\theta) & c_{x} + c_{y} \cot(\theta) & 0 \\ 0 & \frac{f_{y}}{\sin(\theta)} & \frac{c_{y}}{\sin(\theta)} & 0 \\ 0 & 0 & 1 & 0 \end{array} \right]\)

where \(f_{x} = f k_{x}\) and \(f_{y} = f k_{y}\) represent the focal length of the camera in pixels along directions \(x\) and \(y\) respectively.

The five parameters \(\left( \begin{array}{ccccc} c_{x} & c_{y} & f_{x} & f_{y} & \theta \end{array} \right)\) of the matrix \(K\) are called intrinsic parameters of the camera.

In the end, the pinhole camera model is described by five intrinsic parameters \(\left( \begin{array}{ccccc} c_{x} & c_{y} & f_{x} & f_{y} & \theta \end{array} \right)\) and six extrinsic parameters (three for the rotation and three for the translation).