Geometric calibration of a camera or a stereoscopic vision sensor

The tridimensional space of the scene is fitted with its orthonormal reference frame \(R_{w}\). Each of the two cameras has its own orthonormal reference frame: we will call them left camera reference frame \(R_{c}\) and right camera reference frame \(R'_{c}\). Figure 12 illustrates those three reference frames as well as the rigid transformations allowing the expression of a point in another reference frame.

With those conventions, we can write the following relations down:

\(\begin{align*} \tilde{M_{c}} &\cong \mathbf{T} \tilde{M} \\ \tilde{M'_{c}} &\cong \mathbf{T'} \tilde{M} \\ \tilde{M'_{c}} &\cong \mathbf{T}_{s} \tilde{M}_{c} \end{align*}\)

These equations show us that the three transformations are not independent since we can determine one of them by using the two others:

\(\mathbf{T} \cong \mathbf{T}_{s}^{-1} \mathbf{T'}\)

\(\mathbf{T'} \cong \mathbf{T}_{s} \mathbf{T}\)

\(\mathbf{T}_{s} \cong \mathbf{T'} \mathbf{T}^{-1}\)

When a point \(M\) of the scene is simultaneously visible by both cameras, it gives us two points: \(m\) for the left camera and \(m'\) for the right one. Using the geometric model of the camera and the relation of dependence between the three reference frames \(R_{w} , R_{c}\) and \(R'_{c}\), we can write the relations of \(m\) and \(m'\) according to \(M\):

\(\tilde{m} \cong \mathbf{KT} \tilde{M}\)

\(\tilde{m'} \cong \mathbf{K'} \mathbf{T'} \tilde{M} \cong \mathbf{K'}\mathbf{T}_{s} \mathbf{T} \tilde{M}\)