Multiple camera system models are defined as a set of projective camera models. One camera is chosen to be the reference camera. The extrinsics of each camera can be factored into two parts: the transform from the camera’s coordinate system to the reference camera’s coordinate system and the reference camera’s coordinate system to the world coordinate system. Using this notation, a rigid \(n\)-camera system may be described by a single world to system set of extrinsic parameters and \(n-1\) fixed sets of extrinsics parameters describing the relative position of the cameras.

Imatest Calibration Assumptions

When performing a calibration in imatest, the system of cameras is assumed to be rigid. In a rigid camera system, the position and orientation of the cameras are fixed relative to each other, while the system as a whole is allowed to move.

Example

Let \(\mathbf{P}_1\) be the reference camera model and let \(\mathbf{P}_2\) and \(\mathbf{P}_3\) be other cameras in the camera system. The projective camera matrices for these

\(\mathbf{P}_1=\left[\begin{array}{ccc}&&\\&\mathbf{K}_1&\\&&\end{array}\right]\left[\begin{array}{ccc|c}&&&\\&\mathbf{R}_{W\rightarrow1}&&\mathbf{t}_{W\rightarrow1}\\&&&\end{array}\right]=\left[\begin{array}{ccc}&&\\&\mathbf{K}_1&\\&&\end{array}\right]\left[\begin{array}{ccc|c}&&&\\&\mathbf{I}&&\mathbf{0}\\&&&\end{array}\right]\left[\begin{array}{ccc|c}&&&\\&\mathbf{R}_{W\rightarrow1}&&\mathbf{t}_{W\rightarrow1}\\&&&\end{array}\right]\)

\(\mathbf{P}_2=\left[\begin{array}{ccc}&&\\&\mathbf{K}_2&\\&&\end{array}\right]\left[\begin{array}{ccc|c}&&&\\&\mathbf{R}_{W\rightarrow2}&&\mathbf{t}_{W\rightarrow2}\\&&&\end{array}\right]=\left[\begin{array}{ccc}&&\\&\mathbf{K}_2&\\&&\end{array}\right]\left[\begin{array}{ccc|c}&&&\\&\mathbf{R}_{1\rightarrow2}&&\mathbf{t}_{1\rightarrow2}\\&&&\end{array}\right]\left[\begin{array}{ccc|c}&&&\\&\mathbf{R}_{W\rightarrow1}&&\mathbf{t}_{W\rightarrow1}\\&&&\end{array}\right]\)

\(\mathbf{P}_3=\left[\begin{array}{ccc}&&\\&\mathbf{K}_3&\\&&\end{array}\right]\left[\begin{array}{ccc|c}&&&\\&\mathbf{R}_{W\rightarrow3}&&\mathbf{t}_{W\rightarrow3}\\&&&\end{array}\right]=\left[\begin{array}{ccc}&&\\&\mathbf{K}_2&\\&&\end{array}\right]\left[\begin{array}{ccc|c}&&&\\&\mathbf{R}_{1\rightarrow3}&&\mathbf{t}_{1\rightarrow3}\\&&&\end{array}\right]\left[\begin{array}{ccc|c}&&&\\&\mathbf{R}_{W\rightarrow1}&&\mathbf{t}_{W\rightarrow1}\\&&&\end{array}\right]\)

In this case, 

\(\left[\begin{array}{ccc|c}&&&\\&\mathbf{R}_{a\rightarrow b}&&\mathbf{t}_{a\rightarrow b}\\&&&\end{array}\right]\)

is the transformation of points in coordinate system \(a\) to points coordinate system \(b\), and \(W\) represents a world coordinate system.