# Multiple Camera Models

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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.