Sensor Models

Camera

5-parameter OpenCv Distortion Model

The 5-parameter OpenCV Model has the following parameters

$$\boldsymbol{\beta}=\left[\begin{matrix}f&,&c_x&,&c_y&,&k_1&,&k_2&,&p_1&,&p_2&,&k_3\end{matrix}\right]$$

$f$ = focal length, $c_x, c_y$ = image center

$k_1, k_2, k_3$ = Radial distortion parameters, $p_1, p_2$ = Tangential distortion parameters

The full projection model $\mathbf{P}\left(\mathbf{t}^s_{sx_i}\right)$ is given by:

$$\mathbf{P}\left(\mathbf{t}^s_{sx_i}\right) = \left[\begin{matrix}f & 0 \\ 0 & f \end{matrix}\right]\mathbf{p}_d + \left[\begin{array}{c}c_x\\c_y\end{array}\right]$$ $$\mathbf{t}^s_{sx_i} = \left[\begin{array}{c}t_x\\t_y\\t_z)\end{array}\right]$$ $$\mathbf{p}_d = s\mathbf{p}_m + \left(2\mathbf{p}_m\mathbf{p}_m^T + r^2\mathbf{I}\right)\left[\begin{array}{c}p_2\\p_1\end{array}\right]$$ $$s = 1 + k_1r^2 + k_2r^4 + k_3r^6$$ $$r^2 = \mathbf{p}_m^T\mathbf{p}_m$$ $$\mathbf{p}_m = \left[\begin{array}{c}t_x / t_z\\ t_y / t_z\end{array}\right]$$

Gyroscope

Accelerometer