Sensor Models

Camera

Cameras project a point in space resolved in the camera frame into the image space as a pixel with a set of parameters:

$\mathbf{t}^s_{sx}$ is the world point resolved in the camera frame.

$\mathbf{p}$ is pixel location of that point as observed by the camera.

$\boldsymbol{\beta}$ is a vector of intrinsics parameters.

$\mathbf{P}$ is the projection function.

$$\mathbf{p} = \mathbf{P}\left(\mathbf{t}^s_{sx}, \boldsymbol{\beta}\right)$$ $$\mathbf{t}^s_{sx} = \left[\begin{matrix}t_x\\t_y\\t_z\end{matrix}\right]$$

For the derivation of $\mathbf{t}^s_{sx}$, see here.

5-parameter OpenCv Model

Link to API

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

$f$ = focal length

$c_x, c_y$ = image center

$k_1, k_2, k_3$ = Radial distortion parameters

$p_1, p_2$ = Tangential distortion parameters

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

Kannala-Brandt Model

Link to API

$$\boldsymbol{\beta} = \left[\begin{matrix}f&,&c_x&,&c_y&,&k_1&,&k_2&,&k_3&,&k_4\end{matrix}\right]$$

$f$ = focal length

$c_x, c_y$ = image center

$k_1, k_2, k_3, k_4$ = Radial distortion parameters

$$\mathbf{P}\left(\mathbf{t}^s_{sx_i}, \boldsymbol{\beta}\right) = \left[\begin{matrix}f&0\\0&f\end{matrix}\right]\mathbf{p}_d + \left[\begin{matrix}c_x\\c_y\end{matrix}\right]$$ $$\mathbf{p}_d = \frac{\theta_d}{r}\mathbf{p}_m$$ $$\theta_d = \theta + k_1\theta^3 + k_2\theta^5 + k_3\theta^7 + k_4\theta^9$$ $$\theta = \arctan\left(r\right)$$ $$r^2 = {\mathbf{p}_m}^T\mathbf{p}_m$$ $$\mathbf{p}_m = \left[\begin{matrix}t_x / t_z\\t_y/t_z\end{matrix}\right]$$

Gyroscope

Gyroscopes transduce angular velocity motion to a 3-D measurement vector:

$\boldsymbol{\omega}^s_{ws}$ is the gyroscope's angular velocity w.r.t. the world resolved in the gyroscope frame.

$\mathbf{f}$ is measurement vector produced by the gyroscope.

$\boldsymbol{\beta}$ is a vector of intrinsics parameters.

$\mathbf{F}$ is the transduction function.

$$\mathbf{f} = \mathbf{F}\left(\boldsymbol{\omega}^s_{ws}, \boldsymbol{\beta}\right)$$

For the derivation of $\boldsymbol{\omega}^s_{ws}$, see here.

Isotropic Scale

Link to API

$$\boldsymbol{\beta} = \left[\begin{matrix}s\end{matrix}\right]$$

$s$ = scale

$$\mathbf{F}\left(\boldsymbol{\omega}^s_{ws}, \boldsymbol{\beta}\right) = s\boldsymbol{\omega}^s_{ws}$$

Isotropic Scale and Bias

Link to API

$$\boldsymbol{\beta} = \left[\begin{matrix}s&,&b_x&,&b_y&,&b_z\end{matrix}\right]$$

$s$ = scale

$$\mathbf{F}\left(\boldsymbol{\omega}^s_{ws}, \boldsymbol{\beta}\right) = s\boldsymbol{\omega}^s_{ws} + \left[\begin{matrix}b_x\\b_y\\b_z\end{matrix}\right]$$

Accelerometer

Accelerometers transduce the specific force at the sensor's location referenced to an inertial frame into a 3-D measurement vector:

$\mathbf{p}^s_{ws}$ is the specific force observed by the accelerometer w.r.t. the world resolved in the accelerometer's frame.

$\mathbf{f}$ is measurement vector produced by the gyroscope.

$\boldsymbol{\beta}$ is a vector of intrinsics parameters.

$\mathbf{F}$ is the transduction function.

$$\mathbf{f} = \mathbf{F}\left(\mathbf{p}^s_{ws}, \boldsymbol{\beta}\right)$$

For the derivation of $\mathbf{p}^s_{ws}$, see here.

Isotropic Scale

Link to API

$$\boldsymbol{\beta} = \left[\begin{matrix}s\end{matrix}\right]$$

$s$ = scale

$$\mathbf{F}\left(\mathbf{p}^s_{ws}, \boldsymbol{\beta}\right) = s\mathbf{p}^s_{ws}$$

Isotropic Scale and Bias

Link to API

$$\boldsymbol{\beta} = \left[\begin{matrix}s&,&b_x&,&b_y&,&b_z\end{matrix}\right]$$

$s$ = scale

$$\mathbf{F}\left(\mathbf{p}^s_{ws}, \boldsymbol{\beta}\right) = s\mathbf{p}^s_{ws} + \left[\begin{matrix}b_x\\b_y\\b_z\end{matrix}\right]$$

VectorNav

Link to API

$$\boldsymbol{\beta} = \left[\begin{matrix}s_x&s_y&s_z\end{matrix}\right]$$