Inclinometer Assembly Error Calibration and Horizontal Image Correction in Photoelectric Measurement Systems

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. 2018 Jan 16;18(1):248. doi: 10.3390/s18010248

Algorithm 1: Horizontal image correction using the inclinometer.

Rotate the camera coordinate system to the inclinometer coordinate system using the rotation matrix $R_{i c}$ in Equation (1);
Rotate the inclinometer coordinate system to the geodetic coordinate system according to the direction cosine matrix $R_{g i}$ in Equation (6);
Rotate the geodetic coordinate system to the horizontal camera coordinate system by rotating 90° around the $X_{g}$ -axis, the rotation matrix $R_{h}$ can be indicated as
$R_{h} = [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos 90^{\circ} & - \sin 90^{\circ} \\ 0 & \sin 90^{\circ} & \cos 90^{\circ} \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{matrix}] .$ (7)
According to homography matrix theory in multiple-view geometry [9], the homography matrix $H_{12}$ between two images can be simplified as
$H_{12} = K R_{12} K^{- 1},$ (8)
where $K$ is the intrinsic parameter matrix of the camera, which can be expressed as
$K = [\begin{matrix} f_{x} & 0 & c_{x} \\ 0 & f_{y} & c_{y} \\ 0 & 0 & 1 \end{matrix}],$ (9)
where $f_{x}$ and $f_{y}$ are focal lengths in $x$ and $y$ directions, and $c_{x}$ and $c_{y}$ are principal points.

$R_{12}$ is the rotation matrix from image 1 to image 2
$R_{12} = R_{h} R_{g i} R_{i c} = R_{h} R_{z} (α) R_{i} (θ, ϕ) R_{i c} .$ (10)

Substitute Equation (10) into Equation (8); the homography matrix $H_{12}$ can be written as
$H_{12} = K R_{12} K^{- 1} = K R_{h} R_{z} (α) R_{i} (θ, ϕ) R_{i c} K^{- 1} .$ (11)

Substitute Equations (7) and (9) into Equation (11); we have
$H_{12} = [\begin{matrix} f_{x} & 0 & c_{x} \\ 0 & f_{y} & c_{y} \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{matrix}] [\begin{matrix} \cos α & - \sin α & 0 \\ \sin α & \cos α & 0 \\ 0 & 0 & 1 \end{matrix}] R_{i} (θ, ϕ) R_{i c} K^{- 1} .$ (12)

$= [\begin{matrix} f_{x} \cos α + c_{x} \sin α & - f_{x} \sin α + c_{x} \cos α & 0 \\ c_{y} \sin α & c_{y} \cos α & - f_{y} \\ \sin α & \cos α & 0 \end{matrix}] R_{i} (θ, ϕ) R_{i c} K^{- 1} .$ (13)

Finally, the horizontal corrected image can be obtained by multiplying the homography matrix $H_{12}$ with the original image.