カメラ座標

$$ P_c = \left( \begin{array}{c} X_c \ Y_c \ Z_c \end{array} \right) $$

$$ x’ = X_c / Z_c \ y’ = Y_c / Z_c \ r^2 = x’^2 + y’^2 \ \theta = {\rm atan}(r) $$

歪み補正 / Distortion correction

$$ \theta_d = \theta ( 1 + k_1 \theta^2 + k_2 \theta^4 + k_3 \theta^6 + k_4 \theta^8 )\ x’’ = \frac{ \theta_d }{r} x’ \ y’’ = \frac{ \theta_d }{r} y' $$

カメラ座標変換

$$ \left( \begin{array}{c} u \ v \end{array} \right) =

\left( \begin{array}{ccc} f_x & 0 & c_x \ 0 & f_y & c_y \ \end{array} \right)

\left( \begin{array}{c} x’’ \ y’’ \ 1 \end{array} \right) $$

OpenCV

K = np.array([
    [ 1, 0, 320 ],
    [ 0, 1, 240 ],
    [ 0, 0, 1 ],
], dtype=np.float32)
D = np.array([ 1, 0, 0, 0 ], dtype=np.float32)

undistorted_points = np.array([[
    [ 0, 0 ],
    [ 0, 10 ],
    [ 10, 0 ],
    [ 10, 10 ],
]], dtype=np.float32)

distorted_points = cv2.fisheye.distortPoints(undistorted_points, K, D)

distorted_points

array([[[320.     , 240.     ],
        [320.     , 244.65497],
        [324.65497, 240.     ],
        [323.44827, 243.44826]]], dtype=float32)