Fisheye distortion

カメラ座標

Pc=(XcYcZc)P_c = \left( \begin{array}{c} X_c \\ Y_c \\ Z_c \end{array} \right)
x′=Xc/Zcy′=Yc/Zcr2=x′2+y′2θ=atan(r)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

θd=θ(1+k1θ2+k2θ4+k3θ6+k4θ8)x′′=θdrx′y′′=θdry′\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

(uv)=(fx0cx0fycy)(x′′y′′1)\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)