图像点畸变矫正

畸变是发生在世界点通过透镜投影到相机坐标系这个过程中的，所以去畸变操作是在相机坐标系下进行的，一般去畸变流程为：

像素点通过内参矩阵转换到归一化相机平面，基本相机模型为：

\begin{matrix} K = \begin{bmatrix} f_x&0&c_x\\0&f_y&c_y\\0&0&1 \end{bmatrix}P_n\\ P_n = K^{-1} P_{uv} \end{matrix}

在归一化相机平面通过畸变模型去畸变。

Radial-Tangential

畸变矫正实现

\begin{bmatrix} x' \\ y' \end{bmatrix} = \left[ \begin{array}{l} x \frac{(1+K_1r^2+K_2r^4+K_3r^6)}{(1+K_4r^2+K_5r^4+K_6r^6)} + 2P_1xy + P_2(r^2+2x^2) + S_1r^2+S_3r^4 \\ y \frac{(1+K_1r^2+K_2r^4+K_3r^6)}{(1+K_4r^2+K_5r^4+K_6r^6)} + 2P_2xy + P_1(r^2+2y^2) + S_2r^2+S_4r^4 \end{array} \right]

上式为opencv支持的对归一化平面上的点进行畸变校正的过程，包含三种畸变：

径向畸变：[k1 k2 k3]基础的径向畸变多项式系数，[k4 k5 k6]高阶径向畸变多项式系数，一般用于广角镜头
切向畸变：[p1 p2]主要的切向畸变系数，[ $\tau_x$ $\tau_y$]是为兼容内部模型而引入的切向畸变系数，有时也被称为倾斜畸变（Tilting Distortion），它们提供了对切向畸变更精细的描述，尤其是在需要高精度标定的场合，该模型也被称为薄棱镜畸变模型的一部分，但其数学形式与s系数不同。
薄棱镜畸变：[s1 s2 s3 s4]

一般支持参数类型为

[k1 k2 p1 p2]
[k1 k2 p1 p2 k3]
[k1 k2 p1 p2 k3 k4 k5 k6]
[k1 k2 p1 p2 k3 k4 k5 k6 s1 s2 s3 s4]
[k1 k2 p1 p2 k3 k4 k5 k6 s1 s2 s3 s4 $\tau_x$ $\tau_y$]

Equidistant

Kannala-Brandt, Radially Symmetric Model, opencv中的fisheye, kalibr中的 pinhole + equidistant 都是指该模型

畸变矫正实现

\begin{matrix} r^2 = x^2 + y^2\\ \theta = arctan(r)\\ \theta _d = \theta(k_0 + 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 \end{matrix}

代码

CamBase

测试结果

内置算法

============================================================
Radtan Model Test
============================================================
Radtan (Internal) Correctness Test:
Original Point:      (0.3, -0.2)
Distorted Pixel:     (772.204, 271.3)
Final Undistorted:   (0.3, -0.2)
Round-trip Error:    2.271766e-08
[SUCCESS] Low error.

Radtan (Internal) Speed Test (100000 points):
distort() time:   0.295156 ms
undistort() time: 2.670738 ms

============================================================
Equi (Fisheye) Model Test
============================================================
Equi (Internal) Correctness Test:
Original Point:      (0.300000, -0.200000)
Distorted Pixel:     (732.269954, 298.525131)
Final Undistorted:   (0.300000, -0.200000)
Round-trip Error:    1.494683e-16
[SUCCESS] Low error.

Equi (Internal) Speed Test (100000 points):
distort() time:   1.358781 ms
undistort() time: 7.232002 ms

opencv

============================================================
Radtan Model Test
============================================================
Radtan (OpenCV) Correctness Test:
Original Point:      (0.3, -0.2)
Distorted Pixel:     (772.204, 271.3)
Final Undistorted:   (0.3, -0.2)
Round-trip Error:    2.271766e-08
[SUCCESS] Low error.

Radtan (OpenCV) Speed Test (100000 points):
distort() time:   0.297319 ms
undistort() time: 54.243528 ms

============================================================
Equi (Fisheye) Model Test
============================================================
Equi (OpenCV) Correctness Test:
Original Point:      (0.300000, -0.200000)
Distorted Pixel:     (732.269954, 298.525131)
Final Undistorted:   (0.300000, -0.200000)
Round-trip Error:    1.494683e-16
[SUCCESS] Low error.

Equi (OpenCV) Speed Test (100000 points):
distort() time:   1.251700 ms
undistort() time: 37.757270 ms

图像点畸变矫正

Radial-Tangential

Equidistant

代码

CATALOG

FEATURED TAGS