A Fast Calibration Method for Underwater Camera Intrinsic Parameters Based on Refraction Model
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摘要: 针对水下视觉作业时相机内参不准造成的作业精度较低的问题, 提出了一种水下相机快速标定方法。该方法仅需1张包含2组互相正交平行线的图片, 通过求解平行线在图像平面的消失点, 构建与等效焦距相关的正交关系, 从而求解相机的内参。在处理水下成像畸变问题时, 通过考虑二阶径向畸变模型, 并以最小重投影误差为优化目标, 求解畸变系数, 以实现水下图像的畸变校准, 提高内参标定精度。进一步, 通过对比空气中图像和水下目标等效空气图像证明了该方法还原的等效图像具有较高的精度。实验结果表明, 文中所提方法操作简便, 显著降低了相机标定时现场环境的要求, 且能在保证一定精度的同时, 有效提升内参标定速度, 适用于水下相机的标定任务。Abstract: To address the low operational precision caused by inaccurate camera intrinsic parameters in underwater visual tasks, a fast calibration method for underwater cameras is proposed. A single image containing two sets of mutually orthogonal parallel lines is used in the proposed method. By solving the vanishing points of these lines on the image plane, an orthogonal relationship related to the equivalent focal length is established. This enables the intrinsic parameters of the camera to be determined. To address the underwater imaging distortion problem, the distortion coefficients are solved using the second-order radial distortion model with the minimum reprojection error as the optimization objective, thereby achieving the distortion calibration of underwater images and improving the accuracy of intrinsic parameter calibration. Furthermore, the accuracy of the method in restoring images is demonstrated by comparing in-air target images with their equivalent air images of underwater targets. Experimental results indicate that the proposed method is simple to operate, significantly reduces environmental requirements during camera calibration, effectively enhances calibration speed while maintaining a certain level of accuracy, and is suitable for calibration tasks of underwater cameras.
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图 4 基于消失点求焦距的几何示意图
Figure 4. Geometric schematic for finding the focal length based on the vanishing point
图 10 不同位姿的水下标定板图像
Figure 10. Underwater calibration board images for different positions
表 1 基于消失点的水下快速标定结果
Table 1. Underwater rapid calibration results based on vanishing point
参数 计算结果 消失点${W_I}$/px $\left( { - 10442.71, - 291.76} \right)$ 消失点${W_J}$/px $\left( {1763.43, - 5518.66} \right)$ 等效焦距${f_w}$/mm $5.054$ 内参矩阵${\boldsymbol{K}}$ $\left[ {\begin{array}{*{20}{c}} {1347.75}&0&{1024} \\ 0&{1347.75}&{768} \\ 0&0&1 \end{array}} \right]$ 重投影误差/px $0.177$ 计算时间/s $0.0084$ 
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表 2 基于消失点的水下快速标定优化结果
Table 2. Optimization results of underwater fast calibration based on vanishing point
参数 计算结果 畸变系数${\eta _1},{\eta _2}$ $ 0.2126,0.6465 $ 像平面中心点 $\left( {1020.97,763.23} \right)$ 消失点${W_I}$/px $\left( { - 19700.14, - 1150.59} \right)$ 消失点${W_J}$/px $ \left( {1621.54, - 4291.55} \right) $ 等效焦距${f_w}$/mm $6.05$ 内参矩阵${\boldsymbol{K}}$ $\left[ {\begin{array}{*{20}{c}} {1691.7}&0&{1020.97} \\ 0&{1691.7}&{766.23} \\ 0&0&1 \end{array}} \right]$ 重投影误差/px $0.089$ 计算时间/s $1.5808$
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表 3 不同图像的标定结果
Table 3. Calibration results for different images
图片编号 等效焦距/mm 重投影误差$\left( {pixel} \right)$ 计算时间/s (a) 6.1426 0.0250 2.58 (b) 6.3126 0.0598 1.94 (c) 6.4598 0.0823 1.14 (d) 6.0965 0.0249 2.75 平均 6.2529 0.048 2.10
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表 4 基于消失点的空气中快速标定优化结果
Table 4. Optimization results for fast calibration in air based on vanishing point
参数 计算结果 畸变系数 $\left( {0.0015,0.0017} \right)$ 像平面中心点 $\left( {1022.15,767.53} \right)$ 等效焦距${f_a}$/mm $ {\text{4}}{\text{.64}} $ 内参矩阵${\boldsymbol{K}}$ $ \left[ {\begin{array}{*{20}{c}} {1237.1}&0&{1022.15} \\ 0&{1237.1}&{767.53} \\ 0&0&1 \end{array}} \right] $ 重投影误差/px $ {\text{0}}{\text{.067}} $
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表 5 特征点像素值
Table 5. Feature point pixel value
特征点 像素坐标/px $A$ $\left( {811.61,550.41} \right)$ $B$ $\left( {1284.39,608.70} \right)$ $C$ $\left( {780.87,898.99} \right)$ $D$ $\left( {1247.80,928.53} \right)$ $A'$ $\left( {812.91,553.81} \right)$ $B'$ $\left( {1292.70,610.01} \right)$ $C'$ $\left( {781.48,907.39} \right)$ $D'$ $\left( {1255.83,938.12} \right)$
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表 6 等效图像与空气中图像相对误差
Table 6. Relative error between the equivalent image and the image in air
等效图像特征点 X方向相对误差/% Y方向相对误差/% $A'$ 0.162 0.616 $B'$ 0.645 0.215 $C'$ 0.001 0.934 $D'$ 0.643 1.032
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