Wenqi Liu,1
Zhen Zhang,1
Yonggang Gu,2,*
and Chao Zhai2,3
1Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, 443# Huangshan Road, Hefei 230027, Anhui Province, China
2Experiment Center of Engineering and Material Science, University of Science and Technology of China, 443# Huangshan Road, Hefei 230027, Anhui Province, China
The perspective camera model has difficulty handling refracted light in the underwater environment. To achieve accurate and convenient calibration in large underwater scenes, we propose a method based on the underwater refractive camera model in this paper. First, the initial values of the refraction parameters are solved using refraction coplanarity constraints. Then the initial values are optimized nonlinearly using co-point constraints, which simplifies the optimization process of existing methods. In the field of view of ${200}\;{\rm mm} \times {200}\;{\rm mm}$, the experiment results show that the reconstruction accuracy of the proposed method can reach below 0.02 mm, and it is equally effective in the case of sparse calibration.
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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Most of the calculations in this paper are performed in the world coordinate system.
,
Rotation and translation matrix from the world coordinate system to the camera coordinate system.
, ,
Refractive indices of air, glass, and water, respectively.
,
Vertical distance from the camera’s optical center to the refractive surface between glass and air, and glass thickness, respectively.
Unit vector indicating the direction of the normal to the refractive surface, generally pointing to the camera side.
Object point with known coordinate values under the body coordinate system.
, ,
Direction vectors of refracted rays in air, glass, and water, respectively.
,
Intersection points of refracted rays in air–glass and glass–water refracting surfaces, respectively.
Superscript $l$ is associated with the left camera, and superscript $r$ is associated with the right camera, which applies to all symbols and is not repeated in the following.
Table 2.
Experimental Results of Calibration Using 2025 and 49 Corresponding Pointsa
2025 Corresponding Points
49 Corresponding Points
Calibration Result
Our Method
Chen’s Method
Our Method
Chen’s Method
[−0.0519; 0.0133; 0.9986]
[−0.0512; 0.0142; 0.9986]
[−0.0522; 0.0148; 0.9985]
[−0.05415; 0.0248; 0.9982]
0.00°
0.00°
1.93°
99.7360
101.2611
101.8767
146.921
0.0152
0.0212
0.3826
97.7220
99.7428
99.4802
146.3385
0.0205
0.0178
0.3984
msd
−0.0008
−0.0008
−0.0003
0.0054
0.0084
0.0082
0.0084
0.0143
acc
0.0178
0.0174
0.0178
0.0303
time
0.7087
32.3075
0.5067
30.3063
When calculating $\Delta n$, $\Delta d_0^l$, and $\Delta d_0^r$, the data in the second column are used as reference values; time is in seconds; $d_0^l$, $d_0^r$, msd, $s$, acc are in millimeters.
Values are in millimeters.
A-B indicates that the value in that row is the spacing between planes A and B, and so on.
Case 1 indicates the calibration results calculated by our method with 2025 points, Case 2 indicates the calibration results calculated by Chen’s method on 2025 points, Case 3 indicates the calibration results calculated by our method with 49 points, and Case 4 indicates the calibration results calculated by Chen’s method with 49 points.
GT indicates ground truth.
Most of the calculations in this paper are performed in the world coordinate system.
,
Rotation and translation matrix from the world coordinate system to the camera coordinate system.
, ,
Refractive indices of air, glass, and water, respectively.
,
Vertical distance from the camera’s optical center to the refractive surface between glass and air, and glass thickness, respectively.
Unit vector indicating the direction of the normal to the refractive surface, generally pointing to the camera side.
Object point with known coordinate values under the body coordinate system.
, ,
Direction vectors of refracted rays in air, glass, and water, respectively.
,
Intersection points of refracted rays in air–glass and glass–water refracting surfaces, respectively.
Superscript $l$ is associated with the left camera, and superscript $r$ is associated with the right camera, which applies to all symbols and is not repeated in the following.
Table 2.
Experimental Results of Calibration Using 2025 and 49 Corresponding Pointsa
2025 Corresponding Points
49 Corresponding Points
Calibration Result
Our Method
Chen’s Method
Our Method
Chen’s Method
[−0.0519; 0.0133; 0.9986]
[−0.0512; 0.0142; 0.9986]
[−0.0522; 0.0148; 0.9985]
[−0.05415; 0.0248; 0.9982]
0.00°
0.00°
1.93°
99.7360
101.2611
101.8767
146.921
0.0152
0.0212
0.3826
97.7220
99.7428
99.4802
146.3385
0.0205
0.0178
0.3984
msd
−0.0008
−0.0008
−0.0003
0.0054
0.0084
0.0082
0.0084
0.0143
acc
0.0178
0.0174
0.0178
0.0303
time
0.7087
32.3075
0.5067
30.3063
When calculating $\Delta n$, $\Delta d_0^l$, and $\Delta d_0^r$, the data in the second column are used as reference values; time is in seconds; $d_0^l$, $d_0^r$, msd, $s$, acc are in millimeters.
Values are in millimeters.
A-B indicates that the value in that row is the spacing between planes A and B, and so on.
Case 1 indicates the calibration results calculated by our method with 2025 points, Case 2 indicates the calibration results calculated by Chen’s method on 2025 points, Case 3 indicates the calibration results calculated by our method with 49 points, and Case 4 indicates the calibration results calculated by Chen’s method with 49 points.
GT indicates ground truth.
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