A Likelihood-Based Triangulation Method for Uncertainties in Through-Water Depth Mapping
Abstract
:1. Introduction
2. Related Works
3. Methodology
3.1. Proposed Likelihood Triangulation
3.1.1. Line-of-Sight Modeling
- The incidence angle , the angle between the feature vector and the nadir , is calculated as follows:
- Using Snell’s Law, the refraction angle when transitioning from water to air is as follows:
- The adjusted vector to refraction, a function of , , and the refractive index n, is computed by applying a rotation in the plane , with angle :
- The incidence point on the interface, where intercepts the water–air interface, is calculated as follows:
- Finally, the line-of-sight vector , the normalized vector from to , is defined as follows:
3.1.2. Camera Pose Statistical Model
Camera Pose Data
Statistical Model
3.1.3. MLE-Based Triangulation
3.2. Uncertainty Evaluation
3.2.1. Profile Likelihood
3.2.2. First-Order Statistical Tests
3.2.3. Evaluation of Confidence Interval Performance
4. Results
4.1. Simulated Experiments
4.2. Water Column Depth Inference
4.2.1. WCD Uncertainties
4.2.2. Evaluation of Uncertainty Metrics
4.3. Water–Air Interface Height Inference
5. Discussion
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
WCD | Water Column Depth |
WAI | Water Air Interface |
GCP | Ground Control Points |
SfM | Structure from Motion |
MVS | Multi-View Stereo |
RPC | Rational Polynomial Coefficients |
FoV | Field of View |
IFoV | Instantaneous Field of View |
SDB | Satellite-Derived Bathymetry |
GNSS | Global Navigation Satellite System |
INS | Inertial Navigation System |
MLE | Maximum Likelihood Estimation |
LS | Least Square |
CI | Confidence Intervals |
CL | Confidence Level |
Appendix A. Equivalence Between the Wald Test Based on the Expected Fisher Information and the Variance–Covariance Propagation Under Gaussian Errors
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Symbol | Description |
---|---|
Parameter underlying the data (multidimensional) | |
True value of the parameter | |
Parameter of interest (scalar) | |
Nuisance vector parameter (multidimensional) | |
Observed data (multidimensional) | |
Likelihood function | |
Log-likelihood function | |
Profile log-likelihood function | |
Second derivative of the log likelihood with respect to | |
Maximum likelihood estimate of given | |
Partial observed Fisher information | |
Wald statistic (observed) | |
Signed root likelihood ratio statistic | |
, | Significance level, confidence level (CL) |
Interior orientation vector | |
Feature vector | |
Feature point position | |
Incidence point position | |
Line-of-sight vector | |
Line-of-sight quaternion | |
h | Water–air interface height parameter |
n | Refractive index ratio |
, | Incidence angle and refracted angle |
Backward refraction angle | |
Backward line-of-sight vector | |
Water Column Depth (WCD) | |
Nadir vector | |
Parameter of camera position vector | |
Measured camera position vector | |
Variance–covariance matrix of camera position | |
Measured line-of-sight quaternion | |
Bingham orientation parameter | |
Bingham concentration parameter |
Feature Point | Base–Height Ratio (B/H) | Viewing Scenario |
---|---|---|
0.36 | Crossing lines | |
0.36 | Parallel lines | |
0.72 | Parallel lines |
Camera Pose Quality | Position Noise | Attitude Noise | |
---|---|---|---|
Rolling/Pitch | Yaw | ||
Fair | 0.5 m | 0.1° | 0.1° |
Good | 0.05 m | 0.01° | 0.1° |
Excellent | 0.05 m | 0.01° | 0.01° |
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Share and Cite
Ghannami, M.A.; Daniel, S.; Sicot, G.; Quidu, I. A Likelihood-Based Triangulation Method for Uncertainties in Through-Water Depth Mapping. Remote Sens. 2024, 16, 4098. https://doi.org/10.3390/rs16214098
Ghannami MA, Daniel S, Sicot G, Quidu I. A Likelihood-Based Triangulation Method for Uncertainties in Through-Water Depth Mapping. Remote Sensing. 2024; 16(21):4098. https://doi.org/10.3390/rs16214098
Chicago/Turabian StyleGhannami, Mohamed Ali, Sylvie Daniel, Guillaume Sicot, and Isabelle Quidu. 2024. "A Likelihood-Based Triangulation Method for Uncertainties in Through-Water Depth Mapping" Remote Sensing 16, no. 21: 4098. https://doi.org/10.3390/rs16214098
APA StyleGhannami, M. A., Daniel, S., Sicot, G., & Quidu, I. (2024). A Likelihood-Based Triangulation Method for Uncertainties in Through-Water Depth Mapping. Remote Sensing, 16(21), 4098. https://doi.org/10.3390/rs16214098