# Robust Filter-Based Visual Navigation Solution with Miscalibrated Bi-Monocular or Stereo Cameras

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## Abstract

**:**

## 1. Introduction

- A loss of depth estimation coverage: feature matching accuracy is impacted by the wrong calibration, making the 3D environment reconstruction sparse with potential obstacles’ misdetection;
- A loss of accuracy almost impossible to detect during the actual mission. This implies the wrong depth estimation, and so the wrong localization and 3D reconstruction.

## 2. Related Works

## 3. Mitigation of Stereo Calibration Errors

#### 3.1. Background on Linearly Constrained EKF

#### 3.2. Stereo-Based Visual Navigation

#### 3.3. LCEKF Customization for Stereo-Based Visual Navigation

## 4. Experiments

- A standard EKF-based navigation that supposes the given calibration parameters $\mathit{\theta}$ to be good (denoted EKF);
- A standard EKF-based navigation that contains the 6 parameters of $\mathit{\theta}$ in its state vector such that the recalibration is performed on-line (denoted EKFfull).

#### 4.1. State/Measurement Models

#### 4.2. Noises and Miscalibration/Decalibration Types

- (i)
- Reference case: $d\widehat{\mathit{\theta}}=\mathbf{0}$;
- (ii)
- Constant noise: $d\widehat{\mathit{\theta}}=\mathit{\delta}$. This represents the use of the wrong initial calibration, but with no stress during the experiment;
- (iii)
- Sinusoidal noise: $d\widehat{\mathit{\theta}}=\mathit{\delta}sin(t)$. This case represents for example the day/night thermal constraint encountered for planetary exploration rovers;
- (iv)
- Discontinuous noise: $d\widehat{\mathit{\theta}}=\mathit{\delta}H(t)$. This case represents a chock or accident encountered while navigating ($H(\xb7)$ can be a combination of Heaviside functions);
- (v)
- Periodic noise: $d\widehat{\mathit{\theta}}=\mathit{\delta}\mathsf{\Pi}(t)$; This case simulated a strong vibration of the system for a non-perfectly rigid stereo system ($\mathsf{\Pi}(\xb7)$ is a periodic square wave).

- For the IMU: ${\mathbf{Q}}_{acc}={(0.1dt)}^{2}{\mathbf{I}}_{3}$; ${\mathbf{Q}}_{gyr}={(\frac{0.5\pi}{180}dt)}^{2}{\mathbf{I}}_{3}$;
- For the 2D detections: ${\mathbf{P}}_{z}={(0.5)}^{2}{\mathbf{I}}_{2}$.

#### 4.3. SLAM Experiment with IMU

#### 4.3.1. Reference Case Comparison (i)

#### 4.3.2. Noisy Cases (ii), (iii), and (iv)

#### 4.3.3. Periodic Noise Case (v)

#### 4.4. SLAM Experiment Comparison with and without the IMU

#### 4.5. On the Influence of Kalman Covariance Initialization for Calibration Parameters

#### Conclusions

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Szczepanski, M. Online Stereo Camera Calibration on Embedded Systems. Ph.D. Thesis, Université Clermont Auvergne, Clermont-Ferrand, France, 2019. [Google Scholar]
- Blume, H. Towards a Common Software/Hardware Methodology for Future Advanced Driver Assistance Systems; River Publishers: Roma, Italy, 2017. [Google Scholar]
- Longuet-Higgins, H.C. A computer algorithm for reconstructing a scene from two projections. Nature
**1981**, 293, 133–135. [Google Scholar] [CrossRef] - Hartley, R. In Defense of the 8-Point Algorithm. IEEE Trans. Pattern Anal. Mach. Intell.
**1997**, 19, 580–593. [Google Scholar] [CrossRef][Green Version] - Nistér, D. An efficient solution to the five-point relative pose problem. IEEE Trans. Pattern Anal. Mach. Intell.
**2004**, 26, 756–770. [Google Scholar] [CrossRef] [PubMed] - Zhang, Z.; Luong, Q.T.; Faugeras, O. Motion of an uncalibrated stereo rig: Self-calibration and metric reconstruction. IEEE Trans. Robot. Autom.
**1996**, 12, 103–113. [Google Scholar] [CrossRef] - Zhang, Z. A flexible new technique for camera calibration. IEEE Trans. Pattern Anal. Mach. Intell.
**2000**, 22, 1330–1334. [Google Scholar] [CrossRef][Green Version] - Takahashi, H.; Tomita, F. Self-Calibration of Stereo Cameras; ICCV: Vienna, Austria, 1988. [Google Scholar]
- Sun, P.; Lu, N.G.; Dong, M.L.; Yan, B.X.; Wang, J. Simultaneous all-parameters calibration and assessment of a stereo camera pair using a scale bar. Sensors
**2018**, 18, 3964. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dang, T.; Hoffmann, C. Stereo calibration in vehicles. In Proceedings of the IEEE Intelligent Vehicles Symposium, Parma, Italy, 14–17 June 2004. [Google Scholar]
- Dang, T.; Hoffmann, C.; Stiller, C. Self-calibration for active automotive stereo vision. In Proceedings of the IEEE Intelligent Vehicles Symposium, Meguro-Ku, Japan, 13–15 June 2006. [Google Scholar]
- Dang, T.; Hoffmann, C.; Stiller, C. Continuous Stereo Self-Calibration by Camera Parameter Tracking. IEEE Trans. Image Process.
**2009**, 18, 1536–1550. [Google Scholar] [CrossRef] [PubMed] - Mueller, G.R.; Wuensche, H.J. Continuous extrinsic online calibration for stereo cameras. In Proceedings of the IEEE Intelligent Vehicles Symposium, Gothenburg, Sweden, 19–22 June 2016. [Google Scholar]
- Mueller, G.R.; Wuensche, H.J. Continuous stereo camera calibration in urban scenarios. In Proceedings of the IEEE International Conference on Intelligent Transportation, Yokohama, Japan, 16–19 October 2017. [Google Scholar]
- Collado, J.M.; Hilario, C.; de la Escalera, A.; Armingol, J.M. Self-calibration of an on-board stereo-vision system for driver assistance systems. In Proceedings of the IEEE Intelligent Vehicles Symposium, Gothenburg, Sweden, 19–22 June 2016. [Google Scholar]
- De Paula, M.B.; Jung, C.R.; da Silveira, L., Jr. Automatic on-the-fly extrinsic camera calibration of onboard vehicular cameras. Expert Syst. Appl.
**2014**, 41, 1997–2007. [Google Scholar] [CrossRef] - Musleh, B.; Martín, D.; Armingol, J.M.; De la Escalera, A. Pose self-calibration of stereo vision systems for autonomous vehicle applications. Sensors
**2016**, 16, 1492. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rehder, E.; Kinzig, C.; Bender, P.; Lauer, M. Online stereo camera calibration from scratch. In Proceedings of the IEEE Intelligent Vehicles Symposium, Los Angeles, CA, USA, 11–14 June 2017. [Google Scholar]
- Häne, C.; Heng, L.; Lee, G.H.; Fraundorfer, F.; Furgale, P.; Sattler, T.; Pollefeys, M. 3D visual perception for self-driving cars using a multi-camera system: Calibration, mapping, localization, and obstacle detection. Image Vis. Comput.
**2017**, 68, 14–27. [Google Scholar] [CrossRef][Green Version] - Muhovič, J.; Perš, J. Correcting Decalibration of Stereo Cameras in Self-Driving Vehicles. Sensors
**2020**, 20, 3241. [Google Scholar] [CrossRef] [PubMed] - Konolige, K. Small vision systems: Hardware and implementation. In Robotics Research; Springer: Berlin/Heidelberg, Germany, 1998; pp. 203–212. [Google Scholar]
- Crassidis, J.L.; Junkins, J.L. Optimal Estimation of Dynamic Systems; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Poulo, S. Adaptive Filtering, Algorithms and Practical Implementations; Springer: New York, NY, USA, 2008. [Google Scholar]
- Simon, D. Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches; John Wiley & Sons: Hoboken, NJ, USA, 2006. [Google Scholar]
- Spall, J.C.; Wall, K.D. Asymptotic distribution theory for the Kalman filter state estimator. Commun. Stat.-Theory Methods
**1984**, 13, 1981–2003. [Google Scholar] [CrossRef] - Vorobyov, S.A. Principles of minimum variance robust adaptive beamforming design. Signal Process.
**2013**, 93, 3264–3277. [Google Scholar] [CrossRef] - Liu, X.; Li, L.; Li, Z.; Fernando, T.; Iu, H.H. Stochastic stability condition for the extended Kalman filter with intermittent observations. IEEE Trans. Circuits Syst. II Express Briefs
**2016**, 64, 334–338. [Google Scholar] [CrossRef] - Vila-Valls, J.; Chaumette, E.; Vincent, F.; Closas, P. Modelling mismatch and noise statistics uncertainty in linear MMSE estimation. In Proceedings of the European Signal Processing Conference, A Coruna, Spain, 2–6 September 2019. [Google Scholar]
- Hrustic, E.; Ben Abdallah, R.; Vilà-Valls, J.; Vivet, D.; Pagès, G.; Chaumette, E. Robust linearly constrained extended Kalman filter for mismatched nonlinear systems. Int. J. Robust Nonlinear Control
**2021**, 31, 787–805. [Google Scholar] [CrossRef]

**Figure 1.**Stereo vision observation model. A 3D point is reprojected in both cameras. The extrinsic calibration of the stereo pair is defined by the rototranslation $R{T}_{stereo}$.

**Figure 2.**(

**a**) represents the classical map state appended upon landmark initialization. The appended parts correspond to the landmark’s mean and covariance (black) and the cross-variances between the landmark and the rest of the map (gray). (

**b**) represents the appended solution used followed by an update under constraints.

**Figure 3.**Simulated trajectory. This example represents one of the studied 6D trajectories generated along a spline with its associated random 3D point cloud.

**Figure 4.**Ground truth $\mathit{\delta}$ error calibration value for the different experiments. Major effects of decalibration noise are applied on the ${t}_{x}$ translation (the baseline) and on the ${\alpha}_{z}$ rotation (rotation around the baseline axis). Blue, red, and yellow are respectively the values for ${t}_{x}$, ${t}_{y}$, and ${t}_{z}$ and ${\alpha}_{x}$, ${\alpha}_{y}$, and ${\alpha}_{z}$.

**Figure 5.**SLAM without noise (Case i): methods’ comparison. (

**1**) and (

**2**) represent respectively the MSE of the pose and attitude estimation over 100 executions. Full lines are the mean of the estimation, and dotted line represent the 3 sigma values. (

**3**) represents the MSE of the landmark convergence during the simulation.

**Figure 6.**SLAM with constant noise (Case ii): methods’ comparison. Figures (

**1a**) and (

**2a**) represent the MSE of the pose and attitude estimation over 100 executions for the three methods. As the EKF results hide the two other, the same results without the EKF are presented in Figures (

**1b**) and (

**2b**). Full lines are the mean of the estimation, and dotted lines represent the 3 sigma value. Figure (

**3**) represents the MSE of the landmark convergence during the simulation.

**Figure 7.**SLAM with sinusoidal noise (Case iii): methods’ comparison. Figures (

**1a**) and (

**2a**) represent the MSE of the pose and attitude estimation over 100 executions for the three methods. As the EKF results hide the two other, the same results without the EKF are presented in Figures (

**1b**) and (

**2b**). Full lines are the mean of the estimation, and dot lines represent the 3 sigma value. Figure (

**3**) represents the MSE of the landmark convergence during the simulation.

**Figure 8.**SLAM with discontinuous noise (Case iv): methods’ comparison. Figures (

**1a**) and (

**2a**) represent the MSE of the pose and attitude estimation over 100 executions for the three methods. As the EKF results hide the two other, the same results without the EKF are presented in figures (

**1b**) and (

**2b**). Full lines are the mean of the estimation, and dotted lines represent the 3 sigma value. Figure (

**3**) represents the MSE of the landmark convergence during the simulation.

**Figure 9.**SLAM with periodic noise (Case v): methods’ comparison. Figures (

**1a**) and (

**2a**) represent the MSE of the pose and attitude estimation over 100 executions for the three methods. As the EKF results hide the two other, the same results without the EKF are presented in Figures (

**1b**) and

**(2b**). Full lines are the mean of the estimation, and dotted line represent the 3 sigma value. Figure (

**3**) represents the MSE of the landmark convergence during the simulation.

**Figure 10.**SLAM with high amplitude periodic noise (Case v): methods’ comparison. Figures (

**1a**) and (

**2a**) represent the MSE of the pose and attitude estimation over 100 executions for the three methods. The LCEKF results without the EKF and EKFfull are presented in Figures (

**1b**) and (

**2b**). Full lines are the mean of the estimation, and dotted lines represent the 3 sigma value. Figure (

**3**) represents the MSE of the landmark convergence during the simulation. Figure (

**4**) shows the EKFfull calibration parameters’ estimation.

**Figure 11.**SLAM with no noise (Case i): methods’ comparison. Figures (

**1**) and (

**2**) represent the MSE of the pose and attitude estimation over 10 executions for the three methods while using the IMU for prediction. All the methods performed in the same way. Figures (

**3**) and (

**4**) represent the MSE of the pose and attitude estimation over 10 executions for the three methods while using a constant velocity model for prediction (no proprioceptive sensor). It can be seen that without the use of the IMU, both the LCEKF and EKFfull had a positioning error that increased over time (classical observed drift). Nevertheless, the LCEKF is more robust.

**Figure 12.**SLAM with constant noise (Case ii): methods’ comparison. Figures (

**1**) and (

**2**) represent the MSE of the pose and attitude estimation over 10 executions for the three methods while using the IMU for prediction. All the methods performed in the same way. Figures (

**3**) and (

**4**) represent the MSE of the pose and attitude estimation over 10 executions for the three methods while using a constant velocity model for prediction (no proprioceptive sensor). It can be seen that without the use of the IMU, both the LCEKF and EKFfull had a positioning error that increased over time (classical observed drift). Nevertheless, the LCEKF is more robust.

**Figure 13.**SLAM with chock noise (Case iii): methods’ comparison. Figures (

**1**) and (

**2**) represent the MSE of the pose and attitude estimation over 10 executions for the three methods while using the IMU for prediction. All the methods performed in the same way. Figures (

**3**) and (

**4**) represent the MSE of the pose and attitude estimation over 10 executions for the three methods while using a constant velocity model for prediction (no proprioceptive sensor). It can be seen that without the use of the IMU, both the LCEKF and EKFfull had a positioning error that increased over time (classical observed drift). Nevertheless, the LCEKF is more robust.

**Figure 14.**SLAM with periodic square noise: methods’ comparison. Figures (

**1**) and (

**2**) represent the MSE of the pose and attitude estimation over 10 executions for the two methods while using the IMU for prediction. Figures (

**3**) and (

**4**) represent the MSE of the pose and attitude estimation over 10 executions for the two methods while using a constant velocity model for prediction (no proprioceptive sensor). It can be clearly seen that the EKFfull diverged in both case.

**Figure 15.**SLAM without the IMU and smooth sinusoidal noise. With a good random walk for K parameters, the estimation performed properly, making the EKFfull work fine. Note that the LCEKF and EKFfull performed in the same way.

**Figure 16.**SLAM without the IMU and smooth sinusoidal noise. Without random walk for K parameters, the estimation was not performed properly, making the EKFfull diverge. Note that there was no impact on the LCEKF as the K parameters were not estimated.

**Figure 17.**SLAM without the IMU and smooth sinusoidal noise. With underestimated random walk for K parameters, the estimation was not performed properly, making the EKFfull diverge. Note that there was no impact on the LCEKF as the K calibration parameters were not estimated.

**Figure 18.**SLAM without the IMU and smooth sinusoidal noise. With overestimated random walk for K parameters, the estimation was not performed properly, making the EKFfull diverge. Note that there was no impact on the LCEKF as the K calibration parameters were not estimated.

Figure 15 | Figure 16 | Figure 17 | Figure 18 | |
---|---|---|---|---|

Random walk | well estimated | none | underestimated | overestimated |

${\sigma}_{\mathrm{rotation}}^{2}$ | ${10}^{-6}$ rad${}^{2}$ | 0 rad${}^{2}$ | $0.5\times {10}^{-6}$ rad${}^{2}$ | $2\times {10}^{-5}$ rad${}^{2}$ |

${\sigma}_{\mathrm{translation}}^{2}$ | 1 mm${}^{2}$ | 0 mm${}^{2}$ | 0.1 mm${}^{2}$ | 20 mm${}^{2}$ |

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**MDPI and ACS Style**

Vivet, D.; Vilà-Valls, J.; Pages, G.; Chaumette, E. Robust Filter-Based Visual Navigation Solution with Miscalibrated Bi-Monocular or Stereo Cameras. *Remote Sens.* **2022**, *14*, 1470.
https://doi.org/10.3390/rs14061470

**AMA Style**

Vivet D, Vilà-Valls J, Pages G, Chaumette E. Robust Filter-Based Visual Navigation Solution with Miscalibrated Bi-Monocular or Stereo Cameras. *Remote Sensing*. 2022; 14(6):1470.
https://doi.org/10.3390/rs14061470

**Chicago/Turabian Style**

Vivet, Damien, Jordi Vilà-Valls, Gaël Pages, and Eric Chaumette. 2022. "Robust Filter-Based Visual Navigation Solution with Miscalibrated Bi-Monocular or Stereo Cameras" *Remote Sensing* 14, no. 6: 1470.
https://doi.org/10.3390/rs14061470