doi: 10.3934/jimo.2021059

Stereo visual odometry based on dynamic and static features division

College of Missile Engineering, Rocket Force University of Engineering, Xi'an, Shaanxi 710025, China

* Corresponding author: Guangbin Cai

Received  June 2020 Revised  December 2020 Early access  March 2021

Fund Project: The first author is mainly supported by NSSF of China under Grant (No. 61773387)

Accurate camera pose estimation in dynamic scenes is an important challenge for visual simultaneous localization and mapping, and it is critical to reduce the effects of moving objects on pose estimation. To tackle this problem, a robust visual odometry approach in dynamic scenes is proposed, which can precisely distinguish between dynamic and static features. The key to the proposed method is combining the scene flow and the static features relative spatial distance invariance principle. Moreover, a new threshold is proposed to distinguish dynamic features.Then the dynamic features are eliminated after matching with the virtual map points. In addition, a new similarity calculation function is proposed to improve the performance of loop-closure detection. Finally, the camera pose is optimized after obtaining a closed loop. Experiments have been conducted on TUM datasets and actual scenes, which shows that the proposed method reduces tracking errors significantly and estimates the camera pose precisely in dynamic scenes.

Citation: Hui Xu, Guangbin Cai, Xiaogang Yang, Erliang Yao, Xiaofeng Li. Stereo visual odometry based on dynamic and static features division. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021059
References:
[1]

P. F. Alcantarilla, J. J. Yebes, J. Almazán et. al., On combining visual slam and dense scene flow to increase the robustness of localization and mapping in dynamic environments, 2012 IEEE International Conference on Robotics and Automation, Saint Paul, Minnesota, USA, IEEE, 2012. Google Scholar

[2]

Y. AnB. Li and L. Wang, Calibration of a 3D laser rangefinder and a camera based on optimization solution, J. Ind. Manag. Optim., 17 (2021), 427-445.  doi: 10.3934/jimo.2019119.  Google Scholar

[3]

A. AngeliD. Filliat and S. Doncieux, Fast and incremental method for loop-closure detection using bags of visual words, IEEE Transactions on Robotics, 24 (2008), 1027-1037.   Google Scholar

[4]

C. Bibby and I. Reid, Simultaneous localisation and mapping in dynamic environments (SLAMIDE) with reversible data association, Robotics: Science and Systems, Atlanta, Georgia, USA, 2007. Google Scholar

[5]

L. Bose and A. Richards, Fast Depth Edge Detection and Edge Based Rgb-D Slam, IEEE International Conference on Robotics and Automation, Stockholm, Sweden, IEEE, 2016. Google Scholar

[6]

C. CadenaL. Carlone and H. Carrillo, Simultaneous localization and mapping: Present, future, and the robust-perception age, IEEE Transactions on Robotics, 32 (2016), 1309-1332.   Google Scholar

[7]

C. Choi, A. J. Trevor and H. I. Christensen, Rgbd Edge Detection and Edge-Based Registration, 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, Tokyo, Japan, IEEE, 2013. Google Scholar

[8]

A. J. DavisonI. D. Reid and N. D. Molton, MonoSLAM: Real-time single camera SLAM, IEEE Transactions on Pattern Analysis & Machine Intelligence, 29 (2007), 1052-1067.   Google Scholar

[9]

J. EngelV. Koltun and D. Cremers, Direct sparse odometry, IEEE Transactions on Pattern Analysis & Machine Intelligence, 40 (2018), 611-625.   Google Scholar

[10]

J. Engel, T. Schöps and D. Cremers, LSD-SLAM: Large-Scale Direct Monocular SLAM, European Conference on Computer Vision, Springer, Zürich, Switzerland, 2014. Google Scholar

[11]

J. Fan, On the Levenberg-Marquardt methods for convex constrained nonlinear equations, J. Ind. Manag. Optim., 9 (2013), 227-241.  doi: 10.3934/jimo.2013.9.227.  Google Scholar

[12]

C. Forster, M. Pizzoli and D Scaramuzza, SVO: Fast Semi-Direct Monocular Visual Odometry, IEEE International Conference on Robotics and Automation, Hong Kong, China, IEEE, 2014. Google Scholar

[13]

C. ForsterZ. Zhang and M. Gassner, SVO: Semi-direct visual odometry for monocular and multicamera systems, IEEE Transactions on Robotics, 33 (2017), 249-265.   Google Scholar

[14]

J. Fuentes-PacheoJ. Ruiz-Ascencio and J. M. Rendón-Mancha, Visual simultaneous localization and mapping: A survey, Artificial Intelligence Review, 43 (2015), 55-81.   Google Scholar

[15]

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[17]

D. K. Gu and D. W. Zhang, A parametric method to design dynamic compensator for high-order quasi-linear systems, Nonlinear Dynamics, 100 (2020), 1379-1400.   Google Scholar

[18]

C. Kerl, J. Sturm and D. Cremers, Dense Visual Slam for Rgb-D Cameras, 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, Tokyo, Japan, IEEE, 2013. Google Scholar

[19]

D. H. Kim and J. H. Kim, Image-Based Icp Algorithm for Visual Odometry using a Rgb-D Sensor in a Dynamic Environment, Robot Intelligence Technology and Applications, Gwangju, Korea, Springer, 2013. Google Scholar

[20]

D. H. Kim, S. B. Han and J. H. Kim, Visual Odometry Algorithm using an Rgb-D Sensor and Imu in a Highly Dynamic Environment, Robot Intelligence Technology and Applications, Beijing, China, Springer, 2015. Google Scholar

[21]

D. H. Kim and J. H. Kim, Effective background model-based rgb-d dense visual odometry in a dynamic environment, IEEE Transactions on Robotics, 32 (2016), 1565-1573.   Google Scholar

[22]

M. Labbe and F. Michaud, Appearance-based loop closure detection for online large-scale and long-term operation, IEEE Transactions on Robotics, 9 (2013), 734-745.   Google Scholar

[23]

S. Li and D. Lee, Rgb-d slam in dynamic environments using static point weighting, IEEE Robotics and Automation Letters, 2 (2017), 2263-2270.   Google Scholar

[24]

B. LiD. Yang and L. Deng, Visual vocabulary tree with pyramid TF-IDF scoring match scheme for loop closure detection, Acta Automatica Sinica, 37 (2011), 665-673.   Google Scholar

[25]

Y. LiG. Zhang and F. Wang, An improved loop closure detection algorithm based on historical model set, Robot, 37 (2015), 663-673.   Google Scholar

[26]

Z. L. LinG. L. Zhang and E. Yao, Sterero visual odometry based on motion object detection in the dynamic scene, Acta Optica Sinica, 37 (2017), 187-195.   Google Scholar

[27]

M. Lourakis and X. Zabulis, Model-Based Pose Estimation for Rigid Objects, International conference on computer vision systems, St. Petersburg, Russia, Springer, 2013. Google Scholar

[28]

R. Mur-ArtalJ. M. M. Montiel and J. D. Tardós, ORB-SLAM: A versatile and accurate monocular slam system, IEEE Transactions on Robotics, 31 (2015), 1147-1163.   Google Scholar

[29]

R. Mur-Artal and J. D. Tardós, ORB-SLAM2: An opensource slam system for monocular, stereo, and rgbd cameras, IEEE Transactions on Robotics, 335 (2017), 1255-1262.   Google Scholar

[30]

D. NistérO. Naroditsky and J. Bergen, Visual odometry for ground vehicle applications, Journal of Field Robotics, 23 (2006), 3-20.   Google Scholar

[31]

Z. Peng, Research on Vision-Based Ego-Motion Estimation and Environment Modeling in Dynamic Environment, Ph.D. dissertation, Zhejiang University, Hangzhou, China, 2013. Google Scholar

[32]

D. Scaramuzza and F. Fraundorfer, Visual odometry, IEEE Robotics & Automation Magazine, 18 (2011), 80-92.   Google Scholar

[33]

J. Sturm, N. Engelhard, F. Endres et. al., A Benchmark for the Evaluation of RGB-D SLAM Systems, IEEE International Conference on Intelligent Robots and Systems, Vilamoura, Portugal, IEEE, 2012. Google Scholar

[34]

Y. SunM. Liu and M. Q. H. Meng, Improving rgbd slam in dynamic environments: A motion removal approach, Robotics and Autonomous Systems, 89 (2017), 110-122.   Google Scholar

[35]

W. Tan, H. Liu, Z. Dong et. al., Robust Monocular SLAM in Dynamic Environments, IEEE International Symposium on Mixed and Augmented Reality, Adelaide, Australia, IEEE, 2013. Google Scholar

[36]

G. YounesD. Asmar and E. Shammas, Keyframe-based monocular slam: Design, survey, and future directions, Robotics and Autonomous Systems, 98 (2017), 67-88.   Google Scholar

show all references

References:
[1]

P. F. Alcantarilla, J. J. Yebes, J. Almazán et. al., On combining visual slam and dense scene flow to increase the robustness of localization and mapping in dynamic environments, 2012 IEEE International Conference on Robotics and Automation, Saint Paul, Minnesota, USA, IEEE, 2012. Google Scholar

[2]

Y. AnB. Li and L. Wang, Calibration of a 3D laser rangefinder and a camera based on optimization solution, J. Ind. Manag. Optim., 17 (2021), 427-445.  doi: 10.3934/jimo.2019119.  Google Scholar

[3]

A. AngeliD. Filliat and S. Doncieux, Fast and incremental method for loop-closure detection using bags of visual words, IEEE Transactions on Robotics, 24 (2008), 1027-1037.   Google Scholar

[4]

C. Bibby and I. Reid, Simultaneous localisation and mapping in dynamic environments (SLAMIDE) with reversible data association, Robotics: Science and Systems, Atlanta, Georgia, USA, 2007. Google Scholar

[5]

L. Bose and A. Richards, Fast Depth Edge Detection and Edge Based Rgb-D Slam, IEEE International Conference on Robotics and Automation, Stockholm, Sweden, IEEE, 2016. Google Scholar

[6]

C. CadenaL. Carlone and H. Carrillo, Simultaneous localization and mapping: Present, future, and the robust-perception age, IEEE Transactions on Robotics, 32 (2016), 1309-1332.   Google Scholar

[7]

C. Choi, A. J. Trevor and H. I. Christensen, Rgbd Edge Detection and Edge-Based Registration, 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, Tokyo, Japan, IEEE, 2013. Google Scholar

[8]

A. J. DavisonI. D. Reid and N. D. Molton, MonoSLAM: Real-time single camera SLAM, IEEE Transactions on Pattern Analysis & Machine Intelligence, 29 (2007), 1052-1067.   Google Scholar

[9]

J. EngelV. Koltun and D. Cremers, Direct sparse odometry, IEEE Transactions on Pattern Analysis & Machine Intelligence, 40 (2018), 611-625.   Google Scholar

[10]

J. Engel, T. Schöps and D. Cremers, LSD-SLAM: Large-Scale Direct Monocular SLAM, European Conference on Computer Vision, Springer, Zürich, Switzerland, 2014. Google Scholar

[11]

J. Fan, On the Levenberg-Marquardt methods for convex constrained nonlinear equations, J. Ind. Manag. Optim., 9 (2013), 227-241.  doi: 10.3934/jimo.2013.9.227.  Google Scholar

[12]

C. Forster, M. Pizzoli and D Scaramuzza, SVO: Fast Semi-Direct Monocular Visual Odometry, IEEE International Conference on Robotics and Automation, Hong Kong, China, IEEE, 2014. Google Scholar

[13]

C. ForsterZ. Zhang and M. Gassner, SVO: Semi-direct visual odometry for monocular and multicamera systems, IEEE Transactions on Robotics, 33 (2017), 249-265.   Google Scholar

[14]

J. Fuentes-PacheoJ. Ruiz-Ascencio and J. M. Rendón-Mancha, Visual simultaneous localization and mapping: A survey, Artificial Intelligence Review, 43 (2015), 55-81.   Google Scholar

[15]

D.-K. GuG.-P. Liu and G.-R. Duan, Robust stability of uncertain second-order linear time-varying systems, J. Franklin Inst., 356 (2019), 9881-9906.  doi: 10.1016/j.jfranklin.2019.09.014.  Google Scholar

[16]

D.-K. Gu and D.-W. Zhang, Parametric control to second-order linear time-varying systems based on dynamic compensator and multi-objective optimization, Appl. Math. Comput., 365 (2020), 124681, 25 pp. doi: 10.1016/j.amc.2019.124681.  Google Scholar

[17]

D. K. Gu and D. W. Zhang, A parametric method to design dynamic compensator for high-order quasi-linear systems, Nonlinear Dynamics, 100 (2020), 1379-1400.   Google Scholar

[18]

C. Kerl, J. Sturm and D. Cremers, Dense Visual Slam for Rgb-D Cameras, 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, Tokyo, Japan, IEEE, 2013. Google Scholar

[19]

D. H. Kim and J. H. Kim, Image-Based Icp Algorithm for Visual Odometry using a Rgb-D Sensor in a Dynamic Environment, Robot Intelligence Technology and Applications, Gwangju, Korea, Springer, 2013. Google Scholar

[20]

D. H. Kim, S. B. Han and J. H. Kim, Visual Odometry Algorithm using an Rgb-D Sensor and Imu in a Highly Dynamic Environment, Robot Intelligence Technology and Applications, Beijing, China, Springer, 2015. Google Scholar

[21]

D. H. Kim and J. H. Kim, Effective background model-based rgb-d dense visual odometry in a dynamic environment, IEEE Transactions on Robotics, 32 (2016), 1565-1573.   Google Scholar

[22]

M. Labbe and F. Michaud, Appearance-based loop closure detection for online large-scale and long-term operation, IEEE Transactions on Robotics, 9 (2013), 734-745.   Google Scholar

[23]

S. Li and D. Lee, Rgb-d slam in dynamic environments using static point weighting, IEEE Robotics and Automation Letters, 2 (2017), 2263-2270.   Google Scholar

[24]

B. LiD. Yang and L. Deng, Visual vocabulary tree with pyramid TF-IDF scoring match scheme for loop closure detection, Acta Automatica Sinica, 37 (2011), 665-673.   Google Scholar

[25]

Y. LiG. Zhang and F. Wang, An improved loop closure detection algorithm based on historical model set, Robot, 37 (2015), 663-673.   Google Scholar

[26]

Z. L. LinG. L. Zhang and E. Yao, Sterero visual odometry based on motion object detection in the dynamic scene, Acta Optica Sinica, 37 (2017), 187-195.   Google Scholar

[27]

M. Lourakis and X. Zabulis, Model-Based Pose Estimation for Rigid Objects, International conference on computer vision systems, St. Petersburg, Russia, Springer, 2013. Google Scholar

[28]

R. Mur-ArtalJ. M. M. Montiel and J. D. Tardós, ORB-SLAM: A versatile and accurate monocular slam system, IEEE Transactions on Robotics, 31 (2015), 1147-1163.   Google Scholar

[29]

R. Mur-Artal and J. D. Tardós, ORB-SLAM2: An opensource slam system for monocular, stereo, and rgbd cameras, IEEE Transactions on Robotics, 335 (2017), 1255-1262.   Google Scholar

[30]

D. NistérO. Naroditsky and J. Bergen, Visual odometry for ground vehicle applications, Journal of Field Robotics, 23 (2006), 3-20.   Google Scholar

[31]

Z. Peng, Research on Vision-Based Ego-Motion Estimation and Environment Modeling in Dynamic Environment, Ph.D. dissertation, Zhejiang University, Hangzhou, China, 2013. Google Scholar

[32]

D. Scaramuzza and F. Fraundorfer, Visual odometry, IEEE Robotics & Automation Magazine, 18 (2011), 80-92.   Google Scholar

[33]

J. Sturm, N. Engelhard, F. Endres et. al., A Benchmark for the Evaluation of RGB-D SLAM Systems, IEEE International Conference on Intelligent Robots and Systems, Vilamoura, Portugal, IEEE, 2012. Google Scholar

[34]

Y. SunM. Liu and M. Q. H. Meng, Improving rgbd slam in dynamic environments: A motion removal approach, Robotics and Autonomous Systems, 89 (2017), 110-122.   Google Scholar

[35]

W. Tan, H. Liu, Z. Dong et. al., Robust Monocular SLAM in Dynamic Environments, IEEE International Symposium on Mixed and Augmented Reality, Adelaide, Australia, IEEE, 2013. Google Scholar

[36]

G. YounesD. Asmar and E. Shammas, Keyframe-based monocular slam: Design, survey, and future directions, Robotics and Autonomous Systems, 98 (2017), 67-88.   Google Scholar

Figure 1.  Stereo camera model
Figure 2.  Generation of a visual vocabulary tree
Figure 3.  Overview of the proposed algorithm in dynamic scenes
26]">Figure 4.  Classification of the scene flow based on angles [26]
Figure 5.  Invariance of the relative spatial distance of the static points
Figure 6.  Construction of the virtual map points
Figure 7.  Three static features selected by the algorithm
Figure 8.  Dynamic features obtained by the algorithm
Figure 9.  Experiment scene sets
Figure 10.  Experimental results of ORB-VO in lab scenes
Figure 11.  Experimental results of the proposed method in lab scenes
Figure 12.  Loop-closure detection result of the inverse proportional function
Figure 13.  Loop-closure detection result of the negative exponential power function
Figure 14.  Loop-closure detection result of the negative exponential power function
Figure 15.  Comparisons between estimated trajectories and the ground truth in walking sequences
Figure 16.  Comparisons between estimated trajectories and the ground truth in sitting sequences
Table 1.  Translation drift and rotational drift of VO method on TUM dataset
Sequences RMSE of translational drift [m/s] RMSE of rotational drift [$ ^{\circ} $/s]
DVO BaMVO SPW-VO Our Method DVO BaMVO SPW-VO Our Method
sitting-static 0.0157 0.0248 0.0231 0.0112 0.6084 0.6977 0.7228 0.3356
sitting-xyz 0.0453 0.0482 0.0219 0.0132 1.4980 1.3885 0.8466 0.5753
sitting-rpy 0.1735 0.1872 0.0843 0.0280 6.0164 5.9834 5.6258 0.6811
sitting-halfsphere 0.1005 0.0589 0.0389 0.0151 4.6490 2.8804 1.8836 0.6103
walking-static 0.3818 0.1339 0.0327 0.0293 6.3502 2.0833 0.8085 0.5500
walking-xyz 0.4360 0.2326 0.0651 0.1034 7.6669 4.3911 1.6442 2.3273
walking-rpy 0.4038 0.3584 0.2252 0.2143 7.0662 6.3898 5.6902 3.9555
walking-halfsphere 0.2628 0.1738 0.0527 0.1061 5.2179 4.2863 2.4048 2.2983
Sequences RMSE of translational drift [m/s] RMSE of rotational drift [$ ^{\circ} $/s]
DVO BaMVO SPW-VO Our Method DVO BaMVO SPW-VO Our Method
sitting-static 0.0157 0.0248 0.0231 0.0112 0.6084 0.6977 0.7228 0.3356
sitting-xyz 0.0453 0.0482 0.0219 0.0132 1.4980 1.3885 0.8466 0.5753
sitting-rpy 0.1735 0.1872 0.0843 0.0280 6.0164 5.9834 5.6258 0.6811
sitting-halfsphere 0.1005 0.0589 0.0389 0.0151 4.6490 2.8804 1.8836 0.6103
walking-static 0.3818 0.1339 0.0327 0.0293 6.3502 2.0833 0.8085 0.5500
walking-xyz 0.4360 0.2326 0.0651 0.1034 7.6669 4.3911 1.6442 2.3273
walking-rpy 0.4038 0.3584 0.2252 0.2143 7.0662 6.3898 5.6902 3.9555
walking-halfsphere 0.2628 0.1738 0.0527 0.1061 5.2179 4.2863 2.4048 2.2983
Table 2.  RMSE of the ATE of camera pose estimation (m$ ^{-1} $)
Sequences ORB-SLAM2 MR-SLAM SPW-SLAM SF-SLAM Our Method
sitting-static 0.0082 0.0081 0.0073
sitting-xyz 0.0094 0.0482 0.0397 0.0101 0.0090
sitting-rpy 0.0197 0.0180 0.0162
sitting-halfsphere 0.0211 0.0470 0.0432 0.0239 0.0164
walking-static 0.1028 0.0656 0.0261 0.0120 0.0108
walking-xyz 0.4278 0.0932 0.0601 0.2251 0.0884
walking-rpy 0.7407 0.1333 0.1791 0.1961 0.3620
walking-halfsphere 0.4939 0.1252 0.0489 0.0423 0.0411
Sequences ORB-SLAM2 MR-SLAM SPW-SLAM SF-SLAM Our Method
sitting-static 0.0082 0.0081 0.0073
sitting-xyz 0.0094 0.0482 0.0397 0.0101 0.0090
sitting-rpy 0.0197 0.0180 0.0162
sitting-halfsphere 0.0211 0.0470 0.0432 0.0239 0.0164
walking-static 0.1028 0.0656 0.0261 0.0120 0.0108
walking-xyz 0.4278 0.0932 0.0601 0.2251 0.0884
walking-rpy 0.7407 0.1333 0.1791 0.1961 0.3620
walking-halfsphere 0.4939 0.1252 0.0489 0.0423 0.0411
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