RESEARCH

Optimization-Based Estimation and Factor Graphs

Factor graph optimization as a unifying framework for GNSS, INS, carrier phase, Doppler, maps, perception, and integrity — reasoning over time across heterogeneous measurements.

factor graph optimizationFGOGNSS/INSRTKPPP-RTKsmoothingnonlinear optimizationrobust estimationsensor fusion

Pillar lead: Weisong Wen

Factor graph optimization is the mathematical engine that links signals, measurements, maps, sensors, and integrity. We use it as a unifying framework — letting navigation systems reason over time, fuse heterogeneous measurements, and handle outliers more flexibly than classical filtering-only pipelines.

The Extended Kalman Filter condenses all history into one current state with a single linearization and handles outliers poorly; IPNL uses factor graph optimization to jointly optimize a window of states over time and fuse heterogeneous measurements into a robust trajectory.

The foundational paper Factor Graph Optimization for GNSS/INS Integration: A Comparison with the Extended Kalman Filter (Wen, Pfeifer, Bai, Hsu, NAVIGATION 2021) was named the 2024 Most-Cited Paper in NAVIGATION: Journal of the Institute of Navigation. The follow-on work extends FGO into integrity-constrained estimation, smartphone IMU-only indoor SLAM (IEEE TAES 2024), 3D vision-aided GNSS RTK (NAVIGATION 2023), and self-supervised weighting (AutoW, ION GNSS+ 2024). The open-source GraphGNSSLib library is the community-facing artifact of this pillar.

Related Publications

2026 (4)

patent
Hsu, L. T& Wen, W (2026).3D LiDAR Aided GNSS and the Method for NLOS Detection and Correction (CA 3,156,087). Canadian Intellectual Property Office (CIPO).
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Song, B., Giamou, M., Yan, P., Xia, C., & Hsu, L.-T. (2026).Certifiable Alignment of GNSS and Local Frames via Lagrangian Duality. IEEE Robotics and Automation Letters. https://doi.org/10.1109/LRA.2026.3693943
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2025 (15)

book
Hsu, L. T., Zhang, G., & F., N. H. (2025).Chapter: Map-Aided GNSS Positioning. Springer Publishers Inc.
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Hsu, L. T., Zhang, G., & Wen, W. (2025).Principles of Indoor Positioning and Indoor Navigation. Artech House Publishers Inc.
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Hsu, L. T& Wen, W (2025).3D LiDAR Aided GNSS and the Method for NLOS Detection and Correction (ZL 2022 1 0905688.2). China National Intellectual Property Administration (CNIPA).
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Hsu, L. T& Wen, W (2025).3D LiDAR Aided GNSS and the Method for NLOS Detection and Correction (HK 40077647). Hong Kong Intellectual Property Department.
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Bai, S., Wen, W., Su, D., & Hsu, L. T. (2025).Graph-Based Indoor 3D Pedestrian Location Tracking With Inertial-Only Perception. IEEE Transactions on Mobile Computing. https://doi.org/10.1109/tmc.2025.3526196
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Yan, P., Li, Z., Huang, F., Wen, W., & Hsu, L. T. (2025).Fault Detection Algorithm for Gaussian Mixture Noises: An Application in Lidar/IMU Integrated Localization Systems. NAVIGATION: Journal of the Institute of Navigation. https://doi.org/10.33012/navi.684
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Zheng, X., Wen, W., & Hsu, L. T. (2025).Safety-quantifiable line feature-based monocular visual localization with 3d prior map. IEEE Transactions on Intelligent Transportation Systems. https://doi.org/10.1109/tits.2025.3572620
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Liu, X., Wen, W., Huang, F., Gao, H., Wang, Y., & Hsu, L. T. (2025).3-D LiDAR-Aided GNSS NLOS Mitigation for Reliable GNSS-RTK Positioning in Urban Canyons. IEEE Transactions on Instrumentation and Measurement. https://doi.org/10.1109/tim.2025.3629836
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Liu, X., Wen, W., Zhang, L., & Hsu, L. T. (2025).3D LiDAR Aided GNSS NLOS Correction by Direction-of-Arrival Estimation Using Doppler Measurements in Urban Canyons. IEEE Transactions on Intelligent Transportation Systems. https://doi.org/10.1109/tits.2025.3596582
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Huang, F., Zhong, Y., Chen, H., Dongzhe, Su, Jin, Wu, Wen, W., & Hsu, L.-T. (2025).Roadside GNSS Aided Multi-Sensor Integrated System for Vehicle Positioning in Urban Areas. 2025 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). https://doi.org/10.1109/IROS60139.2025.11246475
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Zhong, Y., Hoi-Fung, Ng, Hsu, L.-T., & Wen, W. (2025).GNSS/Map Integration with Batch Factor Graph Optimization for Reliable Trajectory Estimation in Urban Canyon. 2025 IEEE/ION Position, Location and Navigation Symposium (PLANS). https://doi.org/10.1109/PLANS61210.2025.11028209
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Song, B., Wen, W., & Hsu, L.-T. (2025).Certifiably Optimal Satellite Orbit Determination Based on Doppler Measurements for Low-Earth-Orbit Satellites. 2025 IEEE/ION Position, Location and Navigation Symposium (PLANS). https://doi.org/10.1109/PLANS61210.2025.11028259

2024 (23)

patent
Hsu, L. T& Wen, W (2024).3D LiDAR Aided GNSS and the Method for NLOS Detection and Correction (US 12,123,961 B2). U.S. Patent and Trademark Office. https://patents.google.com/patent/US12123961B2
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Zhong, Y., Hu, R., Bai, X., Li, X., Hsu, L. T., & Wen, W. (2024).Enhancing GNSS Positioning Accuracy for Road Monitoring Systems: A Factor Graph Optimization Approach Aided by Geospatial Information. IEEE Transactions on Instrumentation and Measurement. https://doi.org/10.1109/tim.2024.3369156
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Xu, P., Zhang, G., Zhong, Y., Yang, B., & Hsu, L. T. (2024).A Framework for Graphical GNSS Multipath and NLOS Mitigation. IEEE Transactions on Intelligent Transportation Systems. https://doi.org/10.1109/tits.2024.3374819
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Zhang, L., Ng, H. F., Zhang, G., & Hsu, L. T. (2024).Ray-Tracing Correction for GNSS Velocity Estimation Using Doppler Frequency: A Feasibility Analysis. IEEE Transactions on Instrumentation and Measurement. https://doi.org/10.1109/tim.2024.3385033
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Jin, R., Zhang, G., Hsu, L. T., & Hu, Y. (2024).A Survey on Cooperative Positioning Using GNSS Measurements. IEEE Transactions on Intelligent Vehicles. https://doi.org/10.1109/tiv.2024.3397879
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Bai, S., Wen, W., Hsu, L. T., & Yang, P. (2024).Factor Graph Optimization-Based Smartphone IMU-Only Indoor SLAM With Multi-Hypothesis Turning Behavior Loop Closures. IEEE Transactions on Aerospace and Electronic Systems. https://doi.org/10.1109/taes.2024.3430256
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Xia, X., Wen, W., & Hsu, L. T. (2024).Integrity-Constrained Factor Graph Optimization for GNSS Positioning in Urban Canyons. NAVIGATION: Journal of the Institute of Navigation. https://doi.org/10.33012/navi.660
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Zhang, L., Ng, H. F., Zhong, Y., Zhang, G., & Hsu, L. T. (2024).GNSS Doppler Velocity Estimation Aided by 3D Mapping Database. IEEE Transactions on Intelligent Transportation Systems. https://doi.org/10.1109/tits.2024.3486448
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Bai, S., Wen, W., Yue, Yu, & Hsu, L.-T. (2024).Invariant Extended Kalman Filtering for Pedestrian Deep-Inertial Odometry. The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. https://doi.org/10.5194/isprs-archives-xlviii-4-2024-607-2024
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Penghui, Xu& Hsu, L.-T. (2024).AutoW: Self-Supervision Learning for Weighting Estimation in GNSS Positioning. ION GNSS+, The International Technical Meeting of the Satellite Division of The Institute of Navigation. https://doi.org/10.33012/2024.19896
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Hoi-Fung, Ng, Penghui, Xu, Zhong, Y., Zhang, G., Wen, W., & Hsu, L.-T. (2024).DGNSS Corrected Pseudorange and Time-Differenced Carrier Phase (TDCP) Measurements using Differentiable Factor Graph Optimization (DFGO). ION GNSS+, The International Technical Meeting of the Satellite Division of The Institute of Navigation. https://doi.org/10.33012/2024.19908
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Vicenzo, S., Xin, Qi, Bing, Xu, & Hsu, L.-T. (2024).Machine Learning Assisted GNSS Direct Position Estimation for Urban Environments Applications. The International Technical Meeting of the The Institute of Navigation. https://doi.org/10.33012/2024.19567
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Yuan, Li, Liu, X., Wen, W., Hsu, L.-T., Yuan, Y., Bian, G., & Chen, Q. (2024).Factor Graph Optimization Based Multi Epoch Ambiguity Resolution for GNSS RTK and its Evaluation in Hong Kong Urban Canyons. ION GNSS+, The International Technical Meeting of the Satellite Division of The Institute of Navigation. https://doi.org/10.33012/2024.19805
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Yan, P., Wen, W., Huang, F., & Hsu, L.-T. (2024).A Fault Detection Algorithm for LiDAR/IMU Integrated Localization Systems with Non-Gaussian Noises. The International Technical Meeting of the The Institute of Navigation. https://doi.org/10.33012/2024.19564
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Meiling, Su, Wang, B., Ahad, S., Huang, X., Zhang, G., & Hsu, L.-T. (2024).An Adaptive Weighted GNSS/VINS/Wi-Fi RTT-based Seamless Positioning System for Smartphone. 2024 14th International Conference on Indoor Positioning and Indoor Navigation (IPIN). https://doi.org/10.1109/IPIN62893.2024.10786175

2023 (23)

patent
Gong, Z, Bao, X, B, Y., Y, G., Hsu, L. T, & Wen, W (2023).Vehicle Position Enhancement System and Method (车载位置增强系统及方法) (CN 202310289415.4). China National Intellectual Property Administration (CNIPA). https://patents.google.com/patent/CN116660938A
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Wen, W., Bai, X., & Hsu, L. T. (2023).3D Vision Aided GNSS Real-Time Kinematic Positioning for Autonomous Systems in Urban Canyons. NAVIGATION: Journal of the Institute of Navigation. https://doi.org/10.33012/navi.590
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Xu, P., Zhang, G., Yang, B., & Hsu, L. T. (2023).PositionNet: CNN-based GNSS positioning in urban areas with residual maps. Applied Soft Computing. https://doi.org/10.2139/ssrn.4194604
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Huang, F., Wen, W., Zhang, J., Wang, C., & Hsu, L. T. (2023).Dynamic Object-aware LiDAR Odometry Aided by Joint Weightings Estimation in Urban Areas. IEEE Transactions on Intelligent Vehicles. https://doi.org/10.1109/tiv.2023.3338141
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Hsu, L. T., Huang, F., Ng, H. F., Zhang, G., Zhong, Y., Bai, X., & Wen, W. (2023).Hong Kong UrbanNav: An open-source multisensory dataset for benchmarking urban navigation algorithms. NAVIGATION: Journal of the Institute of Navigation. https://doi.org/10.33012/navi.602
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Ng, H. F., Hsu, L. T., & Zhang, G. (2023).Multi-Epoch Kriging-Based 3D Mapping-Aided GNSS and Doppler Measurement Fusion using Factor Graph Optimization. NAVIGATION: Journal of the Institute of Navigation. https://doi.org/10.33012/navi.617
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Zheng, Xi, Wen, W., & Hsu, L.-T. (2023).Tightly-coupled Line Feature-aided Visual Inertial Localization within Lightweight 3D Prior Map for Intelligent Vehicles. 2023 IEEE 26th International Conference on Intelligent Transportation Systems (ITSC). https://doi.org/10.1109/itsc57777.2023.10422196
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Huang, F., Chen, H., Urtay, A., Dongzhe, Su, Wen, W., & Hsu, L.-T. (2023).Roadside Infrastructure assisted LiDAR/Inertial-based Mapping for Intelligent Vehicles in Urban Areas. 2023 IEEE 26th International Conference on Intelligent Transportation Systems (ITSC). https://doi.org/10.1109/itsc57777.2023.10422552
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Bai, X.& Hsu, L.-T. (2023).GVIM: GNSS/Visual/IMU/Map Integration Via Sliding Window Factor Graph Optimization in Urban Canyons. Proceedings of the 36th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2023). https://doi.org/10.33012/2023.19458
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Bai, S., Wen, W., Hsu, L.-T., & Yue, Yu (2023).Factor Graph Optimization-based Indoor Pedestrian SLAM with Probabilistic Exact Activity Loop Closures using Smartphone. 2023 13th International Conference on Indoor Positioning and Indoor Navigation (IPIN). https://doi.org/10.1109/ipin57070.2023.10332510
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Vicenzo, S., Bing, Xu, Dey, A., & Hsu, L.-T. (2023).Experimental Investigation of GNSS Direct Position Estimation in Densely Urban Area. Proceedings of the 36th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2023). https://doi.org/10.33012/2023.19341
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Penghui, Xu, Hoi-Fung, Ng, Zhong, Y., Zhang, G., Wen, W., Yang, Bo, & Hsu, L.-T. (2023).Differentiable Factor Graph Optimization with Intelligent Covariance Adaptation for Accurate Smartphone Positioning. Proceedings of the 36th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2023). https://doi.org/10.33012/2023.19297
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Huang, F., Wen, W., Zhang, G., Dongzhe, Su, & Hsu, L.-T. (2023).Adaptive Multi-Sensor Integrated Navigation System Aided by Continuous Error Map from RSU for Autonomous Vehicles in Urban Areas. 2023 IEEE 26th International Conference on Intelligent Transportation Systems (ITSC). https://doi.org/10.1109/itsc57777.2023.10422216
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Zhang, H., Wen, W., & Hsu, L.-T. (2023).A New GNSS Ambiguity Resolution Method Through Mixed Integer Programming. Proceedings of the 36th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2023). https://doi.org/10.33012/2023.19317
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Zhang, L., Hoi-Fung, Ng, Zhang, G., & Hsu, L.-T. (2023).3D Mapping Aided GNSS Positioning Using Doppler Frequency for Urban Areas. Proceedings of the 36th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2023). https://doi.org/10.33012/2023.19340
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Liu, X., Wen, W., & Hsu, L.-T. (2023).3D LiDAR Aided GNSS NLOS Correction with Direction of Arrival Estimation Using Doppler Measurements. Proceedings of the 36th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2023). https://doi.org/10.33012/2023.19447

2022 (8)

patent
Hsu, L. T& Wen, W (2022).3D LiDAR Aided GNSS and the Method for NLOS Detection and Correction (US 2023/0033404 A1). U.S. Patent and Trademark Office. https://patents.google.com/patent/US20230033404A1
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Wen, W., Zhang, G., & Hsu, L. T. (2022).GNSS Outlier Mitigation via Graduated Non-Convexity Factor Graph Optimization. IEEE Transactions on Vehicular Technology. https://doi.org/10.1109/tvt.2021.3130909
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Zhang, L.& Hsu, L. T. and Zhang T. (2022).A Novel INS/USBL Integrated Navigation Scheme Via Factor Graph Optimization. IEEE Transactions on Vehicular Technology. https://doi.org/10.1109/tvt.2022.3177739
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Meng, Q. and Hsu L. T. (2022).Resilient Interactive Sensor-Independent-Update Fusion Navigation Method. IEEE Transactions on Intelligent Transportation Systems. https://doi.org/10.1109/tits.2022.3150273
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Wen, W.& Hsu, L. T. (2022).AGPC-SLAM: Absolute Ground Plane Constrained 3D LiDAR SLAM. NAVIGATION: Journal of the Institute of Navigation. https://doi.org/10.33012/navi.527
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Liu, X., Wen, W., & Hsu, L.-T. (2022).3D LiDAR Aided GNSS Real-time Kinematic Positioning via Coarse-to-fine Batch Optimization for High Accuracy Mapping in Dense Urban Canyons. Proceedings of the 35th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2022). https://doi.org/10.33012/2022.18545

2021 (10)

patent
Zhang, G& Hsu, L. T (2021).3D map-aided GNSS cooperative positioning method and system (ZL 201910866398.X). China National Intellectual Property Administration. https://patents.google.com/patent/CN110426717B
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Wen, W., Pfeifer, T., Bai, X., & Hsu, L. T. (2021).Factor Graph Optimization for GNSS/INS Integration: A Comparison with the Extended Kalman Filter. NAVIGATION: Journal of the Institute of Navigation. https://doi.org/10.1002/navi.421
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H.F., N.& Hsu, L.T. (2021).3D Mapping Database Aided GNSS RTK and Its Assessments in Urban Canyons. IEEE Transactions on Aerospace and Electronic Systems. https://doi.org/10.1109/taes.2021.3069271
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2020 (7)

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Wen, W., Zhang, G., & Hsu, L. T. (2020).Object Detection Aided GNSS and Its Integration with LiDAR in Highly Urbanized Areas. IEEE Intelligent Transportation Systems Magazine. https://doi.org/10.1109/mits.2020.2994131
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Zhang, G., Wen, W. Xu B., & Hsu, L. T. (2020).Extending Shadow Matching to Tightly-Coupled GNSS/INS Integration System. IEEE Transactions on Vehicular Technology. https://doi.org/10.1109/tvt.2020.2981093
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Zhang, G., F., N. H., & Hsu, L. T. (2020).3D Mapping Database Aided GNSS Based Collaborative Positioning Using Factor Graph Optimization. IEEE Transactions on Intelligent Transportation Systems. https://doi.org/10.1109/tits.2020.2988531

2019 (4)

book
Hsu, L. T., Wen, W., & Zhang, G (2019).Chapter 2 - Vehicular Positioning System. Nova Science Publishers Inc.
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Wen, W., Zhang, G., & Hsu, L. T. (2019).Correcting NLOS by 3D LiDAR and building height to improve GNSS single point positioning. NAVIGATION: Journal of the Institute of Navigation. https://doi.org/10.1002/navi.335
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Wen, W., Bai, X., Kan, Y.C., & Hsu, L. T. (2019).Tightly Coupled GNSS/INS Integration Via Factor Graph and Aided by Fish-eye Camera. IEEE Transactions on Vehicular Technology. https://doi.org/10.1109/tvt.2019.2944680
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