{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T04:05:52Z","timestamp":1770696352830,"version":"3.49.0"},"reference-count":44,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,1,25]],"date-time":"2026-01-25T00:00:00Z","timestamp":1769299200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["42425003"],"award-info":[{"award-number":["42425003"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Dual-LiDAR systems are widely deployed in autonomous driving, yet extrinsic calibration remains challenging in non-overlapping field-of-view (FoV) configurations where correspondence-based methods are unreliable. We propose an engineering-oriented 2.5D calibration framework that estimates horizontal extrinsics (x,y,yaw) via motion-guided planar alignment and then refines them using Gaussian Process Implicit Surfaces (GPIS), which provide continuous and probabilistic surface constraints from spatially disjoint scans. This design avoids calibration targets and reduces dependence on strong scene assumptions, improving robustness under noise and weak structure. Extensive high-fidelity simulation experiments demonstrate centimeter-level lateral accuracy and sub-degree yaw error, consistently outperforming representative motion-based and BEV-based baselines under both clean and noisy settings. To further assess real-world applicability, we conduct a preliminary nuScenes case study by splitting LiDAR scans into front and rear subsets to emulate a non-overlapping dual-LiDAR setup, achieving improved yaw accuracy and competitive lateral precision. Overall, the proposed method serves as a practical refinement stage for non-overlapping dual-LiDAR calibration, with a favorable balance of accuracy, robustness, and engineering feasibility.<\/jats:p>","DOI":"10.3390\/s26030800","type":"journal-article","created":{"date-parts":[[2026,1,26]],"date-time":"2026-01-26T16:13:48Z","timestamp":1769444028000},"page":"800","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["GPIS-Based Calibration for Non-Overlapping Dual-LiDAR Systems Using a 2.5D Calibration Framework"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-5733-3432","authenticated-orcid":false,"given":"Huan","family":"Yu","sequence":"first","affiliation":[{"name":"School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaohong","family":"Zhang","sequence":"additional","affiliation":[{"name":"Chinese Antarctic Center of Surveying and Mapping, Wuhan University, Wuhan 430079, China"},{"name":"Hubei Luojia Laboratory, Wuhan 430079, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ming","family":"Li","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-1567-2929","authenticated-orcid":false,"given":"Desheng","family":"Zhuo","sequence":"additional","affiliation":[{"name":"School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pin","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Computer Science, Wuhan University, Wuhan 430079, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Man","family":"Li","sequence":"additional","affiliation":[{"name":"Dongfeng Motor Corporation Technical Center, Wuhan 430010, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuanyuan","family":"Shi","sequence":"additional","affiliation":[{"name":"Dongfeng Motor Corporation Technical Center, Wuhan 430010, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lyu, X., Liu, S., Qiao, R., Jiang, S., and Wang, Y. (2025). Camera, LiDAR, and IMU Spatiotemporal Calibration: Methodological Review and Research Perspectives. Sensors, 25.","DOI":"10.3390\/s25175409"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Jiao, J., Yu, Y., Liao, Q., Ye, H., Fan, R., and Liu, M. (2019, January 3\u20138). Automatic Calibration of Multiple 3d Lidars in Urban Environments. Proceedings of the 2019 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Macau, China.","DOI":"10.1109\/IROS40897.2019.8967797"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Wei, P., Yan, G., Li, Y., Fang, K., Cai, X., Yang, J., and Liu, W. (2022). CROON: Automatic Multi-LiDAR Calibration and Refinement Method in Road Scene. arXiv.","DOI":"10.1109\/IROS47612.2022.9981558"},{"key":"ref_4","first-page":"8501707","article-title":"An Overlap-Free Calibration Method for LiDAR\u2013Camera Platforms Based on Environmental Perception","volume":"72","author":"Zhang","year":"2023","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_5","first-page":"8502612","article-title":"Targetless Extrinsic Calibration of Multiple Small FoV LiDARs and Cameras Using Adaptive Voxelization","volume":"71","author":"Liu","year":"2022","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_6","first-page":"177","article-title":"BLCC: A Benchmark for Multi-LiDAR and Multi-camera Calibration","volume":"Volume 15520","author":"Ide","year":"2025","journal-title":"MultiMedia Modeling, Proceedings of the 31st International Conference on Multimedia Modeling, MMM 2025, Nara, Japan, 8\u201310 January 2025"},{"key":"ref_7","unstructured":"Velodyne Lidar Inc. (2026, January 15). Velodyne HDL-32E Datasheet, 2018. Available online: https:\/\/www.mapix.com\/wp-content\/uploads\/2018\/07\/97-0038_Rev-M_-HDL-32E_Datasheet_Web.pdf."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Yin, L., Wang, W., Yu, H., and Luo, B. (2020, January 6\u20138). Targetless Extrinsic Calibration for Multi-beam LIDARs with Narrow Overlapping Field of View. Proceedings of the 2020 Chinese Automation Congress (CAC), Shanghai, China.","DOI":"10.1109\/CAC51589.2020.9326755"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1109\/TIM.2017.2757148","article-title":"A Target-Free Automatic Self-Calibration Approach for Multibeam Laser Scanners","volume":"67","author":"Gong","year":"2017","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1177\/02783649922066213","article-title":"Hand-Eye Calibration Using Dual Quaternions","volume":"18","author":"Daniilidis","year":"1999","journal-title":"Int. J. Robot. Res."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Maroli, J.M., Ozguner, U., Redmill, K., and Kurt, A. (2017, January 16\u201319). Automated Rotational Calibration of Multiple 3D LIDAR Units for Intelligent Vehicles. Proceedings of the 2017 IEEE 20th International Conference on Intelligent Transportation Systems (ITSC), Yokohama, Japan.","DOI":"10.1109\/ITSC.2017.8317598"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Zhu, Z., and Liu, J. (2013, January 22\u201323). Unsupervised Extrinsic Parameters Calibration for Multi-beam LIDARs. Proceedings of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013), Hangzhou, China.","DOI":"10.2991\/iccsee.2013.278"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Kulmer, D., Tahiraj, I., Chumak, A., and Lienkamp, M. (2025). Multi-LiCa: A Motion and Targetless Multi LiDAR-to-LiDAR Calibration Framework. arXiv.","DOI":"10.1109\/MFI62651.2024.10705773"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zhou, Y., Han, T., Nie, Q., Zhu, Y., Li, M., Bian, N., and Li, Z. (2024). Adaptive Point-Line Fusion: A Targetless LiDAR\u2013Camera Calibration Method with Scheme Selection for Autonomous Driving. Sensors, 24.","DOI":"10.3390\/s24041127"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1109\/2.30720","article-title":"Using Occupancy Grids for Mobile Robot Perception and Navigation","volume":"22","author":"Elfes","year":"1989","journal-title":"Computer"},{"key":"ref_16","unstructured":"Biber, P., and Strasser, W. (2003, January 27\u201331). The Normal Distributions Transform: A New Approach to Laser Scan Matching. Proceedings of the 2003 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS 2003) (Cat. No.03CH37453), Las Vegas, NV, USA."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1109\/TRO.2021.3078287","article-title":"Robust Odometry and Mapping for Multi-LiDAR Systems with Online Extrinsic Calibration","volume":"38","author":"Jiao","year":"2021","journal-title":"IEEE Trans. Robot."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Shan, T., and Englot, B. (2018, January 1\u20135). LeGO-LOAM: Lightweight and Ground-Optimized Lidar Odometry and Mapping on Variable Terrain. Proceedings of the 2018 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Madrid, Spain.","DOI":"10.1109\/IROS.2018.8594299"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Lin, J., and Zhang, F. (August, January 31). Loam Livox: A Fast, Robust, High-Precision LiDAR Odometry and Mapping Package for LiDARs of Small FoV. Proceedings of the 2020 IEEE International Conference on Robotics and Automation (ICRA), Paris, France.","DOI":"10.1109\/ICRA40945.2020.9197440"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Himmelsbach, M., Hundelshausen, F.V., and Wuensche, H.J. (2010, January 2\u201324). Fast Segmentation of 3D Point Clouds for Ground Vehicles. Proceedings of the 2010 IEEE Intelligent Vehicles Symposium, La Jolla, CA, USA.","DOI":"10.1109\/IVS.2010.5548059"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Jiao, J., Liao, Q., Zhu, Y., Liu, T., Yu, Y., Fan, R., Wang, L., and Liu, M. (2019, January 9\u201312). A Novel Dual-Lidar Calibration Algorithm Using Planar Surfaces. Proceedings of the 2019 IEEE Intelligent Vehicles Symposium (IV), Paris, France.","DOI":"10.1109\/IVS.2019.8814136"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1111\/j.1467-8659.2007.01016.x","article-title":"Efficient RANSAC for Point-Cloud Shape Detection","volume":"26","author":"Schnabel","year":"2007","journal-title":"Comput. Graph. Forum"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s10514-012-9321-0","article-title":"OctoMap: An Efficient Probabilistic 3D Mapping Framework Based on Octrees","volume":"34","author":"Hornung","year":"2013","journal-title":"Auton. Robot."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1309","DOI":"10.1109\/TRO.2016.2624754","article-title":"Past, Present, and Future of Simultaneous Localization and Mapping: Toward the Robust-Perception Age","volume":"32","author":"Cadena","year":"2016","journal-title":"IEEE Trans. Robot."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1618","DOI":"10.1109\/TRO.2025.3532506","article-title":"iKalibr: Unified Targetless Spatiotemporal Calibration for Resilient Integrated Inertial Systems","volume":"41","author":"Chen","year":"2025","journal-title":"IEEE Trans. Robot."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3734","DOI":"10.1109\/TRO.2022.3174476","article-title":"Observability-aware Intrinsic and Extrinsic Calibration of LiDAR-IMU Systems","volume":"38","author":"Lv","year":"2022","journal-title":"IEEE Trans. Robot."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"103071","DOI":"10.1016\/j.inffus.2025.103071","article-title":"Accurate and Automatic Spatiotemporal Calibration for Multi-modal Sensor System Based on Continuous-time Optimization","volume":"120","author":"Li","year":"2025","journal-title":"Inf. Fusion"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.isprsjprs.2023.04.004","article-title":"AFLI-Calib: Robust LiDAR-IMU Extrinsic Self-calibration Based on Adaptive Frame Length LiDAR Odometry","volume":"199","author":"Wu","year":"2023","journal-title":"ISPRS J. Photogramm. Remote. Sens."},{"key":"ref_29","unstructured":"Williams, O., and Fitzgibbon, A. (2006, January 12\u201313). Gaussian Process Implicit Surfaces. Proceedings of the Gaussian Processes in Practice, Bletchley Park, UK."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Mahler, J., Patil, S., Kehoe, B., van den Berg, J., Ciocarlie, M., Abbeel, P., and Goldberg, K. (2015, January 26\u201330). GP-GPIS-OPT: Grasp Planning with Shape Uncertainty Using Gaussian Process Implicit Surfaces and Sequential Convex Programming. Proceedings of the 2015 IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, USA.","DOI":"10.1109\/ICRA.2015.7139882"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"4078","DOI":"10.1109\/TRO.2023.3296982","article-title":"Log-GPIS-MOP: A Unified Representation for Mapping, Odometry, and Planning","volume":"39","author":"Wu","year":"2023","journal-title":"IEEE Trans. Robot."},{"key":"ref_32","unstructured":"Khan, M., Ibuki, T., and Chatterjee, A. (2024). Gaussian Process Implicit Surfaces as Control Barrier Functions for Safe Robot Navigation. arXiv."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"10994","DOI":"10.1109\/LRA.2024.3474517","article-title":"Constraining Gaussian Process Implicit Surfaces for Robot Manipulation via Dataset Refinement","volume":"9","author":"Kumar","year":"2024","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2461","DOI":"10.1109\/LRA.2021.3061356","article-title":"Faithful Euclidean Distance Field from Log-Gaussian Process Implicit Surfaces","volume":"6","author":"Wu","year":"2021","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/978-3-319-07488-7_22","article-title":"GPmap: A Unified Framework for Robotic Mapping Based on Sparse Gaussian Processes","volume":"Volume 105","author":"Mejias","year":"2015","journal-title":"Field and Service Robotics"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Jadidi, M.G., Mir\u00f3, J.V., Valencia, R., and Andrade-Cetto, J. (June, January 31). Exploration on Continuous Gaussian Process Frontier Maps. Proceedings of the 2014 IEEE International Conference on Robotics and Automation (ICRA), Hong Kong, China.","DOI":"10.1109\/ICRA.2014.6907754"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1177\/0278364911421039","article-title":"Gaussian Process Occupancy Maps","volume":"31","author":"Ramos","year":"2012","journal-title":"Int. J. Robot. Res."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Osher, S., and Fedkiw, R. (2003). Level Set Methods and Dynamic Implicit Surfaces, Springer Nature.","DOI":"10.1007\/b98879"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Hendriks, F., Tideman, M., Pelders, R., Bours, R., and Liu, X. (2010, January 15\u201317). Development Tools for Active Safety Systems: Prescan and VeHIL. Proceedings of the 2010 IEEE International Conference on Vehicular Electronics and Safety, QingDao, China.","DOI":"10.1109\/ICVES.2010.5550948"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Nacu, C.R., Fodorean, D., Husar, C., Grovu, M., and Irimia, C. (2018, January 20\u201322). Towards Autonomous EV by Using Virtual Reality and Prescan-Simulink Simulation Environments. Proceedings of the 2018 International Symposium on Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM), Amalfi, Italy.","DOI":"10.1109\/SPEEDAM.2018.8445211"},{"key":"ref_41","unstructured":"Low, K.L. (2004). Linear Least-Squares Optimization for Point-to-Plane Icp Surface Registration, University of North Carolina."},{"key":"ref_42","first-page":"1","article-title":"LOAM: Lidar Odometry and Mapping in Real-Time","volume":"Volume 2","author":"Zhang","year":"2014","journal-title":"Proceedings of the Robotics: Science and Systems, Berkeley, CA, USA, 12\u201316 July 2014"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Tan, Z., Zhang, X., Teng, S., Wang, L., and Gao, F. (2024). A Review of Deep Learning-Based LiDAR and Camera Extrinsic Calibration. Sensors, 24.","DOI":"10.3390\/s24123878"},{"key":"ref_44","first-page":"63","article-title":"Gaussian Processes in Machine Learning","volume":"Volume 3176","author":"Bousquet","year":"2004","journal-title":"Advanced Lectures on Machine Learning"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/3\/800\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T05:14:32Z","timestamp":1769836472000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/3\/800"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,25]]},"references-count":44,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["s26030800"],"URL":"https:\/\/doi.org\/10.3390\/s26030800","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,25]]}}}