{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T18:09:20Z","timestamp":1772042960649,"version":"3.50.1"},"reference-count":59,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,4,10]],"date-time":"2018-04-10T00:00:00Z","timestamp":1523318400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>To overcome the huge volume problem of light detection and ranging (LiDAR) data for the derivation of digital terrain models (DTMs), a least squares compactly supported radial basis function (CSRBF) interpolation method is proposed in this paper. The proposed method has a limited support radius and fewer RBF centers than the sample points, selected by a newly developed surface variation-based algorithm. Those make the linear system of the proposed method not only much sparser but also efficiently solvable. Tests on a synthetic dataset demonstrate that the proposed method is comparable to the smoothing RBF, and far superior to the exact RBF. Moreover, the first is much faster than the others. The proposed method with the RBF centers selected by the surface variation-based algorithm obviously outperforms that with the random selection of equal number. Real-world examples on one private and ten public datasets show that the surfaces of simple interpolation methods including inverse distance weighting, natural neighbor, linear and bicubic suffer from the problems of roughness, peak-cutting, discontinuity and subtle terrain feature loss, respectively. By contrast, the proposed method produces visually appealing results, keeping a good tradeoff between noise removal and terrain feature preservation. Additionally, the new method compares favorably with ordinary kriging (OK) for the generation of high-resolution DTMs in terms of interpolation accuracy, yet the former is much more robust to spatial resolution variation and terrain characteristics than the latter. More importantly, our method is about 4 times faster than OK. In conclusion, the proposed method has high potential for the interpolation of a large LiDAR dataset, especially when both interpolation accuracy and computational cost are taken into account.<\/jats:p>","DOI":"10.3390\/rs10040587","type":"journal-article","created":{"date-parts":[[2018,4,10]],"date-time":"2018-04-10T13:06:08Z","timestamp":1523365568000},"page":"587","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Least Squares Compactly Supported Radial Basis Function for Digital Terrain Model Interpolation from Airborne Lidar Point Clouds"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0995-5308","authenticated-orcid":false,"given":"Chuanfa","family":"Chen","sequence":"first","affiliation":[{"name":"State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China"},{"name":"Shandong Provincial Key Laboratory of Geomatics and Digital Technology of Shandong Province, Shandong University of Science and Technology, Qingdao 266590, China"}]},{"given":"Yanyan","family":"Li","sequence":"additional","affiliation":[{"name":"Shandong Provincial Key Laboratory of Geomatics and Digital Technology of Shandong Province, Shandong University of Science and Technology, Qingdao 266590, China"}]},{"given":"Na","family":"Zhao","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Resources and Environment Information System, Institute of Geographical Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7886-0172","authenticated-orcid":false,"given":"Bin","family":"Guo","sequence":"additional","affiliation":[{"name":"Shandong Provincial Key Laboratory of Geomatics and Digital Technology of Shandong Province, Shandong University of Science and Technology, Qingdao 266590, China"}]},{"given":"Naixia","family":"Mou","sequence":"additional","affiliation":[{"name":"Shandong Provincial Key Laboratory of Geomatics and Digital Technology of Shandong Province, Shandong University of Science and Technology, Qingdao 266590, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,4,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"8631","DOI":"10.3390\/rs70708631","article-title":"Interpolation routines assessment in ALS-derived digital elevation models for forestry applications","volume":"7","author":"Montealegre","year":"2015","journal-title":"Remote Sens."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.cageo.2016.11.006","article-title":"Anti-aliasing filters for deriving high-accuracy DEMs from TLS data: A case study from Freeport, Texas","volume":"100","author":"Xiong","year":"2017","journal-title":"Comput. Geosci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.geomorph.2011.03.012","article-title":"Digital terrain modeling","volume":"137","author":"Wilson","year":"2012","journal-title":"Geomorphology"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1294","DOI":"10.3390\/rs6021294","article-title":"Segmentation-based filtering of airborne LiDAR point clouds by progressive densification of terrain segments","volume":"6","author":"Lin","year":"2014","journal-title":"Remote Sens."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/j.geomorph.2014.03.008","article-title":"High-resolution topography for understanding Earth surface processes: Opportunities and challenges","volume":"216","author":"Tarolli","year":"2014","journal-title":"Geomorphology"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.geomorph.2010.11.003","article-title":"Airborne laser scanning of forested landslides characterization: Terrain model quality and visualization","volume":"126","author":"Razak","year":"2011","journal-title":"Geomorphology"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5669","DOI":"10.1080\/01431160802709237","article-title":"Adaptive mapped least squares SVM-based smooth fitting method for DSM generation of LIDAR data","volume":"30","author":"Shi","year":"2009","journal-title":"Int. J. Remote Sens."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3889","DOI":"10.1080\/01431160500181671","article-title":"LIDAR density and linear interpolator effects on elevation estimates","volume":"26","author":"Anderson","year":"2005","journal-title":"Int. J. Remote Sens."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.cageo.2008.09.001","article-title":"Evaluating error associated with lidar-derived DEM interpolation","volume":"35","author":"Bater","year":"2009","journal-title":"Comput. Geosci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1080\/15481603.2014.980086","article-title":"Effect of point density and interpolation of LiDAR-derived high-resolution DEMs on landscape scarp identification","volume":"51","author":"Chu","year":"2014","journal-title":"GISci. Remote Sens."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"701","DOI":"10.14358\/PERS.76.6.701","article-title":"Effects of topographic variability and Lidar sampling density on several DEM interpolation methods","volume":"76","author":"Guo","year":"2010","journal-title":"Photogramm. Eng. Remote Sens."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1109\/LGRS.2005.848533","article-title":"Simultaneous spline approximation and topographic analysis for lidar elevation data in open-source GIS","volume":"2","author":"Mitasova","year":"2005","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1109\/LGRS.2006.887055","article-title":"Spatial interpolation of elevation data with variable density: A new methodology to derive quality DEMs","volume":"4","author":"Hofierka","year":"2007","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"805","DOI":"10.14358\/PERS.71.7.805","article-title":"Effects of terrain morphology, sampling density, and interpolation methods on grid DEM accuracy","volume":"71","author":"Aguilar","year":"2005","journal-title":"Photogramm. Eng. Remote Sens."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.cageo.2017.08.007","article-title":"Big geo data surface approximation using radial basis functions: A comparative study","volume":"109","author":"Majdisova","year":"2017","journal-title":"Comput. Geosci."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Wendland, H. (2004). Scattered Data Approximation, Cambridge University Press.","DOI":"10.1017\/CBO9780511617539"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/j.geomorph.2016.06.025","article-title":"A robust interpolation method for constructing digital elevation models from remote sensing data","volume":"268","author":"Chen","year":"2016","journal-title":"Geomorphology"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.1109\/TGRS.2017.2758795","article-title":"Robust interpolation of DEMs from Lidar-derived elevation data","volume":"56","author":"Chen","year":"2018","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1023\/A:1018932227617","article-title":"Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration","volume":"11","author":"Beatson","year":"1999","journal-title":"Adv. Comput. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/imanum\/drh021","article-title":"A Krylov subspace algorithm for multiquadric interpolation in many dimensions","volume":"25","author":"Faul","year":"2005","journal-title":"IMA J. Numer. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1876","DOI":"10.1137\/060662083","article-title":"Fast radial basis function interpolation via preconditioned Krylov iteration","volume":"29","author":"Gumerov","year":"2007","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.apnum.2016.11.003","article-title":"Reconstruction of 3D scattered data via radial basis functions by efficient and robust techniques","volume":"113","author":"Crivellaro","year":"2017","journal-title":"Appl. Numer. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/0898-1221(90)90272-L","article-title":"Theory and applications of the multiquadric-biharmonic method. 20 years of discovery 1968\u20131988","volume":"19","author":"Hardy","year":"1990","journal-title":"Comput. Math. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1717","DOI":"10.1137\/S1064827599361771","article-title":"Fast solution of the radial basis function interpolation equations: Domain decomposition methods","volume":"22","author":"Beatson","year":"2001","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_25","unstructured":"Chui, C.K., Schumaker, L.L., and St\u00f6ckler, J. (2002). Fast evaluation of radial basis functions: Methods based on partition of unity. Approximation Theory X: Wavelets, Splines, and Applications, Vanderbilt University Press."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1007\/s11004-011-9346-5","article-title":"Moving surface spline interpolation based on Green\u2019s function","volume":"43","author":"Deng","year":"2011","journal-title":"Math. Geosci."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Fasshauer, G.E. (2007). Meshfree Approximation Methods with MATLAB, World Scientific Publishing.","DOI":"10.1142\/6437"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1823","DOI":"10.1190\/1.1527082","article-title":"Smooth fitting of geophysical data using continuous global surfaces","volume":"67","author":"Billings","year":"2002","journal-title":"Geophysics"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J., Fright, W.R., McCallum, B.C., and Evans, T.R. (2001). Reconstruction and Representation of 3D Objects with Radial Basis Functions, Siggraph\/ACM.","DOI":"10.1145\/383259.383266"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1109\/72.80341","article-title":"Orthogonal least squares learning algorithm for radial basis function networks","volume":"2","author":"Chen","year":"1991","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/BF02519037","article-title":"Least squares surface approximation to scattered data using multiquadratic functions","volume":"2","author":"Franke","year":"1994","journal-title":"Adv. Comput. Math."},{"key":"ref_32","unstructured":"Ohtake, Y., Belyaev, A., and Seidel, H.P. (2004, January 7\u20139). 3D scattered data approximation with adaptive compactly supported radial basis functions. Proceedings of the Shape Modeling Applications, Genova, Italy."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1854","DOI":"10.1111\/j.1467-8659.2010.01653.x","article-title":"Surface reconstruction based on hierarchical floating radial basis functions","volume":"29","author":"Meyer","year":"2010","journal-title":"Comput. Gr. Forum"},{"key":"ref_34","first-page":"501","article-title":"Fast interpolation and approximation of scattered multidimensional and dynamic data using radial basis functions","volume":"12","author":"Skala","year":"2013","journal-title":"WSEAS Trans. Math."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Bjorck, A. (1996). Numerical Methods for Least Squares Problems, SIAM.","DOI":"10.1137\/1.9781611971484"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.1029\/JB076i008p01905","article-title":"Multiquadric equations of topography and other irregular surfaces","volume":"76","author":"Hardy","year":"1971","journal-title":"J. Geophys. Res."},{"key":"ref_37","unstructured":"Pauly, M., Gross, M., and Kobbelt, L.P. (November, January 27). Efficient simplification of point-sampled surfaces. Proceedings of the Conference on Visualization\u201902, Boston, MA, USA."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Saad, Y. (2003). Iterative Methods for Sparse Linear Systems, SIAM.","DOI":"10.1137\/1.9780898718003"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Peckham, R.J., and Jordan, G. (2007). Optimisation of interpolation parameters using cross-validation. Digital Terrain Modelling, Springer.","DOI":"10.1007\/978-3-540-36731-4"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"6623","DOI":"10.1007\/s12517-014-1717-z","article-title":"A generalization of inverse distance weighting method via kernel regression and its application to surface modeling","volume":"8","author":"Chen","year":"2015","journal-title":"Arabian J. Geosci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1044","DOI":"10.1016\/j.cageo.2007.07.010","article-title":"An adaptive inverse-distance weighting spatial interpolation technique","volume":"34","author":"Lu","year":"2008","journal-title":"Comput. Geosci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1080\/02693799008941549","article-title":"Kriging: A method of interpolation for geographical information systems","volume":"4","author":"Oliver","year":"1990","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation, Oxford University Press.","DOI":"10.1093\/oso\/9780195115383.001.0001"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Chiles, J.P., and Delfiner, P. (1999). Geostatistics: Modeling Spatial Uncertainty, Wiley.","DOI":"10.1002\/9780470316993"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2981","DOI":"10.1080\/0143116031000086835","article-title":"Higher-order interpolation of regular grid digital elevation models","volume":"24","author":"Kidner","year":"2003","journal-title":"Int. J. Remote Sens."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3069","DOI":"10.1080\/01431160500057905","article-title":"Estimating the propagation error of DEM from higher-order interpolation algorithms","volume":"26","author":"Shi","year":"2005","journal-title":"Int. J. Remote Sens."},{"key":"ref_47","unstructured":"Sibson, R. (1981). A Brief Description of Natural Neighbour Interpolation, John Wiley & Sons. Interpreting Multivariate Data."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Ledoux, H., and Gold, C. (2005). An efficient natural neighbour interpolation algorithm for geoscientific modelling. Developments in Spatial Data Handling, Springer.","DOI":"10.1007\/3-540-26772-7_8"},{"key":"ref_49","unstructured":"(2018, February 10). Ordinary Kriging-File Exchange. Available online: https:\/\/cn.mathworks.com\/matlabcentral\/fileexchange\/29025-ordinary-kriging."},{"key":"ref_50","unstructured":"(2018, February 10). Experimental (Semi-) Variogram-File Exchange. Available online: https:\/\/cn.mathworks.com\/matlabcentral\/fileexchange\/20355-experimental--semi---variogram."},{"key":"ref_51","unstructured":"(2018, February 10). variogramfit-File Exchange. Available online: https:\/\/cn.mathworks.com\/matlabcentral\/fileexchange\/25948-variogramfit."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.envsoft.2017.05.009","article-title":"Terrestrial laser scanning improves digital elevation models and topsoil pH modelling in regions with complex topography and dense vegetation","volume":"95","author":"Baltensweiler","year":"2017","journal-title":"Environ. Model. Softw."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1002\/esp.1731","article-title":"A comparison of interpolation methods for producing digital elevation models at the field scale","volume":"34","author":"Erdogan","year":"2009","journal-title":"Earth Surf. Process. Landf."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1111\/j.1467-9671.2005.00233.x","article-title":"Modelling the spatial distribution of DEM error","volume":"9","author":"Carlisle","year":"2005","journal-title":"Trans. GIS"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.isprsjprs.2004.05.004","article-title":"Experimental comparison of filter algorithms for bare-Earth extraction from airborne laser scanning point clouds","volume":"59","author":"Sithole","year":"2004","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.geomorph.2007.02.006","article-title":"A new method of surface modelling and its application to DEM construction","volume":"91","author":"Yue","year":"2007","journal-title":"Geomorphology"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1080\/13658810110060442","article-title":"Detecting outliers in irregularly distributed spatial data sets by locally adaptive and robust statistical analysis and GIS","volume":"15","author":"Liu","year":"2001","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1400","DOI":"10.1016\/j.patcog.2014.10.014","article-title":"Outlier detection and robust normal-curvature estimation in mobile laser scanning 3D point cloud data","volume":"48","author":"Nurunnabi","year":"2015","journal-title":"Pattern Recogn."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"3347","DOI":"10.3390\/rs70303347","article-title":"A robust algorithm of multiquadric method based on an improved huber loss function for interpolating remote-sensing-derived elevation data sets","volume":"7","author":"Chen","year":"2015","journal-title":"Remote Sens."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/10\/4\/587\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:00:15Z","timestamp":1760194815000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/10\/4\/587"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,10]]},"references-count":59,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,4]]}},"alternative-id":["rs10040587"],"URL":"https:\/\/doi.org\/10.3390\/rs10040587","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,4,10]]}}}