Journal of Real-Time Image Processing

, Volume 16, Issue 2, pp 459–475 | Cite as

Fast total least squares vectorization

  • Ales JelinekEmail author
  • Ludek Zalud
  • Tomas Jilek
Original Research Paper


This paper proposes a novel algorithm for the vectorization of ordered sets of points, named Fast Total Least Squares (FTLS) vectorization. The emphasis was put on low computational complexity, which allows it to be run online on a mobile device at a speed comparable to the fastest algorithms, such as the Douglas–Peucker (DP) algorithm, while maintaining a much higher quality of the approximation. Our approach is based on the total least squares method, therefore all the points from the cloud contribute to its approximation. This leads to better utilization of the information contained in the point cloud, compared to those algorithms based on point elimination, such as DP. Several experiments and performance comparisons are presented to demonstrate the most important attributes of the FTLS algorithm.


Point cloud Vectorization Least squares Robotics Linear regression 



This work was supported by the Technology Agency of the Czech Republic under the project TE01020197 “Centre for Applied Cybernetics 3”.


  1. 1.
    Lu, Z., Baek, S., Lee, S.: Robust 3D line extraction from stereo point clouds. In: 2008 IEEE Conference on Robotics, Automation and Mechatronics, vol. 00, pp. 1–5, IEEE (2008)Google Scholar
  2. 2.
    Hirose, K., Saito, H.: Fast line description for line-based SLAM. In: Procedings of the British Machine Vision Conference 2012, pp. 83.1–83.11, British Machine Vision Association (2012)Google Scholar
  3. 3.
    Nguyen, V., Gächter, S., Martinelli, A., Tomatis, N., Siegwart, R.: A comparison of line extraction algorithms using 2D range data for indoor mobile robotics. Auton. Robots 23, 97–111 (2007)CrossRefGoogle Scholar
  4. 4.
    Pears, N.: Feature extraction and tracking for scanning range sensors. Robot. Auton. Syst. 33, 43–58 (2000)CrossRefGoogle Scholar
  5. 5.
    Shi, W., Cheung, C.: Performance evaluation of line simplification algorithms for vector generalization. Cartogr. J. 43, 27–44 (2006)CrossRefGoogle Scholar
  6. 6.
    Liu, J., Zhang, J., Xu, F., Huang, Z., Li, Y.: Adaptive algorithm for automated polygonal approximation of high spatial resolution remote sensing imagery segmentation contours. IEEE Trans. Geosci. Remote Sens. 52, 1099–1106 (2014)CrossRefGoogle Scholar
  7. 7.
    Zhao, J., You, S., Huang, J.: Rapid extraction and updating of road network from airborne LiDAR data. In: 2011 IEEE Applied Imagery Pattern Recognition Workshop (AIPR), pp. 1–7, IEEE (2011)Google Scholar
  8. 8.
    Dyken, C., Dæhlen, M., Sevaldrud, T.: Simultaneous curve simplification. J. Geogr. Syst. 11, 273–289 (2009)CrossRefGoogle Scholar
  9. 9.
    Kandal, P., Karschti, S.: Method for simplified storage of data representing forms. Patent US 8787703 B2, 22 July 2014. PrintGoogle Scholar
  10. 10.
    Lange, R., Dürr, F., Rothermel, K.: Efficient real-time trajectory tracking. VLDB J 20, 671–694 (2011)CrossRefGoogle Scholar
  11. 11.
    Popa, I.S., Zeitouni, K., Oria, V., Kharrat, A.: Spatio-temporal compression of trajectories in road networks. GeoInformatica 19, 117–145 (2015)CrossRefGoogle Scholar
  12. 12.
    Werner, M., Schauer, L., Scharf, A.: Reliable trajectory classification using Wi-Fi signal strength in indoor scenarios. In: 2014 IEEE/ION Position, Location and Navigation Symposium—PLANS 2014, pp. 663–670, IEEE (2014)Google Scholar
  13. 13.
    Thiebault, A., Tremblay, Y.: Splitting animal trajectories into fine-scale behaviorally consistent movement units: breaking points relate to external stimuli in a foraging seabird. Behav. Ecol. Sociobiol. 67, 1013–1026 (2013)CrossRefGoogle Scholar
  14. 14.
    Romadi, M., Oulah, R., Thami, H., Romadi, R., Chiheb, R.: Detection and recognition of road signs in a video stream based on the shape of the panels. In: 2014 9th International Conference on Intelligent Systems: Theories and Applications (SITA-14), pp. 1–5, IEEE (2014)Google Scholar
  15. 15.
    Danuser, G., Stricker, M.: Parametric model fitting: from inlier characterization to outlier detection. IEEE Trans. Pattern Anal. Mach. Intell. 20, 263–280 (1998)CrossRefGoogle Scholar
  16. 16.
    Arifoglu, D., Sahin, E., Adiguzel, H., Duygulu, P., Kalpakli, M.: Matching Islamic patterns in Kufic images. Pattern Anal. Appl. 18, 601–617 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rizzardi, M., Troisi, S.: Approximation of irregular polylines by means of a straight-line graph. Appl. Geomat. 3, 171–182 (2011)CrossRefGoogle Scholar
  18. 18.
    Gong, W., Mao, F., Song, S.: Signal simplification and cloud detection with an improved Douglas–Peucker algorithm for single-channel lidar. Meteorol. Atmos. Phys. 113, 89–97 (2011)CrossRefGoogle Scholar
  19. 19.
    Choi, T., Park, C., Do, H., Park, D., Kyung, J., Chung, G.: Trajectory correction based on shape peculiarity in direct teaching manipulator. Int. J. Control Autom. Syst. 11, 1009–1017 (2013)CrossRefGoogle Scholar
  20. 20.
    Hough, P.V.C.: Method and means for recognizing complex patterns. Patent US 3069654 A, 18 Dec 1962. PrintGoogle Scholar
  21. 21.
    Guerreiro, R.F.C., Aguiar, P.M.Q.: Connectivity-enforcing Hough transform for the robust extraction of line segments. IEEE Trans Image Process.: Publ. IEEE Signal Process. Soc. 21, 4819–4829 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ni, K., Armstrong-Crews, N., Sawyer, S.: Geo-registering 3D point clouds to 2D maps with scan matching and the Hough Transform. In: 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 1864–1868, IEEE (2013)Google Scholar
  23. 23.
    Fischler, M.A., Bolles, R.C.: Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Commun. ACM 24, 381–395 (1981)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mirmehdi, M., Palmer, P.L., Kittler, J.: Robust line segment extraction using genetic algorithms. In: Image Processing and Its Applications, 1997, Sixth International Conference on, vol. 1, pp. 141–145, IEEE (1997)Google Scholar
  25. 25.
    Cai, Y., Guo, Q.: Point set generalization based on the Kohonen Net. Geo-Spat. Inf. Sci. 11, 221–227 (2008)CrossRefGoogle Scholar
  26. 26.
    Naouai, M., Narjess, M., Hamouda, A.: Line recognition algorithm using constrained delaunay triangulation. In: Proceedings of the ELMAR, September 2010, pp. 15–17 (2010)Google Scholar
  27. 27.
    Guerreiro, R.F.C., Aguiar, P.M.Q.: Extraction of line segments in cluttered images via multiscale edges. In: 2013 IEEE International Conference on Image Processing, pp. 3045–3048, IEEE (2013)Google Scholar
  28. 28.
    Wenyin, L., Dori, D.: From raster to vectors: extracting visual information from line drawings. Pattern Anal. Appl. 2, 10–21 (1999)CrossRefzbMATHGoogle Scholar
  29. 29.
    Altantsetseg, E., Muraki, Y., Matsuyama, K., Konno, K.: Feature line extraction from unorganized noisy point clouds using truncated Fourier series. Vis Comput. 29, 617–626 (2013)CrossRefGoogle Scholar
  30. 30.
    Douglas, D.H., Peucker, T.K.: Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Cartogr: Int. J. Geogr. Inf. Geovis. 10, 112–122 (1973)CrossRefGoogle Scholar
  31. 31.
    Saalfeld, A.: Topologically consistent line simplification with the Douglas–Peucker Algorithm. Cartogr. Geogr. Inf. Sci. 26, 7–18 (1999)CrossRefGoogle Scholar
  32. 32.
    Ma, J., Xu, S., Pu, Y., Chen, G.: A real-time parallel implementation of Douglas–Peucker polyline simplification algorithm on shared memory multi-core processor computers. In: Proceedings of the ICCASM 2010–2010 International Conference on Computer Application and System Modeling, vol. 4, no. Iccasm, pp. 647–652 (2010)Google Scholar
  33. 33.
    Zhao, Z., Saalfeld, A.: Linear-time sleeve-fitting polyline simplification algorithms. In: Proceedings of AutoCarto, pp. 214–223 (1997)Google Scholar
  34. 34.
    Reumann, K., Witkam, A.P.M.: Optimizing curve segmentation in computer graphics. In: Proceedings of International Computing Symposium, (Amsterdam), pp. 467–472, North-Holland Publishing Company (1974)Google Scholar
  35. 35.
    Yin, J., Carlone, L., Rosa, S., Bona, B.:Graph-based robust localization and mapping for autonomous mobile robotic navigation. In: 2014 IEEE International Conference on Mechatronics and Automation, pp. 1680–1685, IEEE (2014)Google Scholar
  36. 36.
    Arras, K.O., Siegwart, R.: Feature extraction and scene interpretation for map-based navigation and map building. In: Gage, D.W. (ed.) Proceedings of SPIE 3210, Mobile Robots XII, vol. 3210, pp. 42–53 (1998)Google Scholar
  37. 37.
    Rippa, S.: Adaptive approximation by piecewise linear polynomials on triangulations of subsets of scattered data. SIAM J. Sci. Stat. Comput. 13, 1123–1141 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Garland, M., Heckbert, P.S.: Surface simplification using quadric error metrics. In: Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques—SIGGRAPH, no. ’97 May, pp. 209–216, ACM Press, New York, New York, USA (1997)Google Scholar
  39. 39.
    Chen, C., Yan, C., Cao, X., Guo, J., Dai, H.: A greedy-based multiquadric method for LiDAR-derived ground data reduction. ISPRS J. Photogramm. Remote Sens. 102, 110–121 (2015)CrossRefGoogle Scholar
  40. 40.
    Jensfelt, P.: Approaches to mobile robot localization in indoor environments. Ph.D. thesis, KTH (2001)Google Scholar
  41. 41.
    Diosi, A., Kleeman, L.: Uncertainty of line segments extracted from static SICK PLS laser scans. In: Australiasian Conference on Robotics and Automation, p. 10 (2002)Google Scholar
  42. 42.
    Adcock, R.J.: A problem in least squares. The Analyst 5, 53 (1878)CrossRefGoogle Scholar
  43. 43.
    Golub, G.H., van Loan, C.F.: An analysis of the total least squares problem. SIAM J. Numer. Anal. 17, 883–893 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Deming, E.W.: Statistical Adjustment of Data. Dover Publications, Mineola, New York (2011)zbMATHGoogle Scholar
  45. 45.
    Deriche, R., Vaillant, R., Faugeras, O.: From noisy edges points to 3D re-construction of a scene : a robust approach and its uncertainty analysis. Ser. Mach. Percept. Artif. Intell. 2, 71–79 (1992)Google Scholar
  46. 46.
    Hu, X., Li, X., Zhang, Y.: Fast filtering of LiDAR point cloud in urban areas based on scan line segmentation and GPU acceleration. IEEE Geosci. Remote Sens. Lett. 10, 308–312 (2013)CrossRefGoogle Scholar
  47. 47.
    Zalud, L., Kopecny, L., Burian, F.: Orpheus reconnissance robots. In: 2008 IEEE International Workshop on Safety, Security and Rescue Robotics, no. October, pp. 31–34, IEEE (2008)Google Scholar
  48. 48.
    Bailey, T.: Mobile robot localisation and mapping in extensive outdoor environments. The University of Sydney, Ph.d. (2002)Google Scholar
  49. 49.
    Tsardoulias, E., Petrou, L.: Critical rays scan match SLAM. J. Intell. Robot. Syst. 72, 441–462 (2013)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.CEITEC - Central European Institute of TechnologyBrno University of TechnologyBrnoCzech Republic

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