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Robust Estimation

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Geospatial Algebraic Computations

Abstract

In many fields of geosciences such as robotics [413], computer vision [351], digital photogrammetry [538], surface reconstruction [388], computational geometry [336], digital building modelling [48], forest planning and operational activities [386] to list but a few, it is a fundamental task to extract plane features from three-dimensional (3D) point clouds, – i.e., a vast amount of points reflected from the surface of objects collected – using the cutting edge remote sensing technology of laser scanning, e.g., [450]. Due to the physical limitations of the sensors, occlusions, multiple reflectance, and noise can produce off-surface points, thereby necessitating robust fitting techniques. Robust fitting means an estimation technique, which is able to estimate accurate model parameters not only consisting of small error magnitudes in the data set but occasionally large scale measurement errors (outliers). Outliers definition is not easy. Perhaps considering the problem from a practical point of the view, one can say that data points, whose appearance in the data set causes dramatically change in the result of the estimated parameters can be labeled as outliers. Basically, there are two different methods to handle outliers;

  1. (i)

    weighting out outliers

  2. (ii)

    discarding outliers

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References

  1. Awange JL, Grafarend EW (2002) Sylvester resultant solution of planar ranging problem. Allgemeine Vermessungs-Nachrichten 108:143–146

    Google Scholar 

  2. Awange JL, Grafarend EW (2005) Solving algebraic computational problems in geodesy and geoinformatics. Springer, Berlin

    Google Scholar 

  3. Awange JL (2010) GNSS environmental monitoring. Springer, Berlin

    Book  Google Scholar 

  4. Armenakis C, Gao Y, Sohn G (2010) Co-registration of lidar and photogrammetric data for updating building databases. In: ISPRS Archives, Haifa, vol 38, pp 96–100

    Google Scholar 

  5. Borrmann D, Elseberg J, Lingemann K, Nüchter A (2011) The 3D hough transform for plane detection in point clouds: a review and a new accumulator design. 3D Res 02:02003. 3DR EXPRESS

    Google Scholar 

  6. Chen CC, Stamos I (2007) Range image segmentation for modeling and object detection in urban scenes. In 3- D Digital Imaging and Modeling, 2007. (3DIM ’07). Sixth International Conference on, pp. 185–192, Quebec, Canada

    Chapter  Google Scholar 

  7. Chum O, Matas J (2002) Randomized RANSAC with T(d, d) test. In: Proceedings of the British Machine Vision Conference (BMVC ’02), Cardiff, vol 2. BMVA, pp 448–457

    Google Scholar 

  8. Chum O (2008) Optimal randomized RANSAC. IEEE Trans Pattern Anal Mach Intell 30:1472–1482

    Article  Google Scholar 

  9. DalleMole VL, do Rego RLME, Araújo AF (2010) The self-organizing approach for surface reconstruction from unstructured point clouds. In: Matsopoulos GK (ed) Computer and information science, artificial intelligence, “Self-Organizing Maps”. doi:10.5772/9180

    Google Scholar 

  10. Deschaud JE, Goulette F (2010) A fast and accurate plane detection algorithm for large noisy point clouds using filtered normals and voxel growing. In: Proceedings of the International Symposium on 3DPVT, Paris

    Google Scholar 

  11. Diebel JR, Thrun S, Brunig M (2006) A Bayesian method for probable surface reconstruction and decimation. ACM Trans Graph 25(1):39–59

    Article  Google Scholar 

  12. Draelos MT (2012) The kinect up close: modifications for short-range depth imaging. A thesis Master of Science, Electrical Engineering, Raleigh

    Google Scholar 

  13. Faugere JC (2014) Introduction to documentation of the FGb package in Maple. http://www-polsys.lip6.fr/~jcf/Software/FGb/Documentation/index.html

  14. Fernandez JC, Singhania A, Caceres J, Slatton KC, Starek M, Kumar R (2007) An overview of Lidar cloud processing softwares, GEM Center report no. Rep-2007-12-01, Civil and Coastal Engineering Department, University of Florida

    Google Scholar 

  15. Fischler MA, Bolles RC (1981) Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Commun ACM 24:381–395

    Article  Google Scholar 

  16. Gordon SJ, Lichti DD (2004) Terrestrial laser scanners with a narrow field of view: the effect on 3D resection solutions. Surv Rev 37:448–468

    Article  Google Scholar 

  17. Huffel SV, Lemmerling P (2002) Total least square and errors-in-variables modeling: analysis, algorithms and applications. Kluwer Academic, Dordrecht

    Book  Google Scholar 

  18. Huang C-M, Tseng Y-H, Plane fitting methods of Lidar point cloud, Department of Geomatics, National Cheng Kung University, Taiwan, tseng@mail.ncku.edu.tw

    Google Scholar 

  19. Hubert M, Rousseeuw PJ, Van den Branden K (2005) ROBPCA: a new approach to robust principal component analysis. Technometrics 47(1):64–79

    Article  Google Scholar 

  20. Krarup T, Kubik K, Juhl J (1980) Götterdammerung over least squares. In: Proceedings of International Society for Photogrammetry 14-th Congress, Hamburg, pp 370–378

    Google Scholar 

  21. Kukelova Z, Bujnak M, Pajdla T (2008) Automatic generator of minimal problem solvers. In: ECCV’08, Part III, Marseille. Volume 5304 of lecture notes in computer science, pp 302–315

    Google Scholar 

  22. Kukelova Z (2012) Algebraic methods in computer vision. PhD thesis, Center for Machine Perception, Czech Technical University, Prague

    Google Scholar 

  23. Lewis RH (2008) Heuristics to accelerate the Dixon resultant. Math Comput Simul 77(4):400–407

    Article  Google Scholar 

  24. Lakaemper R, Latecki LJ (2006) Extended EM for planar approximation of 3D data. In: IEEE International Conference on Robotics and Automation (ICRa), Orlando, May 2006

    Google Scholar 

  25. Lewis RH (2014) Computer algebra system Fermat. http://home.bway.net/lewis

  26. Lewis RH, Coutsias EA (2007) Algorithmic search for flexibility using resultants of polynomial systems. In: Botana F, Recio T (eds) Automated deduction in geometry. Lecture notes in computer science, vol 4869. Springer, Berlin, pp 68–79

    Chapter  Google Scholar 

  27. Lewis RH, Stiller P (1999) Solving the recognition problem for six lines using the Dixon resultant. Math Comput Simul 49:205–219

    Article  Google Scholar 

  28. Lukács G, Martin R, Marshall D (1998) Faithful leat-squares fitting of spheres, cylinders, cones and tori for reliable segmentation. In: Burkhardt H, Neumann B (eds) Computer Vision, ECCV ’98, vol I. LNCS 1406. Springer, Berlin/Heidelberg, pp 671–686

    Chapter  Google Scholar 

  29. Lictbbau D (2009) Cylinders through five points: computational algebra and geometry. http://library.wolfram.com/infocenter/Conferences/7521/. Accessed 13 Oct 2009

  30. Mitra NJ, Nguyen A (2003) Estimating surface normals in noisy point cloud data. In: Proceeding SCG ’03 Proceedings of the Nineteenth Annual Symposium on Computational Geometry (SoCG’03), 8–10 June, San Diego, ACM, 1-58113-663-3/03/0006, pp 322–328

    Google Scholar 

  31. Nievergelt Y (2000) A tutorial history of least squares with applicatons to astronomy and geodesy. J Comput Appl Math 121:37–72

    Article  Google Scholar 

  32. Norris-Roger M, Behrendt R (2013) From points to products – business benefits from Lidar using ArcGIS, SA Forestry Magazine, 2013 June

    Google Scholar 

  33. Nurunnabi A, Belton D, West G (2012) Diagnostic -Robust statistical analysis for local surface fitting in 3D point cloud data. In: ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, vol I–3, 2012 XXII ISPRS Congress, 25 Aug 2012, Melbourne, pp 269–275

    Google Scholar 

  34. Paláncz B, Somogyi A, Lovas T, Molnár B (2013) Plane fitting to point cloud via Gröbner basis, e-publication, Wolfram Research Information Center, MathSource/8491

    Google Scholar 

  35. Paláncz B (2014) Fitting data with different error models. Math J. 16, 1–22

    Google Scholar 

  36. Poppinga J, Vaskevicius N, Birk A, Pathak K (2006) Fast plane detection and polygonalization in noisy 3D range images. In: International Conference on Intelligent Robots and Systems (IROS), Nice. IEEE

    Google Scholar 

  37. Preisendorfer RW (1988) Principal component analysis in meteorology and oceanography. Elsevier, Amsterdam

    Google Scholar 

  38. Raguram R, Frahm JM, Pollefeys M (2008) A comparative analysis of RANSAC techniques leading to adaptive real-time random sample consensus. In: Computer Vision – ECCV 2008, Marseille. Lecture notes in computer science, vol 5303, pp 500–513

    Google Scholar 

  39. Raguram R, Chum O, Pollefeys M, Matas J, Frahm JM (2013) USAC: a universal framework for random sample consensus. IEEE Trans Pattern Anal Mach Intell 35(8):2022–38. doi:10.1109/TPAMI.2012.257

    Article  Google Scholar 

  40. Rose C, Smith D (2000) Symbolic maximum likelihood estimation with Mathematica. The Statistician 49:229–240

    Google Scholar 

  41. Russeeuw PJ, Van Driessen K (1999) A fast algorithm for the minimum covariance determinant estimator. TECHNOMETRICS 41(3):212–223

    Article  Google Scholar 

  42. Schaffrin B, Wieser A (2008) On weighted total least-squares adjustment for linear regression. J Geod 82(7):415–421. doi:10.1007/s00190-007-0190-9

    Article  Google Scholar 

  43. See Ref. [447]

    Google Scholar 

  44. Sotoodeh S (2006) Outlier detection in laser scanner point clouds. In: IAPRS, Dresden, vol XXXVI/5, pp 297–301

    Google Scholar 

  45. Stathas D, Arabatzi O, Dogouris S, Piniotis G, Tsini D, Tsinis D (2003) New monitoring techniques on the determination of structure deformation. In: Proceedings of the 11th FIG Symposium on Deformation Measurements, Santorini

    Google Scholar 

  46. Stewènius H (2005) Gröbner basis methods for minimal problems in computer vision. PhD thesis, Lund University

    Google Scholar 

  47. Tofallis C (2002) Model fitting for multiple variables by minimizing the geometric mean deviation. In: van Huffel S, Lemmerling P (eds) Total least squares and errors-in-variables modelling: algorithms, analysis and applications. Kluwer Academic, Dordrecht/Boston/London

    Google Scholar 

  48. Wang S, Tseng Y-H, Habib AF (2010) Least-squares building model fitting using aerial photos and Lidar data. Least-squares building model fitting using aerial photos and LiDAR data. In Proceedings of the ASPRS 2010 Annual Conference, San Diego, CA, USA, 26–30

    Google Scholar 

  49. Weingarten JW, Gruener G, Siegwart R (2004) Probabilistic plane fitting in 3D and an application to robotic mapping. IEEE Int Conf 1:927–932

    Google Scholar 

  50. Yang MY, Förtsner W (2010) Plane detection in point cloud data, TR-IGG-P-2010-01, Technical report. Nr.1, Department of Photogrammetry Institute of Geodesy and Geo-information, University of Bonn

    Google Scholar 

  51. Yaniv Z (2010) Random sample consensus (RANSAC) algorithm, a generic implementation. Georgetown University Medical Center, Washington, DC. http://yanivresearch.info/writtenMaterial/RANSAC.pdf. Accessed 29 May 2014

  52. Zuliani M (2012) RANSAC for Dummies, vision.ece.ucsb.edu/ ∼ zuliani

    Google Scholar 

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Authors and Affiliations

  1. Curtin University, Perth, West Australia, Australia

    Joseph L. Awange

  2. Budapest University of Technology and Economics, Budapest, Hungary

    Béla Paláncz

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  1. Joseph L. Awange
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  2. Béla Paláncz
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Awange, J.L., Paláncz, B. (2016). Robust Estimation. In: Geospatial Algebraic Computations. Springer, Cham. https://doi.org/10.1007/978-3-319-25465-4_12

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