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Fitting a Sphere via Gröbner Basis

  • Robert LewisEmail author
  • Béla Paláncz
  • Joseph Awange
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10931)

Abstract

In indoor and outdoor navigation, finding the local position of a sphere in mapping space employing a laser scanning technique with low-cost sensors is a very challenging and daunting task. In this contribution, we illustrate how Gröbner basis techniques can be used to solve polynomial equations arising when algebraic and geometric measures for the error are used. The effectiveness of the suggested method is demonstrated, thanks to standard CAS software like Mathematica, using numerical examples of the real world.

Keywords

Point cloud Outliers SOM Numerical Gröbner basis 

References

  1. 1.
    Franaszek, M., Cheok, G.S., Saidi, K.S., Witzgall, C.: Fitting spheres to range data from 3-D imaging systems. IEEE Trans. Instrum. Measur. 58(10), 3544–3553 (2009)CrossRefGoogle Scholar
  2. 2.
    Józków, G., Thoth, C., Koppanyi, Z., Grejner - Brzezinska, D.: Combined matching of 2D and 3D kinect data to support indoor mapping and navigation. In: ASPRS 2014 Annual Conference, Luisville, Kentucky, USA, 23–28 March 2014Google Scholar
  3. 3.
    Ogundana, O., Coggrave, C., Burguete, R.L., Huntley, J.M.: Fast hough transform for automated detection of spheres in three-dimensional point clouds. Opt. Eng. 0001 46(5), 051002–051002-11 (2007)CrossRefGoogle Scholar
  4. 4.
    Zhou, Z., Guan, Y., Zhan, X., Lu, T.: Robust algorithm for fitting Sphere to 3D point clouds in terrestrial laser scanning. In: The International Archives of Photogrammetry and Spatial Information Science, vol. XXXVII Part B5, Beijing, pp. 519–522 (2008)Google Scholar
  5. 5.
    Molnár, B., Toth, C.K., Detrekõi, A.: Accuracy test of microsoft kinect for human morphological measurements. In: International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, vol. XXXIX - B3, 2012 XXII ISPRS Congress, Melbourne, Australia, 25 August – 01 September 2012Google Scholar
  6. 6.
    Awange, J., Palancz, B., Lewis, R.H., Völgyesi, L.: Mathematical Geosciences. Springer, Heidelberg (2018)CrossRefGoogle Scholar
  7. 7.
    DalleMole, V.L., do Rego, R.L.M.E., Araujo, A.F.R.: The self - organizing approach for surface reconstruction from unstructured point clouds. In: Matsopoulos, G.K. (ed.) Self-Organizing Maps, INTECH, pp. 167–188. Rijeka, Croatia (2010)Google Scholar
  8. 8.
    Sjoberg, J.: Neural Networks 1.2, Mathematica Adds On (2018). https://www.wolfram.com/products/applications/neuralnetworks/

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsFordham UniversityNewYorkUSA
  2. 2.Department of GeoinformaticsBudapest Technical UniversityBudapestHungary
  3. 3.Department of Spatial SciencesCurtin UniversityPerthAustralia

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