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Plane Fitting to Point Cloud via Gröbner Basis

Bela Palancz
Organization: Budapest University of Technology and Economics
Department: Photogrammetry and Geoinformatics
Árpad Somogyi
Tamas Lovas
B. Molnár
Revision date


This notebook presents a new approach for surface reconstruction from point clouds. A fast algorithm with zero complexity to fit a plane to point clouds in Total Least Squares (TLS) sense has been developed using Gaussian-type error distribution. The application of maximum likelihood method results a multivariate polynomial system which can be solved via symbolic computation using reduced Gröbner basis in analytic form. In case of plane fitting when datasets are affected heavily by outliers, this algorithm can be embedded into the RANdom SAmple Consensus (RANSAC) method. In order to increase the effectiveness of this integrated algorithm Self-Organizing Map (SOM) can be employed as data preprocessing method to eliminate the main portion of the noisy data points by creating representative code-book vectors. In this way the total running time can be decreased considerably. The study compares the performance of the suggested algorithm to that of other methods and demonstrates its efficiency on both synthetic data adopted from the literature as well as on real, laser scanned measurements.

*Applied Mathematics > Visualization
*Mathematics > Algebra > Polynomials
*Mathematics > Geometry > Computational Geometry
*Wolfram Technology > Application Packages > Wolfram Research Applications > Neural Networks

Surface reconstruction, plane fitting, point cloud, LiDAR, TLS, Gröbner basis, outliers, RANSAC, SOM
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PlaneFitting.nb (8.4 MB) - Mathematica Notebook
slope.dat (874.9 KB) - Unknown MIME type