Second section describes the tools which are used in the study: the RANSAC algorithm, mathematical models of geometric objects, least square estimation of parameters, and their accuracies. The accuracy of the parameters is estimated in two different ways: i) from the residuals of points from the models and ii) as the standard deviation of parameters’ values from multiple repetitions. This is followed by the analysis of the reliability of the RANSAC with which the coordinates of characteristic points from the scans of spheres and cones can be estimated. RANSAC is then used to detect points which represent a plane in a simulated point cloud. In the last section the influence of data characteristics and input parameters on the reliability of RANSAC are discussed. Some suggestions for practical use and quality estimation of RANSAC are given in the conclusion.
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2 METHODS
RANSAC is used to find points of a point cloud representing objects (or their parts) that can be modeled as geometrical objects. The basic idea behind RANSAC is that the optimal parameters are those describing the mathematical model which covers the greatest amount of points. Firstly, the minimal subset of points needed to uniquely determine a geometrical object is selected randomly from the point cloud. Secondly, the parameters of the model are determined for this subset. The residuals of other points in the point cloud from model are computed. The points in a point cloud are classified as inliers, if respective residuals are smaller than a given threshold and as outliers, if respective residual are greater respectively. The procedure is repeated until the desired number of points is classified as inliers or until a desired confidence level is reached. According to the number of inliers in each repetition the best model is chosen. The algorithm is described in mathematical notation below (Fischler and Bolles, 1981).
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2.1 Description of the RANSAC algorithm
The input data for the algorithm are: m – the number of points needed to uniquely determine the parameters of a chosen mathematical model, t – the threshold, points within it are considered inliers, w – the expected percentage of inliers in a point cloud, p – the confidence level (probability that only inliers will be randomly chosen in at least one of the repetitions), S – the data, a point cloud. The algorithm: 1. Determine a number of iterations N (see section. 2.2). 2. For k 1, ..., N – randomly pick m points from S Sk – determine the parameters of the mathematical model Mk from Sk – Sk* s|s S \ Sk k s t 3. S * {Sk* Sk* max Sk* } k 1,, N
Where k(s) are residuals of points in S from the mathematical model Mk. The result of the algorithm is a set S *. Set S * is the one of S k* which contains the maximal number of points. Tilen Urbančič, Anja Vrečko, Klemen Kregar | ZANESLJIVOST METODE RANSAC PRI OCENI PARAMETROV GEOMETRIJSKIH OBLIK | THE RELIABILITY OF RANSAC METHOD WHEN ESTIMATING THE PARAMETERS OF GEOMETRIC OBJECT | 69-97 |
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