Bayesian approach for circle fitting including prior knowledge

Abstract

The fitting of geometric shapes into measurement data is a frequently occurring task in metrology, computer vision or pattern recognition. Least-Squares methods correspond to the state of the art, but do not make use of existing prior knowledge about the measurement system or the measurement object. By using prior knowledge, the uncertainty of a measurement can be reduced. A simple example for prior knowledge is the diameter of a bore hole, which is always greater than zero. The Bayesian approach offers the possibility to include this prior knowledge for the fitting of geometric shapes. In the following, a Bayesian approach for fitting a circle is presented and compared with the established Total-Least-Squares method.

Bayesian analysis of uncertainties in circle, straight-line and ellipse fitting considering a-priori knowledge − comparative analysis with total-least-squares approaches

“…The following investigation has already been carried out similarly in [10]. In the framework of the present paper, the obtained results are re-examined using additional evaluation metrics to obtain a more comprehensive overview and understanding.…”

“…When comparing the posterior PDFs for the cases considered (see figure 3), it becomes clear that the variances (a measure of the width of the PDF and therefore a measure of the accuracy of the fit) strongly depend on the quality and quantity of the measurement data. If the measurement data represent just a small circular segment as in figure 3 (or standard deviation ŝq ), representing the width of the posterior PDF, can be reduced significantly [10]. For this purpose, Bayesian circle fitting using simulated measurement data is further examined with regard to the reduction of the Bayesian uncertainty.…”

“…For this purpose, Bayesian circle fitting using simulated measurement data is further examined with regard to the reduction of the Bayesian uncertainty. Note that this task represents an extended evaluation of the results already obtained in [10]. This re-examination involves simulating 10,000 measurements each for different-sized circular segments (circular wrapping angle of 55 , 60 , 170 , 180 the ideal circle: Taking a look at the averaged standard deviation R ¯ŝ it can be seen that prior knowledge has a great influence on the determined uncertainty in the case of small circular segments ( 100…”

A novel circle center location method for a large-scale wafer

“…Therefore, it is extremely important to have an accurate and efficient circle center location method in the wafer before the visual inspection of wafer surface defects. Generally, the methods of measuring the circle, or circle detection and location methods can be briefly summarized as contact measurement [3,4] and non-contact measurement [5]. For example, Mayyas [6] developed a three-point inverse kinematic algorithm to measure the radius and roundness of bearing circular arc from three distributed sensors, which is a contact measurement method.…”

A Novel Reconstruction Method for Measurement Data Based on MTLS Algorithm

“…Although the MLS has been widely used due to its good fitting features [ 20 , 21 ], it also has its own drawbacks, especially in approximating curves and surfaces. As a reconstruction method, the local coefficients are obtained by least squares (LS) method which assumes that the error exists in the dependent variable of the measurement data [ 22 , 23 ]. To take the random errors of all variables into account [ 24 , 25 ], Scitovski et al [ 26 ] generalized the traditional algorithms based on the work of Lancaster et al and put forward the Moving Total Least Squares (MTLS) method.…”