Elsevier

Measurement

Volume 145, October 2019, Pages 640-647
Measurement

A general accuracy measure for quality of elliptic sections fitting

https://doi.org/10.1016/j.measurement.2019.06.003Get rights and content

Abstract

Least squares (LS) fitting, the most widespreadly used approach for ellipses, operates by minimizing the sum of squares of some error term measured at each data point. It is not an easy task to measure the accuracy of any fitting method in fitting elliptical sections since they rely on different error criteria. There is no unique criterion valid for any fitting method for quantifying for ellipse fitting. For this reason, there is a need for a general measure that can be used to compare the accuracy of fitted ellipses using different methods. In this work, an error measure is proposed which can be used both to measure the accuracy of any ellipse fitting method and to compare the accuracy of the ellipses fitted with different (i.e. algebraic or geometric) methods. This measure is generated from the widely known orthogonal least squares fitting (OLSF) method revising the computation scheme of initial values for the orthogonal contacting points for an ellipse in the study. This is a flexible error measure since it always computes orthogonal distance residuals between data points and the optimal ellipse and can then be used to compare the performance of different ellipse fitting methods. By computing this measure it is possible to obtain the precision of the ellipse parameters with respect to the orthogonal distance residuals. This measure is applied to the measurement of the outer section of a piston and results indicate the effectiveness of the criterion.

Introduction

Ellipse fitting is one of the standard problems of pattern recognition and ellipse detection in a sparse data set is of importance in various fields of applied science ranging from astronomy to geology. Various algorithms have been proposed from different perspectives. There are two main classes of least squares (LS) methods known as algebraic and geometric fitting for geometric features, which are distinguished by their respective definitions of the corresponding error distances. One category has been referred to as algebraic fitting, where the implicit form of the conic section is used and the residual is minimized. This gives rise to linear LS problems subject to constraints and these can be usually solved efficiently as eigenvalue or singular value problems. The other is geometric fitting, where the errors in the data are minimized. Here, the implicit form of the conic may be used and the problem posed as a constrained LS problem in the parameters. Within the group of LS methods, algebraic and geometric distances refer to the parameters minimized in the second-order polynomial equation representing the ellipse. Algebraic least squares solutions are linear and their solution is relatively easy. Despite advantages in implementation and computational costs, there are disadvantages associated with the accuracy and physical interpretation of fitting parameters [1], [2]. The geometric fitting of ellipse has attracted much attention and it gives a more meaningful parameter to minimize with respect to ellipses.

Estimation of the ellipse parameters is generally performed by defining an error function and minimizing this function. The accuracy and cost of the forecasting process depend on the geometric structure of the error function, whether this function is linear or not. The researchers discussed some error functions and their geometric properties related to the subject. The error distances are defined by deviations from the expected value (i.e., zero) of the implicit equation at each point indicated. The inequality of the equation indicates that the given point is not on the geometric feature (i.e, there are some fitting errors). Most publications related to LS fitting generally relate to the sum of squares of algebraic distances or their versions [3]. By geometric fitting, the error distances are defined with the orthogonal distances from the given points (i.e. point dataset) to the geometric feature to be fitted. Geometric fitting has some advantages that facilitate the interpretation of the fitting parameters and errors [1], [4], [5]. Ahn et al. [5] proposed the orthogonal least square method to overcome the deficiencies of the algebraic fitting. Orthogonal least square fitting method minimizes the sum of the squares of the Euclidian distance defined as the orthogonal distance from the data point to the ellipse. Despite its sensitivity to non-Gaussian noise, LS fitting is probably the most common approach used to estimate ellipse parameters; this is the reason for the computation efficiency as an estimator. It works by minimizing the sum of the squares of some of the error term measured at each data point. For this reason, many different ellipse estimation techniques depend on a reasonable error term. The orthogonal distance measure adopted as the most natural and best error measure of least squares techniques [6] can be used to solve problems related to algebraic fitting, as mentioned earlier. Although there is a closed form solution for the calculation of the orthogonal point for a general ellipse, a numerical instability may occur in the application of the analytical formula [7], [8].

An objective measure of the quality of fit is defined as the sum of the normal distances of all data points to the optimal ellipse. The idea of defining the error function of a data point based on the normal distance of the ellipse is a special case of a more general problem (called function fitting) based on minimizing the orthogonal deviations. Although the best fit, this error function is quite complicated and disadvantageous in terms of calculation. The orthogonal least squares (OLSF) method proposed by Ahn et al. minimizes the sum of squares of the Euclidian distance defined as the ellipse orthogonal distance from the data point. OLSF has a clear geometric interpretation and high accuracy. It uses the iterative Gauss-Newton algorithm and in each iteration, for each data point, the orthogonal contacting points are found repeatedly. The orthogonal contact point is the point on the ellipse that has the shortest distance to the respective data point.

On the other hand it is not an easy task to compare ellipse fitting methods because it is not clear how to measure the “goodness” of a method. There are some choices for an accuracy measure for the fitting methods [1]. Besides, it is essential to test the accuracy (or reliability) of the estimated fitting parameters for a method. To determine which of the best ellipses can be fitted to a given set of points, the squares of distances are typically used, both algebraically and geometrically. The most suitable fit is obtained by using the sum of the orthogonal distances because this criterion represents the actual geometric distance between the points and the fitted ellipse. The LS methods involve the minimization of the ellipse fitting errors using the smallest sum of the squares of the shortest distances from the input data to the fitted ellipse. Fitting residuals can be algebraic or geometric and are not identical. The orthogonal distance is the shortest distance from a data point to an ellipse. The closest point on the ellipse from the given point is called the orthogonal point. To initiate the iterative procedure, an initial guess for the parameter vector must be given, and this initial guess must be carefully performed. As an initialization to start the iteration the use of the circle fitting algorithm was suggested [5].

As it is difficult to test the accuracy of the estimated fitting parameters there is a need to establish a comparable measure valid for different ellipse fitting algorithms. Orthogonal distance measure would be a good choice for this purpose since OLSF is simple and robust nonparametric algorithm for geometric fitting of ellipse [9], [5]. In the study we used this method to compute the orthogonal distance residuals to reveal accuracy aspect of different fitting methods. The computation of initial contacting points strategy for OLSF was developed since the initial guess is a key issue for parameter estimation. One concern of the study is the calculation of initial values for orthogonal point on the ellipse. It was realized that there is an inadequacy for the computation of the initial values for the orthogonal distances in original OLSF method. The original method proposed by Ahn et al. [5] might yield inconsistent results in some cases for the initial values of orthogonal distances for especially inner points of an ellipse in a noisy dataset. One contribution of the study is to revise this computation stage for the estimation of orthogonal contacting points. The main focus of the study is to develop a scheme to compute orthogonal distance residuals for any ellipse fitting algorithm. With this approach the precision of ellipse parameters estimated by algebraic fitting method can be computed with respect to the orthogonal distance residuals. Since this “orthogonal distance” measure can be used for any fitting algorithm it might be called as “orthogonal root mean square” as a general term. The measure can be used for the comparison of the performance of different ellipse fitting methods since always calculates orthogonal distances between data points and the model ellipse. In addition to this by computing the measure it is possible to obtain the precision of the ellipse parameters with respect to the orthogonal distance residuals.

Section snippets

Least squares fitting

Diverse forms of least squares ellipse fitting have been employed in the literature [10], [11], [12]. As previously explained LS fitting minimizes the squares sum of error-of-fit in predefined error measures. As widely accepted there are two main categories of LS fitting problems for geometric features, namely algebraic and geometric fitting.

Computing scheme of the general accuracy measure

A generally used and comparable error measure can be used to compare the performance of different ellipse fitting methods. In order to test the results of the estimation it was thought to be beneficial to compute the orthogonal distance residual values of any ellipse fitting method. This makes it easier to interpret the estimation process for geometrical perspective. In this study, orthogonal distance residuals of algebraic fitting method were computed for this purpose. Kurt and Arslan [15]

Experimental results

As noted earlier the proposed algorithm for our accuracy measure was coded in Matlab environment and given in the Appendix A as pseudo codes. The algorithm was created with the version 7.10.0 of the MatlabR2010a and run on 32 bit processor. Thereafter, the algorithm was also tested on 64 bit processor in the same version. Our measure is a flexible measure since it always computes orthogonal distance residuals between data points and the model ellipse for any ellipse fitting method as previously

Conclusion

An accuracy measure is presented for testing the accuracy of fitting of elliptic sections for any ellipse fitting method. Since orthogonal distance is assumed to be the natural and best error measure in least squares the measure was generated from the widely adopted OLSF method by modifying the computation scheme of initial values for the orthogonal distances. Our accuracy measure can be used to compare the performance of different ellipse fitting methods since it always computes orthogonal

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1

Kocaeli University, Department of Geomatics, Umuttepe Yerleskesi, 41380 Kocaeli, Turkey.

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