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

Fitting standard geometric elements into measurement data using Least-Squares techniques is a common task in signal processing across various technical applications. However, the application of these well-established but purely data-based methods does not consider potentially available prior knowledge about the measurand of interest or the measuring device. Thus, up to this day, additional information is usually left unused beyond a few academic applications. By applying a Bayesian approach, this prior knowledge can be incorporated into the fitting task, potentially leading to a reduction in overall uncertainty and fragility of the evaluation result. In this study, Bayesian models are proposed for incorporating prior knowledge into circular, linear, and ellipse fitting tasks. The general approaches as well as specific results are compared to the established Total-Least-Squares method within the example of the application of the F-operator in surface texture measurement illustrating the practical benefits of the approach.


Introduction
Fitting standard geometric elements such as circles, straight-lines or ellipses (as well as the three-dimensional pendants sphere, plane and ellipsoid) into a set of measured points is a frequently occurring task in many technical applications, e.g. to quantify form deviations between measured data and its underlying nominal geometry [1][2][3][4].
In manufacturing metrology, the consideration and elimination of an underlying nominal shape of the measuring object is an indispensable first step, which is also included in the corresponding standardization of roughness measurement as the 'F-operator ' [5].In this context, the most typical nominal shapes to be removed (and thus to be fitted) are straight lines (planes) and circles (spheres).
The parameters of fitted geometric elements typically are estimated using Ordinary-Least-Squares (OLS) or Total-Least-Squares (TLS) methods [6].However, often prior knowledge from former measurements, of the measuring equipment, or regarding the minimal uncertainty of the parameters to be estimated is available.For example, it may be known that a parameter to be estimated is positive and greater than zero (diameter of a bore hole) or knowledge from manufacturing experience allows much more precise estimates represented by an expected value and an associated standard deviation [7].To this day, this information usually remains unused [6] beyond a few academic applications.This can lead to the fact that tolerances required in production cannot be confirmed based on pure measurement data in some cases and/or manufactured workpieces are incorrectly declared as rejects.
One concept to combine fundamentally different types of information, like measurement data and prior knowledge, is the application of the Bayesian analysis [7][8][9][10][11][12][13][14][15].To perform an analysis that combines information obtained by measurements and prior knowledge, a mathematical expression that allows these fundamentally different types of information to fuse has to be formulated.A concept capable of accomplishing this task is the concept of conditional probability density functions.Uncertainties in the data obtained by measurement equipment can be considered, as well as physical relationships and prior information on the measurands.The theory is universally applicable to most measurement data evaluation tasks, including complex nonlinear adjustments and cases where established Least-Squares approaches fail [12].
This paper is intended to make the Bayesian approach accessible to practical use cases of manufacturing metrology.As a first step, the Bayesian analysis approach is described in general (section 2).Afterwards, the Bayesian approach is applied to the often-used and well-known standard use cases of fitting circles (section 3 as extended investigation based on [10]), straight lines (section 4) and ellipses (section 5) into a set of measurement data.
A comprehensive fundamental review and analysis of many existing alternative geometric form fitting approaches beyond Total-Least-Squares methods, some of which are easier to implement and/or more suitable for specific cases, is covered in [16].The derived models and algorithms of this publication are described and the overall performance of the Bayesian analysis approach is investigated and discussed explicitly and exclusively in comparison to the well-established Total-Least-Squares methods, which do not consider existing prior knowledge [17].In addition, the Bayesian circle fitting approach is applied to a realistic signal processing task in the context of optical roughness metrology (section 6).
Since this work is related to [10], the basics of the Bayesian approach (section 2) as well as its application to circle fitting (section 3) can also partly be found in [10].Furthermore, [18] provides a compact summary of the approaches and results derived in this work.However, for the sake of completeness, these sections are reiterated below.

The Bayesian approach
Bayesian analysis is based on the mathematical description that couples fundamentally different categories of information: measurement data and prior knowledge.This coupling is done by modeling several (conditional) probability density functions (PDF) that can be linked to each other according to Bayes' theorem [19]: The PDF f post ( | ) q h is the so-called 'a-posteriori' PDF (hereafter referred to as posterior).The posterior assigns probabilities to the occurrence of each parameter combination , , , n 1 2 available, whereby each x i represents a measured n-tuple (e.g.two-or three-dimensional spatial coordi- nates x x y , ).By evaluating the posterior, parameter estimates as well as corresponding uncertainties can be determined considering information obtained by measurement data as well as several types of prior knowledge.However, the required posterior cannot be described directly in most cases but by applying Bayes' theorem [6].According to (1), the posterior arises by combining PDFs that are easier to formulate, called the likelihood and a-priori PDF.
The likelihood f like ( | )  h q contains information obtained by pure measurement data processed in a corresponding measurement model.The prior knowledge of the measurands of interest etc. is contained in the a-priori PDF f prior ( ) q (hereafter referred to as priori).The term C post results from the normalization constant in the denominator.Combining likelihood and priori leads to the formulation of the posterior, wherefore the posterior can be seen as the carrier of the entire information available, see figure 1.
For the Bayesian analysis approach, it is therefore of particular importance to model PDFs for likelihood and priori.Exemplary models used in the literature [6, 7, 11] that are adopted here are described in the following sections.

Likelihood f like ( | ) h q
To create the likelihood f , like ( | ) h q the underlying measurement model must first be defined by formulating a functional relationship between acquired measurement data h, model parameters q and presumed measurement deviations x (practical examples of such functions are shown in sections 3, 4 and 5).In (2), a linear relationship common in metrology is assumed as an example, where the result of an (indirect) measurement g , ( ) h q is composed as a functional relationship between the model subjected to the measurement h( ) q and presumed measurement deviations x [20]: Another common assumption used here is that the measurement deviation is multivariate Gaussian distributed 0, , which can be described by s representing the variance of the measurement deviation [21].Combining equations (2) and (3), the likelihood is finally obtained as

A-Priori f prior ( ) q
For information from measurement data to be coupled with prior knowledge about the model parameters, this prior knowledge must also be formulated mathematically as a PDF.It is assumed that prior knowledge is also available as Gaussian distributed random variables, which is adopted from [7].It should be noted that this assumption is a strong simplification of the reality.It cannot simply be assumed that prior knowledge naturally emerges in the form of a mathematically well-defined PDF, not to mention a Gaussian distributed PDF.Nevertheless, this assumption can be used here to analyze and demonstrate the potential of the Bayesian approach in the context of geometric fitting.
The mathematical expression of the priori expands with the number of model parameters , , , n 1 2 q q q ¼ that are included in the underlying model and for which prior knowledge could potentially be available as follows with parameters to be estimated i q and corresponding expected values i,0 q and variances .
Parameter estimates and their associated uncertainties can be determined by evaluating the posterior which bundles measurement data, the underlying measurement model and existing prior knowledge in a single PDF and therefore serves as the carrier of the entire information available.As already mentioned, the posterior arises by combining likelihood (4) and priori (5) according to Bayes' theorem in (1).In general, PDFs (so is the posterior) can be evaluated by applying common evaluation approaches known from statistics.Exemplary characteristic values used within this paper are listed in table 1 [22].
In the following, the Bayesian approach, with its possibility of considering prior knowledge, is described and investigated for the tasks of fitting circles, straight lines and ellipses into sets of measurement data.

Model description
The measurement model for fitting circles is based on the approach that measured data points x y , i i ( ) deviate radially in relation to the underlying ideal circle (orthogonality relationship corresponding to the established Total-Least-Squares fitting approach [17]): Here, R represents the radius of the circle to be fitted and x y , i i ( ) represent the coordinates of the i-th measured data point.Figure 2 illustrates the orthogonality relationship between a measured data point x y , i i ( ) and the circle described by its radius R and center coordinates x y , .
c c ( ) Finally, the distance R h between a data point and the circle's center point is Table 1.Exemplary characteristic values for the evaluation of (a-posteriori) PDFs.

Maximum-A-Posteriori estimation:
described by R x y x y x x y y , , , .10 Combining equations (9) and (10), an expression for x is obtained as .
The deviation x is assumed to be Gaussian dis- tributed 0, xy Assuming prior knowledge of the circle parameters to be Gaussian distributed as well (as described in section 2.2), the expression , , , , .14 Evaluating the posterior f post (carrier of the entire information), circle parameter estimates (values for the radius and center coordinates of the circle that fit best into the set of measurement data according to a specific metric) as well as corresponding uncertainties can be calculated.
Since the measurement model is nonlinear, the integrals in (7) and (8) are not analytically solvable for this case and numerical methods must be applied to fully evaluate the posterior [6, 7].
In the following section, the algorithm for the calculation of the circle parameters including its associated uncertainties is explained and it is shown how resulting uncertainties can be reduced by considering prior knowledge.

Computation algorithm
The measurement model in (9) is nonlinear, in contrast to the approaches described in [6, 7, 11].Numerical methods capable of the evaluation include numerical quadrature or Monte-Carlo methods.Quadrature methods are suitable for numerical integration tasks of low order (dimensions D 3  ) [23].
To solve integrals of higher order (D 3  ) numeri- cally, Monte-Carlo methods are usually preferred [23].For the determination of ( ˆˆˆ) and , , the Bayesian model presented, numerical quadrature is performed using Matlab and its built-in functions for numerical integration [24].The calculation of

MAP q
does not require computationally expensive integration methods but can be determined using Nelder-Mead [25] or Gauss-Newton optimization algorithms [26].
The most efficient method to calculate MMSE q and , 2 ŝq regarding the computational costs, is by numerical quadrature of second order.Since the radius R is linearly integrated into the model function in (11), the dimension of the posterior can be reduced from three to two [6, 7].Besides the benefit of significantly reduced computation time, another advantage is that a reduced two-dimensional posterior can be easily visualized graphically, which allows quick visual impressions of the achieved measurement result quality.However, the resulting terms are very inconvenient to handle, which is why they are not shown here.

Results
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.First, it is generally shown how the posterior behaves, ignoring prior knowledge (posterior without considering prior knowledge is equal to the likelihood).In doing so, three sets of measurement data are simulated according to the following regulation:  (a) N 30  60 For the reasons mentioned above, the dimension of the posterior is decreased from three to two by analytical integration via R.This results in a PDF that is only dependent on the circle center coordinates, but no longer on the radius.
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 By including prior knowledge, uncertainties expressed by the variances , , (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.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  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 ) the influence of prior knowledge on the resulting uncertainty is negligible.Therefore, large uncertainties (represented by broad PDFs as shown in figure 3(a)) can already be significantly reduced by including only poor prior knowledge.

Model description
The measurement model for the straight-line fitting approach is also based on the orthogonality relationship between measured data points x y , i i ) and an ideal straight-line g to be fitted (see figure 5).
For the formulation of the likelihood, the functional relationship between the orthogonal distance of a measured data point x y , i i ( ) and the straight-line to be fitted, analogous to the circle fitting model described above, is derived.For this purpose, the orthogonal distance i x is selected as a formulation using the Hesse normal form describing the straight-line g as with j as the angle of the straight-line's normal vector and u as the straight-line's distance from the origin.Under the condition of orthogonality, a measured data point x y , i i ( ) can be formulated as using the orthogonal distance i x from the measured data point to the ideal straight-line to be fitted.
Eliminating x g and y g by combining (17) and (18) results in the expression of the squared distance

Computation algorithm
The determination of u , , the Bayesian model presented is performed analogous to the circle fitting approach described above using Matlab and its built-in functions for optimization and numerical integration and optimization.
Again, MMSE q and 2 ŝq are calculated by numerical quadrature of second order, although the resulting posterior could be described by a one-dimensional PDF after performing an analytical integration via u.

Results
The performance of Bayesian analyses in straight-line fitting using the model described above is investigated by comparing three different cases of underlying measurement data basis, each with and without consideration of existing prior knowledge.For this purpose, scattered measurement data are simulated according to the following regulation: s The three cases of the straight-line segments to be measured and fitted are imaged in the upper row of figure 6 exemplarily using ten sample points (n 10 s = ).It is assumed that in the case where prior knowledge is available opt 0 j j = and 0.1 5.73  s =  j applies.In contrast, no prior information regarding the straight-line's distance to the origin u is available ( u s  ¥).Similar to the circle fitting simulations per- formed and described above, 1,000 measurements per scenario (three cases visualized in the upper row of figure 6, each without and with considering prior knowledge) are simulated and evaluated.The resulting averaged standard deviations u ¯ŝ and ¯ŝ j as a measure of uncertainties of the estimates for u and j are imaged in the center and lower row of figure 6.
In figure 6 it can be seen that the resulting uncertainties of the Bayesian straight-line fitting procedure increase with the data basis getting poorer, meaning the total segment length and/or number of sample points n s decreasing.In addition, the influ- ence of prior knowledge increases with the data basis getting poorer.It can be concluded that the influence of prior knowledge on straight-line fitting is expected to only be significant in cases, where the data basis is poor and/or more scattered.This observation is in accordance with the findings in the previous section.

Model description
When fitting an ellipse, multiple approaches have been examined for various application scenarios, see e.g.[27].In the chosen approach, five parameters have to be estimated.These include the half-axes lengths a and b, center coordinates x y , , c c ( ) as well as the angle of inclination .
a Furthermore, the orthogonality condition between measured data points and the ellipse to be fitted is also applied in the Bayesian ellipse fitting model derived (see figure 7).Therefore, the argument of the likelihood is the orthogonal distance , x analogous to the circular and straight-line fitting models described above.
The derivation of the functional relationship between measured points x y , , ( ) a and the orthogonal distances x is described in [28].Taking this relationship into account, the following term applies for the description of the likelihood, analogous to previous sections: , , , ,  with  , , , , , , 21 The priori arises according to (5), also analogous to previous sections, as a combination of several Gaussian distributed PDFs for each parameter to be estimated.

Computation algorithm
While the ellipse parameters can still be estimated easily by using the Maximum-A-Posteriori estimator (6), this is no longer the case for the determination of the corresponding Bayesian uncertainties.The determination of the uncertainties of the ellipse parameters requires a multidimensional integration of a 5D PDF.
Since a closed solution of the occurring integrals is not existing and numerical quadrature methods are inefficient when solving higher-dimensional integrals, alternative methods like Monte-Carlo methods are required.Monte Carlo methods are defined by the use of random numbers for solving numerical and nonnumerical problems [29].A fundamental element of all Monte Carlo methods is the sampling of a given distribution, which is used here for evaluating the 5D PDF of the Bayesian ellipse fitting model.
A suitable Monte-Carlo method used within this work is the so-called Metropolis Monte-Carlo method, named after Nick Metropolis.This procedure is described in detail in numerous publications [30,31] and is particularly suitable for high-dimensional evaluation tasks.The Metropolis procedure is based on so-called Random Walks.Starting from a certain initial point, a path is taken through the area of the probability density function.This path is composed of many randomly generated points that make up the sampling.Since a point is always found in the neighborhood of the previous point, successive points are always correlated and as a consequence, a larger number of samples is required to achieve a statistically defined quality of the result.

Results
In this section the performance of the Bayesian ellipse fit is examined based on a single simulated measurement data set assuming different boundary conditions (different levels of available prior knowledge and decreasing measurement data basis).For this purpose, a single set of measurement data is generated according to the following regulation: 60, 40, 10, 20, 70 0, , 1 ) The following cases are investigated to again examine the influence of the segment size: (c) N 90 each considering the following cases of (Gaussian distributed) prior knowledge: (1) No prior knowledge (corresponds to TLS Fit): The achieved results are imaged in figure 8 and the evaluated characteristic values of half axis a are listed in detail in table 2. The histograms in the lower part of figure 8 are also limited to the PDFs of the half-axis a since this ellipse parameter is the most critical one in this example.Since a circle is a special type of ellipse, the ellipse fitting procedure in general delivers similar results to the results presented in previous sections.The smaller the measured ellipse wrapping angle j D is, the more uncertain the fitting results and the more potential influence the consideration of prior knowledge has on the final result.However, in contrast to the circle or straight-line fitting procedure, for five ellipse parameters to be evaluated, the computation parameters of the Monte-Carlo method applied have to be chosen carefully with respect to the underlying PDF as well.One important computation parameter, for example, is the number of PDF sampling points (corresponds to the length of the Random Walk), since the computation time highly depends on it.Other parameters for example influence the scale of the PDF region that is evaluated.These parameters are very critical, especially when the measurement data basis is poor and no prior knowledge is available.In this case the validity of achieved results has to be proven, for example with non-desirable trial & error methods using various computation parameters.However, further  investigation of this method in the context of Bayesian fitting of geometric elements is omitted here.

Application on measured data
In the following, the possibilities of the Bayesian approach for geometry element fitting are illustrated by using a real data example.This is done exemplarily by performing circular fits, since this task frequently occurs in practice [1, 4, 32] and may highly benefit from including prior knowledge into the fitting process.The potential of the Bayesian approach is demonstrated here by the example of a profile roughness measurement using an optical sensor on a cylindrical surface.Optical topography measurement techniques such as confocal microscopy (areal) or confocal chromatic point scanning (profile) are often the method of choice for topography or roughness measurement when larger areas (in the order of mm 2 ) are to be measured and evaluated in a short time, or when the object to be measured is inaccessible for tactile methods [33].
In the evaluation of the acquired measurement data with regard to roughness or waviness, the form removal by the F-operator is a mandatory, standardized procedure.This elementary operation in signal processing for both roughness and waviness evaluation allows the removal of the underlying nominal shape [34].In the example of a roughness measurement on a cylindrical surface, the underlying nominal circular shape must be removed from the signal in the first step, which is classically done by Ordinary-Least-Squares or Total-Least-Squares fitting based on the measurement data.
A well-known phenomenon in optical topography measurements of e.g.circular objects is that measurements of areas with large underlying gradients often show non-evaluable artifacts, in the best case, however, significantly higher measurement deviations than corresponding measurements of areas with small gradients [35].For confocal microscopes for example it has been demonstrated that the height uncertainty is associated with the curvature of the surface [36] and for white-light interferometers phase jumps are possible [37].In general, the various principles of optical surface topography measurement share the characteristic that the higher the gradients to be measured, the higher the measurement uncertainty to be expected (up to non-evaluable artifacts).
The typically applied strategy for fitting circles to such a data set is to exclude the uncertain and potentially artifactual regions of the measured topography from the fitting process.However, this approach, depending on the degree of reduction of the measured data used for fitting, as already pointed out in figures 3 and 4, may again lead to increased uncertainty in the result.In general, the thinner the data base, the higher the expected uncertainty in the circle fitting result.
In the described example, the introduction of a-priori knowledge, in particular about the diameter of the measured object, which is determined by another method or specified by the manufacturer within certain tolerances, can be advantageous for the robustness of the circle fitting process.A possible application of the presented Bayesian approach is illustrated using the example of the characterization of an optical topography measurement instrument for a specific measurement task.
In preparation for a larger series of measurements, e.g. in series manufacturing, the suitability of an existing optical topography measuring instrument for robust topography measurements on cylindrical workpieces is to be investigated.Proof of the capabilities of the measuring instrument is often required in audits or requested by the customer.The following aspects are to be examined: 1. Since the challenges of measuring in the area of large gradients are well known, it is to be investigated how the measurement deviation in the peripheral areas (large gradients) behaves in comparison to the measurement in the central area (small gradients).
2. Further, it is to be examined whether the measurement in the peripheral areas, in addition to increased measurement uncertainty, also shows a systematic characteristic, i.e. whether the topography measurement in the peripheral areas reproducibly provides either too small or too large height values.
The capability study of the measuring device is to be carried out explicitly with the aid of representative real workpieces and not with the aid of calibration standards, i.e. material measures which cannot sufficiently reflect the real measuring task.

Classical approach
First, the characterization is performed using the classical approach.For this purpose, the measured circular arc is reduced to such an extent that the data set, on which the circle fit is based, no longer contains any obvious artifacts in the large gradient region.A Total-Least-Squares circle fit is then applied to the remaining data set (see figure 9(a)), which is subsequently used to eliminate the circular nominal shape from the measured data set.The result of these operations provides the orthogonal distances TLS x between the real measure- ment data and the fitted Total-Least-Squares circle (see figures 9(b) and (c)).The statistical distribution of the residuals follows a Gaussian PDF.
The result of this procedure clearly shows how the optical measurement generally behaves with different underlying gradients (the first aspect under investigation).The low noise in the central area is associated with the area featuring small gradients and the high noise in the peripheral areas results from the larger gradients in these areas.
Regarding the directional tendency of the measurement deviation in the peripheral areas (the second aspect under investigation), no statement can be made based on this result.The reason for this is that the TLS circular fit in this example is primarily aligned to the peripheral regions, since the scatter of the measurement data is particularly high there, thus stronger affects the result of the TLS fit than the central, less noisy region.This property of the TLS circle fit conflicts with the fact that these peripheral regions should not be used, and certainly should not be weighted particularly heavily, to align the entire measurement data set.

Bayesian approach
To investigate whether a systematic characteristic of the measurement deviations in the peripheral area is present, the procedure described above is adapted.First, only the central area of the measurement data is used for the circle fit.Furthermore, instead of the TLS circle fit, a Bayesian circle fit is applied, which incorporates a-priori knowledge about the radius of the measured object.The following parameters are used for the circle fit: Without performing an interpretation or quantitative evaluation of the results shown in figures 10(b) and (c), (a) clear systematic trend of the measurement deviation can be recognized by using the Bayesian approach at this point (2.aspect under investigation).The statistical distribution of the residuals is broader, but also more concentrated around the value 0 m.

Bayes
x » m In this example, the extent of existing prior knowledge significantly influences the information contained in the pure measurement data, although only very rough prior knowledge is assumed and applied.This illustrates the additional benefit of the Bayesian method for uncertain measurement data sets.However, the Bayesian approach presented here also works for cases with very precise or no prior knowledge.This qualifies the Bayesian approach as a versatile method, which permits the consideration of different boundary conditions in a comprehensive uncertainty analysis.

Conclusion and outlook
Fitting geometric standard elements into a set of measurement data is commonly practiced in many technical applications using Least-Squares algorithms.However, this process does not consider potentially existing prior knowledge, which is why these methods remain below their capabilities.For this reason, Bayesian approaches for standard fitting tasks with the possibility of considering potentially existing prior knowledge to reduce uncertainties of the evaluation process are being investigated.The given study showed in detail in which applications, under which conditions and at what computational costs a significant reduction in uncertainty can be achieved, i.e. how high the potential of the Bayesian approach can be estimated for various applications.First, it was demonstrated how information obtained from measurement data can be combined with information based on prior knowledge, in general, using conditional probability density functions which can be evaluated afterwards by well-known statistical parameters.
Subsequently, Bayesian models for the specific tasks of fitting circles, straight-lines and ellipses were adapted or derived, optimized with regard to their performance and examined for potential benefits compared to state-of-the-art methods.For this purpose, measurement data sets representing varying cases were simulated.Using these data, it was investigated to which degree potential prior knowledge might influence resulting uncertainties.The results are summarized and a final qualitative evaluation (from − − through 0 to ++) of computational cost, numerical stability and overall potential benefit compared to Total-Least-Squares methods is given in table 3.In general, the more uncertain the measurement data basis is, the stronger the results can be influenced by any kind of prior knowledge.
In section 6, the Bayesian model for circular fitting was tested for practical suitability using a real data example of optical measurement.Here, as well, the previously formulated hypothesis that uncertainties induced by noisy measurement data can be significantly reduced by including prior knowledge was confirmed.
To summarize, it should be noted that Bayesian modeling may offer benefits compared to established fitting methods since its application can significantly reduce uncertainties or stabilize evaluations by considering prior knowledge.Even trivial assumptions such as the diameter of a bore hole being larger than zero but smaller than the workpiece it is bored through can serve as usable prior knowledge.Since the presented Bayesian method can also be applied in the absence of prior knowledge, it represents a comprehensive method that covers the state-of-the-art method and additionally provides the possibility to include any type of prior knowledge into the analysis.
In upcoming research, a-priori knowledge models used within Bayesian analyses will be addressed in more detail since not every kind of prior knowledge can be easily described by a probability density function, not to mention a Gaussian distributed one, in real-world scenarios.Meeting this challenge will significantly increase the popularity of the Bayesian approach in practical applications.

Figure 1 .
Figure 1.Relationship of likelihood, a-priori and a-posteriori PDFs in the context of Bayesian analysis.

Figure 2 .
Figure 2. Relationship of a measured data point and an ideal circle.

N
The following cases are shown in figure 3 to illustrate the effect of the considered circle segment: (a), the variances of the estimated circle parameters grow large.As the measured circular segment increases, the variances decrease sharply (see figures 3(b)-(d)).

Figure 3
Figure 3 Comparison of three different circle fitting cases: (a) N 30 = measured data points on a circular arc of 60 total j D = (upper plot) and the resulting PDF (lower plot), (b) N 60 = and 120 , total j D =  (c) N 90 = and 180 , total j D =  (d) scale comparison of the expansion of resulting PDFs imaged in (a)-(c).

Figure 4 .
Figure 4. Averaged results for the characteristic values of the posterior PDFs used for radius evaluation with respect to the measured circular arc total j D (without and with consideration of prior knowledge, including 95% confidence interval): (a) Maximum-A-Posteriori estimation, (b) Minimum-Mean-Square-Error estimation, (c) Averaged standard deviation.

Figure 5 .
Figure 5. Relationship of a measured data point and an ideal straight-line.

N
The three cases investigated represent measurements of segments of a straight-line of different total lengths ((a) 5, (b) 10 and (c) 15), each measured by different numbers of sample points n .

Figure 6 .
Figure 6.Different cases of underlying measurement data bases for the Bayesian straight-line fitting approach to be evaluated.Columns: Visualization and evaluation of simulated scattered measurements of a straight-line with a total length of (a) 5, (b) 10 and (c) 15.Upper row: Visualization of the straight-line segments measured, exemplarily using ten sample points (n 10 s = ).Center row: Averaged results for the standard deviation u ¯ŝ of the posterior PDFs used for evaluating the straight-line's distance u to the origin with respect to the number of sample points n s (without and with consideration of prior knowledge of opt 0 j j = with 0.1 s = j ).Lower row: Averaged results for the standard deviation ¯ŝ j of the posterior PDFs used for evaluating the straight-line's angle with respect to the number of sample points n s (without and with consideration of prior knowledge of opt 0 j j = with 0.1 s = j

Figure 7 .
Figure 7. Relationship of a measured data point and an ideal ellipse.

Figure 8 .Table 2 .
Figure 8.Comparison of three different ellipse fitting cases with consideration of different levels of prior knowledge: (a) N 60 = measured data points on an elliptic arc of 120 total j D = (upper plot) and the resulting PDF of the half axis a (lower plot), (b) N 75 = and 150 , total j D =  (c) N 90 = and 180 total j D = .

s
Accordingly, the prior knowledge about the radius of the measured object R s is assumed to be very poor in relation to the relatively low measurement uncertainty in the central measurement area .xyIt has already been shown that even a small amount of prior knowledge can have a significant influence on the result with a poor data base.The result of the determination of the orthogonal distances Bayes x of the real measurement data to the Bayesian circle fit is shown in figure10.The radius determined using the Bayesian method in this example is R 1497.60 m MAP ˆ= m with R ŝ = 9.36 m. m In comparison, a TLS fit on the same reduced database provides a radius of R 1524significantly higher than the radius expected from existing prior knowledge.

Figure 9 .
Figure 9. (a) Measured data set with its corresponding Total-Least-Squares fitted circle, (b) Residuals TLS x as orthogonal distance of measured data points from TLS fitted circle plotted over the circular angle, (c) Normalized histogram of residuals TLSx .

Figure 10 .
Figure 10.(a) Measured data set (black line) and the reduced data set (red dots) with its corresponding Bayesian fitted circle (red line), (b) Residuals Bayes x as orthogonal distance of measured data points from Bayesian fitted circle plotted over the circular angle, (c) Normalized histogram of residuals Bayes x .