Improved Relative-entropy Method for Eccentricity Filtering in Roundness Measurement Based on Information Optimization

In this study, we propose the improved relative-entropy of the ideal circle function to the measured information of the radius error of the workpiece surface to make an eccentricity filtering in roundness measurement. Along with a correct assessment for the parameters of the eccentricity filtering, the extracted information from the measured information is obtained by the minimization of the improved relative-entropy. The case studies show that the information optimization is characterized by decreasing the improved relative-entropy, the extracted information almost coincides with the real information, the improved relative-entropy has a strong immunity to the stochastic disturbance of the rough work piece-surface and the increase of the minimum of the improved relative-entropy counteracts the effect of the stochastic disturbance on the assessment for parameters in eccentricity filtering.


INTRODUCTION
The information entropy is a measure for the uncertainty of an information source (Woodbury and Ulrich, 1998;Xia et al., 2010).The relative entropy as an important concept in the information entropy theory is a measure for the relation of two information sources (Woodbury and Ulrich, 1998;Aviyente and Williams, 2005;Zhang et al., 2008;Molnár and Szokol, 2010;Saravanan et al., 2012;Guan et al., 2012).
So far the relative entropy is widely applied to many areas in science and technology, such as timefrequency analysis (Aviyente and Williams, 2005), fault diagnosis (Guan et al., 2012), models election (Techakesari and Ford, 2012), data processing (Sharma, 2012) and phase retrieval (Soldovieria et al., 2010) and so on.
The minimum relative-entropy theory is one of the popular relative-entropy theories.According to this theory, the smaller the relative-entropy is, the closer the relation of two information sources is; and vice versa.But, the use of the relative entropy relies on the known probability density function of a population studied (Woodbury and Ulrich, 1998;Woodbury, 2004;Molnár and Szokol, 2010).If the probability density function is unknown in advance, the relative entropy becomes ineffective.For this reason, taking the eccentricity filtering in roundness measurement as an example, this study proposes an improved relative-entropy method without any requirement for prior information about probability density functions and the characteristic of the improved relative-entropy function is investigated with the help of information optimization.
In this study, we propose the improved relativeentropy of the ideal circle function to the measured information of the radius error of the workpiece surface to make an eccentricity filtering in roundness measurement.Along with a correct assessment for the parameters of the eccentricity filtering, the extracted information from the measured information is obtained by the minimization of the improved relative-entropy.The case studies show that the information optimization is characterized by decreasing the improved relativeentropy, the extracted information almost coincides with the real information, the improved relative-entropy has a strong immunity to the stochastic disturbance of the rough work piece-surface and the increase of the minimum of the improved relative-entropy counteracts the effect of the stochastic disturbance on the assessment for parameters in eccentricity filtering.

MATHEMATICAL MODEL OF INFORMATION OPTIMIZATION
Assume the real information of the radius error of a workpiece surface is expressed as: (1) where, r = The real information of the radius error of the workpiece surface t = The angle variable The information of a basic circle produced by the pressed deformation of the sensor, j = The harmonic order F j = The amplitude of the jth harmonic and t 0 is the initial phase The measured information of the radius error of the workpiece surface can be obtained by a roundness measuring instrument, as follows: (2) with ( 3) and (4) where, r 1 is the measured information of the radius error of the workpiece surface and a and b are two eccentricity components.
The eccentricity components a and b are of the positioning errors (Adam et al., 2012;Garcia-Plaza et al., 2012) and must be filtered out from the measured information r 1 so that the real information r is extracted and the roundness of the workpiece surface can then be evaluated.For this reason, the improved relativeentropy is defined as: (5) with ( 6) and ( 7) where E is the improved relative-entropy of the ideal circle function to the measured information of the radius error of the workpiece surface, r 00 is the ideal circle function, x(1), x(2) and x(3) are three parameters to be solved and x is the parameter vector.
In equations ( 6) and ( 7), three parameters, x (1), x (2) and x (3), correspond to a, b and R 0 , respectively.According to the minimum relative-entropy theory, if the minimum of the improved relative-entropy, E min , is gained at: (8) then the values of x(1), x(2) and x(3), are, respectively,  the estimated values of a, b and R 0 , as follows: (9) where, x * is the optimum value vector of the three parameters and  * ,  * , and  0 * are, respectively, the optimum values of a, b and R 0 , which satisfy Eq. ( 8).
The information of estimating for the real information of the radius error of the workpiece surface is given by: (10) where, r 0 is the estimated information of the real information of the radius error of the work piece surface, that is, r 0 is the information extracted from the measured information of the radius error of the workpiece surface by means of eccentricity filtering and can be employed to evaluate the roundness of the workpiece surface.
The roundness of the workpiece surface can be obtained as: (11) where,  is the roundness of the workpiece surface.
From Eq. ( 8) and ( 11) the obtained roundness relies on the information extracted from the measured information with the help of the minimum relativeentropy.Therefore, it is a characteristic parameter of the improved relative-entropy based on information optimization for eccentricity filtering.It follows that the roundness evaluating process is a process of decreasing the relative entropy, along with an optimization of the measured information, without any requirement for prior information about probability density functions.

CASE STUDIES AND DISCUSSION
Roundness evaluation of of smooth workpiecesurface with change of relative-entropy: This is Case 1.This case studies the change of the relative-entropy with three parameters, x(1), x(2) and x(3), by a simulation of the roundness evaluating for the measured information of the radius error of the smooth workpiece-surface with the second harmonic interfered  by an eccentricity.The imitated parameters are shown in Table 1.From Eq. ( 1) and ( 2) and Table 1, the real information and the measured information of the radius error of the workpiece surface are imitated, as shown in Fig. 1 and 2, respectively.Obviously, in polar coordinates (t = 0~360˚) the real information r is an ellipse and the measured information r 1 is a cardioid, with a large difference between the two due to the eccentricity interference.
In Cartesian coordinates as shown in Fig. 3 and 4 (t = 0~2 ), the difference between the two can directly be compared and the good result obtained using the method proposed in this study can clearly be found.It can be seen from Fig. 3 and 4 that although the difference between the real information r and the measured information r 1 is very significant, the extracted information r 0 still is perfect, which almost coincides with the real information r.This is because the decreasing relative-entropy based on information optimization for eccentricity filtering makes a correct assessment for the parameters a, b, R 0 , F 2 and , as shown in Table 2.
It is easy to see from Table 2 that the relative errors between the estimated values and the truth values are very small; indicating that the improved relativeentropy based on information optimization is good at eccentricity filtering in roundness measurement, π r δ  without any requirement for prior information about probability density functions.Figure 5 to 7 present the change of the improved relative-entropy E with three parameters, x(1), x(2) and x(3).It is found from Fig. 5 to 7 that the improved relative-entropy E proposed in this study is a complex function of three parameters, x(1), x(2) and x(3) and information optimization is characterized by decreasing the improved relative-entropy E. When the improved where, N(0,  2 ) stands for the normal distribution function with the 0 mean and the s standard deviation, which is employed for the expression of the rough workpiece-surface.The imitated parameters are shown in Table 3.
The real information and the measured information of the radius error of the rough workpiece-surface are shown in Fig. 8 and 9, respectively.The difference between the two can directly be compared and the good result obtained using the method proposed in this study can clearly be found.It can be seen from Fig. 8 to 10 that although the difference between the real information r and the measured information r1 is very significant, the extracted information r0 still is perfect, which almost coincides with the real information r.This is because the decreasing relative-entropy based on  information optimization for eccentricity filtering makes a correct assessment for the parameters a, b, R 0 , F 2 and  , as shown in Table 4.
It is easy to see from Table 4 that the relative errors between the estimated values and the truth values are    similar to that in Fig. 5 to 7 in Case 1, revealing a strong immunity of the improved relative-entropy to the stochastic disturbance of the rough workpiece-surface.This shows that the improved relative-entropy is of high robustness that is discussed below.
Robustness of improved relative-entropy with stochastic disturbance of rough workpiece-surface in roundness evaluation: This is Case 3.This case studies the robustness of the improved relative-entropy with the stochastic disturbance of the rough workpiecesurface in roundness evaluation.The imitated parameters are shown in Table 3 and the results are shown in Tables 5 to 8. Clearly, although the standard deviation s takes the value in the range from 0.0001mm to 0.002mm, the relative errors between the estimated values of the parameters, a, b, R 0 and δr and their truth values still are very small.The price of such a result is the increase of the minimum of the improved relative-entropy with the stochastic disturbance of the rough workpiecesurface in roundness evaluation, as shown in Fig. 14, that is, the increase of the minimum of the improved relative-entropy counteracts the effect of the stochastic disturbance of the rough workpiece-surface on the assessment for parameters, a, b, R 0 and and the real information can soundly be extracted.

CONCLUSION
This study proposes the improved relative-entropy method for the eccentricity filtering in roundness measurement based on information optimization, without any requirement for prior information about probability density functions.The improved relative-entropy is a complex function of the parameters of eccentricity filtering and information optimization is characterized by decreasing the improved relative-entropy.When the improved relative-entropy is approaching to the minimum, the parameter vector takes the optimum value vector and the information of eccentricity filtering is optimized in roundness measurement.
The extracted information from the measured information almost coincides with the real information.This is because the decreasing relative-entropy based on information optimization for eccentricity filtering makes a correct assessment for the parameters.
The improved relative-entropy has a strong immunity to the stochastic disturbance of the rough workpiece-surface.The increase of the minimum of the improved relative-entropy counteracts the effect of the stochastic disturbance on the assessment for parameters in eccentricity filtering and the real information can soundly be extracted.

Fig. 3 :
Fig. 3: Comparison among real information r, measured information r 1 , and extracted information r 0 in Case 1

Fig. 14 :
Fig. 14: Change of minimum of improved relative-entropy with stochastic disturbance of rough work piecesurface in Case 3

Table 1 :
Imitated parameter in case 1

Table 2 :
Result of assessment for parameter in Case

Table 3 :
Imitated parameter in Cases 2 and 3

Table 4 :
Result of assessment for parameter in Case 2 (s = 0.0001 mm)

Table 5 :
Result of assessment for parameter in Case 3 (s = 0.0002 mm)

Table 7 :
Result of assessment for parameter in Case 3 (s = 0.0009 mm)

Table 8 :
Result of assessment for parameter in Case 3 (s = 0.002 mm) The change law of the improved relative-entropy is presented in Fig.11to 13 in this Case and, overall, it is