=Paper=
{{Paper
|id=Vol-1638/Paper47
|storemode=property
|title=Determination of the bilateral filter's parameters for the analysis of surface geometry deviations
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper47.pdf
|volume=Vol-1638
|authors=Vadim A. Pechenin,Mikhail A. Bolotov,Ekaterina R. Stepanova
}}
==Determination of the bilateral filter's parameters for the analysis of surface geometry deviations ==
https://ceur-ws.org/Vol-1638/Paper47.pdf
Image Processing, Geoinformatics and Information Security
DETERMINATION OF THE BILATERAL FILTER'S
PARAMETERS FOR THE ANALYSIS OF SURFACE
GEOMETRY DEVIATIONS
V.A. Pechenin, M.A. Bolotov, E.R. Stepanova
Samara National Research University, Samara, Russia
Abstract. This paper considers a method of determination of the bilateral fil-
ter's parameters while processing geometric measurements. The proposed
method is formulated as an optimization problem that takes into account the fol-
lowing parameters of the excess after filtration: its compliance with the Gaussi-
an distribution, the availability of correlation. The paper describes approvement
of the method to determine the optimal parameters of the bilateral filter when
filtering the form of deviations of the measured points of a flat surface ma-
chined by grinding. Considered method of determination of the bilateral filter's
parameters is implemented in the software package MATLAB and it can be
used in the processing of geometric information obtained from measurements.
Keywords: method, form deviation, bilateral filter, Pearson’s chi-square test,
autocorrelation coefficient, variance.
Citation: Pechenin VA, Bolotov MA, Stepanova ER. Determination of the
bilateral filter's parameters for the analysis of surface geometry deviations.
CEUR Workshop Proceedings, 2016; 1638: 386-392. DOI: 10.18287/1613-
0073-2016-1638-386-392
Introduction
The precision of determining the geometric parameters of measurement object de-
pends on the accuracy of the measurement of parts by coordinate measuring machine
(CMM). The problem of precision is particularly acute when measuring parts have
complex geometric shapes. Measured information consists of a set of points in space.
During processing of the measured data it is often necessary to remove noise repre-
senting the inaccuracy of measuring instrument [1,2]. For modern contact CMM such
error is 0.7-2 µm but when measuring complex curved surfaces or if condition is non-
compliance with normal conditions this measurement error can be greater. When
measuring surfaces with laser and optical means (e.g., T-SCAN-CS and 6 COMET
company Steinbichler Optotechnik GmbH) the magnitude of errors increase to 10-50
µm. The problems associated with noise reduction occur in image processing [3-5].
There are extensive research and a variety of developed noise reduction algorithms
specifically for image processing tasks, they can be used for noise reduction in the
processing of the measured data [6,7].
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For this study the coordinates of points on a flat surface treated by grinding were
measured by the coordinate measuring machine DEA Global Performance 07.10.07.
Deviations from the nominal geometry are calculated at the measured points. Filtra-
tion of calculated deviations at surface points is performed. The aim of the research is
to determine the filter parameters suitable for processing of measured deviations.
1 Calculation of form deviations
Measurement was carried out on a flat surface at 9792 points (153grid points along
the x direction and 64 points in the direction y, figure 1).
Fig. 1. Measurement of a part; 1,2, 3 are flat surfaces
The coordinate system is defined as follows: 1 plane defines the axis OZ; at the inter-
section of the planes 1, 2 and 3 the coordinate system center is calculated; at the inter-
section of the planes 2 and 1 the axis OY is calculated.
Two coordinate systems: roughing and finishing are performed for the most accurate
positioning. The algorithm of positioning is the same in both cases but the planes of
roughing system were measured manually by 4 points on each plane. Finish position-
ing was performed in automatic mode by 9 points for each plane.
Then in the software (PC-DMIS) of CMM the form deviations at given points was
calculated representing a deviation of the measured points from the points on the
nominal surface. The calculated deviations can be characterized by the equation:
I (u ) dF (u ) r . (u ) , (1)
where dF (u ) is the actual form deviation at the point;
r (u ) is the passport value of measurement error, the random component of devia-
tion.
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2 Utilized filter
For the separation of the error on the systematic and random (noise) errors the bilat-
eral filter was used. The bilateral filter was introduced for the first time in [8]. This
filtering technique is used mainly in tasks of image processing [9,10] and it is appli-
cable to noise reduction of measured deviations.
During work of the bilateral filter the value of deviation at the point is calculated (as
for digital images it is the intensity of the pixel) as a weighted medium intensity of the
neighboring values of deviations in a neighborhood of radius r . Weight of neighbor-
ing deviations is changed in accordance with values of the distances between points
(spatial weight) and a value of deviations (the intensity of pixels for images) at points
that is the order filter. The widest application has bilateral filter based on the Gaussian
function. Taking into account that random errors of measurement generally obey a
normal law of distribution [11] Gaussian bilateral filter was used in the work.
The bilateral filter I (u ) for a set of deviations (an image) at the point is defined by the
following equation:
Wc ( p u ) Ws ( I (u ) I ( p) ) I ( p)
ˆI (u ) pN (u ) , (2)
Wc ( p u ) Ws ( I (u ) I ( p) )
pN (u )
where N (u ) is the neighborhood of the point u ;
Wc is a domain filter (Gaussian filter) with parameter c ( Wc ( x) e x /(2c ) );;
2 2
Ws is a rank filter (weighted similarity function) with parameter s
( Ws ( x) e x /(2s ) ).
2 2
In practice (as in our study) the neighborhood N (u ) consists of a set of points q i
which satisfies the inequality:
u qi r , (3)
where r 2 c
The excess of filtering is calculated at each point respectively by the equation:
r (u ) I (u ) Iˆ(u ). (4)
3 Noise analysis
To determine the threshold values of the filter parameters on the basis of pre-position
that the filtered component should be of random nature the linear autocorrelation
coefficients are calculated:
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N y N x n
r (i, j ) r (i n, j )
j 1 i 1
x ( n) , (5)
N y N x n N y N x n
r 2 (i, j ) r 2 (i n, j )
j 1 i 1 j 1 i 1
where x (n) are the linear autocorrelation coefficient in the x direction;
n is a lag of autocorrelation; in the x direction n 1,..., N x 1 .
Similarly the linear coefficient in the y direction is described there is n 1,..., N y 1 .
The null hypothesis lies in the fact that the value of the random component of the
deviation at each i-th point does not correlate to other values, so x (i ) 0 .
To estimate compliance of nature of the calculated excess r of the filtering with
random errors of measurement the test for compliance with the normal law by the
Pearson’s chi-square test (criterion 2 ) was used. According to the null hypothesis
(Н0) the distribution of the array of filtered-off deviations follows the law of normal
distribution and according to the alternative hypothesis that does not correspond to the
normal law.
4 Results
The search problem of the optimal filter parameters { с , s } is formulated as follows:
r min, (6)
with the set of constraints:
2 r 2 ,
critical
(7)
Var r Var ,
critical
where r is the linear autocorrelation coefficient calculated by the equation 5;
2 r is the Pearson’s chi-square test calculated in base of the excess r of the
filtering;
2 is the critical value of the Pearson’s chi-square test for a given belief line of
critical
importance ;
Var r is the variance calculated in base of the excess r of the filtering;
Var is the critical value of the variance, so the passport uncertainty of measuring
critical
instrument is taken in place of it.
The solution was carried out by the method of uniform search.
To determine the random error component the filter with parameters c 1;5 and
s 0.1;3 was used. Figure 2 shows the results of the filtering of the measured form
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deviations at the surface points. In Case F1 parameters c 1 and s 1 are used;
in F2 parameters c 3 and s 0,1 are used; in F3 - c 5 and s 3 are used.
Fig. 2. The results of application of the filter to calculated deviations; a); b) the data after filtra-
tion, c) the noise as a result of filtering
It could be noticed from the results in Figure 1that in the case F3 the surface after the
filtration is smoother, but the magnitude of the random component of the error signif-
icantly exceeds the passport uncertainty of used CMM; it means that we filtered off
the useful signal together with the noise. In any case an additional analysis of the
results is required in order to make a conclusion about the quality of the filter.
According to this criterion only filter with parameters c 3 and s 0,1 is suited.
Autocorrelation coefficients x and y (eq. 4) were also calculated. Table 1 shows
the values of the criterion 2 , as well as the expectations ( ), minimum and maxi-
mum values of autocorrelation coefficients for the series of experiments.
The values of the autocorrelation coefficients indicate a very weak correlation of the
deviations. To estimate the significance of the correlation the Student’s t-test was
used. Despite the fact that the resulting noise (according to criterion 2 ) is strictly
subjected to a normal distribution only in the case F2 it was noticed visually in histo-
grams that distribution in the remaining cases are close to normal. Therefore the re-
sults of the Student’s t-test are applicable to considered noise values [12]. t-test for the
coefficients ( may be x or y ) is calculated by the equation:
t (M 2) M 2 / 1 2 , (8)
where M is the number of points used for calculation of .
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Table 1. Results of the analysis of the filtered-off noise
c s 2 x y
Min Max Min Max
1 0,1 H1 -0,001 -0,013 0,037 0,001 -0,024 0,020
1 1 H1 -0,001 -0,018 0,052 0,001 -0,032 0,029
1 3 H1 -0,001 -0,018 0,052 0,001 -0,032 0,029
3 0,1 H0 0,002 -0,013 0,105 0,003 -0,013 0,051
3 1 H1 0,003 -0,017 0,133 0,004 -0,017 0,065
3 3 H1 0,003 -0,017 0,133 0,004 -0,017 0,065
5 0,1 H1 0,004 -0,014 0,120 0,007 -0,011 0,069
5 1 H1 0,005 -0,018 0,154 0,010 -0,014 0,090
5 3 H1 0,005 -0,018 0,154 0,010 -0,014 0,091
Figure 3 shows how the criteria changes with a lag of n in directions x and y in cases
F1, F2 and F3. A critical value of criterion for sufficiently large M at a significance
level of 0.05 is 1.96 (the horizontal line in the figure).
Fig. 3. The t-test for autocorrelation coefficients
For these samples it may be concluded that increase of parameters c and s in-
creases the probability that the null hypothesis of absence of correlation is incorrect.
5 Conclusion
As a result the filtration of noise in the measurement of form deviation of the preci-
sion surface is performed. The form deviation becomes available in the stage of the
measurement of surface with CMM DEA Global Performance 07.10.07. For the filter-
ing operation the bilateral filter with different parameters c and s were utilized.
The obtained results, as well as their analysis by the criterion 2 and the autocorrela-
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tion function show that use of the bilateral filter allows high-quality noise filtering
and obtaining more reliable data on measured deviations with the parameters c 3
and s 0,1 .
Acknowledgements
This work was financially supported by the Ministry of Education and Science of the
Russian Federation within the framework of the Federal Target Program "Research
and development on priority directions of scientific-technological complex of Russia
for 2014-2020 years". The unique identifier of ASRED is RFMEFI57815X0131.
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