RobustLocallyWeightedRegressionforProfileMeasurementof ... · 2020. 5. 22. · presents an outlier...

Research ArticleRobust Locally Weighted Regression for Profile Measurement ofMagnesium Alloy Tube in Hot Bending Process

Zhibin Yao 12 Rongjun Wang12 Jinning Zhi12 and Qinglu Shi 12

1Heavy Machinery Engineering Research Center of the Ministry Education Taiyuan 030024 Shanxi China2College of Mechanical Engineering Taiyuan University of Science and Technology Taiyuan 030024 Shanxi China

Correspondence should be addressed to Qinglu Shi 327606120qqcom

Received 20 November 2019 Revised 2 March 2020 Accepted 23 April 2020 Published 22 May 2020

Academic Editor Mohamed Shaat

Copyright copy 2020 Zhibin Yao et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Section flattening often occurs in the hot bending process of magnesium alloy tube with large curvature In order to control theforming quality of the tube it is necessary tomeasure the section profile of the magnesium alloy pipe online In this paper the laservision system is used to measure the profile of magnesium alloy tube Due to the influence of the environment and the surfacequality of the pipe there are obviously isolated outliers in the profile data which seriously affects the accuracy and precision of thetube measurement An outlier identification algorithm based on robust locally weighted regression and PakTa criterion isproposed+is algorithm is used to identify the typically isolated outliers in the measurement process and discuss its identificationability Meanwhile it is compared with the moving mean identifier and the Hampel identifier Subsequently the ellipse fitting ofprofile data was carried out and the fitting ellipse parameters and fitting precision of the curved section were obtained At the sametime the fitting results were compared before and after the outliers are eliminated +e experiment proves that the outlieridentification method based on robust locally weighted regression and PakTa criterion can effectively identify outliers in profiledata especially for spot outliers +is algorithm is a robust accurate and efficient outlier identification method which caneffectively improve the laser profile measurement accuracy of the pipe section and has great significance for the quality control ofmagnesium alloy tube

1 Introduction

Magnesium alloy has the advantages of low density and highspecific strength and specific stiffness so it is graduallyreceiving attention from various industries [1ndash4] In par-ticular various types of magnesium alloy tubes are widelyused in aviation aerospace transportation and other fields[5ndash7] However various defects often occur in the hotbending forming process of magnesium alloy tubes withlarge curvatures such as section flattening excessive thin-ning and wrinkle fracture [8ndash11] Section flattening is anunavoidable problem in magnesium alloy tubes with largecurvature As shown in Figure 1 the section flattening of themagnesium alloy tube is mainly due to the action of bendingmoments M1 and M2 the compression stress Fc on theinner side and the tensile stress Ft on the outer side and thepipe is bent due to the uneven force on both sides As a

result the section is deformed to become approximatelyelliptical For the same specification the smaller the bendingradius the larger Ft and Fc and the more obvious the trendof section flattening If it is a bending without a mandrel theflattening is more serious which affects the forming qualityof the pipe In order to control the forming quality of thetube the section profile must be measured online and theflattening of the magnesium alloy section must be detectedin real time

Pipe profile measurement generally uses contact mea-surement and noncontact measurement [12ndash15] Machinevision is an important method in noncontact measurementCompared with traditional manual contact measurementmethods machine vision measurement has high accuracyand fast speed and causes no damage [16ndash19] In this study alaser vision system based on line structure light was used tomeasure the flattening section of magnesium alloy tube

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 7952649 10 pageshttpsdoiorg10115520207952649

caused by the large curvature hot bending to obtain theaccurate profile data of the magnesium alloy tube section

+e online vision measurement of tube bending isstrongly demanded to improve the dimensional accuracyHowever the quality of the profile data deteriorates sharplydue to the environment and surface quality of the pipe Inparticular many isolated outliers appear in the measureddata Outliers generally mean that the observed values aresignificantly different from most observed values that is themeasured data do not obey the statistical distribution law ofthe data [20] In this experiment the outliers appear inisolation as the form of dot or point which are not nec-essarily related to the data quality before and after +ere-fore they are also called isolated outliers Outliers can begenerated for a variety of reasons such as sensor noisechannel interference and human factors [12 21] +eseoutliers will cause data distortion leading to the miscal-culation of model parameters and wrong analysis results+erefore it is necessary to identify and eliminate theoutliers in the laser measurement process to improve theprecision and accuracy of the measurement

Commonly used identification methods for outliersinclude Nair Grubbs Dixon etc [22 23]+ese methods aredifficult to apply in laser profile measurement due to thelimitation of data distribution and data amount +is paperpresents an outlier identification method based on robustlocally weighted regression (RLWR) which is applied tolaser profile detection RLWR is developed from locallyweighted regression (LWR) and belongs to nonparametricestimation Nonparametric estimation is an important re-search direction of modern statistical analysis It can adapt tomore complex nonlinear changes without assuming thespecific form of population distribution and error distri-bution and without directly obtaining data models which ismore flexible robust and widely applied than parameterestimation [24 25]

RLWR is a robust fitting process that integrates localpolynomial estimation and locally weighted regression withexcellent smoothing performance +is method was first

proposed by Cleveland [26] and further elaborated [27 28]subsequently improved by Jacoby [29] and Loader [30] Ithas been gradually applied to different fields of scientificresearch and engineering applications Ma et al [31] used theRLWR to reduce the impact of high-frequency noise onsuperresolution enhancement of multiangle remote sensingimagery Leonor et al [32] minimized the influence of theinhomogeneity effect on tree reradiation pattern by usingRLWR Chen et al [33] proposed an algorithm based onRLWR and robust z-scores for the construction of a pit-freecanopy height models Nurunnabi et al [34 35] and Liu et al[36] used RLWR techniques to study the filtering of 3Dground cloud point data Yu et al [37] applied the method tothe smoothing of combustion kinetics data of pine sawdustbiochar

+is study is based on the robust smoothing perfor-mance of RLWR which is used to smooth the laser mea-surement data and realize the identification of isolatedoutliers At the same time profile fitting was carried out toverify the effectiveness of this algorithm in the profileinspection

+is paper organized as follows Section 2 introduces thelaser vision measurement system and its basic principles andanalyzes the profile data of magnesium alloy tube pointingout the typical outliers in the data In Section 3 the movingaverage algorithm Hampel algorithm and RLWR algorithmare respectively adopted to identify the outliers It focuseson the outlier recognition effect for the RLWR combinedwith PakTa criterion median criterion and quartile crite-rion respectively In Section 4 elliptic fitting of profile datais performed by using the RLWR identification algorithmand relevant elliptic parameters and fitting error are ob-tained at the same time compared with original data Lastlya conclusion is given in Section 5

2 Laser Vision Measurement System andProfile Outliers

21 Laser Vision Measurement System +e laser visionmeasurement system adopted in this experiment mainlyconsists of a line-structured light sensor image processingunit and control unit as shown in Figure 2 +e systemworks as follows the laser source in the sensor projects linestructure light to a measured object +e reflected light iscaptured by the camera in the sensor +e profile data arerecorded and displayed by industrial PC after image pro-cessing +e measuring platform can move along the axialdirection of the pipe which is controlled by the motioncontrol system realizing the profile measurement of eachsection for the pipe

Laser triangulation is the basic principle of line-struc-tured light vision measurement +e line-structured lightprojected by laser source forms a scattered light band on thepipe surface and the scattered light is imaged in CCD array+e distance and coordinate data of the measured point canbe obtained through triangular geometry [38 39]

As shown in Figure 3 the point p(x y z) is a mea-surement point on pipe surface point pprime(u v) is the imagingpoint of p in CCD image plane f is the focal length of

M1

M2

Ft

Ft

FcFc

Figure 1 Force condition of bending process

2 Mathematical Problems in Engineering

camera b is the distance between the center of light sourceand the center of camera lens and θ is the angle between theX axis and the construction line which is formed by themeasured point and the center of light source +e precisecoordinates of p(x y z) on the measured profile can beobtained from the spatial geometry in the figure

22 Typical Profile Outliers Observe the profile data of eachsection obtained by the measuring system +ere are twotypical isolated outliers in the measured profile that is thespot outliers and the point outliers +e spot outliers orspeckle outliers are composed of multiple point outliers asshown on the right side of Figure 4 +e double-pointisolated outliers are shown in the middle of Figure 5

In order to facilitate analysis as shown in Figure 6 thedouble-point outliers in Figure 5 are superimposed in Figure 4Subsequently the recognition ability of different identificationalgorithm is investigated for two types of outliers

3 Identification of Profile Outliers

31 Moving Mean Identifier Recognizes Isolated Outliers

Imageprocessor

Motioncontrolsystem

IndustrialPC

Laser source Camera

Sensor for linearstructured light

Measured bar

Measured profile

Figure 2 +e process of laser vision measurement

Measured object

P (x y x)Z

c

θ b X

Z

Pprime(uv)Light source

uO Lens center

CCD

f X

Figure 3 Principle of triangulation measurementData of typical profile B

Double-point outlier

ndash8 ndash6 0 6ndash10 2ndash2ndash4 4x (mm)

51

515

52

525

53

535

54

545

z (m

m)

Figure 5 Double-point outlier in profile B


51

515

52

525

53

535

54

545

z (m

m)

Figure 6 Typical profile after superposition of outliers

Data of typical profile A

Spot outlier


51

515

52

525

53

535

54

545

z (m

m)

Figure 4 Spot outlier in profile A

Mathematical Problems in Engineering 3

Moving mean identifier or movmean identifier is the easiestway to identify isolated outliers [40 41] +e basic definitionis as follows For the data sequence x1 x2 x3 xn movingwindow length is wl and the outliers are judged by thefollowing equation

yi xi xi minus μi

11138681113868111386811138681113868111386811138681113868le 3σi

outlier xi minus μi

11138681113868111386811138681113868111386811138681113868gt 3σi

⎧⎨

⎩ (1)

where μi is the local mean and σi represents local standarddeviation (LSD) within the moving window +e movingmean identifier is to use the 3σ criterion to judge the outliersin the data window when the length is wl When the dif-ference between the measured value and the local mean isgreater than three times the local standard deviation it isconsidered as an outlier

+e moving mean identifier is used to identify outliers inFigure 6 and the recognition results are shown in Figures 7and 8

+emoving window length is gradually increased Whenthe window length increases to 19 only one point outlier isidentified as shown in Figure 7 When the window length isincreased to 25 the threshold range is reduced in the middleof the profile so that the double-point outliers in the middlecould be completely identified (see Figure 8) However thespot outliers on the right are never identified regardless ofthe length of the moving window

It can be seen from Figures 7 and 8 that the dis-tinguishing threshold at the location of outliers is greatlyincreased due to the existence of outliers in the middle of theprofile With the increase of window length the thresholdcurve in the middle of the profile is gradually smooth and thethreshold range is gradually reduced However the upperand lower threshold ranges on both sides of the profileincrease significantly It indicates that the identificationability of outliers on both sides of the profile decreases withthe increase of window width

32 Hampel Identifier Recognizes Isolated OutliersHampel identifier is a median identification method whichuses the median and absolute median deviation as a robustestimation of the location and distribution of outliers withgood robustness [42ndash45]

Hampel identifier is defined as follows for data se-quences x1 x2 x3 xn the number of neighbors on ei-ther side of is k then the moving window length is 2k + 1and the local median is mi

mi median ximinus k ximinus k+1 ximinus k+2 xi xi+kminus 2(

xi+kminus 1 xi+kminus 1 xi+k1113857(2)

+e local scaled median estimated deviation (MADl) isexpressed as follows

MADl k middot median ximinus k minus mi

11138681113868111386811138681113868111386811138681113868 xi+k minus mi

111386811138681113868111386811138681113868111386811138681113872 1113873 (3)

where k 1(2

radicerfcminus 1(12)) asymp 14826 which is the un-

biased estimation of the Gaussian distribution

yi xi xi minus mi

11138681113868111386811138681113868111386811138681113868le tMADl

outlier xi minus mi

11138681113868111386811138681113868111386811138681113868gt tMADl

⎧⎨

⎩ (4)

When the difference between the measured data and thelocal median is greater than t times MADl the measuredvalue is considered to be an outlier as shown in equation (4)

Hampel identifier is used to recognize the outliers inFigure 6 +e identification effect is observed by changingthe length of moving window when t 3 As shown inFigure 9 when the window length is 5 the threshold range ofidentification is relatively narrow Hampel identifier wasable to identify the double-point outlier in the middle of theprofile but it is unable to identify the spot outliers on theright side At the same time the identification thresholdfluctuates significantly with the spot outliers Furthermorethe misidentification of outliers is observed Besidesinfluenced by the discontinuous data on both sides of theprofile the identification threshold fluctuated greatly andthe endpoint data are identified as outliers

When the window length is increased to 25 as shown inFigure 10 the upper and lower identification threshold onboth sides of the profile increased significantly and thecentral double point outliers can be effectively identified but

2 4ndash4 ndash2 0ndash8 ndash6ndash10 6x (mm)

51515

52525

53535

54545

55

z (m

m)

Original signalOutlier

LowerUpper

Figure 7 Outlier recognition using movmean identifier windowlength 19



LowerUpper

51515

52525

53535

54545

55

z (m

m)



the spot outliers on the right side of the profile are unrec-ognized It is worth noting that the misidentification inFigure 9 no longer appeared Meanwhile several discon-tinuous data at the right end of the profile are identified asoutliers

If the moving window length continues to increase theidentification threshold on both sides of the profile willincrease accordingly Although the double-point outliers canbe identified the spot outliers are still unrecognizable

Compared with the moving mean identifier the Hampelidentifier significantly reduces the threshold range and thefluctuation phenomenon which can effectively identifypoint outliers but it is still unable to identify spot outliers Inaddition the large interval of profile data will increase thedifference between the data and the median resulting in alarge fluctuation for threshold which affects its ability toidentify outliers

33 Recognition of Isolated Outliers Based on RLWR In thissection the RLWR algorithm is combined with PakTacriterion median criterion and quartile criterion to identifythe isolated outliers and choose the appropriate smoothingwindow length and observe its identification effect

331 Identification of Outliers Based on RLWR and PakTaCriterion +e RLWR smoothing algorithm is combined withthe PakTa criterion ie 3σ criterion to identify the laser profileoutliers this algorithm can be referred to as the RLWRPidentifier+e basic approach of this identifier is to smooth theprofile data by using the RLWR algorithm firstly and then theresidual between smoothing data and original data is calcu-lated finally we use the PakTa criterion to identify outliers

+e algorithm of RLWRP identifier is as follows+e measured data sequence is xi yi1113864 1113865 i 1 2 n

+e data model assumes the following

yi g xi( 1113857 + εi (5)

where g(xi) is a smooth function of xi and εi is independentand normally distributed with zero mean and variance

Set the smoothing coefficient as f where 0ltfle 1 andround f middot n to get the data width r r [f middot n] Taking eachobservation point xi as the center select the appropriate f todetermine the smoothing window length wl wl xi plusmn r

Subsequently a weight function is selected for the locallyweighted regression (LWR) LWR typically uses tricubeweight function W(x) for weighted least squares fit definedas

W xi( 1113857

1 minusd xixj( 1113857

maxjisinN xi( )d xixj( 11138571113888 1113889

3⎡⎣ ⎤⎦

3

j isin N xi( 1113857

0 j isin N xi( 1113857

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)

where N(xi) is local neighborhood in smoothing windowwhich is closest to xi and d(xi xj) is the distance between xi

and xj in smoothing window +e value of W(xi) is amaximum for the point closest to xi and reduces to zero forthe point farthest to xi in smoothing window

Use weighted least squares method get estimates ofparameters +e parameters estimates of equation (5) are thevalues of the parameters that minimize

1113944n

i1Wi(x) yi minus g xi( 1113857( 1113857

2 (7)

+e coefficients from each local neighborhood are usedto estimate the fitted values 1113954g(xi) at xi

Generally the RLWR selects the bisquare weight func-tion Q(z) as follows

Q(z) 1 minus |z|21113872 1113873

2 |z|lt 1

0 |z|ge 1

⎧⎨

⎩ (8)

where zi ei(6 middot s) in which ei is the fitting residual ieei yi minus 1113954yi s Median(|ei|) in which s is the median of |ei|

Replace W(xi) with Q(zi) middot W(xi) as new bisquareweight which is used to estimate the new set of RLWRcoefficients by minimizing the error sum of squares

1113944

n

i1Q zi( 1113857W xi( 1113857 yi minus g xi( 1113857( 1113857

2 (9)

+e new RLWR fitting value 1113954yi is calculated usingweighted least squares method Repeat the above steps of



LowerUpper

51515

52525

53535

54545

55

z (m

m)

Figure 10 Outlier recognition using Hampel identifier windowlength 25



LowerUpper

50

51

52

53

54

55

z (m

m)



robust enhancement and the final robust locally weightedfitting value is obtained

Next the outliers are identified according to the 3σcriterion +e residual gi is calculated by using thesmoothing value obtained by the RLWR algorithm and theoriginal measurement data and then we get the mean ofresidual ie μ 1n 1113936

ni1 gi Finally the standard deviation

σ is obtained

σ 1113936 gi minus μ( 1113857

2

n minus 11113890 1113891

12

(10)

When the difference between the residual and the meanis greater than three times the local standard deviation it isconsidered to be an outlier as shown in the followingequation

yi normal data gi minus μ

11138681113868111386811138681113868111386811138681113868le 3σ

outlier gi minus μ1113868111386811138681113868

1113868111386811138681113868gt 3σ

⎧⎨

⎩ (11)

RLWRP identifier is used to identify the outliers in Fig-ure 6 and its identification ability is observed under differentsmoothing windows +e length of smoothing window in-creases gradually from 5+is method shows good recognitioneffect when the length of smoothing window increases to 11

As shown in Figure 11 the blue dot is the original dataand the red dash dot line is the RLWR smoothing curve +eRLWR smoothing curve retains the characteristics oforiginal profile without the risk of excessive smoothing InFigure 12 the distribution trend of the original data is re-moved which is obtained by using the residual between thesmoothing data and the original data and the outliers in theresidual are identified by the 3σ criterion as shown in thebox +e identification result also is plotted in Figure 11 Itcan be seen from the figure that the method successfullyidentifies the double-point outlier and the spot outlierMeanwhile a few discontinuous data at the right end of theprofile are recognized as outliers

332 Identification of Outliers Based on RLWR and MedianCriterion +is section uses RLWR smoothing algorithmcombined withmedian criterion to identify outliers it can becalled the RLWRM identifier +e algorithm is as follows

According to the residual gi obtained in above sectionthe median of the residual sequence is calculated iemg median(gi)

+e scaled median absolute deviation (MAD) is definedas follows

MAD k middot median xi minus median gi( 11138571113868111386811138681113868

11138681113868111386811138681113872 1113873 i 1 2 n

(12)

where k 1(2

radicerfcminus 1(12)) asymp 14826

When the data element is greater than three times MADit is considered to be outlier as shown below

yi xi xi minus mg

11138681113868111386811138681113868

11138681113868111386811138681113868le 3MAD

outlier xi minus mg

11138681113868111386811138681113868

11138681113868111386811138681113868gt 3MAD

⎧⎪⎨

⎪⎩(13)

+e RLWRM identifier is used to identify outliers inFigure 6 +e RLWR smoothing data are used when thesmoothing window length is 11 +e identification effect isshown in Figures 13 and 14 In Figure 13 the blue point isthe residual between RLWR smoothing data and originaldata and the red boxes are the outliers identified by themedian criterion+e identification result and the smoothedvalue are plotted in Figure 14 for observation It can be seenfrom the figure that although the RLWRM identifier canrecognize the double-point outliers and the spot outliersthere are many misidentifications about the isolated outliers

333 Identification of Outliers Based on RLWR and QuartileCriterion +is section uses RLWR smoothing algorithmcombined with quartile criterion to identify outliers it canbe called the RLWRQ identifier Quartile criterion is arelatively robust identification method the algorithm di-vides sorted data into quarters Q1 Q2 and Q3 are theirbreak points Q1 is lower quartile (25 percentile) Q2 ismedian (50 percentile) and Q3 is upper quartile (75 per-centile) +e interquartile range (IQR) is introduced here asa statistic for checking outliers ie IQR Q3 minus Q1

Outliers are defined as elements more than 15 IQRabove Q3 or below Q1

ndash6 0 6ndash8 2ndash2ndash4 4

515

52

525

53

535

54

545

z (m

m)

Original signalSmoothed dataOutliers

Figure 11 Outlier recognition using RLWRP identifier windowlength 11


ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)

ResidualOutliers

Figure 12 Residual of original and smoothed data using RLWRPidentifier


yi xi Q1 minus 15(IQR) legi leQ3 + 15(IQR)

outlier otherwise1113896

(14)

+e RLWRQ identifier is used to identify outliers inFigure 6 Similarly the smoothing window length of RLWRis 11 +e identification results are shown in Figures 15 and16 It can be seen from the figures that the identificationeffect of the RLWRQ identifier is similar to the RLWRMidentifier +is algorithm also has misidentification ofoutliers and identifies more normal data as outliers If thesevalues are removed the profile will not be truly reflectedforming new errors and affecting themeasurement accuracy

4 Profile Fitting and Error Analysis

According to the previous section the RLWRP identifier canobtain better identification results for the isolated outlier+e identification results in Figure 13 are used to removeoutliers and the ellipse fitting experiment is performed bythe least squares method

+e least squares method is one of the most importantmethods of data fitting +e least squares method has thecharacteristics of simple effective and strong applicability+erefore this method is selected to conduct ellipse fittingresearch +is paper chooses the algebraic least squaresmethod to carry out ellipse fitting research which is todetermine the ellipse parameters by measuring the smallestalgebraic distance squared from the fitting ellipse to theellipse

+e elliptic algebraic equation is expressed as follows

x2

+ Axy + By2

+ Cx + Dy + E 0 (15)

According to the principle of least squares method itsobjective function is minimized as follows

F(α) 1113944n

i1[f(A B C D E)]

2

1113944n

i1x2i + Axiyi + By

2i + Cxi + Dyi + E1113872 1113873

2

(16)

To minimize F(α) on the basis of the extreme valueprinciple the following equation exists

ndash6 0 6ndash8ndash10 2ndash2ndash4 4x (mm)


51

515

52

525

53

535

54

545

z (m

m)

Figure 14 Outlier recognition using RLWRM identifier windowlength 11


ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)

Figure 15 Residual of original and smoothed data using RLWRQidentifier

ndash6 0 6ndash8 2ndash2ndash4 4x (mm)


515

52

525

53

535

54

545

z (m

m)

Figure 16 Outlier recognition using RLWRQ identifier windowlength 11


ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)

Figure 13 Residual of original and smoothed data using RLWRMidentifier


zf

zA

zf

zB

zf

zC

zf

zD

zf

zE 0 (17)

+us a linear equation is obtained +en by solving thelinear equations and combining the constraints the values ofthe equation coefficients can be obtained Get the ellipticequation and draw the fitted ellipse Finally get the ellipseequation and draw the fitted ellipse

+e fitting results are compared before and after outliereliminating as shown in Figure 17 Meanwhile the fittingellipse parameters are shown in Table 1

In Figure 17 the blue dots are the original data the boxesare the outliers that are identified the dotted line is the fittedellipse of the original data and the dash-dotted line is thefitting ellipse after removing the outliers

Before removing outliers ldquolowastrdquo is the center of the ellipsefitted with the original data the diameter of the ellipsersquosmajor axis is 117197mm the eccentricity is 09263 and theellipticity is 06233 +e ellipticity is defined as follows

ellipticity Dmax minus Dmin

Dmax (18)

After eliminating the outliers the center of the fittedellipse is ldquo+rdquo the major axis diameter is 158586mm theeccentricity is 06739 and the ellipticity is 02611

At the same time the profile was measured by the co-ordinate measuring machine (CMM) +e measurementdata are shown in Table 1 By comparison it is found that thefitting ellipse with outliers removed is similar to the mea-surement results of CMM

From the above figure and table it can be found that thefitting result of the original profile has deviated from the actualsituation After removing the outliers by using the RLWRPidentifier the fitted ellipse conforms to themeasurement reality

In addition the fitting error before and after removingoutliers is analyzed and the results are shown in Table 2 Forthe original profile the sum of squares due to error (SSE) is71938e minus 06 and the root mean square error (RMSE) is15750e minus 04 After eliminating the outliers the SSE and theRMSE of the fitted ellipse are reduced to 21157e minus 06 and86617e minus 05 respectively It shows that the elimination ofoutliers greatly reduces the fitting error and improves the fittingaccuracy and the measurement results are more accurate

5 Conclusions

In this paper the profile of magnesium alloy tubes with largecurvature is measured by a laser vision system based on linestructure laser +ere are two typical outliers in measurementthat is point outliers and spot outliers For these outliers themoving average method the Hampel method and the RWLRmethod were respectively adopted to identify the outliers ofprofile data And their ability to identify outliers is discussed forthe above methods with different window lengths

+e experiment found that all the above methods couldidentify the isolated point outliers but neither the movingmean method nor the Hampel method could identify theisolated spot outliers In this article the RWLR method isstudied emphatically for the isolated outliers which was

combined with the PakTa criterion the median criterionand the quartile criterion +e research shows that theRWLR smoothing algorithm combined with PakTa crite-rion ie RLWRP identifier can more accurately identifydifferent types of outliers with a lower misidentification rate

At the same time according to the outlier identificationresult of the RWLRP identifier the profile fitting was carriedout by the algebraic least squares method +en the mainparameters of the fitted ellipse are obtained and the fittingerrors are calculated After the comparison and analysis ofthe fitting results before and after the outlier processing it isfound that the data contaminated by outliers will lead to agreat deviation of profile fitting and wrong profile shape

Original data

Outliers

Fitting ellipse oforiginal data

The ellipse center of theoriginal dataFitting ellipse afterremoving outliersThe ellipse center afterremoving outliers

ndash5 0 5 10ndash10x (mm)

40

45

50

55

z (m

m)

Figure 17 Comparison of ellipse fitting result for original profilewindow length 11

Table 1 Elliptical parameters obtained by profile fitting

Ellipticalparameters Original data After removing

outliers+e CMM

dataCentercoordinates

(minus 20536521455)

(minus 21344485674) mdash

Majoraxis (mm) 117197 158586 15683

Minor axis mm) 44145 117173 11378Eccentricity 09263 06739 0688Ellipticity 06233 02611 0283

Table 2 Ellipse fitting error before and after removing outliers

Fitting error Original data After removing outliersSSE 71938e minus 06 21157e minus 06RMSE 15750e minus 04 86617e minus 05MSE 24806e minus 08 75024e minus 09


parameters +e RWLRP identifier is a robust accurate andefficient outlier identification method which can effectivelydeal with outliers in profile data especially for spot outliers+is algorithm is suitable for data cleaning in the linestructure light measurement which can effectively improvethe precision and accuracy of online profile measurement inthe process of hot bending of magnesium alloy tube

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Shanxi EngineeringTechnology Research Center Construction Project(201805D121006) and supported by the Science and Tech-nology Major Project of Shanxi Province (20181102016 and20181102002) and the Natural Science Foundation of ShanxiProvince (201801D121169) and sponsored by the Collabo-rative Innovation Center of Taiyuan Heavy MachineryEquipment (1331 Project)

References

[1] T M Pollock ldquoWeight loss with magnesium alloysrdquo Sciencevol 328 no 5981 pp 986-987 2010

[2] H E Friedrich and B L Mordike Magnesium TechnologyMetallurgy Design Data Applications Springer BerlinGermany 2006

[3] A A Luo ldquoMagnesium casting technology for structuralapplicationsrdquo Journal of Magnesium and Alloys vol 1 no 1pp 2ndash22 2013

[4] S You Y Huang K U Kainer and N Hort ldquoRecent researchand developments on wrought magnesium alloysrdquo Journal ofMagnesium and Alloys vol 5 no 3 pp 239ndash253 2017

[5] B Landkof ldquoMagnesium applications in aerospace andelectronic industriesrdquo Magnesium Alloys and 7eir Applica-tions Wiley-VCH Weinheim Germany pp 168ndash172 2000

[6] W J Joost and P E Krajewski ldquoTowards magnesium alloysfor high-volume automotive applicationsrdquo Scripta Materialiavol 128 pp 107ndash112 2017

[7] L Wang G Fang L Qian S Leeflang J Duszczyk andJ Zhou ldquoForming of magnesium alloy microtubes in thefabrication of biodegradable stentsrdquo Progress in NaturalScience Materials International vol 24 no 5 pp 500ndash5062014

[8] T F Kong X Z Lu and L C Chan ldquoAnalysis and reductionof wrinkling defects for tube-hydroforming magnesium alloycomponents at elevated temperaturesrdquo Materials amp Designvol 173 Article ID 107761 2019

[9] Z-J Tao X-G Fan H Yang J Ma and H Li ldquoA modifiedJohnsonndashCookmodel for NCwarm bending of large diameterthin-walled Tindash6Alndash4V tube in wide ranges of strain rates andtemperaturesrdquo Transactions of Nonferrous Metals Society ofChina vol 28 no 2 pp 298ndash308 2018

[10] Y J Gou Y H Shuang Y Zhou et al ldquoDamage and wallthickness variation of magnesium alloy tube in large curvatureand no-mandrel bending processrdquo Rare Metal Materials andEngineering vol 47 no 8 pp 2422ndash2428 2018

[11] T Hilditch D Atwell M Easton and M Barnett ldquoPerfor-mance of wrought aluminium and magnesium alloy tubes inthree-point bendingrdquo Materials amp Design vol 30 no 7pp 2316ndash2322 2009

[12] J P Bentley Principles of Measurement Systems PearsonEducation London UK 2005

[13] W Liu K Fan P Hu and Y Hu ldquoA parallel error separationmethod for the on-line measurement and reconstruction ofcylindrical profilesrdquo Precision Engineering vol 51 pp 1ndash92018

[14] M Rahayem N Werghi and J Kjellander ldquoBest ellipse andcylinder parameters estimation from laser profile scan sec-tionsrdquo Optics and Lasers in Engineering vol 50 no 9pp 1242ndash1259 2012

[15] C Munzinger G Lanza D Ruch and J Elser ldquoComponent-specific scale for inline quality assurance of spatially curvedextrusion profilesrdquo Production Engineering vol 4 no 2-3pp 193ndash201 2010

[16] Z Gan and Q Tang Visual Sensing and its ApplicationsIntegration of Laser Sensors to Industrial Robots SpringerBerlin Germany 2011

[17] S Egorov A Kapitanov and D Loktev ldquoTurbine bladesprofile and surface roughness measurementrdquo Procedia En-gineering vol 206 pp 1476ndash1481 2017

[18] S Martınez E Cuesta J Barreiro and B Alvarez ldquoAnalysis oflaser scanning and strategies for dimensional and geometricalcontrolrdquo 7e International Journal of AdvancedManufacturing Technology vol 46 no 5ndash8 pp 621ndash629 2010

[19] Y-C Zhang J-X Han X-B Fu and H-B Lin ldquoAn onlinemeasurement method based on line laser scanning for largeforgingsrdquo 7e International Journal of AdvancedManufacturing Technology vol 70 no 1ndash4 pp 439ndash448 2014

[20] D M Hawkins Identification of Outliers Springer BerlinGermany 1980

[21] Y Wang and H-Y Feng ldquoOutlier detection for scanned pointclouds using majority votingrdquo Computer-Aided Designvol 62 pp 31ndash43 2015

[22] N M R Suri N M Murty and G Athithan Outlier De-tection Techniques and Applications Springer Nature BerlinGermany 2019

[23] C C Aggarwal ldquoOutlier analysisrdquo Data Mining SpringerBerlin Germany pp 237ndash263 2015

[24] P M Robinson ldquoNonparametric estimation of time-varyingparametersrdquo Statistical Analysis and Forecasting of EconomicStructural Change Springer Berlin Germany pp 253ndash2641989

[25] S Efromovich Nonparametric Curve Estimation Methods7eory and Applications Springer Berlin Germany 2008

[26] W S Cleveland ldquoRobust locally weighted regression andsmoothing scatterplotsrdquo Journal of the American StatisticalAssociation vol 74 no 368 pp 829ndash836 1979

[27] W S Cleveland and S J Devlin ldquoLocally weighted regressionan approach to regression analysis by local fittingrdquo Journal ofthe American Statistical Association vol 83 no 403pp 596ndash610 1988

[28] W S Cleveland S J Devlin and E Grosse ldquoRegression bylocal fitting methods properties and computational algo-rithmsrdquo Journal of Econometrics vol 37 no 1 pp 87ndash1141988


[29] W G Jacoby ldquoLoess a nonparametric graphical tool fordepicting relationships between variablesrdquo Electoral Studiesvol 19 no 4 pp 577ndash613 2000

[30] C Loader ldquoSmoothing local regression techniquesrdquo Hand-book of Computational Statistics Springer Berlin Germanypp 571ndash596 2012

[31] J Ma J C-W Chan and F Canters ldquoRobust locally weightedregression for superresolution enhancement of multi-angleremote sensing imageryrdquo IEEE Journal of Selected Topics inApplied Earth Observations and Remote Sensing vol 7 no 4pp 1357ndash1371 2014

[32] N R Leonor R F S Caldeirinha T R Fernandes andM G Sanchez ldquoA simple model for average reradiationpatterns of single trees based on weighted regression at60GHzrdquo IEEE Transactions on Antennas and Propagationvol 63 no 11 pp 5113ndash5118 2015

[33] C Chen Y Wang Y Li T Yue and X Wang ldquoRobust andparameter-free algorithm for constructing pit-free canopyheight modelsrdquo ISPRS International Journal of Geo-Infor-mation vol 6 no 7 p 219 2017

[34] A Nurunnabi G West and D Belton ldquoRobust locallyweighted regression for ground surface extraction in mobilelaser scanning 3D datardquo ISPRS Annals of the PhotogrammetryRemote Sensing and Spatial Information Sciences vol II-5W2 pp 217ndash222 2013

[35] A Nurunnabi G West and D Belton ldquoRobust locallyweighted regression techniques for ground surface pointsfiltering in mobile laser scanning three dimensional pointcloud datardquo IEEE Transactions on Geoscience and RemoteSensing vol 54 no 4 pp 2181ndash2193 2016

[36] K Liu W Wang R +armarasa J Wang and Y ZuoldquoGround surface filtering of 3D point clouds based on hybridregression techniquerdquo IEEE Access vol 7 pp 23270ndash232842019

[37] Y Yu X Fu L Yu R Liu and J Cai ldquoCombustion kinetics ofpine sawdust biocharrdquo Journal of 7ermal Analysis andCalorimetry vol 124 no 3 pp 1641ndash1649 2016

[38] G Sansoni M Trebeschi and F Docchio ldquoState-of-the-artand applications of 3D imaging sensors in industry culturalheritage medicine and criminal investigationrdquo Sensorsvol 9 no 1 pp 568ndash601 2009

[39] J Santolaria J J Pastor F J Brosed and J J Aguilar ldquoA one-step intrinsic and extrinsic calibration method for laser linescanner operation in coordinate measuring machinesrdquoMeasurement Science and Technology vol 20 no 4 Article ID045107 2009

[40] T F Townley-Smith and E A Hurd ldquoUse ofmovingmeans inwheat yield trialsrdquo Canadian Journal of Plant Science vol 53no 3 pp 447ndash450 1973

[41] B Bucciarelli L G Taff and M G Lattanzi ldquoA generalizationof the moving meanlowastrdquo Journal of Statistical Computation andSimulation vol 48 no 1-2 pp 29ndash46 1993

[42] F R Hampel ldquoA general qualitative definition of robustnessrdquo7e Annals of Mathematical Statistics vol 42 no 6pp 1887ndash1896 1971

[43] F R Hampel ldquo+e influence curve and its role in robustestimationrdquo Journal of the American Statistical Associationvol 69 no 346 pp 383ndash393 1974

[44] R K Pearson Y Neuvo J Astola and M Gabbouj ldquoGen-eralized hampel filtersrdquo EURASIP Journal on Advances inSignal Processing vol 2016 no 1 p 87 2016

[45] Z B Yao J Q Xie Y Q Tian and Q X Huang ldquoUsinghampel identifier to eliminate profile-isolated outliers in laser

vision measurementrdquo Journal of Sensors vol 2019 Article ID3823691 12 pages 2019


caused by the large curvature hot bending to obtain theaccurate profile data of the magnesium alloy tube section

+e online vision measurement of tube bending isstrongly demanded to improve the dimensional accuracyHowever the quality of the profile data deteriorates sharplydue to the environment and surface quality of the pipe Inparticular many isolated outliers appear in the measureddata Outliers generally mean that the observed values aresignificantly different from most observed values that is themeasured data do not obey the statistical distribution law ofthe data [20] In this experiment the outliers appear inisolation as the form of dot or point which are not nec-essarily related to the data quality before and after +ere-fore they are also called isolated outliers Outliers can begenerated for a variety of reasons such as sensor noisechannel interference and human factors [12 21] +eseoutliers will cause data distortion leading to the miscal-culation of model parameters and wrong analysis results+erefore it is necessary to identify and eliminate theoutliers in the laser measurement process to improve theprecision and accuracy of the measurement

Commonly used identification methods for outliersinclude Nair Grubbs Dixon etc [22 23]+ese methods aredifficult to apply in laser profile measurement due to thelimitation of data distribution and data amount +is paperpresents an outlier identification method based on robustlocally weighted regression (RLWR) which is applied tolaser profile detection RLWR is developed from locallyweighted regression (LWR) and belongs to nonparametricestimation Nonparametric estimation is an important re-search direction of modern statistical analysis It can adapt tomore complex nonlinear changes without assuming thespecific form of population distribution and error distri-bution and without directly obtaining data models which ismore flexible robust and widely applied than parameterestimation [24 25]

RLWR is a robust fitting process that integrates localpolynomial estimation and locally weighted regression withexcellent smoothing performance +is method was first

proposed by Cleveland [26] and further elaborated [27 28]subsequently improved by Jacoby [29] and Loader [30] Ithas been gradually applied to different fields of scientificresearch and engineering applications Ma et al [31] used theRLWR to reduce the impact of high-frequency noise onsuperresolution enhancement of multiangle remote sensingimagery Leonor et al [32] minimized the influence of theinhomogeneity effect on tree reradiation pattern by usingRLWR Chen et al [33] proposed an algorithm based onRLWR and robust z-scores for the construction of a pit-freecanopy height models Nurunnabi et al [34 35] and Liu et al[36] used RLWR techniques to study the filtering of 3Dground cloud point data Yu et al [37] applied the method tothe smoothing of combustion kinetics data of pine sawdustbiochar

+is study is based on the robust smoothing perfor-mance of RLWR which is used to smooth the laser mea-surement data and realize the identification of isolatedoutliers At the same time profile fitting was carried out toverify the effectiveness of this algorithm in the profileinspection

+is paper organized as follows Section 2 introduces thelaser vision measurement system and its basic principles andanalyzes the profile data of magnesium alloy tube pointingout the typical outliers in the data In Section 3 the movingaverage algorithm Hampel algorithm and RLWR algorithmare respectively adopted to identify the outliers It focuseson the outlier recognition effect for the RLWR combinedwith PakTa criterion median criterion and quartile crite-rion respectively In Section 4 elliptic fitting of profile datais performed by using the RLWR identification algorithmand relevant elliptic parameters and fitting error are ob-tained at the same time compared with original data Lastlya conclusion is given in Section 5

2 Laser Vision Measurement System andProfile Outliers

21 Laser Vision Measurement System +e laser visionmeasurement system adopted in this experiment mainlyconsists of a line-structured light sensor image processingunit and control unit as shown in Figure 2 +e systemworks as follows the laser source in the sensor projects linestructure light to a measured object +e reflected light iscaptured by the camera in the sensor +e profile data arerecorded and displayed by industrial PC after image pro-cessing +e measuring platform can move along the axialdirection of the pipe which is controlled by the motioncontrol system realizing the profile measurement of eachsection for the pipe

Laser triangulation is the basic principle of line-struc-tured light vision measurement +e line-structured lightprojected by laser source forms a scattered light band on thepipe surface and the scattered light is imaged in CCD array+e distance and coordinate data of the measured point canbe obtained through triangular geometry [38 39]

As shown in Figure 3 the point p(x y z) is a mea-surement point on pipe surface point pprime(u v) is the imagingpoint of p in CCD image plane f is the focal length of

M1

M2

Ft

Ft

FcFc

Figure 1 Force condition of bending process







Imageprocessor

Motioncontrolsystem

IndustrialPC

Laser source Camera


Measured bar

Measured profile


Measured object

P (x y x)Z

c

θ b X

Z


uO Lens center

CCD

f X




51

515

52

525

53

535

54

545

z (m

m)



51

515

52

525

53

535

54

545

z (m

m)



Spot outlier


51

515

52

525

53

535

54

545

z (m

m)




yi xi xi minus μi

11138681113868111386811138681113868111386811138681113868le 3σi


11138681113868111386811138681113868111386811138681113868gt 3σi

⎧⎨

⎩ (1)











11138681113868111386811138681113868111386811138681113868 xi+k minus mi

111386811138681113868111386811138681113868111386811138681113872 1113873 (3)

where k 1(2



yi xi xi minus mi

11138681113868111386811138681113868111386811138681113868le tMADl

outlier xi minus mi

11138681113868111386811138681113868111386811138681113868gt tMADl

⎧⎨

⎩ (4)





51515

52525

53535

54545

55

z (m

m)


LowerUpper




LowerUpper

51515

52525

53535

54545

55

z (m

m)










yi g xi( 1113857 + εi (5)




W xi( 1113857



3⎡⎣ ⎤⎦

3


0 j isin N xi( 1113857

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)




1113944n

i1Wi(x) yi minus g xi( 1113857( 1113857

2 (7)



Q(z) 1 minus |z|21113872 1113873

2 |z|lt 1

0 |z|ge 1

⎧⎨

⎩ (8)



1113944

n


2 (9)




LowerUpper

51515

52525

53535

54545

55

z (m

m)




LowerUpper

50

51

52

53

54

55

z (m

m)






σ is obtained

σ 1113936 gi minus μ( 1113857

2

n minus 11113890 1113891

12

(10)



11138681113868111386811138681113868111386811138681113868le 3σ


1113868111386811138681113868gt 3σ

⎧⎨

⎩ (11)







11138681113868111386811138681113872 1113873 i 1 2 n

(12)

where k 1(2



yi xi xi minus mg

11138681113868111386811138681113868

11138681113868111386811138681113868le 3MAD

outlier xi minus mg

11138681113868111386811138681113868

11138681113868111386811138681113868gt 3MAD

⎧⎪⎨

⎪⎩(13)





515

52

525

53

535

54

545

z (m

m)




ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)

ResidualOutliers





(14)






x2

+ Axy + By2

+ Cx + Dy + E 0 (15)


F(α) 1113944n

i1[f(A B C D E)]

2

1113944n

i1x2i + Axiyi + By

2i + Cxi + Dyi + E1113872 1113873

2

(16)




51

515

52

525

53

535

54

545

z (m

m)



ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)




515

52

525

53

535

54

545

z (m

m)



ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)



zf

zA

zf

zB

zf

zC

zf

zD

zf

zE 0 (17)






Dmax (18)





5 Conclusions





Original data

Outliers




40

45

50

55

z (m

m)




outliers+e CMM


(minus 20536521455)


Majoraxis (mm) 117197 158586 15683






Data Availability




Acknowledgments


References






















































Imageprocessor

Motioncontrolsystem

IndustrialPC

Laser source Camera


Measured bar

Measured profile


Measured object

P (x y x)Z

c

θ b X

Z


uO Lens center

CCD

f X




51

515

52

525

53

535

54

545

z (m

m)



51

515

52

525

53

535

54

545

z (m

m)



Spot outlier


51

515

52

525

53

535

54

545

z (m

m)




yi xi xi minus μi

11138681113868111386811138681113868111386811138681113868le 3σi


11138681113868111386811138681113868111386811138681113868gt 3σi

⎧⎨

⎩ (1)











11138681113868111386811138681113868111386811138681113868 xi+k minus mi

111386811138681113868111386811138681113868111386811138681113872 1113873 (3)

where k 1(2



yi xi xi minus mi

11138681113868111386811138681113868111386811138681113868le tMADl

outlier xi minus mi

11138681113868111386811138681113868111386811138681113868gt tMADl

⎧⎨

⎩ (4)





51515

52525

53535

54545

55

z (m

m)


LowerUpper




LowerUpper

51515

52525

53535

54545

55

z (m

m)










yi g xi( 1113857 + εi (5)




W xi( 1113857



3⎡⎣ ⎤⎦

3


0 j isin N xi( 1113857

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)




1113944n

i1Wi(x) yi minus g xi( 1113857( 1113857

2 (7)



Q(z) 1 minus |z|21113872 1113873

2 |z|lt 1

0 |z|ge 1

⎧⎨

⎩ (8)



1113944

n


2 (9)




LowerUpper

51515

52525

53535

54545

55

z (m

m)




LowerUpper

50

51

52

53

54

55

z (m

m)






σ is obtained

σ 1113936 gi minus μ( 1113857

2

n minus 11113890 1113891

12

(10)



11138681113868111386811138681113868111386811138681113868le 3σ


1113868111386811138681113868gt 3σ

⎧⎨

⎩ (11)







11138681113868111386811138681113872 1113873 i 1 2 n

(12)

where k 1(2



yi xi xi minus mg

11138681113868111386811138681113868

11138681113868111386811138681113868le 3MAD

outlier xi minus mg

11138681113868111386811138681113868

11138681113868111386811138681113868gt 3MAD

⎧⎪⎨

⎪⎩(13)





515

52

525

53

535

54

545

z (m

m)




ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)

ResidualOutliers





(14)






x2

+ Axy + By2

+ Cx + Dy + E 0 (15)


F(α) 1113944n

i1[f(A B C D E)]

2

1113944n

i1x2i + Axiyi + By

2i + Cxi + Dyi + E1113872 1113873

2

(16)




51

515

52

525

53

535

54

545

z (m

m)



ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)




515

52

525

53

535

54

545

z (m

m)



ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)



zf

zA

zf

zB

zf

zC

zf

zD

zf

zE 0 (17)






Dmax (18)





5 Conclusions





Original data

Outliers




40

45

50

55

z (m

m)




outliers+e CMM


(minus 20536521455)


Majoraxis (mm) 117197 158586 15683






Data Availability




Acknowledgments


References


















































yi xi xi minus μi

11138681113868111386811138681113868111386811138681113868le 3σi


11138681113868111386811138681113868111386811138681113868gt 3σi

⎧⎨

⎩ (1)











11138681113868111386811138681113868111386811138681113868 xi+k minus mi

111386811138681113868111386811138681113868111386811138681113872 1113873 (3)

where k 1(2



yi xi xi minus mi

11138681113868111386811138681113868111386811138681113868le tMADl

outlier xi minus mi

11138681113868111386811138681113868111386811138681113868gt tMADl

⎧⎨

⎩ (4)





51515

52525

53535

54545

55

z (m

m)


LowerUpper




LowerUpper

51515

52525

53535

54545

55

z (m

m)










yi g xi( 1113857 + εi (5)




W xi( 1113857



3⎡⎣ ⎤⎦

3


0 j isin N xi( 1113857

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)




1113944n

i1Wi(x) yi minus g xi( 1113857( 1113857

2 (7)



Q(z) 1 minus |z|21113872 1113873

2 |z|lt 1

0 |z|ge 1

⎧⎨

⎩ (8)



1113944

n


2 (9)




LowerUpper

51515

52525

53535

54545

55

z (m

m)




LowerUpper

50

51

52

53

54

55

z (m

m)






σ is obtained

σ 1113936 gi minus μ( 1113857

2

n minus 11113890 1113891

12

(10)



11138681113868111386811138681113868111386811138681113868le 3σ


1113868111386811138681113868gt 3σ

⎧⎨

⎩ (11)







11138681113868111386811138681113872 1113873 i 1 2 n

(12)

where k 1(2



yi xi xi minus mg

11138681113868111386811138681113868

11138681113868111386811138681113868le 3MAD

outlier xi minus mg

11138681113868111386811138681113868

11138681113868111386811138681113868gt 3MAD

⎧⎪⎨

⎪⎩(13)





515

52

525

53

535

54

545

z (m

m)




ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)

ResidualOutliers





(14)






x2

+ Axy + By2

+ Cx + Dy + E 0 (15)


F(α) 1113944n

i1[f(A B C D E)]

2

1113944n

i1x2i + Axiyi + By

2i + Cxi + Dyi + E1113872 1113873

2

(16)




51

515

52

525

53

535

54

545

z (m

m)



ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)




515

52

525

53

535

54

545

z (m

m)



ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)



zf

zA

zf

zB

zf

zC

zf

zD

zf

zE 0 (17)






Dmax (18)





5 Conclusions





Original data

Outliers




40

45

50

55

z (m

m)




outliers+e CMM


(minus 20536521455)


Majoraxis (mm) 117197 158586 15683






Data Availability




Acknowledgments


References
























































yi g xi( 1113857 + εi (5)




W xi( 1113857



3⎡⎣ ⎤⎦

3


0 j isin N xi( 1113857

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)




1113944n

i1Wi(x) yi minus g xi( 1113857( 1113857

2 (7)



Q(z) 1 minus |z|21113872 1113873

2 |z|lt 1

0 |z|ge 1

⎧⎨

⎩ (8)



1113944

n


2 (9)




LowerUpper

51515

52525

53535

54545

55

z (m

m)




LowerUpper

50

51

52

53

54

55

z (m

m)






σ is obtained

σ 1113936 gi minus μ( 1113857

2

n minus 11113890 1113891

12

(10)



11138681113868111386811138681113868111386811138681113868le 3σ


1113868111386811138681113868gt 3σ

⎧⎨

⎩ (11)







11138681113868111386811138681113872 1113873 i 1 2 n

(12)

where k 1(2



yi xi xi minus mg

11138681113868111386811138681113868

11138681113868111386811138681113868le 3MAD

outlier xi minus mg

11138681113868111386811138681113868

11138681113868111386811138681113868gt 3MAD

⎧⎪⎨

⎪⎩(13)





515

52

525

53

535

54

545

z (m

m)




ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)

ResidualOutliers





(14)






x2

+ Axy + By2

+ Cx + Dy + E 0 (15)


F(α) 1113944n

i1[f(A B C D E)]

2

1113944n

i1x2i + Axiyi + By

2i + Cxi + Dyi + E1113872 1113873

2

(16)




51

515

52

525

53

535

54

545

z (m

m)



ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)




515

52

525

53

535

54

545

z (m

m)



ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)



zf

zA

zf

zB

zf

zC

zf

zD

zf

zE 0 (17)






Dmax (18)





5 Conclusions





Original data

Outliers




40

45

50

55

z (m

m)




outliers+e CMM


(minus 20536521455)


Majoraxis (mm) 117197 158586 15683






Data Availability




Acknowledgments


References




















































σ is obtained

σ 1113936 gi minus μ( 1113857

2

n minus 11113890 1113891

12

(10)



11138681113868111386811138681113868111386811138681113868le 3σ


1113868111386811138681113868gt 3σ

⎧⎨

⎩ (11)







11138681113868111386811138681113872 1113873 i 1 2 n

(12)

where k 1(2



yi xi xi minus mg

11138681113868111386811138681113868

11138681113868111386811138681113868le 3MAD

outlier xi minus mg

11138681113868111386811138681113868

11138681113868111386811138681113868gt 3MAD

⎧⎪⎨

⎪⎩(13)





515

52

525

53

535

54

545

z (m

m)




ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)

ResidualOutliers





(14)






x2

+ Axy + By2

+ Cx + Dy + E 0 (15)


F(α) 1113944n

i1[f(A B C D E)]

2

1113944n

i1x2i + Axiyi + By

2i + Cxi + Dyi + E1113872 1113873

2

(16)




51

515

52

525

53

535

54

545

z (m

m)



ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)




515

52

525

53

535

54

545

z (m

m)



ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)



zf

zA

zf

zB

zf

zC

zf

zD

zf

zE 0 (17)






Dmax (18)





5 Conclusions





Original data

Outliers




40

45

50

55

z (m

m)




outliers+e CMM


(minus 20536521455)


Majoraxis (mm) 117197 158586 15683






Data Availability




Acknowledgments


References



















































(14)






x2

+ Axy + By2

+ Cx + Dy + E 0 (15)


F(α) 1113944n

i1[f(A B C D E)]

2

1113944n

i1x2i + Axiyi + By

2i + Cxi + Dyi + E1113872 1113873

2

(16)




51

515

52

525

53

535

54

545

z (m

m)



ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)




515

52

525

53

535

54

545

z (m

m)



ResidualOutliers

ndash08

ndash06

ndash04

ndash02

0

02

04

06

Resid

ual (

mm

)



zf

zA

zf

zB

zf

zC

zf

zD

zf

zE 0 (17)






Dmax (18)





5 Conclusions





Original data

Outliers




40

45

50

55

z (m

m)




outliers+e CMM


(minus 20536521455)


Majoraxis (mm) 117197 158586 15683






Data Availability




Acknowledgments


References

















































zf

zA

zf

zB

zf

zC

zf

zD

zf

zE 0 (17)






Dmax (18)





5 Conclusions





Original data

Outliers




40

45

50

55

z (m

m)




outliers+e CMM


(minus 20536521455)


Majoraxis (mm) 117197 158586 15683






Data Availability




Acknowledgments


References


















































Data Availability




Acknowledgments


References

















































RobustLocallyWeightedRegressionforProfileMeasurementof ... · 2020. 5. 22. · presents an outlier...

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Transcript of RobustLocallyWeightedRegressionforProfileMeasurementof ... · 2020. 5. 22. · presents an outlier...