Pellet size estimation using spherical fitting

IMTC 2007-IEEEInstrumentation and Measurement TechnologyConference

Warsaw,Poland, May1-3, 2007

Pellet

Size Estimation

Using Spherical Fitting

T.

Andersson, M. J.

Thurley,

0. Marklund

EISLAB,

Departmentof ComputerScience and Electrical Engineering,

Lulea University ofTechnology, 97187 Lulea

E-mail: {tobias.andersson,matthew.thurley,olov.marklund} @ltu.se

URL:www.csee.ltu.se/-tobiasa,www.csee.ltu.se/-mjt Abstract- Evaluation of Spherical Fitting as a technique forsizing

iron orepellets isperformed. Size measurement of pelletin indus-try isusually performedbymanualsamplingandsievingtechniques. Automatic on-line analysis of pellet size would allow non-invasive, frequent and consistent measurement. Previous work has used an as-sumption thatpellets aresphericalto estimatepelletsizes. In this re-search we use a 3D laser camera system in alaboratoryenvironment tocapture 3Dsurfacedataofpelletsand steel balls. Validationofthe 3D data against aspherical model has been performed and demon-stratesthatpelletsare notsphericaland havephysicalstructuresthat aspherical model cannot capture.

Keywords-Sizeestimation, Spherical fitting, Model evaluation, In-dustrialmonitoring, Materialanalysis, Imageanalysis.

I. INTRODUCTION

Pellet's sizes are critical to the efficiency of the blast fur-naceprocess inproduction of steel. Okvist et al. [1] shows how differentpellet size distributions effect the blast furnace process. Overlycoarsepellets effect the blast furnaceprocess negatively andOkvistetal. [1]reports onhowtominimize the effectby operating the furnace with differentparameters. An on-linesystemformeasurementofpelletsizes wouldimprove productivitythrough fast feedback and efficient control of the blast furnace.

Inpelletmanufacturing, manual sampling followed by siev-ing witha squaremesh is used forquality control. The man-ualsamplingisperformedinfrequentlyand istime-consuming.

Fastfeedback ofpellets sizes is desirable.

Blomquist and Wernerson [2] use a statistical model that assumespelletsaresphericalto measurediameter and diameter deviation ofpelletsfrom chordlengths.

Bouajilaet al. [3] estimate size ofgreenpelletswitha 3D lasercamera. Basedontheassumptionthatpelletsare spheri-cal, they applyaspherical smoothingmethodtoobtaina com-plete size distribution of theproduced pellets. Bouajila etal. [3] report that the estimated size distribution correlates well with the reference values.

Inthepresented researchwe use anindustrialprototype3D imagingsystem to capture 3Dsurface data ofpellet pileson a

conveyor. Estimation ofpellet sizes is made basedonthe as-sumption that pelletsare spherical. Anevaluation of the suit-ability of the sphericalassumption ismade.

II. METHODS

Inthis section we outline the methods usedto capturethe 3Dsurface data,segmentit, determinesegmented regions sizes and evaluate thespherical model.

A. Imaging System

Animagingsystem thatcaptures3D surface data has been implemented byMBV Systems [4]. Thesystem is basedon a projected laser line andcameratriangulation[5, triangulation, structuredlight]. Ithasacontinuouswavediode laser with line generating optics and ahigh speed digital cameracapable of 4000framespersecond. The angle between the line ofsight and the laser isapproximately30degrees.

B. CollectionofData

B.1 Pellets

Mechanical sieving is the accepted industry technique for sizing pellets. Asampleof bakedpelletswassieved into 6 size gradings and is shown in table I.

Sieve size(mm)t Weight (kg) % Cum. %

6.3 4.089 12.9 12.9 9 4.755 14.9 27.8 10 12.613 39.6 67.4 11.2 7.493 23.5 91.0 12.5 2.445 7.68 98.7 14 0.4233 1.33 100

t

The lower bound of each sieve sizeincrement

Each size class is captured individually with the imaging system in alaboratory setup. The pellets are spread out on a conveyorbelttomakesurethat the surface of eachpellet isnot occludedby other pellets.

B.2 Steel Balls

The steel balls have a known diameter and have been painted withathinlayer ofmatgreypainttoallow datacapture with theimaging system. The steel balls arepositioned sepa-rately on a tray. The imaging system captures a sample of 45 balls each of size5, 10, 12.7 and 16mm, 30balls of 17.5mm and 15 balls each of 20 and 25mm.

C. SegmentationAlgorithm

Pellet segmentation has been performed by Thurley [6] us-ing amathematical morpholgy implementation [7] forsparse, irregularly spaced 3D surface data.

This technique [6] appliesavariety of algorithmsincluding morphological and linear edge detection, distance transform, local maxima detection and watershedsegmentation.

D. Estimation ofSize

Using the segmented data each pellet is identified and its size may be estimated. Toevaluate the assumption that pel-letsarespherical, pellet size is estimated by fitting aspherical modeltothesegmented data of each pellet.

Todo this consider theequation ofasphere, whichcanbe writtenas seeninequation 1 wherex0,yo andzo arethe coor-dinates for thecenterof the sphere and R is the radius of the sphere.

f

(xo,

yo,

zo,

R)

=

(X

_

Xo)2

+

(y

_

yO)2

+

(z zo)2

-R2 =0

(1) Byusing the partial derivatives as seen in equation 2it is possibleto construct alinearleast-squares problemwith m co-ordinatepoints (xy,

Yi,

Z1),(X2,

Y2,

Z1),. , (xm,

Ym

,Zm)

The solution is given bya= (MM)-1(Mtv) wherevector

ais given by equation4 from which x0, yo, zo and Rcan be determined. a b d J

[

-2x0 2Yo 2zO 2 2 2 xo +Yo +

Z2

(4) R2 E. Model Evaluation

To validate the model we compare estimated values with measured values of steel ball's and pellet's size. The estimated values shouldclearly correlate with measured physical param-etersif thespherical model is valid.

Residuals, which is thepartof the data that the model could notreproduce, mayalso indicate how well the model describes the data. Aspherical model's residualsarecalculatedby equa-tion5, wherexi, yiandziis the coordinates forpoint i.

Ei =

x/(i-xo)2

+

(yi-

yo)2

+

(zi- zo)2

R

(5)

Ljung [8] suggest analysis of basic statistics for residuals andwewilluseroot-mean-square error asshown inequation 6 inthisanalysis where Eiis the residual forpointi andmis the number ofpoints.

Im

RMSError = E2

iil

(6)

Thecomparison of estimated values with known data and the residual analysis give a good indication of how well the modelcapturesthe data.

III. PERFORMANCEOFSPHERICAL FITTING

Theaccuracyof theimagingsystemand thespherical model as a measurementmethod is evaluatedby sizing perfectsteel balls and sievedpellets.

A. Steel Balls

Ox,

0y,

0, azo

Of

0O =0

The linearsystemMa=vis shown inequation 3 where M

isam-by-4 matrix andvisa mlong columnvector.

F

L

-12_x1 -2

_x2

2 Yi 2 Y2 2 z1 2

Z2

2 _ 2 _z2 m Ym m

The model is fittedto each steel ball and the result of esti-mated sizes and residual statistics is shown in table II.

The median of the estimated sizes is close to the known values of the steel balls diameter. The residualanalysisindicate some deviation between data and the model. The median of RMS Error rangefrom 0.107to0.128 for all size classes.

Forperfect steel balls the model and measurement system seems togive goodresults. Thephysical comparisontoknown sizes of the balls isvery good. The residual analysis indicate that there issomedeviation from the model but it is small.

[X

Yi

ZI

1

al

X2

Y2

Z2 1 b

Size* Nbr Size est.* (mm) RMS Errort

(mm) Median IQR° Median IQR°

5 45 5.432 0.308 0.107 0.0133 10 45 10.35 0.165 0.107 0.0073 12.7 45 12.95 0.108 0.106 0.0048 16 45 16.18 0.106 0.119 0.0054 17.5 30 17.63 0.081 0.117 0.0059 20 15 20.03 0.073 0.118 0.0059 25 15 24.95 0.169 0.128 0.0099

* Known diameter of steel ball

*

t

0

Estimated size. Diameter of fitted sphere Root-mean-squareerror. Equation 6

Interquartile range. Range between 25th and 75th

percentile

TABLEII.Result forspherical fittomeasured steel balls. Estimated sizes correlate well with known sizes and the RMS Error estimate is small. The interquartilerangeis small for both estimated diameter and RMS Error.

*

t

0

ine iower

DouncH o0 eacnsievesiLze

increment

Estimated size. Diameter of fittedsphere Root-mean-squareerror. Equation 6

Interquartile range. Range between 25th and 75th

percentile

TABLEIII.Result forspherical fittomeasuredpellets. The estimated size is constantlyoverestimated and theinterquartilerangeisrelatively large.The

RMS Error islarge.

B. Pellets

To evaluate ifaspherical assumption of pellets shape is

ad-equate the spherical model is fitted to eachpellet. The esti-mated diameters and statistics for the residuals for the differ-entclasses are calculated. We presentthe distribution of the

estimated diameters and RMSerrorfor the different classesin

TableIII. The result is also showngraphically in figures 1 and

2using the graphical convention of horizontal box-plots. The centralportion ofabox-plot containsarectangular box.

Inthecenterof this box is ashort thick vertical blackline, this

marks the median value(or 50th percentile) of the data. The leftedge of the rectangular box marks the 25th percentile, and theright edge marks the 75th percentile. The difference

be-tween the 75th and 25th percentile is the interquartile range

(IQR). The IQR isarobust estimate of the spreadof the data.

The circular dotstothe left and rightof each box-plotindicate values that are statistically determined tobe outliers. Values

aredefinedtobe outliers when theyareless than the 25th

per-centile - 1.5IQRorgreaterthan the 75th percentile+ 1.5IQR.

These values correspond to pellets that are particularly

non-spherical. The dashed lines extending tothe left andright of the rectangular box extend to the statistically valid min and

max. The graphs and determination of outlierswerecalculated

using the R statistical analysis package [9].

In figure 1, it is clear that the size estimate for pellets is constantly overestimated. Also, it is important to notice that theinterquartilerangeof the size estimatesis generally larger

than the intervals between the different classes. The classesare

notclearly separable and will notbe suitable for determining pellet size that correspondstosquaremeshsievingtechniques.

The median value of the RMS Error, shown in table III,

range from 0.261 for the smallest size class to 0.366 for the biggestsize class. The interquartilerange is above 0.1 for all

size classes. The RMS Errorclearly indicates that pellets are

notspherical and the distribution of the RMSerrorfor the

dif-ferent classescanbeseeninfigure 2.

The physical comparison to known sizes of pellets

com-bined with the residual analysis indicate that pellets are not spherical. It is importanttonotice that thephysical

compari-sonof the calculated size and the known sieve sizes show that

the estimated sizesarewrongand sensitivetoinput data. Forcomparisonweshow thebox-plots for steel balls in fig-ures 3 and 4 drawn atthe same scale as figure 1 and 2.

An-alyzing figure 3 and 1, it is clear that the distributions for the estimated sizes are very narrow and close tothe known sizes for steel balls. For pellets the distributionsare broad and the

size estimatesare constantly overestimated. Analyzing figure

4and2, it is clear that the RMS Errorsarecomparatively large

forpellets.

Inaddition, the residualsareshown infigure 5 for pellets in

size class 10mm. For comparisonwe show the residuals for

steel balls withadiameter of 10mminfigure 6. It is obvious

that the model doesnotcapturecertainareasofpellets that

de-viate fromaspherical model. The figures show asignificantly

greatervariation of the residuals forpellets than for steel balls and this indicate that the spherical model works well for

per-fect steel balls but do not accountfor all variations inpellet's structure.

Size* Nbr Sizeest.* (mm) RMSErrort

(mm) Median IQR° Median IQR°

6.3 1010 10.34 1.777 0.261 0.134 9 755 11.34 1.553 0.266 0.117 10 867 12.12 1.504 0.274 0.110 11.2 677 13.58 1.794 0.299 0.124 12.5 477 15.02 1.887 0.329 0.133 14 61 16.00 1.690 0.350 0.127

14mm 12.5mm 11.2mm 1Omm -9mm -6.3mm

Sphere Fitting Diameter forPelletsof Various Sizes

o6 6 c L F -- - - DO OOCD (K~D(MD 0 0 0 H d z-h --- 4 1O-D0 0 Io o 25mm -20mm -17.4mm -16mm -12.7mm -1Omm -5mm -5 10 15 20

Calculated Sphere Diameter (mm)

Fig. 1. Distribution of estimated sphere diameters for different size classes. The dashed lines correspondstothe lower bound of each size class.

Sphere Fitting RMS Error for Pellets of Various Sizes

14mm 12.5mm 11.2mm 1Omm -9mm -6.3mm 0.5 1.0

Sphere Fitting Diameterfor Steel Balls of Various Diameters

~~~~~~~~~~~~~II~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

_ II II Ih

5 10 15 20 25

Calculated Sphere Diameter (mm)

Fig. 3. Distribution of estimated sphere diameters for different size classes. The dashed lines correspondstoknown diameters for the balls.

Sphere Fitting RMS Error for Steel Balls of Various Diameter

25mm -20mm -17.4mm -16mm -12.7mm -1Omm -5mm -1.5

Fig. 2. Distribution of estimatederrorof fit for different size classes.

0.5 1.0 1.5

Fig. 4. Distribution of estimatederrorof fit for differentsize classes.

F--- I 1---H 0 0 0 F--- - - -ODO 0 0 0 -F- I ----D 0 0 0 --0 0 -- M- ° k --EE ----dMMOOf(I 00 0 0 0 F--I -- - --MMMoa0DO0 O 000 (IDO O

_-4

_

i

_

I JD

_

13-X~~~~~

Fig.5. Sample of residuals for pellets in class 10 mm. That is pellets of size between 10 and 11.2 mm.

IV. CONCLUSIONS

A study of the adequacy of the assumption that pellets are spherical is made. Pellets are collected andmechanically sieved into different classes. Alsoperfect steel balls with well known propertiesarecollected. Thetwosamplesarecaptured byanimagingsystemthatproduces 3D surface data. The sur-face is segmented and a spherical model fitted to eachpellet and steel ball. Model evaluation basedonphysicalproperties and residualanalysis show that the spherical model works well forperfect steel balls butnotforpellets.

Fig. 6. Sample of residuals for balls with a diameter of 10 mm.

[4] J. E. Larsson, info@mbvsystems.se

[5] "3D scanner: Wikipedia, the free encyclopedia,"

http://en.wikipedia.org/wiki/3d_scanner.

[6] M. J.Thurley,"On-line 3D Surface Measurement of Iron Ore Green Pel-lets," Proc. IEEEConf. Computational Intelligence for Modelling, Con-trol andAutomation,Nov2006.

[7] M. J.Thurley, K. C. Ng, "Identifying, visualizing, and comparing regions inirregularly spaced 3D surface data," Computer Vision and Image

Un-derstanding,vol.98,no.2,pp.239-270,Feb.2005.

[8] L.Ljung, System identification: theoryforthe user, PrenticeHall, 1999,

2.ed,ISBN:0-13-656695-2.

[9] R. Ihaka, R. Gentleman, "R: A Language for Data Analysis and Graph-ics," Journal of Computational and Graphical Statistics, vol.5, no. 3, pp.299-314, 1996, http://www.r-project.org/.

ACKNOWLEDGMENT

Wewishtothank the staffatProcessITInnovations for all time and efforttomake this researchpossible. Wethank John Erik Larsson at MBV-systems for adjustments anddevelopment of the 3D surfaceimagingsystem. WethankKjell-Ove Mickels-sonand Robert Johannson for their invaluableindustry knowl-edge and advice.

[1] L. SundqvistOkvist,A. Dahlstedt, M. Hallin, "The effect on Blast fur-naceProcess ofchanged Pellet Size as a Result ofsegregationin Raw Material Handling," Proc. Ironmaking Conference, pp.167-178, 2001. [2] M. Blomquist, A. Wernerson, "Range camera on conveyor belts:

esti-mating size distribution and systematic errors due toocclusion," Proc. SPIE-TheInternational Societyfor Optical Engineering,vol.3835,pp. 118-126, Sep. 1999.

[3] A. Bouajila, M. Bourassa, J.-A. Boivin, G. Ouellet, T. I. Martinovic,

"On-Line Non-Intrusive Measurement of Greeen PelletDiameter,"Proc. Ironmaking Conference, pp. 1009-1020, 1998.