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Three-dimensional particle image velocimetry: experimental error analysis of a digital angular

stereoscopic system

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1997 Meas. Sci. Technol. 8 1455

(http://iopscience.iop.org/0957-0233/8/12/009)

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Meas. Sci. Technol. 8 (1997) 14551464. Printed in the UK PII: S0957-0233(97)86173-5

Three-dimensional particle imagevelocimetry: experimental erroranalysis of a digital angularstereoscopic system

N J Lawson and J Wu

The GK Williams CRC, Department of Chemical Engineering,The University of Melbourne, Parkville, Victoria 3052, Australia CSIRO, Division of Building, Construction and Engineering, Graham Road,Highett, Victoria 3190, Australia

Received 22 July 1997, accepted for publication 25 September 1997

Abstract. Experimental error analysis of a digital angular stereoscopic PIV systemis presented. The paper firstly describes an experimental rig which includes thedesign of a novel PIV test block for in situ calibration. This allowed the user to setup a static seeded flow volume which was translated in and out of plane to recordPIV images using two megapixel CCD cameras positioned for angular stereoscopicviewing. PIV data were collected for a range of camera angles up to 45 and for arange of flow displacements and processed by cross correlation into a set oftwo-dimensional calibration and flow displacement vectors. These 2D data werethen processed into three-dimensional data by the use of geometric and bicubicspline interpolation algorithms and an error analysis performed on the predicteddisplacements. Results from this analysis have shown optimum systemperformance will be obtained by using camera angles of between 20 and 30 and fnumbers of f 16 and higher. The results have also shown a theoretical prediction ofsystem performance derived in previous work, which considers the ratio of out ofplane to in plane errors, matches to within 8 and 18% of the experimental systemperformance.

1. Introduction

The measurement of fluid flows in engineering hascontinually generated innovative techniques varying froma simple Pitot tube [1] to complex laser techniques [2].Particle image velocimetry (PIV) is a recent but wellestablished, non-intrusive technique for the measurementof mean and instantaneous fluid velocity from a singleplane of interest [3, 4]. PIV in its simplest form involvesplacing a pulsed laser sheet into the measured fluid flowplane and recording the position of seeding particles bythe use of double-exposure photography. This results in atransparency consisting of a large number of particle imagepairs whose average displacement over each small regionof the photograph is a measure of the mean velocity atthe corresponding point in the fluid. From the developedtransparency, a two-dimensional (2D) velocity map is thenextracted by using optical or digital spatial correlationanalysis to determine the particle displacement within eachsmall region of the flow [5, 6].

The application of PIV to increasingly complexfluid flows [7, 8] has resulted in the demand for three-dimensional (3D) measurements and the development

of 3D PIV techniques. A recent review of 3D PIVhighlighted the different methods that had been successfullydemonstrated at that time [9]. These were mainlyholographic PIV (HPIV) [10, 11] and stereoscopic PIVmethods [1216]. More recent methods have includeddual-plane PIV [17, 18] and off axis stereoscopic PIV[19]. If these techniques are considered in terms of theirimaging and data processing systems, HPIV is the mostdemanding in both cases requiring complex optics andadvanced data processing techniques. In contrast dual-plane PIV and stereoscopic PIV only require conventionalcameras and light sheet arrangements although intensiveimage processing is necessary.

Any PIV system will have a level of accuracywhich will determine the validity of its application.Previous authors have investigated in detail the errorcharacteristics of a simple 2D PIV system [2022]. Todate, however, limited work has been carried out on theerror characteristics of 3D PIV systems and has mainlybeen restricted to stereoscopic configurations [12, 13, 19].This previous work has recently been extended by thepresent authors to give a set of theoretical design curves forstereoscopic systems [23]. These characteristics allowed

0957-0233/97/121455+10$19.50 c 1997 IOP Publishing Ltd 1455

N J Lawson and J Wu

Figure 1. Diagram of experimental stereoscopic imaging system.

the overall performance of an angular or translationalstereoscopic system to be analysed for a range of cameraangles and separations.

In what follows, experimental work is presented whichaims to validate the theory outlined in the previous work[23]. The analysis is based on a digital angular stereoscopicsystem which is used to investigate the system performanceover a range of camera angles and fluid displacements.These experimental results are then directly compared tothe theoretical predictions and any discrepancies discussed.

2. Experimental technique

The following section describes the stereoscopic imagingsystem and data acquisition methods used to obtain theexperimental PIV data. The system was based on adirect calibration method by using a design of a PIV testblock developed for the project. This method assumedacceptable linearity between the object and image planesby using an appropriate object distance and resulted ina much simplified image processing routine which coulduse conventional correlation and interpolation routinesfor the calculations. This section also describes thetesting procedures adopted and analysis of the error ratiocharacteristics from experimental error data.

2.1. Imaging system

Figure 1 illustrates the stereoscopic imaging system usedto record the PIV data. In the imaging plane two Kodak

MegaPlus CCD cameras were mounted onto separaterotation stages at an angle to thez axis where1 = and2 = +. These stages were then both mounted ontotwo x translation platforms which were secured onto oneend of an optical breadboard at an appropriate separation.This system allowed fine adjustment of the camera positionsto achieve camera angles up to 45, camera separationsin x up to 600 mm and object distances up to 800 mm.Resolution of the translation stages was 10m and therotation stages 1/10 of a degree. Both cameras had aresolution of 12801024 pixels with 8-bit greyscale outputand an array size of 10 mm 8 mm. All the PIV imageswere recorded using 60 mm Nikon Micro-Nikor lenses. Alens f number off 16 was used for camera angles up to = 30 andf 22 for any camera angles above this. Thesef numbers were based on recommendations for depth offield by Adrian [3].

The object plane consisted of anxz translation stagecentred about the origin O as shown in figure 1 and a lightsheet system connected to a 4 W argon ionlight sourcethrough a fibre-optic delivery. On top of the translationstages a fixed bracket was used to mount the calibrationgrid or PIV test block and provided exact alignment withthe laser sheet for each set of measurements. The light sheetsystem was coupled to the fibre optic through a mountingblock which contained a collimation lens and glass rod.This allowed adjustment of the light sheet divergence andthe light sheet thickness,w, which was eventually set to2.5 mm for all imaging. The object distance,d0, as definedin figure 1 was set to 380 mm for all camera angles. Both

1456

3D PIV

(a)

(b)

Figure 2. (a) Left-hand CCD image of calibration grid.(b) Right-hand CCD image of calibration grid ( = 45).

Figure 3. Typical PIV image from PIV test block ( = 45).

these constants ensured that equal displacements of the testblock in thex andz axes would produce equal percentagesof movement in terms of a percentage of the interrogationregion lengthd, or a percentage of the light sheet widthwat the origin O.

2.2. PIV test blocks

Previous PIV test methods have used glass platescoated with seeding and adhesive with conventional floodillumination for imaging [13, 21]. In the following,however, an alternative test method is presented basedon a design of PIV test block. This method allowsa PIV light source to be used and provides realisticseeding characteristics such as those found in the measuredfluid. The technique also allows the recording of acombination of in and out of plane data and can beadjusted to provide a wide range of particle image sizes andparticle seeding densities if required. In addition the testblocks introduce imperfections such as background scatterand image aberrations that are normally present on PIVrecordings but which are not normally present in computersimulated data.

The PIV test blocks consist of hollow glass spheressuspended in solid clear araldite blocks. The araldite blockswere manufactured by permanently setting Ciba GeigyAraldite resin M with hardener HY6150 into open-top glasscontainers. The glass containers, with internal dimensions20 mm 350 mm 250 mm, were constructed from2 mm thick sheet glass pieces adhered together with siliconesealant. This method gave acceptable viewing propertieswith good non-scratch characteristics. Air bubbles whichformed in the resin during mixing with the hardener wereremoved by placing the mould in a 100 kPa vacuum forone hour before allowing the block to set.

The particles used to seed the test blocks were PottersBallotini SphericelTM hollow glass spheres. They werechosen because of their excellent scattering properties inthe resin. The original 45100m diameter range waswire mesh sieved down to a 4560m size range beforeseeding the resin so as to improve the homogeneity of thesample. Approximate seeding densities were determinedby the amount of seeding weight added to the resin beforemoulding.

2.3. Calibration and data acquisition

A calibration stage is necessary for stereoscopic systemssince the imaging geometry generally causes significantaberrations [12, 13, 16]. The calibration procedure willoutput a space and calibration coefficient map acrossthe image where the calibration coefficients are relatedto the spatial distortion of the image. This distortionbecomes significant when imaging inside a wide fluidlayer and complex stereoscopic equations can be developedto output the calibration coefficients [12]. In this case,however, a simpler directin situ calibration method wasdeveloped based on the PIV test blocks and which reliedon acceptable linearity between the object and image planeby using an appropriate object distance. To this endan object distance of 380 mm was selected and linearitytested at the calibration stage for a range of PIV testblock translations. This approach resulted in a basicimage processing algorithm which used readily availablecorrelation and interpolation routines.

For calibration a grid was placed in the centre of thelight sheet or object plane. This grid allowed alignment of

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N J Lawson and J Wu

the camera positions in thex plane with a grid origin sothat both cameras were imaging a common object plane.Figures 2(a) and 2(b) illustrate the left and right calibrationgrid images which have 5 mm grid spacing inx and yand where the grid origin can clearly be seen. The gridimages also allowed calibration of thex andy axis spacecoordinates by using the data processing software describedin the next section. To obtain the calibration coefficientstwo exposures were recorded from the PIV test block withtranslations between exposures of 200800m in thex orz axes. These displacements corresponded to 7.530% ofthe interrogation region lengthd at O or the light sheetwidth w. The y axis was not required for this analysis.Figure 3 illustrates a typical PIV image obtained from thetest blocks. These data were subsequently processed intoxandz calibration coefficients by cross correlation softwaredescribed in the next section.

For test data acquisition a similar procedure was usedas for thex and z calibration coefficients. In this case,however, in order to obtain suitable data for the erroranalysis, the block was translated by an equal amount inboth thex and z axes. As before the translations chosenwere between 200 and 800m in z and x axes whichcorresponded to a range of 7.530% of the interrogationregion lengthd at O and the light sheet widthw.

2.4. Data processing

Figure 4 illustrates the data processing procedure usedto calculate thex, y and z components of particledisplacement1x, 1y and1z. The procedure can be splitinto three major parts. Each part uses simple correlation andinterpolation algorithms due to the assumption of linearitybetween the object and image planes and the decoupling ofthe x andz characteristics. Three data processing routineswere developed to perform the calculations for each part.The first two involved calibration of the system and thethird the calculation of the three-dimensional displacementcomponents. They axis was not required for the erroranalysis but the relevant equations will still be listed forcompleteness.

In the first routine calibration grid image data wereinitially processed into theX and Y pixel coordinatefunctionsX1(x, y), X2(x, y) andY1(x, y), Y2(x, y). Thesefunctions were produced by performing centroid analysis onthe image across anMg Ng grid at points (Xi1, Y j1 ) and(Xi2, Y

j

2 ) wherei and j are the gridpoint numbers. Thesecentroid pixel positions were then fed the into a bicubicspline interpolation routine ([24], pp 99122). The actualpositions ofx and y in space were determined from thegridpoint data and spacing. A threshold was also used toeliminate background image noise with a value set froma histogram analysis of the images. The resulting bicubicspline interpolation functions were used in all subsequentimage processing to reference any image pixel positionX1,Y1 andX2, Y2 into an actual space positionx and y andvice versa.

The second processing routine was used to produce thex, y and z calibration coefficient functions from the PIVcalibration images. This was done by correlating the pairs

of images on a regularly spacedi, j image grid using thecross correlation functions

R1(, )i,j = +

G1(Xcc1 , Y

cc1 )i,j

H1(Xcc1 , Y cc1 )i,j dXcc1 dY cc1 (1)and

R2(, )i,j = +

G2(Xcc2 , Y

cc2 )i,j

H2(Xcc2 , Y cc2 )i,j dXcc2 dY cc2 (2)where G1(Xcc1 , Y

cc1 )i,j , G2(X

cc2 , Y

cc2 )i,j and H1(X

cc1 ,

Y cc1 )i,j , H2(Xcc2 , Y

cc2 )i,j are the first- and second-exposure

interrogation region images respectively andR1(, )i,j ,R2(, )i,j are the correlation outputs at gridpoint (i, j )[25]. The correlation routine was based on a fast Fouriertransform (FFT) algorithm ([24], pp 530602). A centroidroutine was performed on the three highest peaks in thecorrelation plane and the peak positions stored for post-processing. The interrogation region resolution was setto 64 64 pixels with a 50% overlap and a 4 4 pixelcentroid sample to provide sufficient accuracy [26]. Thepost-processing initially involved validating the vectorsusing a range set by visual inspection of the output vectormaps. Finally thex andz calibration data were calculatedby taking the correlation vector component,R1(, 0)i,jor R2(, 0)i,j , and interpolating and smoothing to reducecorrelation noise as recommended by previous workers[27]. For the y calibration the Y component vectorwas taken,R1(0, )i,j or R2(0, )i,j , and interpolated andsmoothed as before. To convert these data into thex, yand z calibration coefficient functions the correlation datawere initially converted to a pixel mm1 quantity at eachgrid point (Xi1, Y

j

1 ) and (Xi2, Y

j

2 ) such that:

a1(Xi1, Y

j

1 ) =R1(, 0)i,j1xcal

i=1,Mcalj=1,Ncal

(3)

a2(Xi2, Y

j

2 ) =R2(, 0)i,j1xcal

i=1,Mcalj=1,Ncal

(4)

b1(Xi1, Y

j

1 ) =R1(, 0)i,j1zcal

i=1,Mcalj=1,Ncal

(5)

b2(Xi2, Y

j

2 ) =R2(, 0)i,j1zcal

i=1,Mcalj=1,Ncal

i,j

(6)

c1(Xi1, Y

j

1 ) =R1(0, )i,j1ycal

i=1,Mcalj=1,Ncal

(7)

c2(Xi2, Y

j

2 ) =R2(0, )i,j1ycal

i=1,Mcalj=1,Ncal

i,j

(8)

where Mcal , Ncal is the grid size for thea, b and ccoefficients and1xcal , 1ycal and 1zcal are the PIVtest block calibration displacements in thex, y andz axes. These data were fed into a bicubic splineinterpolation routine to produce the calibration coefficientfunctions a1(X1, Y1), a2(X2, Y2), b1(X1, Y1), b2(X2, Y2)and c1(X1, Y1), c2(X2, Y2) which allow estimates of the

1458

3D PIV

Figure 4. Illustration of data processing procedure used to calculate 1x , 1y and 1z particle displacements.

x, y andz calibration coefficients at any pixel position onthe image. All thea1, a2, b1, b2, c1, c2 coefficients andX1, X2, Y1, Y2 interpolation functions were subsequentlystored onto a batch file for use in the third and final dataprocessing routine.

The third processing routine was used to calculate three-dimensional data from actual PIV flow data or in this casetest data recorded from the PIV test blocks. In the first stageof the calculation, the two sets of left and right flow imageswere cross correlated, validated and smoothed using aregularly spacedi, j grid as described previously and storedinto the displacement arrays,1X1(Xi1, Y

j

1 ), 1Y1(Xi1, Y

j

1 )

and1X2(Xi2, Yj

2 ), 1Y2(Xi2, Y

j

2 ) where

1X1(Xi1, Y

j

1 ) = R1(, 0)|i=1,Mfj=1,Nfi,j (9)

1Y1(Xi1, Y

j

1 ) = R1(0, )|i=1,Mfj=1,Nfi,j (10)

1X2(Xi2, Y

j

2 ) = R2(, 0)|i=1,Mfj=1,Nfi,j (11)

1Y2(Xi2, Y

j

2 ) = R2(0, )|i=1,Mfj=1,Nfi,j (12)

and Mf , Nf are the flow grid sizes. These data werealso fed into a bicubic spline interpolation to yield thedisplacement functions1X1(X1, Y1), 1Y1(X1, Y1) and1X2(X2, Y2), 1Y2(X2, Y2). These functions allowed anestimate of the particle image displacement inX orY at any pixel position on the left and right images.

From these displacement functions and the calibrationcoefficient functionsa1(X1, Y1), a2(X2, Y2), b1(X1, Y1),b2(X2, Y2) and c1(X1, Y1), c2(X2, Y2) an estimate of thethree components of particle displacement, (1x, 1y, 1z),can be made at any point (xi, yj ) inside the calibratedflow area using previous simple gas interface stereoscopicderivations as follows [12, 23, 26]:

1x(xi, yj )

= b2(Xi2, Y

j

2 )1X1(Xi1, Y

j

1 ) b1(Xi1, Y j1 )1X2(Xi2, Y j2 )a1(X

i1, Y

j

1 )b2(Xi2, Y

j

2 ) a2(Xi2, Y j2 )b1(Xi1, Y j1 )(13)

1y(xi, yj )

= c1(Xi1, Y

j

1 )1Y1(Xi1, Y

j

1 )+ c2(Xi2, Y j2 )1Y2(Xi2, Y j2 )2

(14)

1z(xi, yj )

= a2(Xi2, Y

j

2 )1X1(Xi1, Y

j

1 ) a1(Xi1, Y j1 )1X2(Xi2, Y j2 )a2(X

i2, Y

j

2 )b1(Xi1, Y

j

1 ) a1(Xi1, Y j1 )b2(Xi2, Y j2 )(15)

where the image coordinates (Xi1, Yj

1 ) and (Xi2, Y

j

2 ) arefound from theX, Y coordinate functions:

Xi1 = X1(xi, yi) (16)

Yj

1 = Y1(xi, yi) (17)Xi2 = X2(xi, yi) (18)Y i2 = Y2(xi, yi). (19)

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N J Lawson and J Wu

2.5. Error analysis

In order to compare the experimental data with thetheoretical error model [23], analysis was carried out onthe output displacement data1x and1z. This consisted ofexamining the rms errorsrms(1x) andrms(1z) in 1x and1z across a regulari, j flow grid with M by N gridpointssuch that:

rms(1x) = 1MN 1xcal

M,Ni=1,j=1

[1xcal 1x(xi, yj )]2

(20)

rms(1z) = 1MN 1zcal

M,Ni=1,j=1

[1zcal 1z(xi, yj )]2.

(21)Errors outside two standard deviations of the data [1xcal1x(xi, yj )] and [1zcal 1z(xi, yj )] were rejected to filterout spurious measurements and 4000 gridpoints were usedfor the analysis. To compare the results to previoustheoretical analysis [23], the experimental error ratio,er ,was found from:

er = rms(1z)rms(1x)

. (22)

The next section will now show the results from theerror analysis for a range of camera angles and fluiddisplacements.

3. Results

Considering the assumption of linearity, we must firstexamine this characteristic for a range of camera anglesand test block translations. Figure 5 shows thea and bcalibration coefficients obtained atx/d0 = 0.1 or the edgeof the image where maximum distortion will occur. In eachcase two sets of test block translations were made from 200to 800m in thex axes andz axes. These results show,for camera angles of = 15 and = 30, that the systemclearly behaves with acceptable linearity in both axes.

If we now examine the individual displacement errorsin x and z it can be seen from figure 6 that significanterrors in thez component begin to occur as the cameraangle is decreased from = 45 to = 10. For examplethe rms error inz at = 45 is 4% rising to a maximumof 13% at = 10. In contrast thex component errorslowly increases with camera angle from around 1% at = 10 to 34% at = 45. These counteracting effectsare caused by the geometric and imaging properties ofthe system. Inz the error will mainly be dependent oncamera geometry since an increase in camera angle willresult in greater imaging sensitivity to az displacement andhence a lower measurement error. Inx, however, greatercamera angles will cause increased imaging problems suchas aberrations with a resultant increase in thex error.Therefore a compromise must be sought between acceptableimaging quality and the requiredz accuracy and these initialresults suggest the optimum range of camera angle to be20 < < 30.

The results in figure 6 also show a dependence onparticle displacement with a greater displacement givingimproved accuracy. These effects can be attributed to thenoise characteristics of the correlation function which hasa random error primarily dependent on particle image size,seeding density and velocity gradient [20, 22]. Thereforebecause the noise is generally independent of particle imagedisplacement, a smaller particle displacement will result ina greater rms % error as shown in the results.

If we now examine the error ratio characteristics forthe system, shown in figure 7, it can be seen that theexperimental trends match the theoretical predictions well.The theoretical trend follows a 1/ tan function [23].These theoretical errors are illustrated more clearly infigure 8 which shows the percentage errors in the predictioncompared with the experimental results for a range ofcamera angles and particle displacements. It can be seenthat the discrepancies in the predictions are worst at thesmallest particle displacement of 7.5% ofd andw with aoverall error of 38%. This error, however, falls to around8 and 18% at the larger particle displacements of 15 and30% of d and w. This improvement is again related tothe noise characteristics of the correlation function andthe resultant fall in rms error with increasing particleimage displacement. The poorer performance with smallerdisplacements is also apparent if the error characteristics inx are examined. These are shown in figure 9 for a rangeof 0.1

3D PIV

Figure 5. Graph to show linearity of system for a range of PIV test block translations and camera angles (x/d0 = 0.1).

Figure 6. Graph to show PIV system performance in terms of x and z rms errors.

a lowerf number hence giving a light power advantage. If,however, sufficient laser power is available and the spatialresolution of the flow structures matches the resolutionprovided by the CCD, these results have shown a relativelycheap 3D PIV system with high measurement performanceis possible by usingin situ calibration in conjunction withsimple processing algorithms.

Previous algorithms developed for particle trackingvelocity (PTV) have suggested an optimum camera angleof = 45 [28] which matches the theoretical errorratio analysis [23] but contradicts the optimum result here

of 20

N J Lawson and J Wu

Figure 7. Graph to show experimental and theoretical error ratios for a range of camera angles and particle displacements.

Figure 8. Graph to show the deviations in the theoretical prediction of error ratio.

reducing its visibility and due to moulding imperfectionswhich formed during the gelling stage. Therefore it mustbe assumed that a thicker block would degrade the imagequality further as found in previous work, when using athick liquid layer [12]. In this situation in order to retainsystem performance a simple solution is to increase thef number of the imaging system [12, 13]. Further testswere carried to study these effects by placing a PIV testblock in the centre of a 600 mm wide water tank andacquiring images using the same object distance. It was

found an additionalf number was necessary to achieve thesame image quality from simple visual inspection duringacquisition. Consequently, higher laser power would berequired when taking measurements in wider liquid flows.These tests though did also demonstrate the flexibility ofthe PIV test block design which can be customized to therig being measured and be used to calibratein situ.

Previous work discussed in detail the problems ofimage registration and calibration in stereo PIV systems[23]. With photographic systems, moire techniques and

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3D PIV

Figure 9. Graph to show the variation of error ratio across the image plane ( = 45).

double imaging onto a single transparency have been usedto overcome the registration uncertainty [12, 13]. In thissystem, however, the use of bicubic spline interpolationswas relied upon on throughout the image processingstage to ensure point to point correspondence of any1X displacements and calibration coefficients. For thesimple uniform flow test images used in this study,these techniques obviously provided acceptable accuracy.If considerably more complex flow regimes are to bemeasured, however, careful choice of the interpolation gridresolution would be required to maintain the accuracy ofthe interpolation functions. This issue was not investigatedin this study and further work is required in this area.

5. Conclusions

This paper has demonstrated and tested a simple design ofdigital angular stereoscopic PIV system. Results from thetests have shown optimum performance will be obtainedby using camera angles of between 20 and 30 and cameraf numbers off 16 and higher. With this set-up and ifmeasuring a uniform flow the in plane and out of plane rmserrors will be 12 and 34% respectively. The results havealso shown a theoretical prediction of system performancederived in previous work [23] which considers the ratioof out of plane to in plane errors matches to within 8and 18% of the experimental system performance providingparticle displacements are restricted between 15 and 30%of the interrogation region length. Error ratios in thiscase were around two. Below this displacement range,the random errors in measurement caused unacceptabledeviation from the theoretical predictions and an increase inimaging resolution was suggested to reduce this problem.

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