characterization of different piv algorithms using the europiv synthetic image generator and

Characterization of Different PIV AlgorithmsUsing the EUROPIV Synthetic Image Generatorand Real images From a Turbulent BoundaryLayer.

J.-M. FOUCAUT, B. MILIAT, N. PERENNE and M. STANISLAS

Laboratoire de Mécanique de Lille, CNRS UMR 8107, Blv. P. Langevin, F59655Villeneuve d'Ascq Cedex, [email protected]

Abstract

To characterize the PIV accuracy, a statistical study has been performed by meansof synthetic and real images. The synthetic images were generated with the Eu-ropiv SIG. They allowed to characterize the systematic and random errors and tooptimize the recording and processing parameter. The influence of the character-istics of particle images like the diameter or the density and of CCD such as thefill ratio or the noise were studied. The algorithm, which gives the best results,was an iterative one using the FFT-based correlation, the sub-pixel window shift-ing technique with a Whittaker interpolation and a three point Gaussian peak-fitting. From a series of real images, the probability density function validated thebenefit of sub-pixel shift. The influence of the velocity gradient and of the out ofplane component have also been investigated. The optimization of experimentalparameter on the basis of this study enables an accurate measurement of the tur-bulent characteristics.

1 Introduction

Particle Image Velocimetry is a measurement technique with a well establishedtheory. Since the pioneering work of Adrian (1991) and Keane and Adrian (1992),many researchers have developped the mathematical basis of this method(Westerweel, 1997, Raffel et al, 1998). The recent improvements due to digital re-cording and analysis, are detailed in Willert and Gharib (1991) and Westerweel(1997). This progress has opened a new route for the assessment of turbulentflows. The study of turbulence with PIV implies a large field and a good spatialresolution to measure a large range of scales, and a high accuracy to measure cor-rectly the small scales. It implies also a careful characterization of the accuracy ofthe method and of its spatial response as compared to the standard tools used inthis field such as Hot Wire Anemometry and Laser Doppler Velocimetry. Even ifthe PIV accuracy has been extensively studied by different authors (Adrian, 1997,Westerweel et al, 1997, Lourenco and Krothapally, 2000), it is not a simple prob-

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lem which is still subject of interest for all users. Recently, Lecordier et al (1999),Nogueira et al (2001), Gui and Wereley (2002) and Scarano and Reithmuller(2000) have proposed new steps of progress with advanced algorithms based onimage interpolation and/or interrogation windows deformation. Scarano (2002)gives a review of iterative processing using image deformation.

Concerning the accuracy, several contributions, using synthetic images, haveallowed to characterize the performance of the method (Raffel et al, 1998,Nogueira et al (2001), Gui and Wereley (2002)). The results of such studies arevery instructive and allow a better understanding of the PIV limitation. The draw-back is that each team which has developed a PIV system, has generally also de-veloped a synthetic image generator. The characteristics of the image generationalgorithm are never completely detailed in the papers, which leads to some inter-rogations about the exploitation of some results (Lourenco, 1988). In the presentcontribution, the Europiv SIG software (Synthetic Image Generator) is used tomake a detailed study of the accuracy of different PIV algorithms. This softwarehas been developed in the frame of the EUROPIV 2 project in order to have acommon tool which allows a true comparison between the PIV algorithms.

To make this study, care was taken to compute systematically the bias errorwhich evidences the well known peak-locking effect (Nogueira et al, 2001) andthe RMS error which gives an estimation of the measurement noise (see Raffel etal, 1998). In a first part, the influence of the image recording features such as theparticle image diameter, the image density, the CCD fill ratio (Westerweel, 1998),the background noise, the out of plane component and the velocity gradient insidethe analysis window are characterized. In the second part, analysis parameterssuch as the detection method, the correlation method and the sub-pixel shift aretested.

Finally, a test was performed on real images of a turbulent flow. These imagesare taken from a database on turbulent boundary layers. The motivation is to opti-mize the processing parameters to get the best PIV accuracy for turbulent flows.

2 Characterization of PIV by means of synthetic images

The generation of the synthetic particle images was done in two steps. In the firststep, two fields of particle image locations were generated. The location of each

point pxr

of displacement vector xδr

is computed randomly in the domain. The

location of the first position is given by 2/xx xp1 δrrr

−= . The second one is

2/xx xp2 δrrr

+= in order to obtain 12x xxrrr

−=δ with a second order accuracy. In

a second step, particle images are computed using the SIG algorithm which is de-scribed in detail in the paper by Lecordier et al (2003). This algorithm allows toprovide images with known characteristics. It takes into account: the laser sheetcharacteristics, the particle size distribution, the image pattern characteristics, theprojection parameters and the CCD characteristics.

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The field size is given in 3D physical dimensions and in 2D pixels. As the SIGcan be also used for stereoscopic PIV, the optical projection is taken into account.The magnification was chosen in order to have a negligible effect of this projec-tion on the result. For the present study, the light sheet profile was chosen as aGaussian. The particle size distribution was taken uniform. The variation of sizeobserved in the images is thus only due to the difference in brightness linked tothe location of the particles in the thickness of the light sheet. The CCD is charac-terized by the fill ratio, which corresponds to the ratio between the active area ofeach CCD sensor and the total surface of this sensor. The image pattern was Gaus-sian. It is integrated on the active area leading to an erf function. In order to be inagreement with other contributions (e.g. Raffel et al, 1998), the standard deviationof the Gaussian σ is considered as di / 4, where di is the particle image diameter.This diameter is then defined by the standard value e-2 of the Gaussian function.In the SIG, a saturation level is required between 0 and 1 (Lecordier et al, 2003).In the present study, it was adjusted to obtain less than 0.05% of saturated pixel.The CCD background mean and noise levels are not added in most of cases.

2.1. Image parameters

In this first step, the image parameters were varied and their influence was stud-ied. The size of the images generated was generally taken as 512x512 pix2 for afield size of 0.1x0.1 m2. This leads to a magnification of 0.2 mm/pix. As explainedin the previous paragraph, the light sheet profile was gaussian, its thickness was 2mm. The analysis was done with 16x16 interrogation windows. The FFT basedcross-correlation was used. A 3 point Gaussian peak-fitting without shift was alsoused in this paragraph. A set of 21 images with displacements from –1 to 1 pixelwith a step of 0.1 pixel was generated for each value of the parameters. The aim isto predict the behavior of the measurement uncertainty when the image parame-ters are varied away from an optimal choice. This optimal set of parameters is de-duced from the literature.

2.1.1 Optimal parameters

In a first step, a set of images corresponding to an optimal case is computed. Thisset should give the best accuracy. The optimal parameters are the following :− a particle image diameter of 2 pixels at e-2, as suggested by Raffel et al (1998),− a fill ratio of 100% (The complete area of the CCD sensor is active.),− a particle concentration of 10 particles per interrogation window (which give

0.039 particle per pixel). This value is close to the optimum obtained by Willertand Gharib (1991),

− a CCD black level of 2 % of the gray level range, no CCD noise,− no gradients and no out-of-plane displacement.

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16 pix

16 pix

Fig. 1 Sample of image, a) 10 particles and b) 20 particles per interrogation window of 16 x16 pix2.

Fig. 1 shows two examples of image generated by the SIG (a) for a density of10 particles and (b) for a density of 20 particles in an interrogation window of16x16 pixels2. Fig. 2 presents the bias and RMS errors in pixels as a function ofthe imposed velocity. Both errors are zero for a zero displacement. The bias errorpresents the classical “peak locking” effect which is a systematic periodic error(see Nogueira et al , 2001). The RMS error is about 10 times larger than the biasand is in agreement with the result of Raffel et al (1998). For a displacement of 0.5pix, the value of the RMS error is of the order of 0.06 pix, which is also in agree-ment with the result of Wereley and Gui (2001).

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Fig. 2 Bias and RMS errors for the optimal case.

2.1.2 Influence of the particle concentration

The first parameter which was varied is the image density. As was shown by Raf-fel et al (1998) or Wereley and Gui (2001), the particle concentration modifies theRMS error. When the number of particles in the interrogation window decreases,it is well known that the RMS error increases. Keane and Adrian (1992) evidencedthat the number of valid detected vectors increases when the particle image den-sity increases. As shown in fig. 3, when the density increases, the particles used to

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build up the correlation peak are more numerous and the RMS error decreases asobserved by Raffel et al (1998) or Wereley and Gui (2001). However, when thedensity increases the peak locking increases, probably due to the number of im-ages cut by the border of the interrogation window which increases (Nogueira etal, 2001). This error is very small with an image density of 5.

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Fig. 3 Bias and RMS errors, influence of particle density.

Fig. 3 shows that image densities between 20 and 40 give smaller RMS errorsbut higher bias errors. These errors seems saturated around an image density 20.Willert and Garib (1991) shown a weak increase of the RMS error for densitieshigher than an optimal value of 10 ppw which is not observed in the present work.In the present study, a concentration of 10 ppw (0.039 particles per pixel) seems tobe a good compromise between the bias and RMS errors and will be kept whenthe influence of other parameters is studied.

2.1.3 Influence of the fill ratio and the particle diameter

It is of interest to check the value of the particle image diameter which minimizesthe errors. The behavior of the errors versus the diameter depends on the fill ratio.In a theoretical study, Westerweel (1998) details the effect of the sensor geometryon the PIV measurement performances. He shows that its influence on errors isnegligible if the particle image diameter is at least two pixels, whatever the fill ra-tio is. As the fill ratio is generally between 50 and 100%, these values and an in-termediate value of 75% were tested. In order to generalize in the case where theintegration on the sensor is not done, a fill ratio of 1% was also studied. Fig. 4 pre-sents the bias and RMS errors for a fill ratio varying from 1% to 100% and a par-ticle image diameter of 2 pixels. As shown by Westerweel (1998), the fill ratio hasroughly no effect on both errors. For diameters smaller than 2 pixels, Westerweelfound a strong influence of the fill ratio on the bias error. He shows that the peaklocking increases with the fill ratio.

Fig. 5 presents the bias and RMS errors for a diameter of 1 pixels and for dif-ferent fill ratios. It can be observed that the behavior of the bias error versus thefill ratio is opposite to Westerweel’s results. This error decreases when the fill ra-tio increases. This difference is probably due to the particle images truncated bythe border of the window which were not taken into account in Westerweel’s theo-retical analysis. For a fill ratio of 100%, the peak locking error is nearly cancelled.

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The same behavior is observed for the RMS error: it decreases as the fill ratio in-creases.

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With a fill ratio of 1% the mean bias error is about five times smaller than themean RMS error. If the diameter is brought down to 0.5 pixel, when the fill rationis higher than 75%, the behavior becomes the same as suggested by Westerweel(1998). For small fill ratios and when the particle images are not located in the ac-tive area of the CCD sensor, the particles with small diameters disappear. This isthe reason why the errors increases when the diameter and the fill ratio decrease.

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To evidence this phenomenon, fig. 6 presents the mean values of the bias andthe RMS errors averaged on the displacements from –1 to 1 pixel. The bias errorbeing symmetrical, the absolute value of this error was averaged. These errors aregiven for fill ratios of 1% and 100%. The mean bias error presents optimal di-ameters of 1 pixel and 2 pixels respectively for the fill ratio of 100% and 1%. Bothcurve are nevertheless fairly flat between 1 and 3 pixels. The mean RMS errorshows optimal diameters close to 2 pixels in both cases (1.8 pix and 2.2 pix re-spectively with a fill ratio of 100% and 1%). As the bias appears significantlysmaller in that region, the best choice of diameter is thus two pixels whatever thefill ratio is. These results are in agreement with those of Raffel et al (1998) con-cerning the RMS error. In fig. 6 a, for a fill ratio of 1%, a change of behavior canbe observed for a diameter of 0.8 pixel. This is probably due to the loss of parti-

PIV Accuracy 169

cles linked to the fill ratio. In conclusion, if the fill ratio is larger than 75%, theoptimal diameter is 2 pixels. As the curve of the RMS errors is relatively flataround 2 pixels, a diameter between 1 and 3 pixels can be acceptable with theclassical algorithm. For fill ratios nearer to 50%, the particle image size shouldmove toward 2 to 3 pixels.

Of course this study does not take into account the effect of micro-lenses placedon the top of the CCD. This effect is not presently modeled in the Europiv SIG butit would be worth to study it.

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Fig. 6 Mean bias and RMS errors versus image particle diameter, fill ratio of 1% and 100%.

2.1.4 Influence of CCD noise

The CCD noise is a white noise added to the image. In the present test, for RMSvalues of the CCD noise (sigma) different from zero, the ratio of sigma over thenoise level was kept constant. Fig. 7 presents the CCD noise effect on the bias andRMS errors. The case sigma = 0 corresponds to the optimal case of paragraph2.1.1. The corresponding background level is 2% of the scale, i.e. 5 grey level fora quantization on 8 bits. The value sigma = 1% corresponds roughly to the noiseof a PCO Sensicam camera. The value sigma = 2 % is close to the characteristicsof a Pulnix TM9701 camera.

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Fig. 7 Mean bias and RMS errors, influence of CCD noise.

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2.1.5 Influence of pixel saturation

Fig. 7 shows that, up to a value of sigma = 2%, the bias error is weakly affected bythe noise. For sigma higher than 2% the peak-locking is amplified and modified.However, the RMS error seems not affected by the noise at least for sigma up to 5%. This is in agreement with the results of Raffel et al (1998). They do not ob-serve strong modification of the RMS error for 5% and 10 % of noise. These re-sults show that the background noise of the standard PIV camera does not affectpresently the measurement accuracy. It must be mentioned that this test does nottake into account the noise due to the spurious light in an actual experiment (back-ground reflection on a model or a wall).

In order to be sure that the complete range of grey level of a camera is used, asmall number of pixel can be saturated. The influence of these saturated pixels ispresented in Fig. 8. The saturation parameter of the SIG varies from 12.5% to50%. Table 1 gives the number Ns of saturated pixels in an image and the corre-sponding percentage Sr of saturated pixel. For Sr less than 0.05% both errors arenot affected by the presence of saturated pixels. Gui and Wereley (2002) did thesame comparison with strongly saturated particle images. They observe no influ-ence on peak locking and a RMS error 4 times larger. The optimal case is obtainedwith a saturation parameter of 25% which gives about 100 saturated pixels in a512 x 512 image (i.e. Sr = 0.035%). For all the tests presented in this study, thesaturation parameter was adjusted to obtain Sr of the order of 0.035%.

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Fig. 8 Mean bias and RMS errors, influence of CCD noise.

Table 1. Number of saturated pixels Ns and saturation ratio Sr versus the saturation pa-rameter of the SIG.

Saturation Ns Sr%12.5 1 0.0004%19 15 0.0057%25 90 0.034%

37.5 1400 0.53%50 4200 1.60%

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2.1.6 Influence of the third component

Two-dimensional two-component (2D2C) PIV allows to measure the in-planecomponents. In the case of experiments and particularly in turbulent flows, there isan out of plane particle motion which can disturb the PIV measurement. In thepresent study, the third component is considered as a Gaussian random distribu-tion with W3 ≈σ . The value of W varies from zero in the optimal case, to amaximum of 5 pixels. In this case, some particles come in and out of the lasersheet between the two exposures, leading to isolated particle images. To improvethe convergence of the results, the field size has to be increased from 512x512 to768x768 pix2. Table 2 gives the number of particles lost in each images.

Table 2. Loss of particle pairs in an interrogation window due to the third velocitycomponent

W (pixel) Percent of lostparticle pairs

W/∆z0

1 2% 0.12 5% 0.23 7% 0.34 10% 0.45 12% 0.5

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Fig. 9 Mean bias and RMS errors, influence of the third component.

In the case of double exposure auto-correlation images, Keane and Adrian(1990) recommended to use W/∆z0 < 0.25 to limit the loss of particle pairs. With alight sheet thickness ∆z0 = 10 pixels, this value is in the middle of the present testrange for W. The bias error presented in fig. 9 is weakly affected by the out ofplane component. On the contrary, the RMS error is strongly influenced by iso-lated particles. This error increases with the third component whatever the dis-placement is. Up to W = 2 pixels the RMS stays within reasonable bounds. Thislimit seems close to this of Keane and Adrian (1990), although the present analy-sis is done with cross-correlation. In a turbulent flow, the value of W can be esti-

mated, using the out of plane turbulent intensity, as tM'w2 2 ∆ where M is the

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magnification in pixel/m. The delay ∆t can then be adjusted to limit W which,from the present study, should verify W/∆z0 < 0.2.

2.1.7 Influence of the velocity gradient

When a turbulent flow is studied close to a wall, PIV measurements have to bedone in a region with a strong velocity gradient. As the PIV method is based onthe hypothesis that the velocity is uniform inside the interrogation windows, if agradient is existing, it can bias the result following two scenarios (Raffel et al,1998). First, as particles are located randomly in the window, it is possible to findlocal distributions of particles leading to a displacement lower or higher than themean looked for. The second scenario appears in regions of larger displacementwhere the probability of isolated particles is higher, leading to a bias toward lowvelocity. These two scenarios contribute mostly to the RMS error. To test the gra-dient effect, a series of 16 images of 4096 x 21 pix2 was generated. The results ofthese images was averaged in order to obtain a good convergence of statistical er-rors. Each image contained particle images corresponding to a range of displace-ment from –1 pix to 1 pix as previously, but on which a constant gradient wasadded. The mean displacement of the gradient was zero and the difference be-tween the top and the bottom of a 16x16 pix2 window was Du. The value Du wasvaried from 0 (optimal case) to 3 pixels. For a density higher than 15 particles perinterrogation area Keane and Adrian (1992) showed that the PIV measurement isnot affected by the gradient if Du is smaller than the particle image diameter (i.e. 2pixels) and does not exceed 3% of the interrogation spot. The results are presentedin fig. 10. As can be seen, the gradient affects rapidly the RMS error. With Du = 1pixel the noise is already doubled and the RMS error for a zero displacement isnot zero any more. The results of Raffel et al (1998) give the same order of mag-nitude. However for Du = 2 the RMS error obtained by Raffel et al is 0.23 insteadof 0.3 in the present paper. This is probably due to the difference in the imagegeneration. When Du is smaller than 1, the bias is not to much affected by the gra-dient. Above this value, a large anti-peak-locking error appears. This limitingvalue is in good agreement with the study of Keane and Adrian (1990) who pre-dict a maximum Du of half of the particle image diameter (2 pixels here).

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Fig. 10 Mean bias and RMS errors, influence of the velocity gradient.

PIV Accuracy 173

A test with a density of 20 particles per interrogation window leads to a reduc-tion of the RMS and an increase of the bias in the same proportion as without gra-dient (fig. 3). The influence of the window size on this results will be discussed inthe part concerning the real images.

2.2 Analysis parameters

The tests performed up to now have allowed to characterize the influence of themain image parameters. This study was done with a standard algorithm. Severalimprovement have been published in the last five years which are supposed to en-hance the accuracy of the PIV measurement (Scarano and Reithmuller, 2000). Toobtain the best from the PIV measurement in turbulent flows, it was of interest tostudy the effect of at least some of these algorithms. This is the aim of the presentstudy. The methods using window deformation have been put aside on purpose, asthey are time consuming. The study is thus focussed on the peak detection algo-rithm, the sub-pixel window shifting and the correlation method. For that purpose,in the following, the image parameters are fixed to the optimal values proposed in2.1.1.

2.2.1 Peak detection algorithm

The peak detection algorithm used here is the standard Gaussian fitting. Wester-weel (1998) has largely shown the advantage of this method. It is generally ap-plied twice on 3 points (Raffel et al, 1998). Ronneberger et al (1998) observed alack of accuracy for the smallest image diameter. They propose a method based ona 2D fitting on 25 points using a non-linear least square interpolation (Levenberg-Marquardt). Both methods have been tested in the present study. Fig. 11 presentsthe bias and RMS errors as a function of the detection method. The 3 points 1Dpeak fitting shows the smallest peak locking error. The same fitting with 5 pointsis the best in terms of RMS error. These 1D methods give the best accuracy for adisplacement parallel to the window side.

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Fig. 11 Mean bias and RMS errors, influence of the peak detection algorithm.

For a displacement at 45°, the results show that the 2D peak fitting using 5x5pix2 is better but only for the RMS. The same test was performed with particles of

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4 pixels in diameter. In that case, the methods using 5 points are better for both er-rors. For particles of 1 pixel, no improvement was observed with the 5 points fit-ting. Ronneberger et al (1998) recommend to use a Fisher transform which has notbeen tested in the present study.

2.2.2 Window shifting

In the study detailed in paragraph 2.1, when the displacement goes to zero the biasand RMS errors generally tend toward zero. This is why a shift of the secondanalysis windows can be used to improve the accuracy. The first idea was to usean integer shift in order to increase the signal to noise ratio (Westerweel et al,1997). In a second step, Lecordier et al (1999) and Wereley and Gui (2002),among other, used an iterative sub-pixel shift by interpolating the interrogationwindow in each image. Wereley and Gui used a bilinear interpolation. FollowingLecordier et al (1999) it is better to increase the order of the interpolator, a bicubicone is also tested here. Scarano and Reithmuller (2000) preferred a Whittaker in-terpolation. They explained that the later is better because it respects the samplingtheorem. Fig. 12 shows a comparison of the errors obtained with an integer shiftand bilinear, bicubic and Whittaker sub-pixel shifts applied on the set of imagescorresponding to the optimal case. It can be observed that the iterative processusing an integer shift does not give an improvement as compared to Fig. 1. Thebilinear interpolation increases the bias error and leads to an anti-peak lockingwith a period doubled due to the use here of a symmetrical shifting technique(both windows are shifted symmetrically). This interpolator does not decrease thenoise. The bicubic shift reduces the RMS but not the bias as compared to the inte-ger shift method. Only the Whittaker interpolation decreases significantly both er-rors.

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These results are not in agreement with those of Wereley and Gui (2002) whofound a large decrease of both errors with a bilinear shift. This is due to the factthat their particle image diameters are of the order of 7 pixels with the same defi-nition as in the present study. It is clear that the optimal diameter obtained in para-graph 2.1 is not the same when a sub-pixel shift is applied. Fig. 13 shows thecomparison of the mean bias and RMS errors obtained with the different interpo-

PIV Accuracy 175

lation algorithms and the integer shift which corresponds to Fig. 6. These errorsare given as a function of the diameter from 1 to 4.5 pixels. As far as the bias erroris concerned, the bilinear interpolator is the less accurate for diameters smallerthan 3.5 pixels. The bicubic shift shows an optimum for a diameter of 4 pixels.The bilinear interpolator behave like the bicubic one, but with a level at least twotimes higher. The best interpolator is the Whittaker one, particularly for a range ofdiameter between 2 and 4 pixels. For diameter smaller than about 1.5 pixels, it isbetter not to interpolate and to use the integer shift. The RMS diagram leads sen-sibly to the same conclusion. The bicubic and Whittaker interpolations show al-most the same behavior. No gain is observable for a diameter of 1 pixel whateverthe method used. The conclusion is that the Whittaker interpolation is the best forparticle diameters from 1.5 to 4 pixels. For diameters smaller than 1.5 pixel, inte-ger shift is recommended. According to Wereley and Gui (2002), for diametershigher than 4.5 pixels, bilinear sub-pixel shift could probably be a good compro-mise.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 1 2 3 4 5Part. im. diameter (pix)

Mea

n bi

as e

rror

(pix

)


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 1 2 3 4 5Part. im. diameter (pix)

Mea

n R

MS

erro

r (p

ix)


Fig. 13 Mean bias and RMS errors versus image particle diameter, influence of windowshifting algorithm.

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1


Bia

s err

or (p

ix)


00.010.020.030.040.050.060.070.080.09

0.1


RM

S er

ror

(pix

)


Fig. 14 Bias and RMS errors, influence of asymmetrical window shifting algorithm.

The use of a symmetrical shift of each window is supposed to provide the bestaccuracy (second order). However, it implies two interpolations and gives a strongmodification of peak locking when bilinear or bicubic shift is used. Fig. 14 pres-ents the same result as Fig. 12 but with a non-symmetrical shift (as in Wereley andGui, 2002). The first window is shifted back by the integer half value of the esti-

Session 3176

mated displacement and the second window is shifted forward from the first win-dow by the total displacement using sub-pixel interpolation. An improvement isobserved mainly for the bilinear and bicubic method.

The anti-peak-locking is always obtained but the period is the same as standardpeak-locking (1 pixel). The Whittaker method presents again the best accuracy. Itcancels practically the bias error and gives the minimal RMS error. This methodof non-symmetrical shift gives an error on the vector location which is at worst of0.5 pixel. This error is not evidenced in the case of a uniform displacement but, inthe case of a turbulent flow, which presents continuous variations of the velocity,it can increase the measurement noise. This is why a symmetrical shift with theWhittaker interpolator is preferred for this kind of flow.

2.2.3 Correlation method

The method to compute the cross-correlation which was used in the optimal case(paragraph 2.1.1) is the classical FFT-based cross correlation. Roth and Katz(2001) recommend the direct correlation computed only in a region of interest, inorder to minimize the computational time. Pust (2000) has also shown some im-provement using direct correlation computations limited or not.

If the location of the peak is almost known in the correlogram, the correlationof two windows of size N x N pix2 can be computed only in an region of interestof size M x M pix2 (Roth and Katz, 2001) following equation:

∑ ∑= =

++×=N,0i N,0j

fg )nj,mi(g)j,i(f)n,m(Φ

with m+i < N+M and n+j < N+M. This amounts to correlate a N x N pix2 windowwith a N+M x N+M pix2 window. The correlogram is then of M x M pix2 size. Thecomputational time is increasing with M. This method will be called DCC (DirectCross-Correlation). It is also possible to limit the computation to m+i < N and n+j< N. The advantage of this method is to not correlate truncated particles (cut bythe border) of the first window with all particles of the second one which tends toincrease the noise (Nogueira et al (2001)). Its drawback is that the correlogram isbiased by a pyramidal function (Raffel et al (1998)). This method will be callLDCC (Limited Direct Cross-Correlation). According to Pust (2000), an im-provement is obtained by normalizing the correlation :

∑ ∑ ∑ ∑∑ ∑

= = = =

= =

−++×−

−++×−

=

N,0i N,0j N,0i N,0j

22N,0i N,0j

fg))n,m(g)nj,mi(g()f)j,i(f(

))n,m(g)nj,mi(g()f)j,i(f(

)n,m(Φ

f and g are the mean gray level respectively of f(i,j) and g(i,j). This normali-zation allows to obtain two additional methods called NDCC (Normalized DirectCross-Correlation) and NLDCC (Normalized Limited Direct Cross-Correlation).

PIV Accuracy 177

Fig. 15 presents the comparison of the four method of direct correlation and of theFFT based Cross-Correlation. The FFT based CC seems less efficient than directcomputation probably due to the periodicity of the signal imposed by FFT whenno padding is used. The LDCC gives better results than DCC. As far as the RMSvalues are concerned, if the displacement is smaller than 0.5 the LDCC is betterthan DCC and if displacement is higher than 0.5 it is the opposite. When the dis-placement goes to zero, the RMS error obtained by DCC does not go to zero. Thisis due to the effect of truncated particles. The normalization does not modify thebias error and decreases slightly the RMS value.

-0.02

-0.01

0

0.01

0.02

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1


Bia

s err

or (p

ix)

FFT based CCDCCNDCCLDCCLNDCC 0

0.010.020.030.040.050.060.070.080.09

0.1


RM

S er

ror

(pix

)

FFT based CCDCCNDCCLDCCLNDCC

Fig. 15 Bias and RMS errors, influence of correlation method.

The same test was performed using the Whittaker sub-pixel shift. The resultsare presented in Fig. 16. As far as the bias error is concerned, all the methods givealmost the same result, the direct computation does not present any improvement.The RMS is almost the same for the limited direct computation with and withoutnormalization and the FFT cross-correlation. The direct correlation shows a RMSerror saturated at about 0.07 pixel. When normalized, the value is about 0.05 pixel.This saturation is again due to the truncated particle images.

-0.02

-0.01

0

0.01

0.02

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1


Bia

s err

or (p

ix)


00.010.020.030.040.050.060.070.080.09

0.1


RM

S er

ror

(pix

)


Fig. 16 Bias and RMS errors, influence of correlation method, Whittaker sub-pixel shift.

2.2.4 Conclusion

From the present analysis, it comes out that the PIV processing method whichshould be adopted for real images of a turbulent flow is a FFT-based cross-

Session 3178

correlation, using iterative sub-pixel shift based on the Whittaker interpolation. Ifthe particle image diameters have been adjusted around 2 pixels, the 1D Gaussianpeak-fitting algorithm is the best suited.

3 Characterization of PIV by Means of Real Images

In this paragraph, the PIV characterization is done on the basis of real images. Anexperiment has been carried out in a boundary layer wind tunnel. The Reynoldsnumber, based on momentum thickness, is 7800. The boundary layer thickness isabout 0.3 m and the free stream velocity is 3 m/s. The PIV records were obtainedin a plane normal to the wall and parallel to the flow. The x axis is horizontal andthe y axis vertical. The camera used was a PCO Sensicam. Its CCD has a size of1280 x 1024 pixels2 of 6.7 x 6.7 µm2 each. The images were recorded at a rate of 8frame-pairs per second by means of the LAVISION software. A 200 mm focallens was used with f# = 11. The magnification was 0.21 which gives in wall units asize of 300 x 230. The whole flow was seeded with Poly-Ethylene Glycol parti-cles. Their diameter is about 1 µm. The light sheet was generated by a BMI pulsedYAG laser with 2 x 250 mJ of energy at 12 Hz. The sheet thickness in the field ofview was about 1 mm corresponding to 8 wall units and ∆z0 = 32 pixels. Follow-ing Adrian (1991), the depth of field was about 7 mm which avoids any problemto focus the particle images. In this configuration, the Airy disk diameter is of theorder of 17 µm which gives a size of the order of 2.8 pixels for the particle images(Adrian,1997). The particle images number was estimated of order of 0.03 parti-cles per pixel. This value is a little bit smaller than the optimal value which shouldbe around 0.04. The analysis was made by the FFT based cross-correlation methodwith integer shift and sub-pixel shift using Whittaker interpolation with a 3 pointGaussian peak fitting in each direction. An overlapping of 50% was used. Thedelay between both laser pulses was chosen to obtain a mean dynamic range of theorder of 8 pixels (150 µs). With this range the turbulence intensity was of 1.6 pix-els. A total of 200 fields were recorded to compute statistics.

3.1 Peak locking

In order to evidence the peak-locking effect, a Probability Density Function hasbeen computed from the difference between the displacement in pixel and its inte-ger value. This PDF is computed from the vectors of a single velocity field. Theanalysis was done with 16 X 16 interrogation windows, leading to about 30 000vectors in the field. The results are presented in Fig. 17,a which compares the it-erative PIV analysis with an integer shift and a Whittaker sub-pixel shift. As canbe seen the integer shift analysis builds up a strong peak-locking which is can-celed by the Whittaker interpolation. Fig. 17,b shows a comparison of PDF of theu component of the velocity fluctuation. This PDF is computed from each vectorline located at 100 wall units of the 200 velocity field. The PIV analysis is per-formed with the same methods as in Fig. 17.a. These results are also compared

PIV Accuracy 179

with Hot Wire Anemometry which shows that the PDF of the flow is Gaussian atthis wall distance.

0%

1%

2%

3%

4%

5%

6%

7%

8%

0 0.2 0.4 0.6 0.8 1Decimal part of the displacement (pix)

PDF

integer shiftWhittaker subpixel shift

0

0.1

0.2

0.3

0.4

0.5

0.6

-4 -3 -2 -1 0 1 2 3 4u/u'

PDF(

u)

Integer shiftWhittakerHWA

Fig. 17 PDF of the decimal part of displacement in pixel (a) and of the velocity inm/s (b).

The results with the integer shift analysis show a strong peak locking effect.These oscillations are cancelled with the Whittaker sub-pixel shift which is veryclose to the hot wire results. To study the effect on the noise, spectra are computedfrom the same data (200 lines located at y+ = 100). The comparison presented inFig. 18,a does not show a significant difference between both analysis method.This allows to conclude that the peak-locking evidenced by the PDF function is arandom noise which level is negligible as compared to the signal in the Fourierspace. It can be observed that, at the smallest wave numbers, the spectrum has aslope of –5/3, which is a standard turbulence one. At the highest wave numbersthe slope is –2 which is typically linked to the noise (Foucaut and Stanislas, 2002).

1E-07

1E-06

1E-05

1E-04

100 1000 10000

k (rad/m)

E11

(m3 /s

2 )

Integer shiftWhittaker

Slope -2Slope -5/3

0

0.5

1

1.5

2

2.5

3

3.5

0 50 100 150 200y+

Tur

bule

nt fl

uctu

atio

ns

WhittakerInteger shift

u'+

v'+

(a) (b)Fig. 18 Turbulence spectra at y+ = 100 (a) and turbulent fluctuation profiles (b).

Fig. 18,b presents the fluctuation profiles u’ and v’ (in wall units). They areclearly not influenced by the peak-locking. In conclusion, the peak-locking effectstrongly occurs when an integer shift iterative analysis method is applied. It is de-tectable in the PDF function and can be removed with a sub-pixel shift based onthe Whittaker interpolation. Nevertheless, with the dynamic range used in the pre-sent experiment, it does not modify the mean velocity and the turbulent fluctuationprofiles.

Session 3180

3.2 Influence of a velocity gradient

In a boundary layer, when the wall distance goes to zero, the velocity profileshows a strong gradient. Fig. 19,a shows the mean velocity profiles computedfrom the 200 fields compared with hot wire anemometry. In this figure, the PIVanalysis is done with 64 x 64, 32 x 32 and 16 x 16 interrogation windows whichallow to obtain respectively about 8000, 16000 and 32000 samples to average. Astandard overlapping of 50% is used. In the log region, the PIV profiles are ingood agreement with Hot Wire Anemometry. Close to the wall, they separate.This separation occurs at different wall distances, increasing with the interrogationwindow size. The separation distances y+ are about 7.7, 13.5 and 22 respectivelyfor window sizes of 16, 32 and 64 pixels. Fig. 19,b shows the mean differencebetween the particle image displacement at the top and at the bottom of the inter-rogation window, as a function of y+. This difference is estimated from the veloc-ity gradient which is computed from the hot wire profiles by centered differenceand multiplied by the window size. The value obtained is comparable to that of Duin paragraph 2.1. Fig. 19,b shows that the separation distance y+ of each analysiscorresponds to Du of the order of 1.6 pixel. This value does not depend on thewindow size and is not far from half the particle image diameter (1.4 pixels).

0

5

10

15

20

25

1 10 100 1000y+

U+ HWA

64 x 64 32 x 32 16 x 16u+ = y+ 0

0.20.40.60.8

11.21.41.61.8

2

1 10 100

y+

Du

(pix

)

16x1632x3264x64

7.7 13.5 22

(a) (b)Fig. 19 Mean velocity profiles (a) and mean velocity gradient (b).

To further investigate this point, Fig. 20.a presents the results from Fig. 5.33 ofRaffel et al (1998). This is the RMS error versus the displacement gradient in thecases of several window sizes and particle densities. These results were obtainedwith synthetic images. As in paragraph 2.1, the RMS error increases with the gra-dient but no universality can be observed. If these data are plotted as a function ofDu instead of the gradient (Fig. 20.b), it appears that the RMS error is not de-pendent on the window size but only on the particle image density. This is in goodagreement with the result obtained above. Even if the results of Fig. 10 are ob-tained with an image density of 10 particles/window, their behavior, also plottedin Fig. 20.b, is in agreement with the Raffel et al results. The RMS error with avalue Du =1.6 pixel is then of the order of 0.2 pixels. The reference length scale isthus clearly the particle image diameter, as indicated by Keane and Adrian (1990).Their criterion based on the interrogation spot size is not verified by these results.

PIV Accuracy 181

3.3 Influence of the third component

In a turbulent flow, there is a third component which is not measured in 2D2C PIVbut can affect the measurement as seen in paragraph 2.1.4. In the flow studiedhere, the third component is random and it can be estimated from the transverseturbulence intensity which is about 1.5 times the wall friction velocity. This givesa value of W of the order of 1.8 pixel. The equivalent light-sheet thickness isabout 32 pixel units. This leads to W/∆z0 = 0.05 which is far below the thresholdof 0.2.

0.01

0.10

1.00

10.00

0.00 0.05 0.10 0.15 0.20

grad u (pix/pix)

RM

S er

ror

16 x 16 32 x 3232 x 32 64 x 6464 x 64

5 part. per IW 20 part. per IW

0.00.20.40.60.81.01.21.41.6

0 2 4 6 8

Du (pix)

RM

S er

ror

16 x 16 32 x 3232 x 32 64 x 6464 x 64 present results

5 part. per IW 20 part. per IW

(a) (b)Fig. 20 RMS error as a function of the velocity gradient (a) and of the maximum displace-ment in the window due to the gradient (b).

3.4 Influence of the window size

As observed in Fig. 18,a the PIV spectrum presents a slope of –2 for the highestwave numbers. This slope is due to the noise (Foucaut and Stanislas, 2002). If theinterrogation window size is increased, the number of particle images used in thecorrelation increases and the noise decreases. Fig. 21,a compares spectra from 16x 16, 32 x 32 and 64 x 64 PIV analysis and from Hot wire anemometry which isconsidered as the most accurate one. The PIV spectra begin at a wave number ofthe order of 160 rad/m which corresponds to 2π divided by the length of field. Asthe field is small, the lowest wave numbers present in the HWA spectrum are notresolved. The highest wave number of the PIV spectra decreases when the win-dow size increases. A cut-off frequency kc can be introduced which corresponds toa sinc function due to the windowing effect (Foucaut and Stanislas (2002)). Thevalue kc is 2.8/X where X is the window size. As can be observed, the cut-offwave numbers with 32 x 32 and 16 x 16 are largely higher than the maximal wavenumber of the hot wire spectrum. Even if the 64 x 64 analysis spectrum is not incomplete agreement with the HWA one, its cut-off wave number is comparable tothe maximum wave number of the HWA spectrum. The difference is due to themeasurement noise of PIV. When the window size decreases this noise level in-creases. Fig. 21,b which presents the turbulent fluctuation profiles with the sameanalysis as Fig. 21,a, is in agreement with the previous result. In this figure, as thewindow size decreases, the fluctuations are in less and less agreement with the

Session 3182

HWA results. This effect which is very sensitive for y+ < 100, is due to the noise.The behavior is the same for the longitudinal u’ and the normal v’ fluctuation pro-files. Fig. 22,a, b and c show samples of fluctuation fields obtained by PIV analy-sis respectively with 16 x 16, 32 x 32 and 64 x 64 windows. In the 16 x 16 case,the turbulent information is blurred by the noise. When the window size increasesthe turbulent structures appear more and more above the noise. This results are incomplete agreement with the spectra and the turbulent fluctuation profiles of Fig.21. It can be concluded that for the study of a turbulent flow, it is important to takeinto account the spectral information to choose the PIV analysis window size.

1E-08

1E-07

1E-06

1E-05

1E-04

1E-03

1E-02

1E+00 1E+01 1E+02 1E+03 1E+04 1E+05

k (rad/m)

E11

(m3 /s

2 )

HWA16x1632x3264x64

kc64kc32

kc16

0

0.5

1

1.5

2

2.5

3

3.5

0 20 40 60 80 100y+

Tur

bule

nt fl

uctu

atio

ns

16 x 1632 x 3264 x 64HWA

u'+

v'+

(a) (b)Fig. 21 Turbulence spectra at y+ = 100 (a) and turbulent fluctuation profiles (b).

100 125 150 175 200x+

0

10

20

30

40

50

60

70

80

y+

1 m/s

100 125 150 175 200

x+

0

10

20

30

40

50

60

70

80

y+

1 m/s

100 125 150 175 200

x+

0

10

20

30

40

50

60

70

80

y+

1 m/s

(a) (b) (c)Fig. 22 Fluctuation velocity field, region of interest analyzed with 16 x 16 (a),

32 x 32 (b) and 64 x 64 (c) window size.

4 Conclusions

PIV is now widely accepted as a reliable method which allows to obtain quantita-tive information about the spatial structure of a flow. To characterize the accuracyof this method, a statistical study was performed by means of synthetic and realimages. This synthetic images based study allows to characterize the bias andRMS errors and to optimize the parameters for recording and analysis. The influ-ence of the recording parameters was studied with a classical analysis by FFT-based cross-correlation with a 3 points Gaussian peak fitting. The effect of particle

PIV Accuracy 183

concentration was discussed. With an optimal value around 0.04 particle per pixel,when the number of particles increases the bias increases and the RMS error de-creases (and vice versa). An optimal diameter around 2 pixels is confirmed for fillratios larger than 75%. A small number of saturated pixel (Sr < 0.035%) presentdoes not increase the errors.

The presence of a background noise modifies and increases the bias error, butfor values significantly larger than those of a contemporary PIV camera. An ac-ceptable variation of the displacement due to a velocity gradient in the window ishalf the image diameter. This result is validated by the study on real images. Theacceptable out of plane displacement is of the order of 20% of the thickness of thelight sheet.

Using these optimal parameters, different algorithms were studied. A peak fit-ting with 3 and 5 points in 1D and 2D was tested. This shows that the 1D/3 pointsGaussian is accurate enough. Iterative algorithms using a sub-pixel shift with dif-ferent interpolators were also tested. The Whittaker interpolator showed the bestresults for a range of particle image diameters between 2 and 4 pixels. The bicubicinterpolator shows optimal performance around 4 pixels. Different correlationmethods were also tested. As expected, the FFT-based correlation gives an accu-racy comparable to the direct computation. Based on this synthetic image study,an analysis was performed with the FFT-based correlation and a 1D Gaussianpeak fitting on 3 points on real images from a turbulent boundary layer flow. Theiterative Whittaker sub-pixel shift was compared to the integer shift. The Whit-taker interpolation allows to remove the peak locking errors. If the dynamic rangeis large enough to resolve the turbulent fluctuations, its effect does not improvethe RMS error and thus the measurement noise. The turbulent fluctuation profileswere not modified when this interpolator was used. The study of the influence ofthe mean velocity gradient led to the same criteria as with the synthetic images.Finally, the present study showed that the sub-pixel shift using an accurate inter-polator like the Whittaker one is not enough to reduce the measurement noise. Aspectral analysis showed that, with interrogation windows of 16x16 pix2 the PIVnoise is of the same order of magnitude as the signal. To improve the accuracy, itis clear that an algorithm with sub-pixel shift is not enough. Nevertheless, the op-timization of experimental parameter on the basis of this study enables an accuratemeasurement of the turbulent statistics. The next step will be to study the im-provement brought by advanced algorithms using image deformation techniques(Scarano 2002). This will be addressed by the PIV challenge 2003 (Stanislas et al,2004).

Acknowledgements

This work has been performed under the EUROPIV2 project. EUROPIV2 (Ajoint program to improve PIV performance for industry and research) is a collabo-ration between LML UMR CNRS 8701, DASSAULT AVIATION, DASA, ITAP,CIRA, DLR, ISL, NLR, ONERA and the universities of Delft, Madrid, Olden-burg, Rome, Rouen (CORIA URA CNRS 230), St Etienne (TSI URA CNRS 842),

Session 3184

Zaragoza. The project is managed by LML UMR CNRS 8701 and is funded bythe European Union within the 5th frame work (Contract N°: G4RD-CT-2000-00190).

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characterization of different piv algorithms using the europiv synthetic image generator and

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