Characterization of Different PIV Algorithms
Using the EUROPIV Synthetic Image Generator
and Real images From a Turbulent Boundary
Layer.
J.-M. FOUCAUT, B. MILIAT, N. PERENNE and M. STANISLAS
Laboratoire de Mécanique de Lille, CNRS UMR 8107, Blv. P. Langevin, F59655
Villeneuve d'Ascq Cedex, jean-marc.foucaut@ec-lille.fr
Abstract
To characterize the PIV accuracy, a statistical study has been performed by means
of synthetic and real images. The synthetic images were generated with the Europiv SIG. They allowed to characterize the systematic and random errors and to
optimize the recording and processing parameter. The influence of the characteristics of particle images like the diameter or the density and of CCD such as the
fill 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 shifting technique with a Whittaker interpolation and a three point Gaussian peakfitting. From a series of real images, the probability density function validated the
benefit of sub-pixel shift. The influence of the velocity gradient and of the out of
plane component have also been investigated. The optimization of experimental
parameter on the basis of this study enables an accurate measurement of the turbulent characteristics.
1 Introduction
Particle Image Velocimetry is a measurement technique with a well established
theory. 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 recording and analysis, are detailed in Willert and Gharib (1991) and Westerweel
(1997). This progress has opened a new route for the assessment of turbulent
flows. The study of turbulence with PIV implies a large field and a good spatial
resolution to measure a large range of scales, and a high accuracy to measure correctly the small scales. It implies also a careful characterization of the accuracy of
the method and of its spatial response as compared to the standard tools used in
this field such as Hot Wire Anemometry and Laser Doppler Velocimetry. Even if
the 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-
164 Session 3
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 on
image 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, have
allowed to characterize the performance of the method (Raffel et al, 1998,
Nogueira et al (2001), Gui and Wereley (2002)). The results of such studies are
very instructive and allow a better understanding of the PIV limitation. The drawback is that each team which has developed a PIV system, has generally also developed a synthetic image generator. The characteristics of the image generation
algorithm are never completely detailed in the papers, which leads to some interrogations about the exploitation of some results (Lourenco, 1988). In the present
contribution, the Europiv SIG software (Synthetic Image Generator) is used to
make a detailed study of the accuracy of different PIV algorithms. This software
has been developed in the frame of the EUROPIV 2 project in order to have a
common tool which allows a true comparison between the PIV algorithms.
To make this study, care was taken to compute systematically the bias error
which evidences the well known peak-locking effect (Nogueira et al, 2001) and
the RMS error which gives an estimation of the measurement noise (see Raffel et
al, 1998). In a first part, the influence of the image recording features such as the
particle image diameter, the image density, the CCD fill ratio (Westerweel, 1998),
the background noise, the out of plane component and the velocity gradient inside
the analysis window are characterized. In the second part, analysis parameters
such as the detection method, the correlation method and the sub-pixel shift are
tested.
Finally, a test was performed on real images of a turbulent flow. These images
are taken from a database on turbulent boundary layers. The motivation is to optimize 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 first
step, two fields of particle image locations were generated. The location of each
r
r
point x p of displacement vector δ x is computed randomly in the domain. The
r
r
r
location of the first position is given by x1 = x p − δ x / 2 . The second one is
r
r
r
r
r
r
x2 = x p + δ x / 2 in order to obtain δ x = x 2 − x1 with a second order accuracy. In
a second step, particle images are computed using the SIG algorithm which is described in detail in the paper by Lecordier et al (2003). This algorithm allows to
provide images with known characteristics. It takes into account: the laser sheet
characteristics, the particle size distribution, the image pattern characteristics, the
projection parameters and the CCD characteristics.
PIV Accuracy 165
The field size is given in 3D physical dimensions and in 2D pixels. As the SIG
can 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 projection on the result. For the present study, the light sheet profile was chosen as a
Gaussian. The particle size distribution was taken uniform. The variation of size
observed in the images is thus only due to the difference in brightness linked to
the location of the particles in the thickness of the light sheet. The CCD is characterized by the fill ratio, which corresponds to the ratio between the active area of
each CCD sensor and the total surface of this sensor. The image pattern was Gaussian. It is integrated on the active area leading to an erf function. In order to be in
agreement with other contributions (e.g. Raffel et al, 1998), the standard deviation
of 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 studied. The size of the images generated was generally taken as 512x512 pix2 for a
field size of 0.1x0.1 m2. This leads to a magnification of 0.2 mm/pix. As explained
in the previous paragraph, the light sheet profile was gaussian, its thickness was 2
mm. The analysis was done with 16x16 interrogation windows. The FFT based
cross-correlation was used. A 3 point Gaussian peak-fitting without shift was also
used in this paragraph. A set of 21 images with displacements from –1 to 1 pixel
with a step of 0.1 pixel was generated for each value of the parameters. The aim is
to predict the behavior of the measurement uncertainty when the image parameters are varied away from an optimal choice. This optimal set of parameters is deduced from the literature.
2.1.1 Optimal parameters
In a first step, a set of images corresponding to an optimal case is computed. This
set 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 Willert
and 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 x
16 pix2.
Bias error (pix)
Fig. 1 shows two examples of image generated by the SIG (a) for a density of
10 particles and (b) for a density of 20 particles in an interrogation window of
16x16 pixels2. Fig. 2 presents the bias and RMS errors in pixels as a function of
the imposed velocity. Both errors are zero for a zero displacement. The bias error
presents 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 bias
and is in agreement with the result of Raffel et al (1998). For a displacement of 0.5
pix, the value of the RMS error is of the order of 0.06 pix, which is also in agreement with the result of Wereley and Gui (2001).
0.02
0.08
0.01
R 0.07
MS0.06
err
0.05
or
0
-1 -0.8 -0.6 -0.4 -0.2 0
-0.01
0.2 0.4 0.6 0.8
Optimal case
1
(pi 0.04
x) 0.03
0.02
Optimal case
0.01
0
-0.02
Exact displacement (pix)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Exact displacement (pix)
1
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 Raffel et al (1998) or Wereley and Gui (2001), the particle concentration modifies the
RMS error. When the number of particles in the interrogation window decreases,
it is well known that the RMS error increases. Keane and Adrian (1992) evidenced
that the number of valid detected vectors increases when the particle image density increases. As shown in fig. 3, when the density increases, the particles used to
PIV Accuracy 167
build up the correlation peak are more numerous and the RMS error decreases as
observed by Raffel et al (1998) or Wereley and Gui (2001). However, when the
density increases the peak locking increases, probably due to the number of images cut by the border of the interrogation window which increases (Nogueira et
al, 2001). This error is very small with an image density of 5.
0.04
0.12
0
-1 -0.8 -0.6 -0.4 -0.2
-0.02
-0.04
0
0.2 0.4 0.6 0.8 1
5 part. images per IW
10 part. images per IW
20 part. images per IW
40 part. images per IW
Exact displacement (pix)
RMS error (pix)
Bias error (pix)
0.1
0.02
0.08
5 part. images per IW
10 part. images per IW
20 part. images per IW
40 part. images per IW
0.06
0.04
0.02
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Exact displacement (pix)
1
Fig. 3 Bias and RMS errors, influence of particle density.
Fig. 3 shows that image densities between 20 and 40 give smaller RMS errors
but 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 densities
higher 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 to
be a good compromise between the bias and RMS errors and will be kept when
the 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 minimizes
the 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 geometry
on the PIV measurement performances. He shows that its influence on errors is
negligible if the particle image diameter is at least two pixels, whatever the fill ratio is. As the fill ratio is generally between 50 and 100%, these values and an intermediate value of 75% were tested. In order to generalize in the case where the
integration on the sensor is not done, a fill ratio of 1% was also studied. Fig. 4 presents the bias and RMS errors for a fill ratio varying from 1% to 100% and a particle image diameter of 2 pixels. As shown by Westerweel (1998), the fill ratio has
roughly no effect on both errors. For diameters smaller than 2 pixels, Westerweel
found a strong influence of the fill ratio on the bias error. He shows that the peak
locking increases with the fill ratio.
Fig. 5 presents the bias and RMS errors for a diameter of 1 pixels and for different fill ratios. It can be observed that the behavior of the bias error versus the
fill ratio is opposite to Westerweel’s results. This error decreases when the fill ratio increases. This difference is probably due to the particle images truncated by
the border of the window which were not taken into account in Westerweel’s theoretical analysis. For a fill ratio of 100%, the peak locking error is nearly cancelled.
168 Session 3
The same behavior is observed for the RMS error: it decreases as the fill ratio increases.
0.35
0.15
0.3
RMS error (pix)
Bias error (pix)
0.1
0.25
0.05
0
-0.05
0.2
0.15
-1 -0.8 -0.6 -0.4 -0.2
0
-0.1
-0.15
Fill ratio 1 %
Fill ratio 50 %
Fill ratio 75 %
Fill ratio 100 %
0.2 0.4 0.6 0.8 1
Fill ratio 1 %
Fill ratio 50 %
Fill ratio 75 %
Fill ratio 100 %
0.1
0.05
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Exact displacement (pix)
Exact displacement (pix)
1
Fig. 4 Bias and RMS errors, influence of fill ratio, particle diameter 2 pixels.
With a fill ratio of 1% the mean bias error is about five times smaller than the
mean RMS error. If the diameter is brought down to 0.5 pixel, when the fill ration
is 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 active area of the CCD sensor, the particles with small diameters disappear. This is
the reason why the errors increases when the diameter and the fill ratio decrease.
0.35
0.15
0.3
0.25
0.05
0
-0.05
-0.1
-0.15
RMS error (pix)
Bias error (pix)
0.1
Fill ratio 1 %
Fill ratio 50 %
Fill ratio 75 %
Fill ratio 100 %
0.2
0.15
-1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
Fill ratio 1 %
Fill ratio 50 %
Fill ratio 75 %
Fill ratio 100 %
Exact displacement (pix)
1
0.1
0.05
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Exact displacement (pix)
1
Fig. 5 Bias and RMS errors, influence of fill ratio, particle diameter 1 pixels.
To evidence this phenomenon, fig. 6 presents the mean values of the bias and
the RMS errors averaged on the displacements from –1 to 1 pixel. The bias error
being symmetrical, the absolute value of this error was averaged. These errors are
given for fill ratios of 1% and 100%. The mean bias error presents optimal diameters of 1 pixel and 2 pixels respectively for the fill ratio of 100% and 1%. Both
curve are nevertheless fairly flat between 1 and 3 pixels. The mean RMS error
shows optimal diameters close to 2 pixels in both cases (1.8 pix and 2.2 pix respectively with a fill ratio of 100% and 1%). As the bias appears significantly
smaller in that region, the best choice of diameter is thus two pixels whatever the
fill ratio is. These results are in agreement with those of Raffel et al (1998) concerning the RMS error. In fig. 6 a, for a fill ratio of 1%, a change of behavior can
be 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%, the
optimal diameter is 2 pixels. As the curve of the RMS errors is relatively flat
around 2 pixels, a diameter between 1 and 3 pixels can be acceptable with the
classical algorithm. For fill ratios nearer to 50%, the particle image size should
move toward 2 to 3 pixels.
Of course this study does not take into account the effect of micro-lenses placed
on the top of the CCD. This effect is not presently modeled in the Europiv SIG but
it would be worth to study it.
0.25
M
ea 0.2
n
R 0.15
M
S
err 0.1
or
(pi 0.05
)
0
0.14
Mean bias error (pix)
0.12
Fill ratio 100%
Fill ratio 1%
0.1
0.08
0.06
0.04
0.02
0
0
1
2
3
Part. im. diameter (pix)
Fill ratio 100%
Fill ratio 1%
0
4
1
2
3
Part. im. diameter (pix)
4
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 RMS
values of the CCD noise (sigma) different from zero, the ratio of sigma over the
noise level was kept constant. Fig. 7 presents the CCD noise effect on the bias and
RMS errors. The case sigma = 0 corresponds to the optimal case of paragraph
2.1.1. The corresponding background level is 2% of the scale, i.e. 5 grey level for
a quantization on 8 bits. The value sigma = 1% corresponds roughly to the noise
of a PCO Sensicam camera. The value sigma = 2 % is close to the characteristics
of a Pulnix TM9701 camera.
Level 2%, sigma 0
Level 2%, sigma 1%
Level 4%, sigma 2%
Level 6%, sigma 3%
Level 8%, sigma 4%
Level 10%, sigma 5%
0.03
Bias error (pix)
0.02
0.1
0.08
0.06
0.01
0
-0.01 -1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
Level 2%, sigma 0
Level 2%, sigma 1%
Level 4%, sigma 2%
Level 6%, sigma 3%
Level 8%, sigma 4%
Level 10%, sigma 5%
0.04
0.02
-0.02
-0.03
-0.04
RMS error (pix)
0.04
0
Exact displacement (pix)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Exact displacement (pix)
Fig. 7 Mean bias and RMS errors, influence of CCD noise.
1
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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 by
the 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 observe strong modification of the RMS error for 5% and 10 % of noise. These results show that the background noise of the standard PIV camera does not affect
presently the measurement accuracy. It must be mentioned that this test does not
take into account the noise due to the spurious light in an actual experiment (background reflection on a model or a wall).
In order to be sure that the complete range of grey level of a camera is used, a
small number of pixel can be saturated. The influence of these saturated pixels is
presented in Fig. 8. The saturation parameter of the SIG varies from 12.5% to
50%. Table 1 gives the number Ns of saturated pixels in an image and the corresponding percentage Sr of saturated pixel. For Sr less than 0.05% both errors are
not affected by the presence of saturated pixels. Gui and Wereley (2002) did the
same comparison with strongly saturated particle images. They observe no influence on peak locking and a RMS error 4 times larger. The optimal case is obtained
with a saturation parameter of 25% which gives about 100 saturated pixels in a
512 x 512 image (i.e. Sr = 0.035%). For all the tests presented in this study, the
saturation parameter was adjusted to obtain Sr of the order of 0.035%.
Bias error (pix)
0.02
0.01
0.07
0.06
0
-0.01
saturation 12,5%
saturation 19%
saturation 25%
saturation 37,5%
saturation 50%
0.08
saturation 12,5%
saturation 19%
saturation 25%
saturation 37,5%
saturation 50%
-1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
RMS error (pix)
0.03
0.05
0.04
0.03
0.02
0.01
-0.02
0
-0.03
-1
Exact displacement (pix)
-0.8
-0.6
-0.4 -0.2
0
0.2
0.4
Exact displacement (pix)
0.6
0.8
1
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 parameter of the SIG.
Saturation
12.5
19
25
37.5
50
Ns
1
15
90
1400
4200
Sr%
0.0004%
0.0057%
0.034%
0.53%
1.60%
PIV Accuracy 171
2.1.6 Influence of the third component
Two-dimensional two-component (2D2C) PIV allows to measure the in-plane
components. In the case of experiments and particularly in turbulent flows, there is
an out of plane particle motion which can disturb the PIV measurement. In the
present study, the third component is considered as a Gaussian random distribution with 3σ ≈ W . The value of W varies from zero in the optimal case, to a
maximum of 5 pixels. In this case, some particles come in and out of the laser
sheet between the two exposures, leading to isolated particle images. To improve
the convergence of the results, the field size has to be increased from 512x512 to
768x768 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 velocity
component
W (pixel)
Percent of lost
W/∆z0
particle pairs
1
2%
0.1
2
5%
0.2
3
7%
0.3
4
10%
0.4
5
12%
0.5
Bias error (pix)
0.02
0.01
W
W
W
W
W
W
=
=
=
=
=
=
0.12
0 pix
1 pix
2 pix
3 pix
4 pix
5 pix
0.1
0.08
0.06
0
-0.01
-1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
W
W
W
W
W
W
0.04
0.02
-0.02
-0.03
RMS error (pix)
0.03
0
Exact displacement (pix)
=
=
=
=
=
=
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Exact displacement (pix)
0 pix
1 pix
2 pix
3 pix
4 pix
5 pix
1
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 a
light sheet thickness ∆z0 = 10 pixels, this value is in the middle of the present test
range for W. The bias error presented in fig. 9 is weakly affected by the out of
plane component. On the contrary, the RMS error is strongly influenced by isolated particles. This error increases with the third component whatever the displacement is. Up to W = 2 pixels the RMS stays within reasonable bounds. This
limit seems close to this of Keane and Adrian (1990), although the present analysis is done with cross-correlation. In a turbulent flow, the value of W can be estimated, using the out of plane turbulent intensity, as 2 w '2 M∆t where M is the
172 Session 3
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 be
done in a region with a strong velocity gradient. As the PIV method is based on
the hypothesis that the velocity is uniform inside the interrogation windows, if a
gradient 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 find
local distributions of particles leading to a displacement lower or higher than the
mean looked for. The second scenario appears in regions of larger displacement
where the probability of isolated particles is higher, leading to a bias toward low
velocity. These two scenarios contribute mostly to the RMS error. To test the gradient effect, a series of 16 images of 4096 x 21 pix2 was generated. The results of
these images was averaged in order to obtain a good convergence of statistical errors. Each image contained particle images corresponding to a range of displacement from –1 pix to 1 pix as previously, but on which a constant gradient was
added. The mean displacement of the gradient was zero and the difference between the top and the bottom of a 16x16 pix2 window was Du. The value Du was
varied from 0 (optimal case) to 3 pixels. For a density higher than 15 particles per
interrogation area Keane and Adrian (1992) showed that the PIV measurement is
not affected by the gradient if Du is smaller than the particle image diameter (i.e. 2
pixels) and does not exceed 3% of the interrogation spot. The results are presented
in fig. 10. As can be seen, the gradient affects rapidly the RMS error. With Du = 1
pixel the noise is already doubled and the RMS error for a zero displacement is
not zero any more. The results of Raffel et al (1998) give the same order of magnitude. However for Du = 2 the RMS error obtained by Raffel et al is 0.23 instead
of 0.3 in the present paper. This is probably due to the difference in the image
generation. When Du is smaller than 1, the bias is not to much affected by the gradient. Above this value, a large anti-peak-locking error appears. This limiting
value is in good agreement with the study of Keane and Adrian (1990) who predict a maximum Du of half of the particle image diameter (2 pixels here).
0.6
0.03
0.5
Bias error (pix)
0.02
0.4
0.01
0.3
0
-0.01 -1 -0.8 -0.6 -0.4 -0.2
-0.02
-0.03
-0.04
RMS error (pix)
0.04
0
0.2 0.4 0.6 0.8
Du =
Du =
Du =
Du =
Du =
Exact displacement (pix)
0
0.5 pix
1 pix
2 pix
3 pix
1
0.2
Du =
Du =
Du =
Du =
Du =
0
0.5 pix
1 pix
2 pix
3 pix
Raffel et al, Du = 2 pix
Raffel et al, Du = 1 pix
0.1
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Exact displacement (pix)
Fig. 10 Mean bias and RMS errors, influence of the velocity gradient.
1
PIV Accuracy 173
A test with a density of 20 particles per interrogation window leads to a reduction of the RMS and an increase of the bias in the same proportion as without gradient (fig. 3). The influence of the window size on this results will be discussed in
the part concerning the real images.
2.2 Analysis parameters
The tests performed up to now have allowed to characterize the influence of the
main image parameters. This study was done with a standard algorithm. Several
improvement have been published in the last five years which are supposed to enhance the accuracy of the PIV measurement (Scarano and Reithmuller, 2000). To
obtain the best from the PIV measurement in turbulent flows, it was of interest to
study the effect of at least some of these algorithms. This is the aim of the present
study. The methods using window deformation have been put aside on purpose, as
they are time consuming. The study is thus focussed on the peak detection algorithm, 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 in
2.1.1.
2.2.1 Peak detection algorithm
The peak detection algorithm used here is the standard Gaussian fitting. Westerweel (1998) has largely shown the advantage of this method. It is generally applied twice on 3 points (Raffel et al, 1998). Ronneberger et al (1998) observed a
lack of accuracy for the smallest image diameter. They propose a method based on
a 2D fitting on 25 points using a non-linear least square interpolation (LevenbergMarquardt). Both methods have been tested in the present study. Fig. 11 presents
the bias and RMS errors as a function of the detection method. The 3 points 1D
peak fitting shows the smallest peak locking error. The same fitting with 5 points
is the best in terms of RMS error. These 1D methods give the best accuracy for a
displacement parallel to the window side.
Bias error (pix)
0.02
0.01
0
-0.01 -1 -0.8 -0.6 -0.4 -0.2
-0.02
-0.03
-0.04
-0.05
0
0.2 0.4 0.6 0.8
1D, 3 pixels
1D, 5 pixels
2D, 3 pixels
2D, 5 pixels
Exact displacement (pix)
1
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
RMS error (pix)
0.05
0.04
0.03
1D, 3 pixels
1D, 5 pixels
2D, 3 pixels
2D, 5 pixels
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Exact displacement (pix)
1
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 5x5
pix2 is better but only for the RMS. The same test was performed with particles of
174 Session 3
4 pixels in diameter. In that case, the methods using 5 points are better for both errors. For particles of 1 pixel, no improvement was observed with the 5 points fitting. Ronneberger et al (1998) recommend to use a Fisher transform which has not
been 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 bias
and RMS errors generally tend toward zero. This is why a shift of the second
analysis windows can be used to improve the accuracy. The first idea was to use
an 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 interrogation
window in each image. Wereley and Gui used a bilinear interpolation. Following
Lecordier et al (1999) it is better to increase the order of the interpolator, a bicubic
one is also tested here. Scarano and Reithmuller (2000) preferred a Whittaker interpolation. They explained that the later is better because it respects the sampling
theorem. Fig. 12 shows a comparison of the errors obtained with an integer shift
and bilinear, bicubic and Whittaker sub-pixel shifts applied on the set of images
corresponding to the optimal case. It can be observed that the iterative process
using an integer shift does not give an improvement as compared to Fig. 1. The
bilinear interpolation increases the bias error and leads to an anti-peak locking
with a period doubled due to the use here of a symmetrical shifting technique
(both windows are shifted symmetrically). This interpolator does not decrease the
noise. The bicubic shift reduces the RMS but not the bias as compared to the integer shift method. Only the Whittaker interpolation decreases significantly both errors.
Integer shift
Bilinear subpixel shift
Whittaker subpixel shift
Bicubic subpixel shift
Bias error (pix)
0.04
0.02
0
-0.02 -1 -0.8 -0.6 -0.4 -0.2
-0.04
-0.06
-0.08
-0.1
0
0.2 0.4 0.6 0.8
Exact displacement (pix)
1
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
Integer shift
Bilinear subpixel shift
Whittaker subpixel shift
Bicubic subpixel shift
RMS error (pix)
0.1
0.08
0.06
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Exact displacement (pix)
1
Fig. 12 Bias and RMS errors, influence of symmetrical window shifting algorithm.
These results are not in agreement with those of Wereley and Gui (2002) who
found a large decrease of both errors with a bilinear shift. This is due to the fact
that their particle image diameters are of the order of 7 pixels with the same definition as in the present study. It is clear that the optimal diameter obtained in paragraph 2.1 is not the same when a sub-pixel shift is applied. Fig. 13 shows the
comparison 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 errors
are given as a function of the diameter from 1 to 4.5 pixels. As far as the bias error
is concerned, the bilinear interpolator is the less accurate for diameters smaller
than 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 two
times higher. The best interpolator is the Whittaker one, particularly for a range of
diameter between 2 and 4 pixels. For diameter smaller than about 1.5 pixels, it is
better not to interpolate and to use the integer shift. The RMS diagram leads sensibly to the same conclusion. The bicubic and Whittaker interpolations show almost the same behavior. No gain is observable for a diameter of 1 pixel whatever
the method used. The conclusion is that the Whittaker interpolation is the best for
particle diameters from 1.5 to 4 pixels. For diameters smaller than 1.5 pixel, integer shift is recommended. According to Wereley and Gui (2002), for diameters
higher than 4.5 pixels, bilinear sub-pixel shift could probably be a good compromise.
Integer shift
Bilinear subpixel shift
Whittaker subpixel shift
Bicubic subpixel shift
0.05
Integer shift
Bilinear subpixel shift
Whittaker subpixel shift
Bicubic subpixel shift
0.16
Mean RMS error (pix)
Mean bias error (pix)
0.06
0.04
0.03
0.02
0.01
0
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
1
2
3
Part. im. diameter (pix)
4
5
0
1
2
3
Part. im. diameter (pix)
4
5
0.08
0.06
Bias error (pix)
0.04
Integer shift
Bilinear subpixel shift
Whittaker subpixel shift
Bicubic subpixel shift
0.02
0
-0.02 -1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
-0.04
-0.06
-0.08
Exact displacement (pix)
1
RMS error (pix)
Fig. 13 Mean bias and RMS errors versus image particle diameter, influence of window
shifting algorithm.
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
Integer shift
Bilinear subpixel shift
Whittaker subpixel shift
Bicubic subpixel shift
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Exact displacement (pix)
1
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 best
accuracy (second order). However, it implies two interpolations and gives a strong
modification of peak locking when bilinear or bicubic shift is used. Fig. 14 presents the same result as Fig. 12 but with a non-symmetrical shift (as in Wereley and
Gui, 2002). The first window is shifted back by the integer half value of the esti-
176 Session 3
mated displacement and the second window is shifted forward from the first window by the total displacement using sub-pixel interpolation. An improvement is
observed mainly for the bilinear and bicubic method.
The anti-peak-locking is always obtained but the period is the same as standard
peak-locking (1 pixel). The Whittaker method presents again the best accuracy. It
cancels practically the bias error and gives the minimal RMS error. This method
of non-symmetrical shift gives an error on the vector location which is at worst of
0.5 pixel. This error is not evidenced in the case of a uniform displacement but, in
the 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 the
Whittaker 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, in
order to minimize the computational time. Pust (2000) has also shown some improvement using direct correlation computations limited or not.
If the location of the peak is almost known in the correlogram, the correlation
of two windows of size N x N pix2 can be computed only in an region of interest
of size M x M pix2 (Roth and Katz, 2001) following equation:
Φ fg ( m , n ) =
∑ ∑ f ( i , j ) × g( i + m , j + n )
i =0 ,N j =0 ,N
with m+i < N+M and n+j < N+M. This amounts to correlate a N x N pix2 window
with a N+M x N+M pix2 window. The correlogram is then of M x M pix2 size. The
computational time is increasing with M. This method will be called DCC (Direct
Cross-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 by
the border) of the first window with all particles of the second one which tends to
increase the noise (Nogueira et al (2001)). Its drawback is that the correlogram is
biased by a pyramidal function (Raffel et al (1998)). This method will be call
LDCC (Limited Direct Cross-Correlation). According to Pust (2000), an improvement is obtained by normalizing the correlation :
Φ fg ( m , n ) =
∑ ∑ ( f ( i , j ) − f ) × ( g( i + m, j + n ) − g( m,n ))
i =0 ,N j =0 ,N
∑ ∑ ( f ( i , j ) − f ) × ∑ ∑ ( g( i + m, j + n ) − g( m,n ))
2
i =0 ,N j =0 ,N
2
i =0 ,N j =0 ,N
f and g are the mean gray level respectively of f(i,j) and g(i,j). This normalization allows to obtain two additional methods called NDCC (Normalized Direct
Cross-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 the
FFT based Cross-Correlation. The FFT based CC seems less efficient than direct
computation probably due to the periodicity of the signal imposed by FFT when
no padding is used. The LDCC gives better results than DCC. As far as the RMS
values are concerned, if the displacement is smaller than 0.5 the LDCC is better
than DCC and if displacement is higher than 0.5 it is the opposite. When the displacement goes to zero, the RMS error obtained by DCC does not go to zero. This
is due to the effect of truncated particles. The normalization does not modify the
bias error and decreases slightly the RMS value.
Bias error (pix)
0.01
0
-1 -0.8 -0.6 -0.4 -0.2
0
1
FFT based CC
DCC
NDCC
LDCC
LNDCC
-0.01
-0.02
0.2 0.4 0.6 0.8
RMS error (pix)
0.02
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
FFT based CC
DCC
NDCC
LDCC
LNDCC
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Exact displacement (pix)
Exact displacement (pix)
1
Fig. 15 Bias and RMS errors, influence of correlation method.
0.02
FFT based CC
DCC
NDCC
LDCC
LNDCC
Bias error (pix)
0.01
0
-1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
-0.01
-0.02
Exact displacement (pix)
1
RMS error (pix)
The same test was performed using the Whittaker sub-pixel shift. The results
are presented in Fig. 16. As far as the bias error is concerned, all the methods give
almost the same result, the direct computation does not present any improvement.
The RMS is almost the same for the limited direct computation with and without
normalization and the FFT cross-correlation. The direct correlation shows a RMS
error 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.
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
FFT based CC
DCC
NDCC
LDCC
LNDCC
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Exact displacement (pix)
1
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 which
should be adopted for real images of a turbulent flow is a FFT-based cross-
178 Session 3
correlation, using iterative sub-pixel shift based on the Whittaker interpolation. If
the particle image diameters have been adjusted around 2 pixels, the 1D Gaussian
peak-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. An
experiment has been carried out in a boundary layer wind tunnel. The Reynolds
number, based on momentum thickness, is 7800. The boundary layer thickness is
about 0.3 m and the free stream velocity is 3 m/s. The PIV records were obtained
in a plane normal to the wall and parallel to the flow. The x axis is horizontal and
the y axis vertical. The camera used was a PCO Sensicam. Its CCD has a size of
1280 x 1024 pixels2 of 6.7 x 6.7 µm2 each. The images were recorded at a rate of 8
frame-pairs per second by means of the LAVISION software. A 200 mm focal
lens was used with f# = 11. The magnification was 0.21 which gives in wall units a
size of 300 x 230. The whole flow was seeded with Poly-Ethylene Glycol particles. Their diameter is about 1 µm. The light sheet was generated by a BMI pulsed
YAG laser with 2 x 250 mJ of energy at 12 Hz. The sheet thickness in the field of
view was about 1 mm corresponding to 8 wall units and ∆z0 = 32 pixels. Following Adrian (1991), the depth of field was about 7 mm which avoids any problem
to focus the particle images. In this configuration, the Airy disk diameter is of the
order 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 particles per pixel. This value is a little bit smaller than the optimal value which should
be around 0.04. The analysis was made by the FFT based cross-correlation method
with integer shift and sub-pixel shift using Whittaker interpolation with a 3 point
Gaussian peak fitting in each direction. An overlapping of 50% was used. The
delay between both laser pulses was chosen to obtain a mean dynamic range of the
order of 8 pixels (150 µs). With this range the turbulence intensity was of 1.6 pixels. 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 has
been computed from the difference between the displacement in pixel and its integer value. This PDF is computed from the vectors of a single velocity field. The
analysis was done with 16 X 16 interrogation windows, leading to about 30 000
vectors in the field. The results are presented in Fig. 17,a which compares the iterative PIV analysis with an integer shift and a Whittaker sub-pixel shift. As can
be seen the integer shift analysis builds up a strong peak-locking which is canceled by the Whittaker interpolation. Fig. 17,b shows a comparison of PDF of the
u component of the velocity fluctuation. This PDF is computed from each vector
line located at 100 wall units of the 200 velocity field. The PIV analysis is performed 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 at
this wall distance.
8%
0.6
7%
0.5
6%
PDF(u)
5%
PDF
Integer shift
Whittaker
HWA
0.4
4%
3%
0.3
0.2
2%
integer shift
Whittaker subpixel shift
1%
0.1
0
0%
0
0.2
0.4
0.6
0.8
Decimal part of the displacement (pix)
-4
1
-3
-2
-1
0
u/u'
1
2
3
4
Fig. 17 PDF of the decimal part of displacement in pixel (a) and of the velocity in
m/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 very
close to the hot wire results. To study the effect on the noise, spectra are computed
from the same data (200 lines located at y+ = 100). The comparison presented in
Fig. 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 a
random noise which level is negligible as compared to the signal in the Fourier
space. It can be observed that, at the smallest wave numbers, the spectrum has a
slope of –5/3, which is a standard turbulence one. At the highest wave numbers
the slope is –2 which is typically linked to the noise (Foucaut and Stanislas, 2002).
3.5
Slope -5/3
Slope -2
1E-06
Integer shift
Whittaker
1E-07
100
1000
k (rad/m)
(a)
10000
Turbulent fluctuations
1E-05
3
2
E11 (m /s )
1E-04
3
u'+
2.5
2
v'+
1.5
Whittaker
Integer shift
1
0.5
0
0
50
100
y+
150
200
(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 are
clearly not influenced by the peak-locking. In conclusion, the peak-locking effect
strongly occurs when an integer shift iterative analysis method is applied. It is detectable in the PDF function and can be removed with a sub-pixel shift based on
the Whittaker interpolation. Nevertheless, with the dynamic range used in the present experiment, it does not modify the mean velocity and the turbulent fluctuation
profiles.
180 Session 3
3.2 Influence of a velocity gradient
In a boundary layer, when the wall distance goes to zero, the velocity profile
shows a strong gradient. Fig. 19,a shows the mean velocity profiles computed
from the 200 fields compared with hot wire anemometry. In this figure, the PIV
analysis is done with 64 x 64, 32 x 32 and 16 x 16 interrogation windows which
allow to obtain respectively about 8000, 16000 and 32000 samples to average. A
standard overlapping of 50% is used. In the log region, the PIV profiles are in
good agreement with Hot Wire Anemometry. Close to the wall, they separate.
This separation occurs at different wall distances, increasing with the interrogation
window size. The separation distances y+ are about 7.7, 13.5 and 22 respectively
for window sizes of 16, 32 and 64 pixels. Fig. 19,b shows the mean difference
between the particle image displacement at the top and at the bottom of the interrogation window, as a function of y+. This difference is estimated from the velocity gradient which is computed from the hot wire profiles by centered difference
and multiplied by the window size. The value obtained is comparable to that of Du
in paragraph 2.1. Fig. 19,b shows that the separation distance y+ of each analysis
corresponds to Du of the order of 1.6 pixel. This value does not depend on the
window size and is not far from half the particle image diameter (1.4 pixels).
25
15
U+
HWA
64 x 64
32 x 32
16 x 16
u+ = y+
10
5
0
1
10
100
y+
(a)
1000
Du (pix)
20
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
16x16
32x32
64x64
1
7.7 13.5 22
10
100
y+
(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 of
Raffel et al (1998). This is the RMS error versus the displacement gradient in the
cases of several window sizes and particle densities. These results were obtained
with synthetic images. As in paragraph 2.1, the RMS error increases with the gradient but no universality can be observed. If these data are plotted as a function of
Du instead of the gradient (Fig. 20.b), it appears that the RMS error is not dependent on the window size but only on the particle image density. This is in good
agreement with the result obtained above. Even if the results of Fig. 10 are obtained with an image density of 10 particles/window, their behavior, also plotted
in Fig. 20.b, is in agreement with the Raffel et al results. The RMS error with a
value Du =1.6 pixel is then of the order of 0.2 pixels. The reference length scale is
thus 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 PIV
but can affect the measurement as seen in paragraph 2.1.4. In the flow studied
here, the third component is random and it can be estimated from the transverse
turbulence intensity which is about 1.5 times the wall friction velocity. This gives
a value of W of the order of 1.8 pixel. The equivalent light-sheet thickness is
about 32 pixel units. This leads to W/∆z0 = 0.05 which is far below the threshold
of 0.2.
1.00
5 part. per IW 20 part. per IW
16 x 16
32 x 32
32 x 32
64 x 64
64 x 64
0.10
0.01
0.00
0.05
0.10
0.15
0.20
grad u (pix/pix)
(a)
RMS error
RMS error
10.00
1.6 5 part. per IW
16 x 16
1.4
32 x 32
1.2
64 x 64
1.0
20 part. per IW
32 x 32
64 x 64
present results
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
Du (pix)
(b)
Fig. 20 RMS error as a function of the velocity gradient (a) and of the maximum displacement 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 highest
wave numbers. This slope is due to the noise (Foucaut and Stanislas, 2002). If the
interrogation window size is increased, the number of particle images used in the
correlation increases and the noise decreases. Fig. 21,a compares spectra from 16
x 16, 32 x 32 and 64 x 64 PIV analysis and from Hot wire anemometry which is
considered as the most accurate one. The PIV spectra begin at a wave number of
the order of 160 rad/m which corresponds to 2π divided by the length of field. As
the field is small, the lowest wave numbers present in the HWA spectrum are not
resolved. The highest wave number of the PIV spectra decreases when the window size increases. A cut-off frequency kc can be introduced which corresponds to
a sinc function due to the windowing effect (Foucaut and Stanislas (2002)). The
value kc is 2.8/X where X is the window size. As can be observed, the cut-off
wave numbers with 32 x 32 and 16 x 16 are largely higher than the maximal wave
number of the hot wire spectrum. Even if the 64 x 64 analysis spectrum is not in
complete agreement with the HWA one, its cut-off wave number is comparable to
the maximum wave number of the HWA spectrum. The difference is due to the
measurement noise of PIV. When the window size decreases this noise level increases. Fig. 21,b which presents the turbulent fluctuation profiles with the same
analysis as Fig. 21,a, is in agreement with the previous result. In this figure, as the
window size decreases, the fluctuations are in less and less agreement with the
182 Session 3
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 profiles. Fig. 22,a, b and c show samples of fluctuation fields obtained by PIV analysis 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 increases
the turbulent structures appear more and more above the noise. This results are in
complete 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 take
into account the spectral information to choose the PIV analysis window size.
1E-02
3.5
3
2
E11 (m /s )
1E-04
1E-05
Turbulent fluctuations
HWA
16x16
32x32
64x64
1E-03
1E-06
1E-07
kc64
1E-08
1E+00
1E+01
1E+02
kc16
kc32
1E+03 1E+04
3
u'+
2.5
2
v'+
1.5
1
16 x 16
32 x 32
64 x 64
HWA
0.5
0
1E+05
0
k (rad/m)
20
40
(a)
y+
60
80
100
(b)
Fig. 21 Turbulence spectra at y+ = 100 (a) and turbulent fluctuation profiles (b).
1 m/s
1 m/s
80
70
70
70
60
60
60
50
50
50
40
y+
80
y+
y+
1 m/s
80
40
40
30
30
30
20
20
20
10
10
0
100
125
150
x+
175
200
0
100
10
125
150
x+
175
200
0
100
125
150
x+
175
200
(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 quantitative information about the spatial structure of a flow. To characterize the accuracy
of this method, a statistical study was performed by means of synthetic and real
images. This synthetic images based study allows to characterize the bias and
RMS errors and to optimize the parameters for recording and analysis. The influence of the recording parameters was studied with a classical analysis by FFTbased 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 decreases (and vice versa). An optimal diameter around 2 pixels is confirmed for fill
ratios larger than 75%. A small number of saturated pixel (Sr < 0.035%) present
does not increase the errors.
The presence of a background noise modifies and increases the bias error, but
for values significantly larger than those of a contemporary PIV camera. An acceptable variation of the displacement due to a velocity gradient in the window is
half the image diameter. This result is validated by the study on real images. The
acceptable out of plane displacement is of the order of 20% of the thickness of the
light sheet.
Using these optimal parameters, different algorithms were studied. A peak fitting with 3 and 5 points in 1D and 2D was tested. This shows that the 1D/3 points
Gaussian is accurate enough. Iterative algorithms using a sub-pixel shift with different interpolators were also tested. The Whittaker interpolator showed the best
results for a range of particle image diameters between 2 and 4 pixels. The bicubic
interpolator shows optimal performance around 4 pixels. Different correlation
methods were also tested. As expected, the FFT-based correlation gives an accuracy comparable to the direct computation. Based on this synthetic image study,
an analysis was performed with the FFT-based correlation and a 1D Gaussian
peak fitting on 3 points on real images from a turbulent boundary layer flow. The
iterative Whittaker sub-pixel shift was compared to the integer shift. The Whittaker interpolation allows to remove the peak locking errors. If the dynamic range
is large enough to resolve the turbulent fluctuations, its effect does not improve
the RMS error and thus the measurement noise. The turbulent fluctuation profiles
were not modified when this interpolator was used. The study of the influence of
the 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 interpolator like the Whittaker one is not enough to reduce the measurement noise. A
spectral analysis showed that, with interrogation windows of 16x16 pix2 the PIV
noise is of the same order of magnitude as the signal. To improve the accuracy, it
is clear that an algorithm with sub-pixel shift is not enough. Nevertheless, the optimization of experimental parameter on the basis of this study enables an accurate
measurement of the turbulent statistics. The next step will be to study the improvement 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 (A
joint program to improve PIV performance for industry and research) is a collaboration between LML UMR CNRS 8701, DASSAULT AVIATION, DASA, ITAP,
CIRA, DLR, ISL, NLR, ONERA and the universities of Delft, Madrid, Oldenburg, Rome, Rouen (CORIA URA CNRS 230), St Etienne (TSI URA CNRS 842),
184 Session 3
Zaragoza. The project is managed by LML UMR CNRS 8701 and is funded by
the European Union within the 5th frame work (Contract N°: G4RD-CT-200000190).
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