4th International Symposium on Particle Image Velocimetry
Göttingen, Germany, September 17-19, 2001
PIV’01 Paper 1013
New source of peak locking related to the window size: analysis
and its removal
J. Nogueira, A. Lecuona and P. A. Rodríguez
Abstract Particle Image Velocimetry data can be affected by the peak-locking effect. Two sources of this
undesirable effect have been documented. They are related to the pixel extension of the particle images and to the
peak-fitting algorithm used in the correlation domain. In this work, a third source, independent to these two ones is
studied. This source is related to the truncation of particles by the interrogation window borders. Its effect is
especially important in those Super-Resolution systems that are based on iteratively reducing the size of the
interrogation window. This is because the smaller the interrogation is, the larger becomes the fraction of particles
truncated by the borders.
The paper is devoted to a discussion on the reasons why the truncation of particles in the border of the window
produces peak locking. The scope includes Fast Fourier Transform algorithms as well as Direct Correlation
algorithms. Furthermore, the ways to avoid this source of peak locking are offered as a straight consequence. This
leads to specific proposals to modify the algorithms in the last steps of Multigrid Super-Resolution PIV systems in
order to prevent the peak-locking effect. The performances of the proposed solutions are verified using both
synthetic and real PIV images.
1
Introduction
Digital PIV systems based on local correlation can take advantage of a discrete offset of the interrogation window to
increase the measurement range and the signal to noise ratio (Westerweel et al. 1997). Consequently, most of the
advanced PIV algorithms include this feature in one way or another. This offset is defined by the integer part of the
local average displacement, aside errors. This leads to a situation where the residual displacement to be measured
between the two interrogation windows is included in the ± 0.5 pixel range. This will be the scenario in which this
work was developed, this means dealing with subpixel displacements. The error that introduces a systematic
deviation towards zero displacement will yield peak locking.
The magnitude of peak locking in the results can be easily ascertained by inspection of the velocity histogram
from a wide region of the image. It appears as a spurious concentration of measurements corresponding to integer
number of pixel displacements. Recently two sources of this peak-locking effect have been identified and
characterised (Prasad et al. 1992, Lourenco and Krothapalli 1995) and (Westerweel 1998). In this paper, a third one
is described. This new source has been identified in recent research of the authors (Nogueira, Lecuona and
Rodríguez 2001). Here a detailed explanation of this source will be presented along with the ways to avoid it in Fast
Fourier Transform (FFT) based PIV and Direct Correlation (DCPIV) systems.
The main difference between these two systems lays on the calculation of the correlation coefficients. The price
to pay for working in the frequency domain is the periodicity that the FFT algorithm assumes in the spatial domain.
This introduces spurious contributions in the entire field of coefficients excluding the central one. DCPIV avoids
this by directly calculating in the spatial domain, but at the expense of an increase in computing time (Utami et al.
1991 and Huang et al. 1993; among others).
This work focuses on cross-correlation PIV systems, although extension to autocorrelation systems is
straightforward.
J. Nogueira, A. Lecuona P. A. Rodríguez. Universidad Carlos III de Madrid, Madrid, Spain
Correspondence to:
Prof. Antonio Lecuona, Departamento de Ingeniería Mecánica. Universidad Carlos III de Madrid
C/ Butarque 15,
28911, Leganés. Madrid, Spain, E-mail: lecuona@ing.uc3m.es
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PIV’01 Paper 1013
2
Preliminary concepts
In order to lay down a nomenclature and to introduce the reasoning in a one-dimension space, which will be
extensively used along the paper, some preliminary concepts will be presented in this section.
A remark about the nomenclature in this work is that the diameter for any gaussian profile is associated with e-2
times the central value (after Prasad et al. 1992; among others).
2.1
DC and FFT algorithms
As it is well known, correlation PIV systems compute the correlation coefficient for each possible displacement,
indicating the largest value the displacement between both interrogation windows. Generally, this calculation is
performed according to the following expression.
F/2
Clm =
∑ f ( ξ ,η ) ⋅ g( ξ + l ,η + m )
ξ ,η =− F / 2
F/2
∑
(1)
F/2
∑ g ( ξ + l ,η + m )
f ( ξ ,η )
2
ξ ,η =− F / 2
2
ξ ,η=− F / 2
f and g are the grey values of the two images to correlate, a and b respectively. F is the side length of the
interrogation window, ξ and η are the orthogonal coordinates from the window center and l and m are the
displacement associated to each calculation of the correlation coefficient.
In what follows visual sketches of the DC and FFT approach to this calculation are given.
DC algorithm: to compute expression (1) an interrogation window is selected in image a and it is locally
correlated moving it along image b. This procedure is sketched in figure 1 in a 1D version, using continuous grey
level profiles, which do not reduce generality at all. This translates into a continuous 1D correlation function.
Window
in
image a
DC calculation of the correlation coefficient as a function of the displacement
Correlation coefficient
C
Grey levels in image a
Displacement
Grey levels in image b
Fig. 1. Sketch of a DC PIV interrogation.
FFT Correlation: the previous computation is time consuming. Consequently, most of recent PIV systems resort
to the Fast Fourier Transform to calculate the correlation coefficients. The price to pay for the computation load
reduction is the periodicity that the system assumes in the spatial domain. To compute expression (1) through the
FFT algorithm, two interrogation windows have to be selected (one in image a and one in image b) and are
correlated moving one respect to the other. This procedure is sketched in figure 2.
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PIV’01 Paper 1013
Window
in
image a
FFT calculation of the correlation coefficient as a function of the displacement
Correlation coefficient C
Grey levels in image a
Window
in
image b
Displacement
Grey levels in image b
Fig. 2. Sketch of an FFT Correlation PIV interrogation.
2.2
Measurement errors due to particle truncation in the case of null displacement between images
Considering the case of null displacement and no noise, images a and b are identical. This could lead to think that
error in this measurement would be zero. A close inspection of both procedures shows that this is not the case, at
least for DC. The presence of particles truncated at the borders of the interrogation window and the limited
resolution of the images precludes an exact result.
DC algorithm: In this case, the behavior of the correlation computing is sketched in figure 3.
Window
in
image a
Grey levels in image a
Correlation
coefficient
Subpixel
interpolation
Error
Grey levels in image b
−∆s
0
∆s
Displacement
Fig. 3. Sketch of a DC PIV interrogation showing error at null particle displacement. ∆s corresponds to the smallest displacement
used for the calculation of the correlation coefficients (typically one pixel) thus giving a discrete value for the correlation
coefficient C.
At the correct displacement the correlation coefficient is 1, as can be observed, although at both sides
asymmetrical values are obtained. This introduces an error in any subpixel-accuracy algorithm for peak fitting,
giving displacements different to the correct one. This error effect at zero displacement can be observed in the
results of the works of Huang et al. (1997) and Fincham and Spedding (1997). In this last work, an asymmetrical
peak fitting was proposed to reduce this problem. This solution is effective for large particles, performing optimally
for 6 pixels diameter particles, following the authors. In the present paper, a different solution is proposed giving
good results also with small particles.
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PIV’01 Paper 1013
FFT correlation: This case is depicted in figure 4, in a similar way as in figure 3.
Window
in
image a
Grey levels in image a
Correlation
coefficients
Window
in
image b
Grey levels in image b
Subpixel
interpolation
−∆s
∆s
0
Displacement
Fig. 4. Sketch of a DC PIV interrogation. ∆s corresponds to the smallest displacement used for the calculation of the correlation
coefficients (typically one pixel) thus giving a discrete value for C.
As it has been commented, the FFT algorithm introduces a spurious periodicity in the calculation of the
correlation coefficients. However, due to the symmetry of the algorithm and the equal size of both windows
everything done on image a is also done on image b. The consequence of this is that for the particular case of null
displacement no error is added to the measurement, as all the spurious contributions in the process are added
symmetrically. In the procedure sketch in figure 4, the value of the correlation coefficient at both sides is zero. This
is a distinctiveness of the example chosen. With any other image, the symmetry would still be present.
3
Measurement errors due to particle truncation in presence of displacement between images
In this case, the reasoning is easier for the FFT case, so it will be analyzed before, changing the presentation order
used above.
3.1
FFT correlation:
Peak locking. When the displacement to be measured s is from -0.5 to 0.5 pixels, as mentioned in the
introduction, peak locking appears. To illustrate the situation figure 5 shows a particular case, where now the
displacement in not 0, but -∆s.
Window
in
image a
Grey levels in image a
Window
in
image b
Correlation
coefficients
Subpixel
interpolation
Error
Grey levels in image b
−2∆s
−∆s
Displacement
Fig. 5. Sketch of a FFT PIV interrogation showing peak locking.
4
0
PIV’01 Paper 1013
Figure 5 depicts an error that tends to 0 thus it translates into peak locking. Generalization of this particular case
to every situation is straightforward, but not obvious. In order to support that this peak locking is general, figure 6
shows a chart of the error obtained in a numerical simulation, for any particle position within the interrogation
window and for any displacement in the mentioned range. The error departs from 0 only when the particle starts to
be truncated, thus the surface has a central plateau. Owing to the negative value of the surface slope for a fixed
particle position, the error is towards 0. This kind of peak locking will be present in any case with discrete offset of
the interrogation windows.
Particle diameter = 6 pixels; Int. window 16x16 pixels
0.2
0.1
0
-0.2
-0.4
10
0.4
12
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
8
Particle position
(pixels)
6
4
2
-2
-0.3
0
-12
-10
-8
-6
-4
-0.1
Error = Meas. - Real disp.
(pixels)
0.4
0.3
Real disp.
(pixels)
Fig. 6. Error due to the truncation of particles by window borders in a conventional FFT PIV. It is shown as a function of the
position of the particle in the interrogation window (average between the particle positions in image a and b) and as a function of
the displacement to be measured. Gaussian particles.
For different particle diameters and window sizes the behavior is similar; more details can be found in Nogueira,
Lecuona and Rodríguez (2001).
Removal of the error: subpixel offset
In figure 6, it can be observed that the error is not larger than the displacement to measure. Accordingly, if a
subpixel offset is applied to the interrogation window, accumulating the corrections found as displacements through
iterative interrogations, Mathematics shows that the procedure converges to 0 displacement. This avoids the particle
truncation error, owing to the single line of null error in figure 6 for any particle position. The accumulated
correction amounts the measured displacement.
This reasoning conforms a theoretical explanation to the experimental results reported by Lecordier et al. (1999)
and Scarano and Riethmuller (2000) where it is shown that offset reduces the peak-locking effect.
Subpixel offset requires interpolation of the image grey levels, introducing an error that is studied elsewhere.
3.2
DC algorithm:
Once the FFT case has been analyzed, the basis to analyze the more complex case of DC are set.
Peak locking. In this case, differences appear in respect to the FFT case. If the window is chosen in image a of
figure 5, DC yields a very similar output, a high value at -∆s, cero at -2∆s and an intermediate value at null
displacement. The particular values would change between the FFT case and the DC case, owing to the different
value of the denominator in expression (1).
Focusing now in the case in figure 7, the window in image a together with the corresponding one in image b,
represent for the FFT case a situation with the same displacement and with opposite position of the particle within
the interrogation window. It can easily be reasoned that the same output depicted in figure 5 would come out, as
figure 6 shows. In this situation the analysis performed by DC gives a different output, as can be observed in figure
7. There the correlation value at -2∆s is not cero, evidencing a loss of symmetry.
5
PIV’01 Paper 1013
Window
in
image a
Grey levels in image
a
Correlation
coefficients
Subpixel
interpolation
Error
Grey levels in image b
−2∆s
−∆s
0
Displacement
Fig. 7. Sketch of a DC PIV interrogation showing peak locking.
A numerical simulation generalizes this reasoning for different displacements and positions within the
interrogation window. The chart in figure 8 shows the output.
Particle diameter = 2 pixels; Int. window 4x4 pixels
0.4
0.2
0
-0.4
1.67
2.00
0.20
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
0.33
1.00
1.33
Particle position
(pixels)
-0.8
0.67
-0.33
-0.6
0.00
-2.00
-1.67
-1.33
-1.00
-0.67
-0.2
Error: Meas. - Real disp.
(pixels)
0.8
0.6
Real disp.
(pixels)
Fig. 8. Error due to the truncation of particles by window borders, in a DC PIV. It is shown as a function of the position of the
particle in the interrogation window and the displacement to be measured.
In this scenario, the subpixel-offset solution proposed for the FFT case to avoid the error would not work
properly for the DC case. The main reason for this is the lack of symmetry in the DC algorithm.
Removal of the error: subpixel offset + Symmetrical Direct Correlation (SDC).
Trying to obtain better measurements from iterative procedures like the one proposed in the FFT case, an
important issue would be to have a cero error line at zero displacement, like in figure 6. The lack of this line in
figure 8 comes from the effect already observed in figure 3. The objective then is to generate a symmetrical
algorithm keeping the advantages of the DC.
An effective way to fulfil this, with a couple of PIV images, a and b, is to add the contributions from two sources:
1.
2.
The correlation of an interrogation window in image a along image b.
The correlation of an interrogation window in image b along image a.
Both contributions can be combined either by adding the correlation coefficients or simply by averaging the
displacements from the two sources. Such a system will be called from now on SDCPIV (Symmetrical Direct
Correlation PIV). The chart analogous to Fig. 8 for this case is depicted in Fig. 9. In this figure another effect
appears, the inverse peak locking. That is, for particles near the origin, the error has the opposite sign as when the
particles are far from the origin.
6
PIV’01 Paper 1013
Inverse peak locking comes from the chosen correlation and peak-fitting algorithms, and it is not related to the
truncation of particles, as can be deduced from figure 9, where the central plateau now is warped in the opposite
director to the “wings”. In some cases, this inverse peak-locking source partially hides the peak locking coming
from the truncation of particles, as it is commented in Section 4.
Particle diameter = 2 pixels; Int. window 4x4 pixels
0.1
0.05
0
-0.1
1.67
2.00
0.20
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
0.33
1.00
1.33
Particle position
(pixels)
-0.2
0.67
-0.33
-0.15
0.00
-2.00
-1.67
-1.33
-1.00
-0.67
-0.05
Error: Meas. - Real disp.
(pixels)
0.2
0.15
Real disp.
(pixels)
Fig. 9. Error due to the truncation of particles by window borders in a SDCPIV system. It is shown as a function of the position
of the particle in the interrogation window and the displacement to be measured. (The direct and inverse peak locking
corresponds, also, to the statistic response of DC PIV)
Under this situation, the discrete offset of the interrogation windows, using previous measurements in an iterative
way, reduces the error like in an FFT PIV, giving better results, as will be commented in section 4. This procedure
reduces the peak-locking error as well as the anti-peak-locking error as far as both are of equal or smaller magnitude
than the real displacement to measure, and this is the case in figure 9. In short, the way to deal with the error from
truncation of particles in DC is to apply SDCPIV + iterative subpixel offset of the interrogation window.
4.
Test on synthetic PIV images.
To check the performance of the modifications proposed in Section 3, several implementations using synthetic
images have been accomplished. The cases selected were of uniform displacement, due to the nature of the effect
under study. The shape of the particles in the images followed a 2D gaussian profile (the discrete grey values of the
pixels corresponded to integration of the continuous function). To aim exclusively at the error produced by
truncation of particles, no noise was added, neither variation on the illumination profile nor size spread of particles.
No out of plane displacement was implemented and, as a result, no missing particles related to this effect appear.
The particles were randomly located. Overlapping particles added their intensity up to the saturation of the grey
level range (the grey level at the centre of an isolated particle corresponds to half the 8 bit dynamic range). Outlier
vectors were not taken into account in the performance tests.
Figs. 6, 8 and 9 show an approximate linear dependence between the error and the real displacement. This
favourable circumstance allows concluding that the magnitude of the errors is similar when there is a small gradient.
4.1.
FFT PIV.
From figure 6 it can be concluded that the measurement is smaller than the real displacement, but the error has
different amplitudes depending on the position of the particle. This gives two contributions to the final error: a bias
(systematic) error as peak locking and an additional random error. Due to the approximate linear dependency
between the theoretical error and the real displacement, both contributions have been scaled with the later. Fig. 10
depicts several cases of conventional PIV, where no actions were taken to avoid the error arising from the truncation
of particles. Both contributions are shown: the bias average error, showing appreciable peak locking, and the
random error. The later is presented as the root mean square (rms) of the deviation in respect to the bias. The cases
were selected to show influences coming from secondary parameters, as follows. The seeding density ranged from a
mean distance between particle images δ of one diameter to three diameters. The diameter of particles d was given
values of 2, 4 and 8 pixels. The plots presented in Fig.10 correspond to a displacement of 0.25 pixels. In the range of
displacements from 0.1 to 0.4 pixels the differences in the results were verified to be smaller than 20% of the
measurement error and the qualitative behaviour was identical. Thus, figure 10 and the following ones are
7
PIV’01 Paper 1013
representative of what happens in general. It can be verified that for F/d < 7, the errors are of the same order of
magnitude than the subpixel displacement to measure.
Peak-locking
0.5
rms around bias
rms (error-bias)/ Real disp.
Bias error/ Real disp.
0.8
0.7
0.4
0.6
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0.0
-0.1
0.0
1
3
5
7
1
3
L /d
F/d
5
7
L /d
F/d
Fig. 10. Bias and rms around bias errors in conventional FFT PIV for several cases: clear symbols δ = d, and filled symbols δ =
3d. Triangles d = 2 pixels, circles d = 4 pixels and squares d = 8 pixels.
Fig. 11 depicts results for the same cases, but allowing for a subpixel window offset, corrected accumulating for
20 iterations (with no window size change). The peak-locking effect is much reduced, as it was predicted. The
combination of both errors is reduced for every case, but the rms around bias is not always smaller for small
windows. This phenomenon is more patent for lower seeding densities and smaller particles. The reason for the
growth of the rms error around bias will be further commented in Section 4.3
When using the FFT, it is customarily applied over interrogation windows with side lengths of 2n pixels.
Nevertheless, Gui and Merzkirch (1998) describe how any window size is easily manageable. Consequently, the
data in Figs. 10 and 11 do not present any restrictions on the window size. To be consistent on the study of errors,
pure FFT was applied even in the cases where the side was not 2n in size.
Peak-locking
0.5
rms around bias
rms (error-bias)/ Real disp.
Bias error/ Real disp.
0.8
0.7
0.4
0.6
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0.0
-0.1
0.0
1
3
5
7
1
L /d
F/d
3
5
7
L /d
F/d
Fig. 11. Bias and rms around bias errors in measurements after 20 iterations, allowing for a subpixel offset of the interrogation
window in a FFT PIV. Same code for symbols than in figure 10.
4.2.
DC PIV.
A first test was performed using DC PIV with a three points symmetrical gaussian peak fitting. Cases with 8 and 4
pixels particle diameters would show smaller errors by the use of asymmetrical peak fitting. Nevertheless, the 2
pixels particle diameter cases would not show improvement, as the correlation peak diameter is too small. These
cases (triangles in Fig. 12) will significantly improve by the use of SDCPIV with subpixel offset. Additionally,
when SDCPIV is used, iterating with subpixel displacement of the interrogation window, no appreciable increase in
accuracy is expected by the use of asymmetrical peak fitting with large particles. Consequently, asymmetrical peakfitting algorithm is not needed in SDCPIV.
8
PIV’01 Paper 1013
Peak-locking
0.5
rms around bias
rms (error-bias)/ Real disp.
Bias error/ Real disp.
0.8
0.7
0.4
0.6
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0.0
-0.1
0.0
1
3
5
7
1
3
L /d
F/d
5
7
L /d
F/d
Fig. 12. Bias and rms around bias errors in DC PIV with symmetrical peak-fitting algorithm. Same code for symbols than in
figure 10.
For DC PIV, it is shown in Fig. 5 and 6 that, for d = 2 pixels, the peak locking due to particle truncation is
partially hidden by inverse peak locking. Therefore, the rms error around the bias is enlarged, as can be seen in Fig.
12 versus Fig. 10.
At this point, it is interesting to remark that the results presented in Figs. 10 and 12 are in accordance with the
errors observed by Huang et al. (1997). This work shows similar results for the case d = 2.8 pixels, F/d = 7.5 and δ =
1.9d.
From Fig. 8 it was already concluded that in a DC PIV system, iterations with a subpixel window offset would
not converge to zero displacement, neither would reduce the final uncertainty, in the case of small particle images.
By the use of SDCPIV, the recovery of symmetry permits to obtain fully profit from the use of iterations with
subpixel offset, yielding more accurate results. It reduces not only the peak locking due to particle truncation but
also the inverse peak locking caused by the peak fitting and correlation calculation algorithms.
Here, a remark can be raised. The calculation of the DC can be performed subtracting the mean grey level in the
interrogation window so that f and g in expression (1) both have positive and negative values, centred on cero. This
operation usually leads to boosting of anti-peak locking. The result is a smaller bias error due to statistical
cancellation of peak and anti-peak locking. As it gives better results, it is a common design rule (Fincham and
Spedding 1997 and Huang et al. 1997; among others) and has been the used in figures 8, 9, 12. Nevertheless, this
detail is not needed when the peak locking is corrected by SDC + subpixel offset. Consequently, the performance in
figure 13 was obtained without subtracting the mean level. Using this alternative, peak locking is more apparent in
the first steps, but iterations balance for this and the result is about 15% lower in uncertainty. The data presented in
Fig. 13 correspond to SDCPIV with symmetrical peak fitting, without mean grey level subtraction, and allowing for
5 iterations.
Peak-locking
0.5
rms around bias
rms (error-bias)/ Real disp.
Bias error/ Real disp.
0.8
0.7
0.4
0.6
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0.0
-0.1
0.0
1
3
5
7
1
L /d
F/d
3
5
7
L /d
F/d
Fig. 13. Bias and rms around bias errors in SDCPIV measurements with symmetrical peak-fitting algorithm, without mean grey
level subtraction and after 5 iterations. Same code for symbols than in figure 10.
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PIV’01 Paper 1013
4.3.
Additional observation.
In conventional FFT PIV the rms error around bias increases as iteration proceeds, for small windows, but not in
SDCPIV. An explanation follows. The images interrogated by both systems were the same, so the only difference
comes from the performance of either correlation coefficient calculations. This comes out of the artefacts created by
the FFT. For F = 8 pixels, each of the 4 neighbours to the maximum correlation coefficient incorporate 12% of
artefacts in their value. This does not happen in the DC calculation. The consequence is that for small windows there
is a remarkable difference in performance. This suggests the use SDCPIV, instead of FFT, if accuracy is pursued.
With small window sizes, the increase in computation time seems affordable.
5.
Test on real PIV images.
In this section, the performance of the proposed solutions is evaluated on real images. For this purpose, and to
further support what has been exposed so far, small particle images were selected. The flow field corresponds to a
∼ 10.5 m/s free air jet in open stagnant atmosphere. It was seeded with 4µm alumina particles. With a time between
laser pulses of 10 µs and a spatial scale of 3,64 pixels/mm, this corresponds to 0.38 pixel displacements. The particle
image diameter is around 1 pixel due to the diffraction limited spot of the optics, 2.44·(1+M)·f#·λ ~ 0.8 pixel. M is
the magnification, f# is the f-number of the lenses and λ is the light wavelength. As an example, Fig. 14 shows a
close up of one of the PIV images. The apparent horizontal distortion is due to a non-unity aspect ratio of the CCD
camera (DANTEC DoubleImage 700).
Fig. 14. Close-up of a 72x50 pixel zone of the real PIV image used for performance test of the different algorithms
Although a uniform displacement zone was selected, the turbulence brings in some contribution to the rms of the
measurements. The plot depicted in Fig. 15 was obtained processing the images with different sizes of the
interrogation windows. As in the synthetic cases, the FFT PIV was allowed to perform 20 iterations, while the
SDCPIV needed only 5 for reaching the same level of correction, due to its higher signal to noise ratio. A behaviour
similar to the one observed for synthetic images can be observed for the mean value of the displacement. For a
certain interrogation window size, the iterative algorithms show smaller peak locking. The remaining peak locking
is coherent with the sensor limitations. This source of peak locking (Westerweel 1998) is not related to truncation of
particles at the window borders.
On the evaluation of the behaviour of the rms, some considerations have to be raised. Here the contribution of the
velocity fluctuations seems important. This produces the expected decay with large interrogation window sizes,
which is due to spatial averaging. Nevertheless, the iterative case of FFT PIV shows a remarkable increase in rms
for small windows, in respect to any other system. This could be attributed to the observation presented in Section
4.3, highlighting the suitability of SDCPIV for small windows.
10
rms around Average
Peak-locking
0.8
0.9
rms (Meas.-Aver.)/ 0.38p.
(0.38p.-Measurement)/ 0.38p.
PIV’01 Paper 1013
0.7
0.8
0.6
0.7
0.5
0.6
0.4
0.5
0.3
0.4
0.2
0.3
0.1
0.2
0.0
0.1
-0.1
0.0
4
8
12
16
20
L (pixels)
F
4
8
12
16
20
L (pixels)
F
Fig. 15. Average and rms around average of measurements on a real image. 0.38 pixel is assumed as the value of the real
displacement for normalisations. Iterative algorithms (filled symbols) and straight ones (clear symbols). FFT (triangles) and DC
(circles) algorithms.
6.
Conclusions.
The errors produced by the truncation of particles at the interrogation window borders have been identified,
characterised and the way to reduce them detailed.
It has been established that, in FFT PIV, allowance for subpixel offsets of the interrogation windows reduces the
uncertainty, especially when dealing with small F/d ratios. This has already been implemented by Lecordier et al.
(1999) and Scarano and Riethmuller (2000). Here, a theoretical explanation to that wise experimental observation is
offered.
It has been established that, in DC PIV systems, the implementation of the SDCPIV alternative, jointly with
subpixel offsets of the interrogation windows, offers a reduction in uncertainty, especially when applied on images
with particles a few pixels in diameter. This is also recommended for the last steps in Multigrid PIV systems.
Applying these modifications, peak locking from truncation of particles is removed. This way, the subpixel
accuracy of peak-fitting algorithms is no worth a debate from this point of view (at null real displacement there is no
difference between the ones already in use). Peak locking from this source is no longer relevant as the iterative
procedure converges even if peak locking is present in the iterations.
The debate about interpolation of discrete values of the correlation is thus transferred to the grey level
interpolation in the PIV images for image offset and, eventually, distortion. Error coming from this source is an
open question, although it has been shown in this paper that its order of magnitude is smaller than the one coming
from the truncation of particles. The geometrical limitations of the sensor are still to be taken into account, as they
represent a real loss of information that cannot be recovered once the image is sampled.
It is not expected that Hybrid (Cowen and Monismith 1997) and LFCPIV (Nogueira, Lecuona and Rodríguez
1999) high-resolution systems are affected by the peak-locking source presented in this paper, due to their unique
conception.
Acknowledgements.
This work has been partially funded by the Spanish Research Agency grant DGICYT TAP96-1808-CE and PB950150-CO2-02, and under the EUROPIV 2 project (A JOINT PROGRAM TO IMPROVE PIV PERFORMANCE
FOR INDUSTRY AND RESEARCH) is a collaboration between LML URA CNRS 1441, DASSAULT
AVIATION, DASA, ITAP, CIRA, DLR, ISL, NLR, ONERA, DNW and the universities of Delft, Madrid (Carlos
III), Oldenburg, Rome, Rouen (CORIA URA CNRS 230), St Etienne (TSI URA CNRS 842), Zaragoza. The project
is managed by LML URA CNRS 1441 and is funded by the CEC under the IMT initiative (CONTRACT N°:
GRD1-1999-10835). Their contribution is greatly appreciated.
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