4th International Symposium on Particle Image Velocimetry
Göttingen, Germany, September 17-19, 2001
PIV’01 Paper 1010
Proposals on the design of 2D multigrid PIV systems: application
of dedicated weighting functions and Symmetric Direct
Correlation.
A. Lecuona, J. Nogueira and P. A. Rodríguez
Abstract Multigrid PIV is a path in the search of high-resolution systems. It follows an iterative scheme that uses
the information of initial evaluations to adapt the system to obtain better ones. This is performed by reducing the
size of the interrogation windows and by redefining the location in the PIV images. In this procedure, two sources of
error can significantly affect the final measurement quality. One is related to the large interrogation windows while
the other is related to the small ones. For large interrogation windows, the error coming from the spatial frequency
response can be changed in form and reduced, allowing this improvement substantially better iterative procedures.
For small interrogation windows, the error coming from the truncation of particles at the interrogation window
borders can be much reduced. This is possible through the implementation of specific algorithms. The proper
combination of both improvements can clearly enhance the performance of Multigrid systems. Due to the
complementarity of the situations in which these two sources of error appear, the discussion covers topics of interest
for any correlation PIV system. This paper gives details on both sources of error and proposes ways to reduce them.
Both studies include tests in synthetic and real PIV images.
1
Introduction
PIV has been established as an important experimental tool for research as well as for industry. Basic Particle Image
Velocimetry is able to describe 2 components of the velocity along a two-dimensional domain (2D 2C PIV).
Nevertheless, in many cases there is still a gap between the information contained in the images and what is
extracted from them by current PIV systems.
Recently, some of the effort on the development of the technique is focused on the extraction of all the
information obtainable from the images, including the development of super-resolution or high-resolution systems.
Generally, all these advanced systems have a core algorithms characterized for being iterative. This means that
the information from initial evaluations is used to tune or adapt the system and obtain better ones.
Focusing on correlation PIV, two main branches for advanced 2D algorithms can be mentioned: LFCPIV and
Multigrid PIV. Information about the former can be found in Nogueira, Lecuona and Rodríguez (1999) and
Rodríguez, Nogueira and Lecuona (2001 in this meeting proceedings). About Multigrid PIV, information can be
found in Soria (1996), Hart (1999), Lecordier et al. (1999), Scarano and Riethmuller (2000) as well as in many
others. This paper is focused in launching some proposals that can increase the accuracy of Multigrid PIV.
In this technique, the iterative cycle is used for two purposes:
1. To reduce the size of the interrogation windows thus increasing the spatial resolution. This is at the expense of
probably a lower signal to noise ratio (s/n) and involving a higher measurement uncertainty. The drawbacks are
specially relevant for the windows small enough to approach the limits of PIV. At intermediate steps
progression towards smaller interrogation windows tries not only to increase the spatial resolution, but also to
increase s/n. This eventually brings an optimum window size. Using large windows, the dispersion of
displacements within the window leads to a low s/n. If they are too small, the signal contained could be not
sufficient.
2. To redefine the interrogation windows location in the PIV images, using the previously measured displacement
field. This is in order to diminish the displacement of particles between correlating windows and thus augment
s/n.
The iterative attribute of Multigrid systems is fully confirmed when there is a decision point to repeat previous
interrogations with a different setup and the same window size. This is so also when the original images are
modified compensating the deformations to improve s/n and the interrogation repeated. Otherwise, the system is just
progressive.
A. Lecuona, J. Nogueira, 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
1
PIV’01 Paper 1010
The proposals reported in this paper will cover large as well as small windows phases. Consequently, this paper
has two main sections (sections 2 and 3) where the topic of respectively large and small windows will be discussed.
2
Errors associated to iterative PIV with large windows.
It is evident that an interrogation window mixes the particle displacements field features that are contained in it.
This is a fact that cannot be avoided. Nevertheless, the way in which these small-scale features are mixed can
generate a qualitative difference in the final output.
2.1
Small scales mixing in usual correlation PIV.
The correlation function does not differentiate the position of individual particles within the interrogation window.
Consequently, each particle contributes in an identical way to the correlation peak. The build up of a correlation
peak is the result of the aggregation of a certain number of particle peaks of similar displacement. Thus, the height
of the resulting peak is related to the number of particles. Other effects that shape the correlation peak are
differences in particle diameter and intensity and the presence of a small dispersion in the displacements.
These considerations imply that the measurement is a weighted average of a group of particle with similar
displacements, and it is usually biased toward the most frequent particle displacement within the window.
The so obtained measurement has to be associated to a point in the PIV measurement plane. This point is usually
the center of the interrogation window. For certain spatial wavelengths, like the one represented by the sinusoidal
displacement field, s(x,y), depicted in figure 1a, this measured displacement, is opposite to that at the center of the
window. Other frequencies have the same effect.
In a Multigrid system, this effect originates an offset of the next step interrogation window in the opposite
direction to what corresponds to its center, thus reducing the s/n in the following step and eventually leading to a
spurious vector or outlier.
To generalize the idea depicted in figure 1a, figure 1b presents the PIV amplitude response, r, to 1D sinusoidal
fields of different spatial wavelengths, λ. The plot also differentiates among different window sizes. This figure is
just an idealization because the real value of r in a PIV depends on more parameters than just the frequency, as
commented before. However, the zones where r < 0 (that corresponds to the phenomenon depicted in figure 1a) are
statistically fully representative. r < 0 corresponds to a π phase change, meaning a reversal in output.
This response to different wavelengths leads to draw conclusions based on the Fourier spatial frequency
decomposition of the displacement field. The main drawback of such an analysis is the sometimes non-linear
character of the PIV peak construction and detection process, just above introduced, which signifies a more complex
behavior than the inherent linear character of the Fourier analysis. Nevertheless, during the multipass process of
iterative PIV systems the statistical behavior is similar, in most cases, to the conclusions that can be drawn from a
Fourier analysis, at least qualitatively. This will be shown in the next subsection.
Apart from the paradigm of Fourier decomposition, it has to be to keep in mind that what is presented in figure 1
will happen for any particle displacement distribution having a similar shape.
1.2
1.0
16 pixels window side
0.8
32 pixels window side
0.6
r
64 pixels window side
0.4
0.2
⇒
0.0
Displacement
field, s(x,y).
−0.2
Measured
displacement
−0.4
∞
a
b
64.00
32.00
21.33
16.00
12.80
λ (pixels)
10.67
9.14
8.00
Fig. 1. a) Example of measurement error in the interrogation window of a conventional PIV system containing a single high
spatial frequency in the displacement field. b) Amplitude frequency response r as a function of 1D spatial wavelength λ. The
intervals with r < 0 indicate where the effect depicted in figure 1a is present. (Note that the scale of λ is not linear; actually, it
corresponds to a linear scale in frequency).
2
PIV’01 Paper 1010
2.2
Effects of negative frequency response in Multigrid PIV.
One of the effects of the frequency intervals where r < 0, as commented in the previous section, is a shift of the
interrogation window in the opposite direction to what should be performed, thus reducing the s/n and eventually
leading to a spurious vector or outlier. This is more relevant when the size of the interrogating window is reduced in
the next interrogation step.
Another effect is related to the use of compensation of the particle pattern deformation together with the
Multigrid process. This seems to be the recent evolution of the Multigrid systems, following the works of Fincham
and Delerce (2000), Scarano and Riethmuller (2000). In these cases the intervals with r < 0 establish that if a
correction of the particle pattern is attempted with this erroneous information, the deformation increases instead of
decreasing. A diverging process may be triggered if successive iterations are implemented.
The basic condition for the unstable growth is that the grid sampled displacement field contains the required
(r < 0) frequencies. Following figure 1b and taking into account the Nyquist criteria, the grid sampled displacement
field will contain unstable frequencies only if the grid node spacing, ∆, is smaller than half the square interrogation
window side length F. To illustrate about this behavior, a pair of synthetic images of a field with uniform
displacement s = 3.6 pixels, uniform particle diameter d = 2 pixels and average particle separation δ = 1.4 pixels
were analyzed. The results are plotted in figure 2.
These synthetic images contain absolutely no frequencies related to the instability but the noise due to
discretization of gray levels. It can be seen that a Multigrid system with compensation of the particle pattern
deformation, iterating with F = 16 pixels and ∆ = 8 pixels, do not show divergence after 45 iterations. The same
system with ∆ = 4 pixels shows it clearly.
Since unstable frequencies are not contained in the displacement field, the divergence, although explosive, only
shows up after a substantial number of iterations. This can be observed in figure 2.
Evolution of error with iterations
0.7
Multigrid PIV with compensarion of part.
pattern and no weigh. func. (F = 16, ∆ = 4)
rms(e ) (pixels)
0.6
0.5
Multigrid PIV with compensarion
of part. pattern and no weigh.
func. (F = 16, ∆ = 8)
0.4
0.3
0.2
0.1
0.0
1
5
9
13
17
21
25
29
33
37
41
45
Number of iterations
Fig. 2. Example of divergence for some systems with compensation of the particle pattern deformation.
In order to further check that this instability is due to the intervals with r < 0, some test were performed with real
images. The test consisted in checking the frequency domain of the measured displacements after several iterations.
The first frequencies that diverged in these situations were the ones predicted, this is, large negative values for r.
2.2
Avoidance of negative frequency response.
As stated in the beginning of Section 2, the mixing of small features is unavoidable. However, the way this mixing
happens can be modulated. One way to do this is to apply a weighting function on the gray levels of the
interrogation window. A weighting function that avoids r < 0 is depicted in figure 3 as expression (1).
3
PIV’01 Paper 1010
8-9
7-8
6-7
5-6
4-5
3-4
2-3
1-2
0-1
0.50
 ξ
υ 2 ( ξ ,η ) = 9 4
 F

2
−4
 η
ξ
+ 1 4
 F
F

2
−4

η
+ 1 (1)

F

0.25
υ2
0.00
-0.25
-0.50
-0.25
0.00
0.25
ξ /F
η /F
-0.50
0.50
Fig. 3. Weighting function designed to avoid negative frequency responses. ξ and η are coordinates with origin at the center of
the interrogation window and F is the length of its side. For more details see Nogueira, Lecuona and Rodríguez (1999).
Using this weighting function the correlation coefficient computation would follow expression (2). In this
situation, the results equivalent to those depicted in figure 1 are depicted in figure 4. In general, the avoidance of
negative frequency response allows obtaining a correctly oriented displacement at the center of the interrogation
window in cases like the one depicted in figure 4a. This would allow for correct shifting of interrogation windows in
further iterations and would avoid the instability of a system with compensation of the particle pattern deformation.
This last point has been checked in synthetic as well as in real images. As an example, the LFCPIV method that uses
the proposed weighting can be mentioned. It gives a rms of the error ≈ 0,03 pixels in the case of figure 2 and after 45
iterations.
F/2
Clm =
∑υ( ξ ,η ) f ( ξ ,η ) ⋅ υ( ξ ,η )g( ξ + l ,η + m )
ξ ,η =− F / 2
F/2
F/2
ξ ,η =− F / 2
ξ ,η = −F / 2
(2)
∑υ 2( ξ ,η ) f 2( ξ ,η ) ∑υ 2( ξ ,η )g 2( ξ + l ,η + m )
1.2
1.0
0.8
64 pixels window side
weighted with expresion (1)
0.6
r
0.4
0.2
⇒
0.0
Displacement
field, s(x,y).
−0.2
Measured
displacement
−0.4
∞
a
b
64.00
32.00
21.33
16.00
12.80
λ (pixels)
10.67
9.14
8.00
Fig. 4. a) Example of measurement in an interrogation window containing a single high spatial frequency in the displacement
field after the application of the weighting function of expression (1). b) Frequency response as a function of spatial 1D
wavelength using the same weighting function. (Note that the scale of λ is not linear; actually, it corresponds to a linear scale in
frequency).
Unfortunately, the application of a weighting function induces some erroneous slip in the measurement
(Nogueira, Lecuona and Rodriguez, 1999). This slip is larger with smaller windows, making inadvisable its use for
windows smaller than 32 pixels.
4
PIV’01 Paper 1010
2.3
Summary of considerations and performance check.
The following table shows the situation in relation to the negative frequency response error:
No weighting + ∆ < F/2
No weighting + ∆ > F/2
Weighted (F > 32) +
any ∆
Multigrid PIV with window r < 0 ⇒ reduction of s/n ratio for
shift
further iterations
r < 0 ⇒ reduction of s/n ratio r > 0
Multigrid PIV with
r < 0 ⇒ divergent error (if several
compensation of the particle iterations are performed with the
pattern deformation.
same unstable freq.)
r < 0 ⇒ reduction of s/n ratio r > 0 and no
for further iterations
for further iterations
instability.
To check the importance of the negative frequency response errors when only a few iterations are carried out,
some performance checks were run. The synthetic images selected for the test had δ = 4.5 pixels (i.e. 4/(π·δ2) ~ 0.06
ppp (particles per pixel)). The particle images shape were gaussian, being the diameter associated to e-2 times the
peak value d = 4 pixels. Where particles overlap, the corresponding intensities were added. 5% of particles had no
second image to correlate due to out of plane velocity or any other optical effect. These images were selected to
allow direct comparison with other super-resolution systems, like the mentioned LFCPIV in Nogueira, Lecuona and
Rodríguez (1999) or Rodríguez, Lecuona and Nogueira (2001) (in this meeting proceedings).
Five different multigrid systems have been tested. In two of them, the size of the interrogation windows had the
following values: 64 pixels for the first iteration, 32 pixels for the second one and 16 pixels for the last three ones. ∆
was successively 16, 8 and 4 pixels for the three iterations left. The smallest λ tested was 20 pixels, thus the three
last iterations were not likely to trigger small wavelength instabilities. The difference between these two systems is
the application of the weighting function to one of them in the first two iterations.
A third system used the weighting function in all the iterations. Due to the problems of this weighting with small
windows, the iterations with F = 16 were substituted by F = 32 in this system.
The remaining two systems correspond to references from others work (Jambunathan et al. 1995 and Scarano and
Riethmuller 2000). These two systems only uses F = 16 in all the iterations.
The results for these options are compared in figure 5. The oscillation around the reference lines of the Multigrid
system with no weighting function is caused by the effect depicted in figure 1, when operating on the different
window sizes. Only the system corresponding to the work of Scarano and Riethmuller (2000) corresponds to
different test images, as the data have been directly taken from their work.
It is now clear that when the instability is not allowed to grow (large grid spacing or few iterations), the error
from r < 0 still affects the accuracy. The detailed analysis of the proposed multigrid system, with no weighting
function measuring the point of wavelength λ = 21.3 pixels of figure 5, on the synthetic images defined in the
introduction, is as follows:
- In the first iteration (F = 64 pixels), figure 1b shows that r is slightly negative. The result is that this first iteration
gives in some places a displacement opposite to the one under measurement. If we analyze the root mean square
value (rms) of the error, e, in the measurement, normalized with the rms of the signal s to be measured, the value
obtained is: rms(e)/rms(s) = 1.07. This value is consistent with a measured field slightly opposite to the real one.
- In the second iteration (F = 32 pixels), figure 1b shows a highly negative value for r. The result in error terms is
rms(e)/rms(s) = 1.34. This is coherent with a measurement with opposite sign to the original one.
- In the three following iterations (F = 16 pixels) r > 0. The result is a reduction of the error in each step, finishing
with rms(e)/rms(s) = 1.19. Following iterations would lead to worse measurements due to accumulation of errors
from other sources.
5
PIV’01 Paper 1010
Displacement field: s = 2sin(2πx /λ x ) (pixels)
1.2
Multigrid PIV+compens. (F :64,32,3x16; ∆ :16,8,3x4)
1
rms(e )/rms(s )
Multigrid PIV+compens.+weight in first 2 steps
(F :64,32,3x16; ∆ :16,8,3x4)
(Multigrid LFCPIV in the first two steps)
0.8
0.6
0.4
0.2
0
Multigrid LFCPIV (F :64,4x32; ∆ :16,8,3x4)
20
25
30
35
40
45
50
55
λ x (pixels); (δ =4.5 pixels).
60
Fig. 5. Performance of the five iterations multigrid systems described in the text, along with the method from Jambunathan et al
(1995) (thin continuous line) and Scarano and Riethmuller (2000) (thin discontinuous line).
It must be accepted that the parameters defined here for a Multigrid PIV are somewhat arbitrary, but the benefits
of the application of the weighting function for steps with windows ≥ 32 pixels are now evident.
The test cases have been chosen with care, so that there is no appreciable instability excitation. This implies that
most of the benefits remain even when there is no compensation of the particle pattern deformation. This result will
be explained more deeply elsewhere.
Another useful conclusion of the comparison here presented is that better results can be obtained with weighted
32 pixels windows rather than non-weighted 16 pixels windows.
3
Errors associated to iterative PIV with small windows
It should be remarked that the application of windows smaller than 16 pixels on real images is still a subject under
development. Many recent applications of advanced multigrid systems do not use smaller windows. An example is
Scarano and Riethmuller (2000). In that paper, devoted to multigrid PIV, the processing of real images is performed
with windows of 32 and 16 pixels.
Nevertheless, there is ongoing research on systems especially designed to deal with smaller windows, Fincham
and Spedding (1997) show good measurements with small windows, but the requirement for a large particle
diameter arises (optimum diameter around 6 pixels). Nogueira, Lecuona and Rodríguez (2001a) and (2001b in this
meeting proceedings) show a promising way to deal with small windows free of this requirement. This is by the use
of Symmetric Direct Correlation (SDC) algorithms. Here some more details are given about such a system.
3.1
Selection of the correlation algorithm:
The correlation function with large interrogation windows is time consuming. Consequently, most of PIV systems
turn to the Fast Fourier Transform (FFT) instead of the Direct Correlation (DC) calculation, in order to save time.
The use of FFT ignores completely the image out of the interrogation windows, substituting it by a periodical field.
The result is the introduction of a spurious contribution in the calculation of the correlation coefficient
corresponding to any not null displacement. This reduces the signal to noise ratio (s/n). Fig. 6 depicts the mechanism
of this source of error, which is more relevant the smaller the window is.
CORRELATION OF INTERROGATION WINDOWS
FROM IMAGES a AND b:
SPURIOUS CONTRIBUTION
TO CORRELATION IN FFT
ONE PIXEL
Fig 6. Error introduced by the periodicity of the FFT.
6
PIV’01 Paper 1010
As this section is devoted to small interrogation windows, it is possible to afford the increase of computing time
of the DC. Using it, the reduction of s/n associated to the periodicity of the FFT is avoided.
Nevertheless getting rid of this source of error does not mean to have a better performing algorithm; other sources
of error have to be checked for. Detailed studies like Huang et al (1997) or the ones from the authors, mentioned in
the introduction of this section, show that for small particles the performance of DC algorithms is better than usual
FFT ones, but not better than some variants of the later, like FFT + discrete offset. One of the main reasons for that
is the error coming from the truncation of particles at the window borders.
The way to avoid it in FFT, by a discrete offset of the interrogation windows, is not applicable to DC. The reason
lays on the lack of symmetry of the algorithm. This lack of symmetry is depicted in Fig. 7. In that figure, it can be
seen that the correlation peak, although peaking in the right position, is not symmetric. As this correlation
coefficient is calculated at discrete increments, this introduces error in the output of the subpixel peak-fitting
algorithm. In DC this source of error is present even if there is no displacement of particles, misleading iterative
offset algorithms.
Image a
Image b
DC-PIV
1
0.5
Correlation coef.
1
Gray level
Gray level
Interrogation
window
1
0.5
0
0
-3 -2 -1
0
1
2
3
Spatial pos. (pix.)
0.8
0.6
0.4
0.2
0
-3 -2 -1
0
1
2
3
Spatial pos. (pix.)
-1
-0.5
0
0.5
1
Displacement (pix.)
Fig 7. Even at zero displacement, DC-PIV losses the correlation peak symmetry if there is truncation of particles at the window
borders.
To avoid the error a new algorithm is proposed: Symmetrical Direct Correlation (SDC). This algorithm consists
in a slight modification of the Direct Correlation.
To force symmetry everything done in image a in respect to image b must be done in image b in respect to image
a and both outputs have to be averaged. A way to fulfill this is to add the contributions from the following two
sources:
1. The correlation of an interrogation window from image a locally correlated along image b.
2. The correlation of an interrogation window from image b locally correlated along image a.
Once the symmetry has been obtained, no error appears for cero displacements and a standard procedure with
subpixel interrogation window offset can be used. This has proven to give good results in tests over synthetic as well
as real images (Nogueira, Lecuona and Rodriguez (2001a) and (2001b)).
A table summarizing these alternatives for small interrogation windows follows. In this table, a qualitative
description is given that is based in the quantitative results of the mentioned works.
s/n reduction because of spurious Error due to truncation of
periodicity of the algorithm
particles
FFT
Present
Medium (peak locking + rms
error)
FFT + discrete offset of
Present
Low (rms error)
interrogation window
DC
Absent
Medium (rms error + small
peak locking)
DC + discrete offset of
Absent
High (rms error + small peak
interrogation window
locking)
SDC
Absent
Medium (rms error + small
peak locking)
SDC + discrete offset
Absent
Very Low (rms error)
of Interrogation
window
7
Overall performance
Normal
Improved
Normal
Normal
Improved
Highly Improved
PIV’01 Paper 1010
4
Conclusions.
Recent studies, here reported, are giving design rules for the development and refinement of multigrid PIV. The
results are applicable to large interrogation windows as well as small windows. Some of this design rules can be
summarized as follows:
If no weighting function is used for the correlation calculation, the growth of unstable modes should be avoided.
This can be performed by keeping the ratio between grid node spacing ∆ and window size in a way that unstable
frequencies are not set to the same values for consecutive iterations.
The use of the weighting function described in expression (1), avoids instability for all wavelengths, but can only
be used with large windows (F ≥ 32 pixels).
The use of this weighting function for large windows gives a further advantage; this is, better positioning of
subsequent windows.
For small windows, the use of Direct Symmetrical Correlation (DSC) plus a discrete window offset improves the
performance of the system.
Acknowledgements
This work has been partially funded by the Spanish Research Agency grant DGICYT TAP96-1808-CE, PB95-0150CO2-02 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°: GRD11999-10835)
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