INSTITUTE OF PHYSICS PUBLISHING
MEASUREMENT SCIENCE AND TECHNOLOGY
Meas. Sci. Technol. 13 (2002) 963–974
PII: S0957-0233(02)33465-9
Analysis and alternatives in
two-dimensional multigrid particle image
velocimetry methods: application of a
dedicated weighting function and
symmetric direct correlation
J Nogueira, A Lecuona, U Ruiz-Rivas and P A Rodrı́guez
Departamento de Ingenierı́a Térmica y de Fluidos, Universidad Carlos III de Madrid,
c/ Butarque 15, 28911-Leganés, Madrid, Spain
Received 5 February 2002, in final form 22 April 2002, accepted for
publication 10 May 2002
Published 20 June 2002
Online at stacks.iop.org/MST/13/963
Abstract
Multigrid particle image velocimetry (PIV) is an open path in the search for
high-resolution PIV methods. It is based on an iterative scheme that uses the
information of initial processing to adapt the method parameters in order to
improve the measurements. This is mainly performed by reducing the size of
the interrogation windows and shifting them. In multigrid PIV, two sources
of error can significantly affect the final measurement quality: (1) the error
coming from the amplitude response of the initial large interrogation
windows to spatial frequencies; (2) the error originating from the truncation
of particles at the borders of the final small interrogation windows. By
applying weighting functions and using symmetric direct correlation both
errors can be reduced, respectively. These techniques have been separately
tested in the past, but a joint implementation has not yet been analysed. This
task is fulfilled and both sources of error are further clarified. For this
purpose, a one-dimensional single wavelength displacement field is used.
This gives us the opportunity to analyse the non-linear behaviour of PIV,
together with the influence of basic parameters on it. In addition to this, the
multigrid method, so far described, is enhanced by compensation of the
particle pattern deformation. The metrological performance of this
advanced method is tested using synthetic images and the results are
compared with those delivered by established PIV methods. Coherence
between these results and those obtained in a real image is also detailed.
Keywords: fluid flow velocity, spectral characterization, multigrid PIV, local
field correction PIV, perturbed and forced jets
1. Introduction
The scientific community and industrial developers recognize
particle image velocimetry (PIV) as an important tool for
measuring velocity in fluid flows. Nevertheless, even the
basic configuration designed to extract only two components
0957-0233/02/070963+12$30.00
© 2002 IOP Publishing Ltd
of the velocity along a two-dimensional domain (2D 2C PIV)
is still under active evolution, mainly by attempts to expand
its capabilities and to reduce the measurement uncertainty.
In the domain of correlation PIV an important aim in
this research activity is the development of higher spatial
resolution techniques. The use of PIV for measuring in
Printed in the UK
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J Nogueira et al
turbulence is recent, but on this topic small-scale description
requires such high resolution, together with a careful spectral
characterization and the removal of bias errors when possible.
The advanced PIV methods that focus on high resolution
are iterative. Relying on initial measurements, they tune
or adapt the method to obtain better ones. In correlation
PIV, two main branches of advanced 2D methods deserve a
mention: local field correction PIV (LFC-PIV) and multigrid
PIV. Information about the former can be found in Nogueira
et al (1999, 2001b). It consists of an iterative PIV method with
subpixel image distortion and shift. Its originality consists of
achieving spatial resolutions far smaller than the size of the
smallest interrogation window. This is possible thanks to the
iterative procedure and by using an appropriated weighting
window that avoids improper response for wavelengths smaller
than the interrogation window.
Information about multigrid PIV can be found in Soria
(1996), Hart (1999), Scarano and Riethmuller (2000) and
Lecordier et al (2001), among others. It is also an iterative PIV
method. In this case, the use of image distortion is optional.
The method achieves high resolution by reducing the size of
the interrogation window with an appropriate shift based on
the previous iteration.
A comparison of the metrological qualities of these
methods can be obtained by comparing figures 1 and 18 in
the review by Scarano (2002). Care should be taken in this
comparison as both horizontal and vertical axes respectively
display inverse concepts. In the paper by Nogueira et al
(2001b) and in this present paper, further evolutions of LFCPIV and multigrid PIV are offered, together with a common
base comparison.
A third way of obtaining high resolution is the use of
hybrid methods. These start by using large interrogation
windows, switching at a certain point to particle tracking
velocimetry (PTV); see Keane et al (1995) and Cowen and
Monismith (1997), among others. The considerations of the
advantages and disadvantages of using PTV are out of the scope
of this paper.
In this paper we mainly focus on launching some proposals
that can increase the accuracy of contemporary multigrid PIV.
In order to evaluate the gains obtained, a comparison with
LFC-PIV seems customary.
In order to establish a common base for comparison, in this
paper peak fitting is always performed using two orthogonal
three-point Gaussian fits.
Although the basic procedure of multigrid PIV has been
documented elsewhere, some notes about it are given below to
establish a framework for further discussion.
In this technique, the iterative cycle of PIV processing is
used for two purposes:
(1) To obtain higher spatial resolution. This is achieved
by reducing the size of the interrogation windows. The
price to pay is accepting the possibility of a lower
signal-to-noise ratio (s/n) and the corresponding higher
measurement uncertainty. The reason for this is a
reduction of the number of correlating particles. This
drawback is especially relevant for those windows small
enough to approach the working limits of PIV. At
intermediate steps in the progression towards smaller
interrogation window sizes, the methodology tries not
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only to increase the spatial resolution, but also to
increase the s/n by reducing the dispersion of particle
displacements within the interrogation window. These
opposing considerations eventually bring an optimum
window size.
(2) To diminish the loss-of-pairs due to the in-plane
displacement of particles between correlating windows.
This is achieved by redefining the location of the
interrogation windows in the PIV images, using the
previously measured displacements. The concept has
been called ‘window shifting’ (see Westerweel et al
(1997), among others). The consequence is an increase
in the s/n and, theoretically, a null average displacement
in the interrogation window at the end of a repetitive
processing. This requires subpixel window shifting and
thus image interpolation.
In addition to this, some advanced methods use ‘image
distortion’ techniques for the same purpose, i.e. s/n ratio
enhancement (Fincham and Delerce (2000), Scarano and
Reithmuller (2000), among others). Image distortion
has already been introduced by Huang et al (1993) at
each window and by Jambunathan et al (1995) over the
whole image. Details on how to perform this distortion
can be found in both works and in Nogueira (1997)
and Nogueira et al (1999), among others. Using this
technique, an attempt is made to reduce the remaining
particle displacements in the interrogation window after
image shift, assuming a continuous local displacement
field between the centres of the interrogation windows,
i.e. ‘grid nodes’. Thus, non-uniform displacements
within the interrogation window are compensated for
by the information obtained in previous interrogations.
Ideally, after a sufficient number of repetitive image
distortion processings, a negligible displacement between
correlating windows remains, thus yielding the maximum
possible s/n.
This technique also requires image
interpolation.
The iterative attribute of multigrid methods is fully
confirmed when there is a point where a decision is taken
on whether to repeat previous interrogations with a different
set-up and the same window size or to progress reducing the
window size. This is the same when the original images are
also deformed. Otherwise, the method is just progressive, in a
hierarchy towards high resolution.
In the study reported here, we do not address the optimum
strategy in the iterative multigrid process, but underpin some
consideration towards optimization, including the number of
iterations and the way to preclude error growth.
The proposals offered in this paper cover large as well as
small window interrogation steps. This is especially relevant
as one of the advantages of multigrid PIV is the low number
of iterations, making all the steps relevant.
2. Synthetic images
In order to compare metrological figures of the proposals,
synthetic images have been used. In these images, the
displacement field, sy (x, y), corresponds to a one-dimensional
(1D) single wavelength λx along the x-direction:
sy = 2 sin(2π x/λx )(pixels).
(1)
Analysis and alternatives in 2D multigrid PIV methods
(a)
(b)
(a)
(b)
Measured
Displacement
Displacement
Real
Position x
Position x
Figure 2. A scheme of PIV displacement distributions in the
measurement of a sinusoidal 1D field: (a) displacement to be
measured; (b) output.
Figure 1. Close-ups of synthetic images (40 × 40 pixels each):
(a) image of type 1; (b) image of type 2.
Amplitudes larger than two pixels have not been
considered in this study, owing to the now current procedure
of shifting the interrogation windows. The out-of-plane loss
of particles, related to large displacements, is considered by
directly eliminating some of them.
Varying the spatial wavelength λx allows us to evaluate
the amplitude and phase response to a monochromatic and
simple signal. Owing to the non-linear characteristics of the
PIV processing, additional considerations are introduced in the
paper, when necessary.
The synthetic images formed by randomly located
particles are discretized assuming an 8 bit square pixel
sensor. The shape of the imaged particles is Gaussian,
and it is integrated with unity fill factor over each pixel
surface (Westerweel 1998). Where particles overlap, the
corresponding intensities are added, thus neglecting laser
interference effects.
The remaining parameters that define the images divide
them into two categories:
• Type 1 images. These are purposely designed to test the
performance of the techniques in respect to the spatial
wavelength amplitude response. For this purpose, they
contain no additional noise. The mean distance between
particle images is small, δ = 2 pixels, i.e. 4/(π δ 2 ) ∼ 0.3
particles per pixel area (ppp). For small enough particles,
the imaged diameter is determined by the diffraction
limit of the lens, rather than the geometrical particle
diameter. This implies a shift towards monodispersivity
of the particle image size distribution. According to this,
the e−2 diameter of the Gaussian particle images was
simplified to have a uniform value of d = 2 pixels.
Although the luminosity of each particle was considered
uniform, overlapping gives a reasonable inhomogeneity,
see figure 1(a).
This type of image is comparable with those obtained
seeding air with micrometre-size oil droplets in large wind
tunnel facilities.
• Type 2 images. It is also important to consider the effect
of noise sources and other difficulties. To fulfil this,
at least partially, and to allow direct comparison with
previous works (Nogueira et al 1999, 2001b) a second
type of image has been designed. The mean distance
between the particle images is larger, δ = 4.5 pixels, i.e.
4/(πδ 2 ) ∼ 0.06 ppp in order to create low s/n areas. The
diameter of the particle images is random, ranging from
d = 2 to 4 pixels. Their random peak intensity ranges
from 0 to 92% of the maximum grey level. Both intensity
and diameter are allowed to randomly and independently
differ up to 10% within each particle pair, which takes into
account effects coming from particle laser scattering and
laser pulse inhomogeneities. Five per cent of particles
have no second image to correlate, simulating out-ofplane displacement. A continuous background grey level
reaches up to 20% of the maximum intensity range and
it is smoothly distributed over large zones (∼60 pixels).
This tries to simulate stray light and room illumination.
This type of image is similar to those obtained in small
laboratory facilities. In these facilities, it is usual
to seed water with pollen, spores or plastic particles.
This, together with smaller magnifications, leads to large
particle image diameters.
Figure 1 depicts a close-up of both types of images.
3. Errors related to iterative PIV with large windows
In a single-step straight PIV processing, correlation and
peak detection processes average the particle displacements
contained in the interrogation window in a complex non-linear
way. Actually, the way in which they are averaged can generate
a substantial difference in the output. In a generic way it is said
that correlation PIV represents the most frequent displacement.
In this section we study the different situations concerning the
mentioned averaging, in order to clarify these concepts.
To evaluate the nature of the averaging process, it is useful
to analyse the response of the PIV to 1D single wavelength
sinusoidal displacement fields, such as those proposed in
expression (1). The window side length F will normalize the
wavelength.
Monte Carlo simulations have been performed to reveal
the details of the averaging process, the results of which will
be presented later. The output of the simulations has shown a
non-linear functional dependence with the input. The degree
of this non-linearity depends on the length scales F , d, δ, λx , sy
and pixel size.
To point out the mechanics of this phenomenon, figure 2
depicts a scheme for a certain, but representative, condition.
It corresponds to an idealization of the measurement obtained
when F /λx < 1. For F /λx > 1 this effect also appears, but in
a more complex way, as it is combined with other effects that
will be later analysed in due detail.
The output shows a deviation over the sinusoidal input
waveform, making it look more squared. This deviation grows
with F /λx when F /λx < 1. From the point of view of
Fourier decomposition, this implies the appearance of negative
965
amplitude third harmonic, besides smaller higher-order terms.
The rationale behind this is as follows.
The flatness of the measured waveform at its top
and bottom is because the PIV may not average all the
displacements within the interrogation window. Actually, it
does not take into account the contribution of particle pairs
whose correlation peak does not overlap with the main one,
and take partially into account those which partially overlap.
The result is that, when the main correlation peak comes
from the maximum sinusoidal displacement, other different
displacements within the interrogation window affect the result
less than they would in the case of a moving average.
Small particles deliver narrow correlation peaks, thus
reducing overlapping and as a result increasing the abrupt
slope change between positive and negative maximum
displacements, see figure 2(b). Increasing the deviation from
the sinusoidal input is the effect obtained and consequently
exacerbating the non-linear effect of PIV. In the limit of very
small pixel-resolved particle diameters, a cluster of correlation
peaks would appear, hence precluding the much-invoked
averaging effect of correlation PIV. The output for a sinusoidal
input would be something similar to a square waveform with
notches.
On the other hand, when the particles are larger than
the displacements within the interrogation window, one can
imagine the situation where all the individual peaks fully
overlap. In this situation, the ensemble correlation peak
represents a particle-weighted average of the displacements
within the window. For this reason, the PIV response to a
spatial waveform is frequently referred to as being similar to
that of the moving average.
Further analysing this averaging of displacements within
the interrogation window, an additional source of error appears
for F /λx > 1, which is specifically important for iterative
processing.
Nogueira et al (1999 and 2001b) have documented that
PIV interrogation windows, without any grey level weighting,
introduce a negative amplitude response for several frequency
intervals. Processing the synthetic images proposed in
section 2 allows a Monte Carlo analysis of this issue. The
response of a single-step straight PIV (not using window
weighting) has been evaluated as the ratio of the amplitude
at wavelength λx , in a Fourier decomposition of the output
waveform, over the sinusoidal input amplitude. Figure 3 shows
the results obtained. This figure also shows the amplitude
response of the moving average with filter width F , as a
reference.
In this figure, the behaviour of the PIV is as described
above, in the interval 0 F /λx < 1, which is marked with
black lines. Out of this interval, one can observe the negative
amplitude response, also mentioned above. This response
is analogous to that of the moving average. This means a
phase inversion response (shift of π radians). For larger
values of F /λx (not depicted in figure 3) successive cycles
of positive and negative amplitude response appear, matching
those of the moving average. In terms of error, these negative
response intervals mean that the root mean square of the error,
rms(e), is larger than that of the signal amplitude, rms (sy ).
rms (e)/rms (sy ) > 100% means the possibility of obtaining
errors larger than 100% in the final output.
966
F und am en ta l wa vele ngth re lative a m plitude
J Nogueira et al
1 .0
0 .8
0 .6
0 .4
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0 .0
A
-0 .2
C
B
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-0 .6
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0 .0
0.2
0 .4
0 .6
0 .8
1.0
1 .2
1 .4
1 .6
1.8
2 .0
N ormalised window size F / λ x
Figure 3. The 1D amplitude responses of single-step PIV without
any weighting function: circles, F = 64; squares, F = 32; open
symbols, type 1 images; filled symbols, type 2 images; full curve,
moving average amplitude response. The squares with a dashed
outline indicate the displacement distribution within the
interrogation window, not to scale.
It is obvious that this phenomenon also occurs for particle
displacement distributions having a shape similar to the
sinusoid (square, triangle, etc).
The mechanisms of correlation peak formation, introduced above, explain the layout of the data. A particle pair
with displacement sp contributes to the main correlation peak,
with displacement sm , only if their difference is less than the
radius considered in the peak fitting algorithm, R, plus the pair
peak waist wp :
|sm − sp | < R + wp .
(2)
This condition does not consider spurious contribution to the
peaks.
Alternatively, it is said that correlation PIV represents the
most frequent particle displacement. This is a more vague
condition than that indicated in expression (2), coinciding with
it somewhere. However, both arguments justify the higher than
moving average amplitude response found in figure 3.
This mechanism also justifies what happens in the region
between points of null response, A and C in figure 3. There
is more aggregation of pair peaks with the opposite sign to the
one corresponding to the centre of the window, thus giving a
false displacement with a maximum value near point B.
Figure 3 shows that the response obtained in type 2 images
is smaller than that obtained in type 1 images. This is caused by
the larger particle diameters used in these images, allowing the
averaging effect to increase in the peak aggregation process, as
indicated previously. Furthermore, the noise content smears
the peaks extending the averaging effect, thus improving
measurement from the linear theory point of view. This means
introducing fewer harmonics and thus less spectral distortion.
On the other hand, its linear low-pass effect is higher.
The smaller response of the 32 pixel windows in respect
to the 64 pixel windows is because, for the same F /λx , λx is
smaller, which leads to larger peak overlapping. This results
from the closer spurious peaks correlating false particle pairs.
Analysis and alternatives in 2D multigrid PIV methods
3.1. Effects of the phase inversion on the amplitude response
of multigrid PIV
Figure 3 shows that the negative amplitude response introduces
errors larger than 100% at certain wavelengths (vertical
distance to 1). As a result, a multigrid PIV method using
windows with F = 64 pixels will introduce this error between
λx = 64 and 32 pixels, besides other smaller wavelength
intervals. Analogously, windows with F = 32 will give the
same effect for wavelengths between λx = 32 and 16 pixels.
These scales are common in flows diagnosed by PIV.
This error gives an initial window shift opposite to the real
one, which can fool the adaptation of the method in search of a
Displacement field: s y = 2sin(2πx /λ x ) (pixels)
1.2
F =32 followed by F =16
1
rms(e )/rms(s )
Here, the fundamental wavelength λx has been used to
measure the amplitude response. Other choices, such as the
use of the overall amplitude, would give different figures, but
obviously the behaviour is the same. Actually, this option
would yield a smaller amplitude response, not revealing the
non-linear effect. This is of relevance for turbulence spectral
assessment.
Another relevant effect is that the amplitude response of
PIV does not decrease with F /λx as fast as a low-pass filter,
such as the moving average. This means that noise, or small
wavelength content, in the displacement field could introduce
in the output an appreciable erroneous influence, of special
concern in the intervals of negative amplitude response.
A main consequence is that spatial derivatives can contain
large errors for F /λx > 0.5 in single-step unweighted PIV
processing, even with no extra consideration to peak locking
(Westerweel 1998, Nogueira et al 2001a).
The effects of particle diameter and of peak fitting radius
found so far make the amplitude response curves such as
figure 3 not unique; on the contrary, they are dependent on
basic parameters of images and processing.
If there are gradients in the vertical displacement,
∂sy /∂y = 0, the effects presented here are modified, as when
∂sx /∂y = 0 or/and ∂sx /∂x = 0. The extension of the
modification is complex to describe, but responds to the same
background described here.
In summary, the simple approach to single-step
unweighted correlation PIV processing just described
highlights its non-linear behaviour. A boost effect of the
large spatial gradients and a damp effect of the small gradients
has been described, analysed, and the originating mechanism
explained.
Structures in a flow field are undoubtedly more
complicated than 1D sinusoids. A study of these remains to
be performed. Nevertheless, if the local shape is similar to
a sinusoid, this makes what has been described here directly
applicable. Expression (2) can be considered as a design rule
for PIV implementation.
In what follows, image distortion techniques are
considered. In addition to the advantages offered by this
technique, its effect on the non-linearity described up to now is
relevant. Image distortion iteratively reduces the displacement
to be measured, eventually converging to a null displacement
to be measured. In this way the response of the method after
a sufficient number of iterations progressively approaches a
linear response. Window shifting alone does not offer this
possibility.
F =64 followed by F =16
0.8
1 step, F =16
0.6
0.4
0.2
2 steps, F =16
0
20
25
30
35
40
45
50
λ x (pixels); (δ =2 pixels).
55
60
65
Figure 4. The relative error shown as a function of wavelength for
different two-step PIV methods, with image distortion and without
any weighting function. A single-step method has been added for
comparison; type 1 images.
better measurement; as a result reducing the s/n and eventually
leading to a spurious vector or outlier. Even without the arousal
of outliers a deleterious effect appears. Figure 4 shows that, for
the mentioned intervals, the use of large windows worsens the
measurement, even when small windows are used afterwards.
To perform this test, the grid distance has been set to 8
pixels for all the iterations. The results vary with different grid
distances, but the qualitative behaviour is the same.
This figure shows that, in both intervals of negative
amplitude response, a single iteration with a small window
performs better than processing first with a large window
and afterwards with a small window. However, sometimes
the use of large windows is necessary to obtain robustness
of the method against difficulties with seeding, illumination
or noise, i.e. the resistance to produce outliers. In other
words, robustness is the resistance to lock in a false peak.
Besides this, a large window can cope with large displacements
but not small displacements. Window shifting and window
deformation using the previous measured displacement, in
principle, can make feasible the following measurement step
with a smaller window. This way of processing means
extending the measuring range and increasing the s/n, thus
enhancing robustness.
The multigrid methods tested in figure 4 include image
distortion.
The behaviour of the methods would be
qualitatively similar with window shifting only, nevertheless
the image distortion amplifies the effect of a good or a bad
measurement. This means that avoiding image distortion
would reduce the effect of the error in cases with large
windows, but would also reduce the good performance in the
case of two steps with F = 16 pixels. Another consideration
to be taken into account is that the presence of outliers could
also amplify the error in the case of image distortion. In the
test presented here (d = 2 pixels, displacement <2 pixels), the
s/n was large enough to avoid outliers in the first iteration.
It can be concluded that, in a conventional multigrid
method, large windows may introduce robustness, but often
at the expense of accuracy, due to the negative amplitude
response.
3.2. Error amplification and instabilities
Multigrid methods, processing through several steps with
image distortion, present an additional effect which comes
967
(a)
(b)
∆
F
F
Figure 5. Two extreme ways of implementing the compensation of
the particle pattern deformation: (a) distortion using only the grid
nodes of the window corners; (b) using all the grid nodes. For
clarity (b) displays only some displacement vectors.
from the negative amplitude response. It produces a distortion
of the PIV images in the opposite direction to the correct
one. This increases the deformation between the correlating
images instead of decreasing it. Hence, this can lead to a
divergent instability, unless countermeasures are taken, such
as those documented in Nogueira et al (1999, 2001b). These
works conclude that a proper selection of the ratio between
the window size F and the grid-node spacing avoids the
instability. The criterion there presented, Fn 2n , is here
proposed again, but now expanded in its application, allowing
it to operate on two consecutive steps, n and n + 1, thus being
less restrictive and yielding the same effect:
Fn+1 2n
(3)
where n is the step number.
In successive steps, if the criterion of expression (3) is
held, the error due to the negative response appears, but no
instability is triggered. This is because for F 2 the
compensation of the particle pattern deformation introduces
the aliased wavelengths as error, instead of the unstable
wavelengths. These aliased wavelengths fall outside the
unstable range.
If the hierarchy of successive steps does not follow the
criterion in expression (3), there is another way of avoiding
the instability, by relying on the same theoretical background.
This consists of implementing the deformation of the image
for each interrogation window just by a reduced number of
grid nodes, instead of the more detailed piecewise deformation
using all the grid nodes. Figure 5 depicts two extreme ways of
doing this. Obviously, the first way implies a low-pass effect
when compared to the second option. The application of lowpass finite impulse response averaging filters on every step to
the displacement field is another alternative, but also brings a
loss in high-resolution capability.
In this study we also indicate that the joint use of some
of these resources, such as image distortion and low-pass
averaging filters, has to be implemented with care, as their
combination can introduce in the final output stable artefacts,
instabilities apart.
3.3. Avoidance of negative amplitude response
There is an unproblematic way of avoiding the negative
amplitude response, contrasting with the cloudy horizon drawn
for the compensation of the particle pattern deformation in the
previous subsection. As stated at the beginning of section 3,
the averaging of small features in the displacement field brings
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Fundamental wavelength rel. amp.
J Nogueira et al
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Normalised window size F /λx
Figure 6. The amplitude response of single-step PIV with the
weighting of expression (4). The dashed curve represents the
moving average and the full curve represents the moving average
weighted with expression (4). The symbols are the same as those in
figure 3.
some problems. However, modulating the way this averaging
occurs is possible, thus alleviating the negative aspects. One
approach to doing this is to apply a weighting function on the
grey level distributions, f and g, of both interrogation window.
A weighting function υ, designed to avoid negative amplitude
response, has already been developed by the authors (Nogueira
1997, Nogueira et al 1999). This function is
2
2
ξ η
η
ξ υ 2 (ξ, η) = 9 4 − 4 + 1 4 − 4 + 1
F
F
F
F
(4)
where ξ F /2 and η F /2 are orthogonal coordinates with
origins at the centre of the interrogation window.
Using this weighting function, the computation of the
correlation coefficient, Clm , has to follow expression (5):
Clm =
F /2
F /2
ξ,η=−F /2
ξ,η=−F /2
υ(ξ,η)f (ξ,η)·υ(ξ,η)g(ξ +l,η+m)
.
F /2
2
2
ξ,η=−F /2 υ (ξ,η)g (ξ +l,η+m)
υ 2 (ξ,η)f 2 (ξ,η)
(5)
Here, l and m are the image displacements.
Figure 6 shows the equivalent results to those depicted in
figure 3, but using this weighting function.
Although this figure plots only for F /λx 2.0, no
negative amplitude response has been detected when using the
weighting function of expression (4). In addition to this, there
are no null amplitude response wavelengths. Actually, the
amplitude response, r, of the moving average weighted with
this function is always positive:
sin(π F /λx )
36
1
−
r=
π F /λx
(π F /λx )2 (π F /λy )2
sin(π F /λy )
× 1−
.
(6)
π F /λy
This window shows both a higher cut-off frequency and a
smaller low-pass order than the unweighted window, which is
favourable in terms of amplitude response.
The absence of negative amplitude response allows for
correct shifting and/or deformation of interrogation windows
in advancing steps, therefore avoiding the instability described
Analysis and alternatives in 2D multigrid PIV methods
Displacement field: s y = 2sin(2πx /λ x ) (pixels)
0.6
0.6
0.5
Unweighted multigrid
Unweighted multigrid
rms(e )/rms(s )
rms(e )/rms(s )
Displacement field: s y = 2sin(2πx / λ x ) (pixels)
0.7
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
Weighted multigrid
Weighted multigrid
0
0
20
25
30
35
40
45
50
λ x (pixels); ( δ=2 pixels).
55
60
65
Figure 7. The relative error of weighted and unweighted multigrid
PIV methods applied on type 1 images, seven steps. The main text
gives the details.
in the previous subsection. This point has been thoroughly
checked in synthetic as well as in real images.
Unfortunately, the application of a weighting function
induces a small but accumulative erroneous slip in the
measurement (Nogueira et al 1999), which shows up after a
large number of steps. This slip is larger with smaller windows,
making inadvisable its use for windows with F < 32 pixels.
4. Comparison between weighted and unweighted
multigrid PIV down to F = 16
In this section we report on the results of testing the proposal
of weighting the interrogation windows. A conventional
multigrid method has been implemented for comparison and
is labelled ‘unweighted’ in figures 7 and 8. It includes
the compensation of the particle pattern deformation using
bilinear velocity interpolation between consecutive grid nodes,
as in Nogueira et al (1999), and using biparabolic grey level
interpolation.
We have selected F = 16 pixels in all the iterations. The
use of larger windows introduces errors within the tested range
of frequencies, as figure 4 depicts, and it is advisable only when
a high robustness is required.
The iteration is formed by six steps of = 8 pixels,
followed by one of = 4. In this way there is no error
amplification for any wavelength, even smaller than 16 pixels,
because of expression (3).
It must be accepted that the parameters defined here for the
multigrid PIV are somewhat arbitrary. Nevertheless, the setup has been chosen with care, so that the performance yield is
representative of the different possibilities. For a certain single
wavelength, error minimization would lead to a different setup, but better overall performance is difficult for a conventional
multigrid method.
This method has been compared with a weighted method
that uses the same grid distances as the conventional multigrid
example and the same number of steps. F = 64 is used in the
first iteration, followed by F = 32 (dashed curves in figures 7
and 8). The large size weighted window warrants that, in
addition to the increase in accuracy, which is observable in the
figures, there is a high robustness.
Another weighted scheme was tested in this campaign.
It corresponds to the same window sizes, but the first step of
= 8 pixels is followed by six steps of = 4. Owing to the
20
25
30
35
40
45
50
55
60
65
λ x (pixels) ; ( δ=4.5 pixels).
Figure 8. The relative error of weighted and unweighted multigrid
PIV methods applied on type 2 images, seven steps. The main text
gives the details.
CORRELATION OF
INTERROGATION WINDOWS
FROM IMAGES a AND b:
FFT SPURIOUS
CONTRIBUTION TO
CORRELATION
Displacement >
– 1 pixel
Figure 9. Error mechanism introduced by the periodicity of the FFT.
weighting, there is no amplification of the error and the grid
distance can be freely chosen. Again, an additional increase in
accuracy, due to a closer sampling, is evident (lower full curve
in figures 7 and 8).
It is obvious that these later advantageous techniques need
a larger computing time, because of the introduction of the
weighting function. Note that part of the denominator in
expression (4) changes when l and m change and thus has to
be calculated the same number of times than the numerator. In
addition, the larger number of vectors in the second weighted
case increases the computing time as well.
5. Errors related to iterative PIV with small
windows
The application of windows smaller than 16 pixels on real
images still seems to be a subject under development. Several
recent applications of advanced multigrid methods do not use
small windows, an example is Scarano and Riethmuller (2000).
In that paper, devoted to multigrid PIV, the authors process real
images with windows of 32 and 16 pixels.
Nevertheless, there is ongoing research on methods
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, giving an optimum diameter of around 6 pixels.
Nogueira et al (2001a) have introduced a way to
successfully deal with small windows and small particles.
The algorithm presented there is the symmetric direct
correlation (SDC). It has been proven that this procedure gives
good results in tests over synthetic as well as over real images,
using subpixel offset of the interrogation windows. Here, it
969
J Nogueira et al
Image a
Image b
0.5
Correlation coef.
Gray level
Gray level
1
1
0.5
0
0
-3 -2 -1
0
1
2
DC-PIV
1
Interrogation
window
3
0.6
0.4
0.2
0
-3 -2 -1
Spatial pos. (pix.)
0.8
0
1
2
3
-1
Spatial pos. (pix.)
-0.5
0
0.5
1
Displacement (pix.)
Figure 10. A 1D example showing that, even at zero displacement, DC-PIV loses the correlation peak symmetry if there is a truncation of
particles at the window borders. Single particle truncated at the border of a 5-pixel window.
Image a
Image b
SDC-PIV
1
1
0.5
1
Interrogation
window
Correlation coef.
Gray level
Gray level
Interrogation
window
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.)
Figure 11. An example of the symmetry recovery of SDC-PIV with a single particle truncated at the window border and null displacement.
Filled and open symbols depict the correlation function of the two DC methods. The full curve depicts the average.
will be evaluated with the compensation of the particle pattern
deformation instead of the simpler subpixel offset.
In section 5.1 we give a brief description of the underlying
principles, as an introduction to discussing the results of the
tests on its performances. In this section we also evaluate the
increase in accuracy with respect to state-of-the-art algorithms.
5.1. Selection of the correlation algorithm for small windows
The correlation function with a pair (a and b) of large
interrogation windows is time consuming. In order to save
time, most PIV methods turn to the fast Fourier transform
(FFT), instead of the direct correlation (DC) calculation.
The use of FFT completely ignores the image out of the
interrogation windows, substituting it by a periodic patch
of windows. The effect is the introduction of a spurious
contribution in the calculation of the correlation coefficient
for any non-null displacement. This reduces the s/n. Figure 9
depicts the mechanism of this source of error, which is more
relevant the smaller the window is.
With small interrogation windows, it is possible to afford
the increase in computing time of the DC. Using it avoids the
reduction of the s/n associated to the periodicity of the FFT.
Nevertheless, getting rid of this source of error does not
necessarily mean having a better performing algorithm; other
sources of error have to be checked. Detailed studies such as
Huang et al (1997) show that DC produces errors, even when
evaluating null displacements between two identical images.
One of the main reasons for this is the truncation of particles
at the window borders (Nogueira et al 2001a).
The way to avoid this error in FFT, using a discrete shift of
the interrogation window, is useless for DC with a symmetrical
970
peak-fitting function or with small particles (Nogueira et al
2001a).
The reason for the error lies with the lack of symmetry of
the DC algorithm itself, which leads to an asymmetric peak.
Figure 10 shows that the correlation peak of a single 2 pixel
wide particle, truncated at the border, although peaking in the
right position, is not shape symmetric. As the correlation
coefficient is calculated at discrete increments, this introduces a
bias error in the output of the subpixel peak-fitting algorithm.
With DC, this source of error is present even if there is no
displacement of the particles, misleading iterative window
shifting algorithms.
We have proposed a new algorithm to avoid the error:
SDC (Nogueira et al 2001a). This algorithm consists of a
slight modification of DC, as follows.
To force symmetry everything done in the first image, a,
in respect to the second image, b, must be done in image b in
respect to image a, and both outputs have to be averaged.
Once recovering symmetry, no error appears for cero
displacements as sketched in figure 11.
Furthermore, the symmetry of the algorithm allows for
iterative subpixel offset dealing with the error coming from
particle truncation (Nogueira et al 2001a), as in the FFT case.
It should be pointed out that the convergence of iterative
methods with image deformation under ideal conditions is
cero displacement for the cases of FFT and SDC, presenting
no truncation error. Other ways to deal with the truncation
error include weighting windows, such as the one already
introduced for large windows, or masks that erase particles at
the borders (Lourenco and Krothapalli 2000). This last option
needs identification of particles, as in PTV, which is not always
Analysis and alternatives in 2D multigrid PIV methods
Displacement field: s y = 2sin(2π x / λ x ) (pixels)
0.2
Weighted + 3 DC steps
rms(e )/rms(s )
rms(e )/rms(s )
Weighted + FFT steps
0.6
Weighted multigrid
Weighted + 3 FFT steps
0.1
LFC-PIV
0.5
Weighted + SDC steps
0.4
Weighted multigrid (only large windows)
0.3
0.2
0.1
Weighted + 3 SDC steps
LFC-PIV
0
0
20
25
30
35
40
45
50
55
60
65
λ x (pixels); ( δ=2 pixels).
Displacement field: s y = 2sin(2πx /λ x ) (pixels)
0.15
Weighted + FFT steps
0.1
Weighted + SDC steps
Weighted multigrid (only large windows)
Optimum
measurement
0.05
0
5
10
LFC-PIV
15
20
25
30
35
0
5
10
15
20
25
30
35
40
45
50
Number of steps
Figure 12. The relative error of small windows processing type 1
images with multigrid schemes, weighting at large windows steps
and with several options at small ones. Ten steps. The large window
method, LFC-PIV, is included for comparison.
rms(e )/rms(s )
Displacement field: s y = 2sin(2πx /λ x ) (pixels)
0.7
40
45
50
Number of steps
Figure 13. The evolution of the relative error shown as a function of
the number of iteration steps. The multigrid schemes of figure 12
have been allowed to progress for 50 steps. Type 1 images, λx = 31
pixels.
feasible for small δ. The advantages and disadvantages of PTV
are studied elsewhere.
5.2. SDC implemented with compensation of the particle
pattern deformation
The SDC is proposed here for small interrogation windows.
As commented in section 3 it is not advisable to weight such
windows and thus they will be used unweighted. Therefore, to
test SDC together with the compensation of the particle pattern
deformation, the extra care to include is the restriction from
expression (3).
Figure 12 presents the results of a test performed on type 1
images. This test takes as landmark the ‘weighted multigrid’
method from figure 7 (lower full curve, corresponding to one
step of F = 64, = 8 followed by six steps of F = 32,
= 4). After these steps, the multigrid process is completed
in three different ways:
(i) with three further steps of SDC using F = 8 and
= 4, giving the performance line marked with hollow
diamonds;
(ii) the same three steps but with FFT giving slightly worse
results;
(iii) with three steps of DC, clearly inappropriate, as predicted.
Figure 14. The evolution of the relative error as a function of the
number of iteration steps. The multigrid schemes of figure 12 have
been allowed to progress for 50 steps. Type 2 images, λx = 31
pixels.
In addition, figure 12 shows the performance of the LFCPIV method, with F = 64 and = 4, for comparison.
Besides the information contained in figure 12, it is of
interest to observe the behaviour of the different schemes with
the number of processing steps. For this purpose, in the
following tests the number of iterative steps is large. In all
the methods considered, a control algorithm avoids excessive
growth of errors and allows for a fair comparison with other
techniques, such as LFC-PIV. This is the same algorithm
that LFC-PIV uses; this means refusing correction of the
measurement in a processing step if the peak of the local
correlation coefficient and those of at least five neighbours are
all reduced (Nogueira et al 2001b). This has almost no effect
for a small number of steps, but improves the measurement
when the number is large. In LFC-PIV, this algorithm tries to
approach the minimum error. When it is reached, the number
of iterations defines the optimum measurement. Figure 13
indicates this point. There, one can observe the evolution of
the error with the number of iterations for the case of λx = 31
pixels. The scale of the error has been zoomed in order to
appreciate the differences among the methods. It seems clear
that the reduction of window size allows us to reach less error
faster, but the weighted multigrid that only uses large windows
(F = 64 for the first step and 32 afterward) achieves a similar
performance after several further iterations.
For type 2 images, the results, shown in figure 14, are
somehow different. The performance of those methods that
reduce the window size down to F = 8 yield an error increase
as soon as this size is applied in step eight. On the other hand,
the results shown here are in some way coherent with those in
figure 13. Both figures show that the larger the interrogation
window is, the lower the error can be. This occurs after a
larger number of steps and with appropriate weighting for large
windows.
We can attribute the difference between figures 13 and 14
either to the larger distance between particles or to the
noise in type 2 images. To distinguish both sources, we
measured images similar to those of type 1, but with δ = 4.5
pixels. The results were qualitatively equal to figure 13,
reproducing almost the same graph, but with 30% higher
error. Consequently, the difference in the error growth rate,
depicted in figure 14, is attributed to the noise sources detailed
in section 2.
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J Nogueira et al
2 pixels
Figure 15. Close-ups of the two real images to correlate (40 × 40
pixels).
= 5 pixels
Figure 17. Flow field measured with LFC-PIV within the square
zone in figure 16. It acts as reference for the performances of other
methods. The grid distance corresponds to 4 pixels. The local
correlation coefficient indicates the presence of twelve false vectors
(1.2% of the total number of vectors). A 11.3 pixel mean
displacement has been subtracted.
Figure 16. General structure of the real flow. Only the zone within
the square is used in the test. A 4.7 pixel mean displacement has
been subtracted.
In summary, the use of weighted windows in a multigrid
method offers a decrease in measurement uncertainty. Care
should be taken when using image deformation with grid
distances smaller than F . In these cases, the error of all the
options here proposed decreased up to a certain number of
steps, reaching a minimum, to grow afterwards. This error
grows more rapidly with noisy images and small windows.
A method based on large weighted windows (without the
reduction of size associated to multigrid) reaches the same
level of uncertainty, but giving even better results for large
numbers of steps. However, none of methods can match the
approximately half uncertainty figure of LFC-PIV on both
types of synthetic images.
Although generalization of these conclusions for < F is
not possible with the available data, they seem to be consistent
with the mechanics explained here and we have not found cases
where different conclusions arise.
The main advantage of small window sizes over other
advanced techniques based on large windows is the reduction
of computing time. On the other hand, higher noise content
in the images seems to compromise its application. This may
indicate a loss in robustness for difficult images.
6. Coherence when processing real images
As the displacement field is unknown in real images, no
direct evaluation of the errors is possible. Nevertheless, the
coherence or likelihood of the results in real images with those
obtained in synthetic ones can be evaluated.
972
The real PIV image chosen for this test corresponds to a
study on the near field of co-flowing round water jets subjected
to simultaneous axial forcing and azimuthal perturbation.
This locks jet instabilities and reduces out-of-plane particle
losses in meridian symmetry planes. The diameter of the
inner nozzle, D, is 15 mm, the mean flow velocity, umean , is
0.69 m s−1 and the velocity of the outer flow, u0 , is 0.12 m s−1 .
Consequently, the Reynolds number, ReD = D(umean −u0 )/ν,
is 7000. The frequency of the axial forcing is 10 Hz.
The seeding used for the PIV measurements is inertized
licopodium spores, which have a diameter around 20 µm and
are neutrally buoyant. A DANTEC DoubleImage 700 camera
was used. Its field of view was set around 45 × 35 mm,
corresponding to the 768 × 484 pixels CCD sensor. The time
between pulses was set to 700 µs. More details about the
experiment and the PIV method can be read in Ruiz-Rivas
et al (2002) and Rodrı́guez et al (2001). Close-ups of the
experimental PIV images are depicted in figure 15, showing a
similar layout as the type 2 synthetic image in figure 1(b).
The particular case examined here corresponds to the
small flow inhomogeneities in the potential zone of the jet
core, shown in the square in figure 16.
The coherence test is based on the results presented in
section 5, supported by the similarity of the images in figures 1
(type 2) and 15. There, LFC-PIV has given accurate and robust
results. Its output after 80 iterations, depicted in figure 17, will
be used as reference.
The deviation between this vector field and those
calculated with other methods will suit as an approximation
of the error. It is marked with an asterisk, rms (e)∗ . In this
case the reference displacement field shows rms (s) = 0.9
pixels, slightly smaller than the amplitude chosen for the
synthetic cases. Figure 18 depicts the results of this evaluation.
After 80 steps, this plot would show null error, due to the
arbitrary reference chosen.
This figure shows a similar behaviour to that depicted
in figure 14, thus implying coherence with the results using
synthetic images, although it seems to correspond to images
with a higher noise content.
Analysis and alternatives in 2D multigrid PIV methods
Weighted + FFT steps
& Weighted + SDC steps
0.8
rms(e )*/rms(s )
In order to ascertain the validity of the conclusions
drawn from synthetic 1D displacement images, a real image
with similar particle layout, but with turbulent displacement
distribution, was prepared. The results obtained in the real
image are coherent with those in the synthetic images.
Displacement field: real
0.9
0.7
0.6
0.5
0.4
Weighted multigrid (only large windows)
0.3
Acknowledgments
0.2
0.1
LFC-PIV
0
0
5
10
15
20
25
30
35
40
45
50
Number of steps
Figure 18. The evolution of the relative error as a function of the
number of iteration steps for the methods studied in this paper. They
correspond to the analysis of the real image indicated in
figures 15–17.
7. Summary and conclusions
The methodology of using 1D sinusoidal waves offers the
possibility of studying the linear and non-linear responses of
PIV, giving hints to further progress in the understanding of its
behaviour.
It has been shown that a negative amplitude response
increases the measurement uncertainty for some wavelength
intervals in single-step unweighted PIV implementation. In
standard multigrid PIV methods, the effect reduces the signalto-noise ratio, thus increasing the possibility of outliers. Even
more, it can induce instability in methods incorporating image
distortion and image shifting techniques, thus leading to
outliers.
Specific ways of implementing the hierarchy of the
progression towards smaller window size and smaller grid
node distance are proposed to avoid this instability. The
use of a proper weighting function offers a more robust
and uncomplicated alternative, although valid only for large
windows. This technique eliminates negative amplitude
response and avoids other errors. No restrictions remain in the
full use of image distortion technique to increase the spatial
resolution.
Under favourable conditions for successfully applying
unweighted windows of 16 × 16 pixels (i.e. evaluating the
response for wavelengths larger than 20 pixels in flow fields
that do not produce outliers for such windows) the results with
weighted windows of 32 × 32 pixels with the same number of
processing steps are clearly better. Figures 7 and 8 illustrate
this point.
The use of this weighting function with large windows
gives a further advantage; i.e. a better positioning of subsequent
windows, hence contributing to the robustness of the method.
Smaller windows of 8 × 8 pixels offer a substantial
reduction of computing time. When almost no noise is present,
they approach the better accuracy of large weighted windows in
fewer steps, see figure 12. In these tests, the systems using SDC
algorithms performed better than those based in FFT or DC.
A reasonable noise content severely reduces the
performance of such small windows. Besides this, a reduction
in robustness is anticipated.
Another main conclusion is that larger weighted windows
offer the potential of a lower error figure, but this shows up
after a substantial number of iterating steps.
This work has been partially funded by the Spanish Research
Agency grant AMB1999-0211, DPI2000-1839-CE, Feder
1FD97-1256 in collaboration with the Universidad de Las
Palmas de Gran Canaria and under the EUROPIV 2 project (a
Joint Programme 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 no GRD1-199910835).
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