INSTITUTE OF PHYSICS PUBLISHING
MEASUREMENT SCIENCE AND TECHNOLOGY
Meas. Sci. Technol. 12 (2001) 1911–1921
PII: S0957-0233(01)23191-9
Local field correction PIV, implemented
by means of simple algorithms, and
multigrid versions
J Nogueira, A Lecuona and P A Rodrı́guez
Department of Mechanical Engineering, Universidad Carlos III de Madrid, c/Butarque 15,
28911-Leganés, Madrid, Spain
Received 21 March 2001, in final form 11 June 2001,
accepted for publication 21 August 2001
Published 9 October 2001
Online at stacks.iop.org/MST/12/1911
Abstract
Local field correction particle image velocimetry (LFCPIV), which was first
presented in 1997, is the only correlation PIV method able to resolve flow
structures smaller than the interrogation window. It presents advantages
over conventional systems and thus offers an alternative in the field of
super-resolution methods. Improvements of the initial version are likely to
promote its application even further. The issues defining some of these
possible improvements were already indicated in the paper that originally
introduced LFCPIV, but not developed. This work presents refinements and
also simplifications of the technique, so that it can be applied using current
algorithms of advanced correlation PIV systems. Furthermore, these
refinements reduce the measurement error and enlarge the range of
application of LFCPIV. In particular, the application of the system is no
longer constrained to images with mean distances between particles larger
than 4 pixels. Besides that, the use of interrogation windows smaller than in
its previous version is evaluated. This allows multigrid LFCPIV
implementations. The results show how multigrid LFCPIV can obtain better
measurements than can the usual multigrid PIV, but still the refined version
of the LFCPIV technique performs even better, at the expense of a larger
computing time. The performance of these methods is evaluated for
synthetic and real images. This includes examples in which the ability to
cope with gradients in velocity, gradients in seeding density and the
presence of boundaries is highlighted.
Keywords: LFCPIV, multigrid PIV, super-resolution PIV, wingtip vortex
1. Introduction
Basic particle image velocimetry (PIV) is able to describe
two components of the velocity along a two-dimensional
domain (2D 2C PIV). It has been established as an important
development method for research as well as for industry.
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. Much of the effort put into
development of the technique has recently been focused on the
extraction of all the information obtainable from the images,
including the development of super-resolution systems. On the
0957-0233/01/111911+11$30.00
© 2001 IOP Publishing Ltd
other hand, the development of robust algorithms able to cope
with especially difficult situations (i.e. large velocity gradients,
seeding inhomogeneities, the presence of boundaries, etc.) has
been identified as a priority in research. Generally, all these
advanced algorithms are characterized as 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.
(i) Multigrid PIV. The iterative cycle is used to reduce the
size of the interrogation windows and also to redefine
Printed in the UK
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J Nogueira et al
their location in the PIV images. When the size of
the interrogation window is reduced, a fractional pixel
offset of the interrogation window is essential in order to
reduce peak locking. This was experimentally found by
Lecordier et al (1999) and Scarano and Riethmuller (2000)
and later explained theoretically by Nogueira et al (2001).
This last work also reported the necessity of great care in
the last steps, recommending to switch to a symmetrical
direct correlation algorithm (SDC).
(ii) LFCPIV, see Nogueira et al (1999). The interrogation
windows are fixed in size and location, but after each
iteration the image is redefined through compensation of
the particle pattern deformation caused by the velocity
gradients in the flow field. This implies both displacement
and deformation of the image. It is performed using
the displacement field from the previous evaluation. The
theoretical end of this process is when both images fully
coincide, thus yielding the highest correlation coefficient.
It presents the ability to resolve small structures with
large interrogation windows. Owing to the weighting,
the compensation and the iterative procedure, the limit
of this resolution is constrained only by the Nyquist
criteria applied to the mean distance between particles.
The convergence of the process for every wavelength is
detailed in Nogueira et al (1999). Of course, the grid
node spacing should also be set to satisfy the Nyquist
criteria and, even in this situation, the noise present in
a PIV image restricts the number of useful iterations
and thus the real achievable resolution. Further insight
into this issue is offered later in the paper. The system
is a development of precursory techniques (Huang et al
1993, Jambunathan et al 1995) that were intended to cope
with large velocity gradients. These systems lack the
high spatial resolution of LFCPIV owing to an instability
related to high spatial frequency. In LFCPIV a proprietary
weighting function was defined in order to avoid this. The
combination of these two capabilities in LFCPIV (i.e. the
ability to cope with large velocity gradients and to resolve
small structures in the flow) results in a very robust highresolution technique.
It is relevant that, in both branches, an interpolation of
the grey level image is needed, either to compensate for the
deformation of the particle pattern or to produce a fractional
pixel displacement. In addition to that, LFCPIV needs to
include a weighting function in the correlation algorithm.
In section 2 of this work it will be stated that this is also
desirable in multigrid PIV, leading to the conclusion that the
aforementioned weighting is worthwhile for both kinds of
system.
In a few words, to implement advanced 2D algorithms in
correlation PIV, in addition to the usual FFT based correlation
algorithm, any of these three other ones have to be used:
(i) interpolation of the grey level image,
(ii) application of a weighting function and
(iii) using symmetrical direct correlation (SDC) or the
equivalent for the last steps of multigrid PIV.
Of these three, the former two will be the only ones
needed for the implementation of LFCPIV described here.
1912
Consequently, this refinement with respect to the previous
version (Nogueira et al 1999) avoids the use of more
complicated algorithms and, thus, avoids the corresponding
requirements.
Owing to the relation between the present work and the one
just mentioned and to allow direct comparison, the synthetic
images used here will be the ones from that work if not specified
otherwise. In these images, the mean distance between particle
images is δ = 4.5 pixels (i.e. 4/(π δ 2 ) ∼ 0.06 particles per
pixel). The mean diameter of particle images is d = 4 pixels.
Their shape is Gaussian, being the diameter associated with
e−2 times the maximum value. Where particles overlap, the
corresponding intensities are added. 5% of particles have
no second image to correlate because of their out-of-plane
velocity.
2. Errors and instabilities associated with image
correction in multigrid PIV
The idea of relocating the information of the original image in
order to optimize the recognition of the PIV correlation peak
is not new. Keane and Adrian (1993) proposed the use of
a discrete offset between the pair of interrogation windows.
Also, the correction of the particle pattern deformation was
implemented, in each interrogation window, by Huang et al
(1993). Later, Jambunathan et al (1995) presented a more
efficient computing method by correcting the whole image
after each iteration. Both reported instabilities in their iteration
and, to avoid them, were forced to use a lowpass filter and/or
reduce the number of iterations to a few. It was not until the
work of Nogueira (1997) and Nogueira et al (1999) that the
source of the instability was identified. These works also
introduced the way to avoid it without losing high-spatialfrequency information. A detailed description can be found
in the mentioned works. Here, only a sketch of this relevant
issue is drawn.
2.1. Sources of error and instability
To start with, it is of interest to remind the reader that
the measurement in a usual PIV interrogation window
is related to the most frequent particle displacement,
independently of its position. The contributions of the various
displacements within the interrogation window have different
effects depending on the ratio between the range of these
displacements and the diameter of the particle images. If this
ratio is smaller than unity, the peaks in the correlation domain
overlap and a weighted average is obtained. For larger ratios,
the overlapping of peaks is reduced, so the bias towards the
most frequent displacement is increased. An extreme case
would correspond to a step discontinuity in the flow field where
two separated peaks could appear.
As a consequence, for certain spatial wavelengths like
the one represented by the sinusoidal displacement field, s,
depicted in figure 1, the measured displacement at the centre
of the interrogation window is opposite to the central one.
In a multigrid system, this effect causes a displacement
of the interrogation window in the opposite direction to that
which corresponds to its centre, thus reducing the signal-tonoise ratio and possibly leading to the formation of a spurious
Local field correction PIV
⇒
Measured
displacement
Displacement
field, s.
Figure 1. An example of measurement error in the interrogation
window of a conventional PIV system due to high-spatial-frequency
content in the displacement field.
Multigrid PIV with compensarion of part.
pattern and no weigh. func. (F = 16, ∆ = 4)
0.6
0.5
Multigrid PIV with compensarion
of part. pattern and no weigh.
func. (F = 16, ∆ = 8)
0.4
0.3
0.2
LFCPIV
0.1
0.0
1
1.2
5
9
13
17
21
25
29
33
37
41
45
Number of iterations
1.0
Figure 3. Examples of divergence for some systems with
compensation for the particle pattern deformation. (F and are
given in pixels.)
16 pixels window side
0.8
32 pixels window side
0.6
r
Evolution of error with iterations
0.7
rms(e ) (pixels)
⇒
Instability in
case of iterative
compensation
of particle
pattern
deformation
64 pixels window side
0.4
0.2
0.0
−0.2
−0.4
∞
64.00
32.00
21.33
16.00
12.80
10.67
9.14
8.00
λ (pixels)
Figure 2. The frequency response as a function of the spatial
wavelength. The zones with r < 0 indicate zones with the effect in
figure 1. (Note that the scale of λ is not linear, actually it
corresponds to a linear scale in frequency.)
vector or outlier. To generalize the idea depicted in figure 1,
figure 2 presents the frequency response, r, to 1D sinusoidal
fields of various spatial wavelengths, λ, for several window
sizes.
Figure 2 is just an idealization, because the quantitative
value of r depends on more parameters than just the frequency.
However, the zones where r < 0 (which corresponds to the
phenomenon depicted in figure 1) are fully correct.
These zones are the source of the instability, as figure 1
indicates. If a correction of the particle pattern is attempted
with this erroneous information, the deformation increases
instead of decreasing. A diverging process is triggered if
successive iterations are implemented.
2.2. Effects in multigrid PIV
Summarizing the previous section, the error from negative
frequency response leads to two effects.
(i) In a conventional multigrid system it leads to a reduction
of the signal-to-noise ratio instead of an increase. This
boosts the possibility of obtaining outliers, specially if the
interrogation window in the following iteration is smaller.
(ii) In a multigrid system with correction of the particle pattern
deformation (Fincham and Delerce 2000, Scarano and
Riethmuller 2000), if no weighting function is used (like
in the works by Huang et al (1993) and Jambunathan
et al (1995)) there is an source of instability to take into
account.
In this section, the magnitude and effect of this error are
analysed, as is the question of whether the use of an appropriate
weighting function would improve the results.
This analysis is presented in two parts. First, the necessary
conditions for the instability to develop are studied. After that,
the uncertainty due to this error, even when the instability is
not allowed to evolve, is analysed for various wavelengths.
The basic condition for the unstable growth is that the
grid-sampled displacement field contains the required (r < 0)
frequencies. The low resolution of CCD cameras gives this
condition except for very simple flows. Besides that, image
noise and image discretization introduce frequencies that could
be troublesome. Last but not least, particles act as samplers
of the flow field, introducing aliasing if they are too far
apart. Nevertheless, following figure 2 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 length of the side of the square
interrogation window, F . To illustrate this behaviour, a pair
of synthetic images of a field with uniform displacement of
3.6 pixels, d = 2 pixels and δ = 1.4 pixels was analysed. The
results are plotted in figure 3.
These synthetic images contain absolutely no frequencies
related to the instability except for the noise due to
discretization of grey levels. It can be seen that a multigrid
system with compensation of the particle pattern deformation,
iterating with F = 16 pixels and = 8 pixels, does not exhibit
divergence after 45 iterations. The same system with = 4
pixels exhibits it clearly. The performance of the LFCPIV
described in section 3 is also depicted for comparison.
Even when this instability is not allowed to grow (large
grid spacing or few iterations), the error from r < 0 affects
the accuracy. To analyse this accuracy, let us define the
usual parameters for a multigrid PIV implementation. A
representative example is a five-iteration algorithm. The size
of the interrogation windows in this test will successively have
the following values: 64 pixels for the first iteration, 32 pixels
for the second one and 16 pixels for the last three. The
space between interrogation grid nodes will be successively
16 pixels, 8 pixels and three times 4 pixels. Correction of the
particle pattern deformation will be applied after each iteration.
Later on, a discussion of the variations of these parameters is
offered.
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J Nogueira et al
The concepts introduced in figures 1 and 2 help us to
understand the output after each iteration. The detailed
analysis of the proposed multigrid system measuring a
sinusoidal 1D field of wavelength λ = 21.3 pixels, on the
synthetic images defined in the introduction, is as follows.
0.50
0.25
υ2
0.00
-0.25
-0.50
-0.25
0.00
ξ/F
0.25
-0.50
0.50
η/F
Figure 4. The weighting function designed to avoid negative
frequency responses. For details see Nogueira et al (1999).
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 )
(i) For the first iteration (F = 64 pixels), figure 2 shows
that r is slightly negative. The result is that this first
iteration gives in some places a displacement opposite to
the one being measured. If we analyse the root mean
square (RMS) value of the error, e, in the measurement,
normalized with respect to the RMS of the signal 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.
(ii) For the second iteration (F = 32 pixels), figure 2
shows a highly negative value for r. The result in error
terms is RMS(e)/RMS(s) = 1.34. This is consistent
with a measurement with opposite sign to the original
displacement field.
(iii) In the three following iterations (F = 16 pixels) the value
of r is positive. The result is a reduction of the error
in each step, finishing with RMS(e)/RMS(s) = 1.19.
Further iterations would lead to worse measurements due
to accumulation of errors from other sources.
8-9
7-8
6-7
5-6
4-5
3-4
2-3
1-2
0-1
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
2.3. Solutions for multigrid PIV
The conclusion from the test in the previous section is that the
effect of the error mentioned can be relevant. One solution for
this would be to use only 16 pixel by 16 pixel interrogation
windows. This would avoid the build-up of error down to
λ = 16 pixels. The performance of such a system has already
been studied with different implementations by Jambunathan
et al (1995) and Scarano and Riethmuller (2000). But the
use of just small windows for every step in a system reduces
considerably the robustness against outliers as well as the
dynamic range of measurable displacements.
Another solution, recommended here (and already
proposed by the authors in Rodrı́guez et al (2000)), is to use
an appropriate weighting function to avoid the source of error
depicted in figures 1 and 2. The weighting function designed
for LFCPIV is able to perform this task. This weighting
function is presented below and depicted in figure 4:
2
2
ξ ξ η
η
υ 2 (ξ, η) = 9 4 − 4 + 1 4 − 4 + 1 (1)
F
F
F
F
where ξ and η are coordinates with their origins at the centre
of the interrogation window and F is the length of its side.
Unfortunately, the application of a weighting function
induces some erroneous slippage into the measurement
(Nogueira et al 1999). This slippage is larger with smaller
windows, making its use for windows with sides smaller than
32 pixels inadvisable. Consequently, for the five-iteration
system analysed in this section, the weighting function is used
only for the first two iterations.
Another option, after using this weighting function, υ, in
the first two steps, is to implement also in the last three steps
a 32 pixel by 32 pixel window with the weighting function,
instead of 16 by 16 without it. This procedure can be defined
as multigrid LFCPIV.
1914
0
Multigrid LFCPIV (F :64,4x32; ∆ :16,8,3x4)
20
25
30
35
40
45
50
λ x (pixels); (δ =4.5 pixels).
55
60
Figure 5. The performance of the five-iteration multigrid systems
described in the text, together with those of the methods from
Jambunathan et al (1995) (thin full line) and Scarano and
Riethmuller (2000) (thin broken line).
The results for all these options are compared in figure 5.
In figure 5, the two thin lines plotted as references correspond to
the two mentioned systems that only use 16 pixel interrogation
windows. The oscillation around the reference lines of the
multigrid system without a weighting function is caused by
the effect depicted in figure 1 when one is operating with the
two different window sizes.
Details on the images used for this evaluation can be found
in Nogueira et al (1999), except for the one from Scarano
and Riethmuller (2000), which was directly taken from that
reference.
It must be accepted that the parameters defined here for
multigrid PIV are arbitrary, but the benefits of the application
of the weighting function for steps with windows larger than
30 pixels are obvious. It should be remarked that most of
these benefits remain even when there is no compensation for
the particle pattern deformation.
Another useful conclusion is that better results can
be obtained with weighted 32 pixel windows rather than
unweighted 16 pixel windows.
It should be remarked that the application of windows
smaller than 16 pixels on real images is still a subject under
development. Many actual 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 on a side.
Local field correction PIV
Nevertheless, there is ongoing research on systems
especially designed to deal with smaller windows, Fincham
and Delerce (1997) show that good measurements can be
obtained with small windows, but the requirement for a large
particle diameter arises (the optimum diameter is around
6 pixels). Nogueira et al (2001) show a promising way to deal
with small windows without this requirement, by the use of
SDC algorithms. Besides the need for appropriate algorithms,
the robustness of the system decreases strongly with decreasing
size of interrogation window. These facts encourage the full
development of a system with resolution smaller than the size
of the interrogation window, thus not requiring size reduction.
3. A refinement of the original LFCPIV method
Up to now, the only method able to obtain resolutions smaller
than the interrogation window is LFCPIV. In this section, a
refinement of the previous version is described.
3.1. Initial considerations
The starting point of any cross-correlation PIV system
is a pair of images of the particle pattern, a and b,
separated by a known time interval. Cross correlation of
the corresponding interrogation windows, at the measurement
points, approximates the displacement field of the particles on
going from one image to the other.
The subsequent action of any iterative system is to use
the information of this measurement to adapt the system in
such a way as to obtain better ones. In particular, in a
system with compensation for the particle pattern deformation,
a third image b∗ is obtained by deforming and shifting b with
the information of the approximate displacement field. This
reduces the relative distortion, thus increasing the signal-tonoise ratio. Consequently, further analysis by cross correlating
a and b∗ yields information to reduce the error of the
approximated displacement field. This allows corrections that
provide successive images b∗ from b to feed the iterative cycle.
Details on the image processing can be found in Jambunathan
et al (1995) and Nogueira et al (1999).
To be able to iterate without instability due to the effect
depicted in figure 1, the weighting function depicted in
figure 4 has to be used (as noted in the previous section).
Unfortunately this introduces a small error that accumulates
through the iterations and starts to be significant after about
15–20 iterations. To avoid this error, a complex algorithm
was designed at the birth of LFCPIV. The price to pay was a
reduction of the field of application of the system to images
with δ > 4 pixels.
Many PIV images are shot in industrial wind tunnels,
with a heavily seeded flow and with a strong need for high
resolution. These images present small distances between
particles and, actually, this means that one makes better use
of the limited CCD sensor resolution. This leads to the search
for new solutions in order to deal with the error introduced by
the weighting function. Here a new one without reduction of
the field of application is proposed. Furthermore, the results
obtained with this refinement reduce the uncertainty of the
measurement, particularly at high spatial frequencies.
One last consideration is that, to obtain a second-order
approximation of the velocity field, the distortion of the particle
pattern can be compensated in both images a and b, giving
a ∗ and b∗ . This way, only half of the deformation has to
be compensated in each image, resulting in a system with an
increased ability to cope with large velocity gradients.
3.2. A new approach to dealing with the error introduced by
the weighting function
Observation of the measurement slippage due to the weighting
function leads to the conclusion that it is generally small, due
to statistical cancellation in the interrogation window. This
means that the locations affected by significant slippage are
scarce. Consequently, a solution is to freeze the measurement
for the few nodes with significant slippage in the ongoing
iteration. This way the evolution of these nodes to give a
worse measurement instead of a better one is avoided.
This is the objective of the system here proposed.
The detection of significant slippage is performed in an
approximate way, but the results obtained demonstrate that
it is a significant improvement. The nodes frozen in a certain
iteration are those with slippage large enough to be detected.
The detection is based on the increase or decrease of the
local correlation coefficient after each compensation of the
particle pattern deformation. Several sources of noise mask
the detection of slippage. The main one is the influence of
neighbouring nodes on the change of the local correlation
coefficient. A specific procedure was designed to deal with
this issue, which is described in detail in what follows.
3.3. The refined system
A detailed description of the refined system is specified through
the following steps.
(i) Calculation of local coefficients. The value of the local
correlation coefficient in a window with F = 2 is
calculated for each grid point. This value will be used
later to search for slippage in the displacement values.
This window corresponds to the region of influence of
each grid node.
(ii) Initial processing of the images. This step is carried out as
in usual cross-correlation PIV. The image is divided into
windows (usually larger than the ones in the previous step)
and these are cross-correlated to find the displacement
peaks. The only difference is that the weighting window
depicted in figure 4 is used. The resulting expression for
the cross-correlation coefficients, Clm , is specified below:
F /2
Clm =
υ(ξ, η)f (ξ, η)υ(ξ, η)g(ξ + l, η + m)
×
ξ,η=−F /2
F /2
υ 2 (ξ, η)f 2 (ξ, η)
ξ,η=−F /2
×
F /2
−1/2
υ 2 (ξ, η)g 2 (ξ + l, η + m)
(2)
ξ,η=−F /2
where f (ξ, η) and g(ξ, η) are the grey-level maps of the
interrogation windows belonging respectively to the first
and second images to be correlated. υ(ξ, η) is the square
root of the values depicted in figure 4.
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J Nogueira et al
The evolution of the error in these iterations is qualitatively
similar to that of its former version (decreasing fast at the
beginning to increase slowly after a minimum). To decide
when to stop the iterations in the case F = 64, an improved
method has been implemented through the following two
modes of operation.
(i) Mode 1 corresponds to normal iterative PIV operation.
All the nodes are considered for evolution. When the
number of worsening nodes is more than half the number
of improving ones, the system changes to mode 2. The
rationale behind this is the need to detect that a large
enough number of nodes have already converged.
(ii) In mode 2 only the nodes with local correlation coefficients
below the average, at the time of mode change, are
considered for evolution. When the number of worsening
nodes is more than the number of improving ones the
system changes to mode 1. This mode complements
mode 1 insofar as it freezes the nodes that have already
converged, allowing the more slowly evolving ones to
proceed.
If no further iteration is made in mode 1 or 2, the system
stops.
In the case F = 32, the slippage introduced by the
weighting function makes it advisable to take into account
more conservative considerations. A reasonable option is to
limit the number of iterations to a few (for example ten). In
both cases (F = 64 and F = 32) the differences from the
minimum value of RMS(e)/RMS(s) were smaller than 0.05.
1916
Displacement field: s = 2sin(2πx /λ x ) (pixels)
0.7
Former LFC-PIV (F = 64 pix.)
0.6
rms(e )/rms(s )
(iii) Compensation for the particle pattern deformation.
Correcting the particle pattern deformation gives images
a ∗ and b∗ . In the system presented here, the interpolation
applied to obtain the grey levels is biparabolic.
(iv) Recalculation of local coefficients. The local correlation
coefficients are obtained by windowing images a ∗ and b∗ .
These values are compared with the ones obtained in
step (i). A lower value may mean detection of significant
slippage or simply influence by neighbouring slips. In
consequence, of the coefficients that worsen, only those
surrounded by at least another five that also worsen, among
the eight neighbours, are considered to contain significant
slippage. The evolution of the nodes with significant
slippage is avoided during this cycle.
(v) Validation and interpolation of displacements to avoid
intermediate false measurements. This step avoids
obvious outliers in intermediate cycles. Any proven
validation and interpolation algorithm would be valid. In
particular, the ones applied here are those from Nogueira
et al (1997).
(vi) Compensation for the particle pattern deformation. With
the approximation to the particle pattern displacement so
far obtained, a new pair of images, a ∗ and b∗ , is obtained
again from a and b. The local coefficients relating a ∗
and b∗ are stored like in step (i).
(vii) Further processing on the images a ∗ and b∗ . These two
images are fed into step (ii). The measurement obtained
is a correction to the previously estimated displacement
field. By adding this correction, a new approximation
to the particle pattern deformation is obtained. This is
supplied to step (iii), defining the iterative loop.
0.5
Refined LFC-PIV (F = 32 pix.)
0.4
Refined LFCPIV (F = 64 pix.)
0.3
0.2
0.1
0
20
25
30
35
40
45
50
λ x (pixels); (δ =4.5 pixels).
55
60
Figure 6. Performances of LFCPIV systems for the displacement
fields indicated. Recent improvements in the mode of operation
have shown lower error figures (Rodrı́guez et al 2001).
The resulting method described so far requires only two
types of algorithms: interpolation of grey levels and correlation
calculation. Use of algorithms on validation and interpolation
of vectors is a valuable option and almost essential for intermediate steps, if the noise or the velocity gradients are large.
A detail that further increases the accuracy of the system
and that will be applied in this paper is to symmetrize
the correlation algorithm. An alternative formulation to
expression (2) can be obtained by swapping f and g. The
resulting measurement of displacement can be averaged with
the one from expression (2). The performance gives an error
reduction of ∼3%. The price to pay is that the computing time
must be doubled.
It is also of interest to note that, when the displacements
to be measured are smaller than 0.5 pixels (typically after the
third iteration or so), only the central correlation coefficient
plus its four closest neighbours have to be calculated. For
this case a direct correlation is more efficient than a FFT
algorithm. Nevertheless, usual application of the system for
high resolution requires more than 100 iterations with of the
order of four pixels, making this a considerable computational
load. It requires, like its previous version, a time of the order
of 10 + (n − 5)/5 times what is required for a conventional
system (n > 5 being the number of iterations).
4. Evaluation of synthetic images
The synthetic images used in section 2 were also used here to
allow direct comparison with the results in that section and the
results from Nogueira et al (1999). In this case the performance
of the system described in section 3 is plotted, in figure 6, for
F = 32 pixels and F = 64 pixels. Also the performance of
the previous version of LFCPIV is plotted. = 4 pixels was
used in all processing.
It can be observed that the case of F = 64 pixels
always performs better. Images with the same attributes but
d = 2 pixels and δ = 1.41 pixels (0.64 particles per pixel)
were processed with even better results. This observation
emphasizes that the restriction of application to images with
δ > 4 pixels has been eliminated.
Up to this point, the system has been detailed and initially
tested. However, the data shown so far are not enough for a
clear picture of the system’s behaviour. More information is
Local field correction PIV
AVERAGED DISPL. PROFILE
< 0.5F
Displacement
field
⇒
4
Measured
displacement
Figure 7. An example of measurement error in the interrogation
window of a conventional PIV system due to non-periodic small
features in the displacement field.
Displacement (pixels)
F
1
0
2
4
6
8
10 12 14 16
Position (pixels)
rms AROUND REAL VALUE
4
rms around real value (pixels)
Some tests were carried out to show that the effectiveness
of the LFCPIV system is not due to these issues. The
conclusions are detailed below.
The presence of a weighting function reduces the effective
size of the window. Nevertheless, in all the tests carried
out, all the weighed windows behaved more robustly than
did windows half the size with no weighting applied. This
means that the number of outliers was smaller. Therefore,
the ratio (effective size)/(real size) can be considered larger
than a half. In comparison with this reduction of effective
size, the increase in spatial resolution is much larger. The
weighted windows in a LFCPIV system are able to resolve
features in the flow with spatial wavelengths smaller than F /4.
Consequently, the increase of resolution cannot be attributed
to the reduction of the effective size. Moreover, figure 6 shows
that a larger window performs better than a smaller one in terms
of accuracy. Although some other effects have to be taken into
account, this indicates that the benefits in terms of resolution
are not coupled to a reduction of effective size.
Concerning the issue of periodicity in the test flow fields,
in order to check whether the periodicity of the field is helping
the algorithm to work, several tests were performed using nonperiodic fields. All of them gave similar results to those using
periodic fields, showing that the capability of the LFCPIV
system to achieve high resolution is not a consequence of the
existence of periodicity in the flow.
Before commenting on some of these tests, it is interesting
to point out some considerations about non-periodic fields.
The foreseeable detectability threshold of a perturbation in a
homogeneous field is depicted in figure 7, for a conventional
PIV system. If the area covered by the perturbation is smaller
than 0.5F 2 , its peak in the correlation domain would be smaller
than that belonging to the unperturbed field. Consequently,
if the two peaks do not overlap, the perturbation would not
be detected. If they overlap, the one associated with the
unperturbed field would be dominant.
Instead of this threshold, a LFCPIV system has a value
of ∼0.21F 2 , thanks to the weighting function. The same
reasoning can be applied for a 2D discontinuity. In addition to
this, an iterative system like LFCPIV would not lose track of the
feature in cases in which the peak of the unperturbed flow in the
correlation domain overlaps with the peak corresponding to the
2
0
needed in order to confirm that the increase of accuracy, with
respect to that of a conventional PIV system, should not be
erroneously attributed to the following two features:
(i) the reduction of the effective size of the window and
(ii) the periodicity in the test displacement fields.
3
3
2
1
0
0
2
4
6
8
10 12 14 16
Position (pixels)
Figure 8. Performances of two systems for measurements of small
non-periodic features in the displacement field. Thick line, field to
be measured; circles, LFCPIV measurements (F = 64 pixels); and
squares, multigrid PIV (F = 16 pixels). The RMS error of the
various measurements around the real value is also depicted, to give
a complete picture.
perturbation. Examples of this circumstance appear when the
step depicted in figure 7 has a finite slope or when the increment
in displacement is smaller than d. In these cases, the iterations
would lead the system to the right solution. To illustrate this
effect, a pair of synthetic images of a zero displacement field
crossed by a perturbation with a parabolic profile (laminar
Poiseuille 2D flow) was generated and analysed. The width of
this perturbation is eight pixels and the maximum value of the
displacement is four pixels. The images were generated with
δ = 1.4 pixels and d = 2 pixels. These images were analysed
using the LFCPIV system with F = 64 pixels. This means
that the width of the perturbation is much smaller than the
foreseeable threshold of detectability defined above. However,
the finite slope introduces information in the larger peak that
slowly guides the iteration to the right solution. A multigrid
system with F = 16 pixels was implemented for comparison.
This system is equivalent to the ones in the caption of figure 5.
The results are depicted in figure 8.
Another example of a non-periodic field, that highlights
other features, such as the presence of boundaries and
seeding density inhomogeneity, follows. Analogously to the
previous case, it concerns a parabolic profile with a maximum
displacement of four pixels and the same values for δ and d.
In this case the width of the profile is 16 pixels and there is no
seeding outside it. A couple of 64 pixel by 64 pixel images to
be correlated are depicted in figure 9.
1917
J Nogueira et al
Figure 9. An example of 64 by 64 interrogation windows, a and b,
of a displacement field that corresponds to a horizontal Poiseuille
flow. The width of the flow is 16 pixels.
AVERAGED DISPL. PROFILE
Displacement (pixels)
4
Figure 11. An image of the central zone of a wingtip vortex
analysed in section 5 (provided by the DLR).
3
5. Application to real images
2
1
0
0
2
4
6
8
10 12 14 16
Position (pixels)
rms AROUND REAL VALUE
rms around real value (pixels)
3
2
1
0
0
2
4
6
8
10 12 14 16
Position (pixels)
Figure 10. Performances of two systems for measurements of small
non-periodic features and boundaries in the displacement field.
Thick line; field to be measured; Circles; LFCPIV measurements
(F = 64 pixels); and squares; multigrid PIV (F = 16 pixels).
It can be seen that no reflection of light was included at
the boundaries. Nonetheless, the case is tough due to the tendency of conventional systems to delocalize the information
from particles inside the interrogation window. The performances of the same two systems are depicted in figure 10.
One last observation is that, in the LFCPIV system
presented here, even in the case of losing the information
on a feature that could be resolved with the grid spacing, the
local coefficient defined in the previous section would inform
us about it. With this information, the time between pulses
can be reduced and the feature would be more easily tracked.
Therefore the LFCPIV system has a high robustness besides
a high accuracy, beyond those of other current high-resolution
techniques.
1918
As an example of application of this version of LFCPIV to
real images, a typical example from large-scale aerodynamic
facilities has been chosen. The particular case corresponds to
the core of a wingtip vortex generated in the DNW wind tunnel
with a fixed model of half an airplane wing, corresponding to
a wingspan of 3.5 m, in a test section of 6 m × 8 m. The PIV
image to be analysed was obtained with the DLR’s standard
PIV system comprising Nd:YAG lasers (2 × 600 mJ), Laskin
nozzle-type aerosol generators and a high-sensitivity cooled
‘cross-correlation’ camera (1280 × 1024 pixels). The time
between images was 2 × 10−5 s. This image, which is one
example out of several thousand PIV recordings, has been
obtained by the DLR within the EC-funded EUROWAKE
project dedicated to the investigation of wake vortices. In
the frame of our work it is studied rather from the point of
view of PIV evaluation methods than of drawing conclusions
with respect to the behaviour of wake vortices. Further
detailed information about this experiment can be found in
Kompenhans et al (1999).
For clarity, only the analysis of the centre of the image
will be presented here (216 × 208 pixels). One image of this
zone is depicted in figure 11.
As has been noted, this image presents a number of
characteristics associated with large industrial facilities that
made it interesting to test this version of LFCPIV on it. In
some zones, δ is smaller than two pixels. This allows us
to show that the new version of LFCPIV has no intrinsic
limitations regarding the distance between particle images.
Simultaneously, it presents high gradients in seeding density
(from 0.6 to 0.1 particles per pixel) and velocity (up to
3.2 × 104 s−1 ). This allows us to show the ability of the system
to deal with such difficulties.
The measurement is performed with F = 64 and = 8
pixels. Processing with conventional PIV systems gives
around 60–100 outliers near the core. The measurement
performed with LFCPIV, depicted in figure 12, gives no final
outliers. On the other hand, three of the central measurements
are questionable due to its low local correlation coefficient
(<0.5). In the zone where these measurements are located, the
Local field correction PIV
: 6 pixels.
(a)
(b)
Figure 14. SNRs for data in figure 12 (>2, white; 2 to 1, grey; <1,
black). (a) Only a discrete offset of interrogation windows.
(b) Compensation for the particle pattern deformation.
Figure 12. A measurement carried out with LFCPIV on the image
in figure 11 and its pair ( = 8 pixels).
Vorticity (1/s)
21000-22500
19500-21000
18000-19500
16500-18000
15000-16500
13500-15000
12000-13500
10500-12000
9000-10500
7500-9000
6000-7500
4500-6000
3000-4500
1500-3000
0-1500
-1500-0
25
22
19
16
13
10
1
7
6
11
column
4
16
21
row
use interpolation of the image grey levels. It is true that
interpolation means a degradation of the quality of the
image. Sometimes, this fact is a source of doubts about the
convenience of its application. The image analysed in this
section suits as an example to show that the benefits of the
compensation for the particle pattern deformation can be much
more significant than the possible degradation. To illustrate
this, the signal-to-noise ratio (SNR) of local 32 by 32 windows
will be studied.
The SNR is usually plotted as the ratio (signal
peak)/(highest noise peak) in the correlation domain. Here,
the differences in seeding density among parts of the image
would fool this indicator because of the differences in the mean
value over which the peaks arise. To allow a more uniform
comparison along the image, an alternative ratio is proposed:
signal peak − average value
.
highest noise peak − average value
(3)
1
26
Figure 13. A vorticity plot of the vectors depicted in figure 12. The
centre peak has been cut (white feature in the centre) due to the
questionable nature of the results and to clarify the presentation.
velocity gradients along streamlines are too large to consider
the time between images appropriate (∼2 × 104 s−1 ). This
means that the system is failing only where the measurement
is not valid for a reason external to the algorithms, which is the
time between pulses. This difficulty may also be combined
with a low seeding density (owing to centrifugal migration)
and the possible existence of an out-of-plane displacement.
It should also be remarked that the displacement field is
not as smooth as it seems visually in figure 12. Figure 13
shows that there is a ring of vorticity of characteristic width
of the order of 32 pixels that is clearly described. This again
highlights the capability of the system to obtain measurements
of features smaller than the interrogation window, even under
adverse conditions. Whether the vorticity peaks in this ring
are caused by peak locking is not clear, but revealing them is
a consequence of the high resolution capability of the method
in this case.
The vorticity algorithm applied here is a usual circulation
filter of size 2. This allows comparison with standard
methods. Nevertheless, more advanced algorithms can be
found in Lecuona et al (1998).
One last issue about real images can be remarked upon.
As was stated is section 1, all the advanced 2D PIV algorithms
In this expression the average value is obtained in the
correlation domain using all the values smaller than the
smallest of the two peaks defined. Using the displacement
field presented in figure 12, this signal-to-noise parameter is
plotted in figure 14 for each grid node, allowing comparison
between two cases:
(i) a discrete offset of windows and
(ii) compensation for the particle pattern deformation.
In figure 14, SNR < 1 means that the noise peak is higher
than the signal peak. This could just indicate a different value
from what has been considered as the true displacement (based
on the solution from LFCPIV). About this, it is important to
mention that the highest peak belonged to a clear outlier in
more than 50% of cases, reinforcing the meaning of SNR < 1.
A similar analysis was performed for 16 by 16 and 64 by
64 interrogation windows with even more evident differences
between the two methods.
To further illustrate this effect, figure 15 shows a set
of interrogation windows and the corresponding correlation
domains. These interrogation windows come from the grid
node of coordinates (9, 9) with its origin in the lower left-hand
corner of figure 12.
6. Conclusions
A refined LFCPIV method that uses only current algorithms
of any advanced 2D PIV system has been developed and its
1919
J Nogueira et al
(I)
image a
image b
(II)
image a*
image b*
Figure 15. The effect of compensation. (I) From left to right: interrogation window from image a, interrogation window from image b and
corresponding correlation. Only a discrete offset of interrogation windows has been used. (II) The same as before but with compensation for
the particle pattern deformation.
metrological quality tested. The new system presents some
advantages over the previous version. There is no restriction
on the mean distance between particles. Smaller interrogation
windows can be used. It provides a lesser measurement
uncertainty than that of its predecessor. The result is a
robust high-resolution system able to cope with large seeding
densities and velocity gradients.
It has been shown that the provision of high resolution
is not coupled either to a decrease in the effective size of the
interrogation window or to the periodicity of the displacement
fields on which the previous version was tested.
A multigrid version that yields results better than standard
multigrid systems down to 16 pixel by 16 pixel interrogation
windows has also been presented. The price to pay is an
increase in computing time by a factor of six.
Recent systems based on windows smaller than 16 pixels
by 16 pixels, like the SDC algorithms mentioned, probably
will give good results in the future. They can always be
implemented after a LFCPIV system instead of a multigrid
system down to 16 pixels by 16 pixels. This version would
obtain benefits from the robustness against outliers of LFCPIV.
Although both SDCPIV and LFCPIV algorithms have
already presented clear improvements, ways for further
refinement of their concepts still remain to be investigated.
Interpolation of the PIV image is needed both for multigrid
systems and for LFCPIV. It has been shown that, when high
velocity gradients are present, the SNR can be significantly
increased thanks to the interpolation. It should be noted
that the interpolation in each iteration is always based on the
original image and there is no interpolation of images that have
already been interpolated, for that would reduce substantially
the quality of the image.
1920
Acknowledgments
This work has partially been funded by the Spanish Research
Agency grant DGICYT TAP96-1808-CE, PB95-0150-CO202 and under the EUROPIV 2 project (a joint programme
to improve PIV performance for industry and research),
which is a collaboration among 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) and Zaragoza. The project
is managed by LML URA CNRS 1441 and is funded by the
CEC under the IMT initiative (contract GRD1-1999-10835).
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