13th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 26–29 June, 2006
New resolution limits in PIV image processing
Holger Nobach1,2,3 and Eberhard Bodenschatz1,2,4
1
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA
2
Max-Planck-Institut für Dynamik und Selbstorganisation, Göttingen, Germany
3
holger.nobach@nambis.de
4
eberhard.bodenschatz@ds.mpg.de
Abstract To achieve improved spatial resolution in PIV image processing without loosing the robustness of large interrogation areas, the image deformation is a potential method. Instabilities of
the iterative correction of the velocity estimates can be avoided by applying appropriate weighting
schemes to the interrogation areas. Recent investigations identified remaining errors, which occur even
if the iteration process is stable. This paper investigates the reason for this effect and how it can be
controlled.
1. Introduction
Since the very early days of the PIV technique, the question of the obtainable accuracy and spatial
resolution has been of essential interest. With rising interest in flows with velocity gradients in the
past years, the ability to follow spatial changes of velocity has become especially important. Since
correlation-based PIV processing is a statistical method, averaging over image subspaces (interrogation
areas), the achievable spatial resolution is coupled inherently to the size of these interrogation areas.
An obvious option to improve the resolution is to use smaller interrogation areas (“super-resolution”
methods). Unfortunately, the PIV cross-correlation technique leads to outliers if the information
content of the interrogation areas is decreased.
In order to avoid outliers, the interrogation areas must be chosen rather conservatively compared to the
sampling density given by the particle distribution. Thus, the achievable resolution lags the principally
available information. With step-wise decreased interrogation area sizes (“multi-grid” methods), and
interrogation area shifts based on the previous velocity estimates [8], the estimation procedure is more
stable. However, the achievable resolution fundamentally depends on the size of the interrogation
areas in the final iteration step. Also sub-pixel interrogation area shifts [2] and interrogation area
deformations [5] (Fig. 1) do not improve the spatial resolution substantially.
2. Image Deformation Method
With image deformation techniques [1, 7] (Fig. 2) the situation can improve, since the deformation’s
degree of freedom is related to the grid of velocity estimates. With a high overlap of neighboring interrogation areas in combination with an iterative correction of residual displacements of the estimated
velocity field the spatial resolution is governed by the grid spacing, rather than the nominal resolution
of the interrogation area size.
Figure 3 shows an example of an simulated velocity field and one of the corresponding PIV images.
Figure 4 shows the estimated velocity field and one deformed PIV image, which ideally matches the
deformed second image. The interrogation areas have a size of 32 × 32 pixel and overlap by 50%
(velocity grid spacing of 16 × 16 pixel).
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13th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 26–29 June, 2006
Figure 1: Interrogation area shift and deformation
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13th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 26–29 June, 2006
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Figure 3: Simulated velocity vector field and PIV image (straight lines are overlayed)
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13th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 26–29 June, 2006
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Figure 5: Estimated velocity vector field and the development of corrections and deviations
3. Instabilities and Ambiguities
To improve the spatial resolution without loosing the robustness of the large interrogation areas, the
overlapping fraction of neighboring interrogation areas must be increased. Unfortunately, repeating
the processing with an overlap of 75% (velocity grid spacing of 8 × 8 pixel) yields an unstable iteration
with exponentially increasing deviations (Fig. 5).
The reason for these instabilities is the top-hat profile of the interrogation windows (top-hat weighting
function to crop the interrogation area from the image), which yields negative values of the frequency
response function in certain ranges of spatial frequencies [3, 6] (Fig. 6). Due to the wrong sign the
displacement between the images is increased instead of decreased and the velocity pre-estimate loop
is even worse for the next iteration. Here, the iterative application of the correction procedure leads
to exponentially increasing deviations.
Of course, by applying a low-pass filter, which removes all components of the estimated velocity field
at those frequency ranges, the instability can be removed. However, the characteristics of this filter
must be adapted to the size of the interrogation window. This filter inturn limits the spatial resolution,
which defeats the intention for improval.
A superior solution is the application of appropriate weighting schemes to the interrogation windows,
which have frequency responses with only positive values [3] (Fig. 7). This avoids the instability
of the iterative optimization, while the resolution is unbounded. Because the frequency response of
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13th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 26–29 June, 2006
weighting (windowing) factor
frequency response
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Figure 6: Top-hat weighting function to crop the interrogation area from the image and it’s frequency
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Figure 7: Top-hat weighting function to crop the interrogation area from the image and it’s frequency
response function
the velocity estimation procedure is positive for all frequencies, the distance between the estimated
velocity and the correct velocity decreases with each iteration step and finally vanishes.
Theoretically, the use of weighted interrogation windows seems to be the ideal tool for PIV image
processing. Unfortunately, the PIV measurement system and its image processing are highly nonlinear, yielding characteristics that are significantly different from linear signal processing theory
described above. In the recent past, a remaining instability has been observed [3, 4] that seems to be
consistent with the non-linearity of the estimation process. Figure 8 shows an example with obvious
deviations of some vectors. The similarity of this behavior of the iterative process to the instabilities
above leads to the suggestion to use a certain low-pass filtering even here.
However, this interpretation is misleading, since the magnitude of correction decreases monotonically,
indicating that the procedure is actually stable. It simply converges towards the wrong values. Even
if a low-pass filter can reduce the fluctuations, the average will remain biased. An interruption of
the correction process after a reasonable number of iterations cannot avoid this bias error as well,
since the divergence rate and starting point of divergence are not predictable. Interestingly, image
interpolation errors do not significantly influence the divergent behavior of the iterative process, all
shown simulations have been done without image interpolations. This was achieved by regenerating the
entire scene in each iteration step with correct values at the sub-pixel positions assuming a Gaussian
shape of particle images with a diameter of 3 pixels, random particle intensities, but the same intensity
for each individual particle in the consecutive images, no pixel integration, linear overlay of particle
images and no quantization. Re-calculations with quantized and interpolated images did not show a
significant difference, even with the rather simple bi-linear interpolation.
Therefore, not the image interpolation but the distribution of available information in the images
must be the reason for this deviation. Indeed the divergent behavior mainly depends on the density
and homogeneous distribution of particles in the image. The more information about the flow field
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13th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 26–29 June, 2006
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Figure 8: Estimated velocity vector field and the development of corrections and deviations
Figure 9: Ambiguity of local deformation
is available inside the interrogation areas for the local velocity estimation, the less is the remaining
bias. However, even if the bias becomes smaller with higher particle densities, it cannot be removed
completely.
The particular reason for this bias is an ambiguity of local deformations (Fig. 9). On the one hand, the
local velocity estimates (center vector in Fig. 9) are averaged over the entire interrogation area (outer
square). On the other hand, the velocity estimate influences only the deformation of the center part
(solid gray background). Therefore, deviations of the particular velocity estimate can be compensated
by the surrounding velocity estimates without changing the averaged displacement. In this case, the
iterative step of the PIV procedure estimates the mean displacement inside the entire interrogation
area to be zero, and does not correct the erroneous pre-estimate.
Due to the weighting function of the interrogation area, the average is dominated by information
close to the particular velocity estimate. Therefore, the effect of the ambiguity decreases, if enough
information (number of particles) is available close to the center of the interrogation area. In this case,
the velocity estimate defines the deformation of exactly that region, which is mainly used for the next
iterative velocity correction. Therefore, the bias decreases with a higher particle density. However, it
cannot be avoided entirely.
This effect can be removed by setting the mean displacement of the deformation to the estimated
velocity, i. e., by adding a constant displacement vector to all vector nodes participating in the deformation of the given interrogation area. In that case, the bias is instantly gone, as can be seen in
Fig. 10 and the iteration is stable even without using the weighting scheme with the interrogation
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13th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 26–29 June, 2006
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Figure 10: Estimated velocity vector field and the development of corrections and deviations
areas. Unfortunately, the correction of the mean displacement inside the interrogation error hinders
the iterative correction of the local velocity estimates and introduces an averaging low-pass filter to
the estimation procedure. Since the characteristic frequencies of this filter is related to the size of the
interrogation area, it effectively works similar to the stabilization filter without the interrogation area
weighting. Consequently, also the spatial resolution is comparable, and, finally, no advantage remains
compared to the classical PIV procedures.
4. Conclusions
With appropriate weighting schemes applied to the interrogation areas, the image deformation PIV
processing technique can be stabilized effectively even for a high percentage of interrogation area
overlap. The potential of this technique to improve the spatial resolution of the PIV image processing
is substantially limited by ambiguities of local deformations, leading to biased velocity estimates. Even
if the remaining bias decreases with an increasing amount of locally available information (particle
density and distribution), it is still present. This holds even if low-pass filters are applied, which can
damp the fluctuations, but cannot avoid the tendency towards biased estimates. This also holds for the
image deformation technique without interrogation area weighting. This ambiguity and the resulting
bias vanish immediately, if the mean displacement of the interrogation areas during the deformation
process is corrected towards the estimated velocity value. Unfortunately, this correction hobbles the
degree of freedom of the deformation and, finally, limits the achievable spatial resolution to that of
classical PIV.
In summary, the results of this investigation, the only useful option to substantially improve the
spatial resolution of the PIV process is to reduce the size of the interrogation areas. This requires
an appropriate particle density to ensure an acceptable probability of outliers and hence, a carefully
prepared experiment. The image processing itself cannot improve the spatial resolution.
Even if an improvement of the spatial resolution would be possible for given PIV images, this would
not be useful: In PIV image processing, the derivation of mean displacements averaged over the interrogation areas is a feature to improve the robustness and the accuracy in comparison to the tracking
of individual particles. The size of the interrogation areas is chosen such that enough information
(number of particle images) is available inside the interrogation areas to ensure an acceptably small
probability of outliers. If the spatial resolution of the PIV processing is increased, the advantage is
lost. It is not sufficient to resolve all flow structures at the new resolution with the same robustness
and accuracy as before, since the available information is unchanged. This has the same effect as
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REFERENCES
13th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 26–29 June, 2006
smaller interrogation area sizes, and thus there is no remaining advantage of the image deformation
technique compared to the window shift and deformation.
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