Hardware Unit for Edge Detection with Comparative Analysis of Different Edge ...
DNW_apriori_error est_report
1. A-priori error estimation of
Particle Image Velcoimetry measurements
Internship Report (Aug 2015 - Nov 2015)
Submitted by: Ankur Kislaya
DNW Supervisor: Dr. Bart van Rooijen, Instrumentation Scientist
Department of Instrumentation and Controls
GERMAN DUTCH WIND TUNNELS (DNW)
Noordostpolder, the Netherlands
3. Nomenclature
↵1 Camera angle for Right Camera [deg]
↵2 Camera angle for Left Camera [deg]
Zo Laser Sheet Thickness [px]
✏bias Bias error [px]
✏rand Random error [px]
Standard Deviation [ ]
d⌧ Particle Image diameter [px]
Io Gaussian Intensity profile [ ]
IP Average Peak Intensity [ ]
NI Image Density [ppp]
Rh Auto-Corelation peak height [ ]
RP Maximum correlation peak height [ ]
wc correlation peak width [px]
A Interrogation Window Size [px2]
M Magnification Factor [mm/px]
N Number of Samples [ ]
PSD Power Spectral density [db]
Re Reynolds Number [ ]
1 Introduction
Flow visualization techniques are used to ob-
tain diagnostic information about the fluid flow
around a wind tunnel model. Since most flu-
ids of interest (air for aerodynamics, water for
hydraulics) are optically transparent, recogniz-
ing their motion requires the fluid to be tagged
by particles that scatter light when illuminated.
The widespread use of flow visualization tech-
niques is owed to a number of benefits: it pro-
vides the description of the flow field without
complicated data reduction and analysis, enables
the validation of numerical methods, and it aids
in the development and verification of new theo-
ries of fluid flow.
Flow visualization is a very old technique to
understand the fluid-structure interaction which
goes back until the era of Leonardo Da Vinci in
16th Century. His famous hand drawings were
perhaps the worlds first accounted documents to
use this technique as scientific tool to study com-
plex problem of turbulent flows. Today, methods
such as oil/smoke visualization, laser doppler ve-
locimetry(LDV), hot wire anemometer(HWA),
shadowgraphy, schlieren method, infrared ther-
mography(IR) o↵ers a range of technique to ad-
dress the flow problem in hand.
Particle Image Velocimetry is an optical and
quantitative method of flow visualization. The
development of PIV during the last two decades
has been characterized by the replacement of ana-
log recording and evaluation techniques by digi-
tal techniques. The PIV technique measures the
velocity of a fluid element indirectly by means of
the measurement of the displacement of tracer
particles. Therefore the flow must be seeded with
tracer particle before the start of the experiment.
A high power light source for the illumination
of the microscopic tracer particles is required
to provide the photographic film or the video
sensor with su cient exposure of scattered light.
The advantage of PIV is its ability to spatially
resolve all three component of the velocity vector
in a two-dimensional plane. The main di↵erence
between PIV and other techniques of flow visu-
alization is that PIV produces two-dimensional
or even three-dimensional quantitative vector
fields, while most other techniques measure the
velocity at a given point or provide solely quali-
tative information. As with any measurement,
PIV cannot measure the velocity vector with
infinite accuracy. Di↵erences between the actual
flow field and measured flow field will always
be present due to the complexity of the tech-
nique. Errors are introduced through multiple
sources such as: calibration error, errors in ex-
Page 2 of 20
4. perimental setup, seeding density, particle image
diameter & displacement gradient, out of plane
displacement, etc.
The error in the estimation of the displacement
in PIV is typically considered to be around 0.1
pixel in magnitude8. But the error level can vary
in magnitude depending on experimental inputs
for a specific PIV test. Accurately predicting
the error levels is also a problem that is faced
by German Dutch Wind Tunnels (DNW). Every
test is di↵erent from the previous test, however
the customers would like to know the expected
error level to validate their CFD results. This
served as a motivation to not only predict the
errors beforehand but also to quantify them for
the parameters which has the most influence
on the PIV experiments namely particle image
diameter, image density, displacement gradient
and out-of-plane motion2.
The aim of uncertainty quantification is esti-
mating a possible value of the error magnitude.
Two approaches are possible, namely a-priori
and a-posteriori uncertainty quantification. A-
priori uncertainty quantification aims at provid-
ing an estimate of the magnitude of error even
before performing the experiment. A-posteriori
uncertainty quantification on the other hand
is done to assess the uncertainty in the vector
field of the measured velocity. Contrarily to the
a-priori approach that provides only general in-
formation on the limitations of an interrogation
algorithm, the a-posteriori approach aims at es-
timating the uncertainty of a specific velocity
field9. Recently, four approaches of uncertainty
quantification have been developed: the uncer-
tainty surface method (US)2, the particle dispar-
ity method (PD)9, the peak ratio method (PR)4
and the correlation statistics method(CS)3. For
uncertainty quantification, methods like PR, PD
and CS method depends on the parameters de-
rived from the actual correlation plane to evalu-
ate error.However, US method does not require
actual correlation plane from the measured dis-
placement.
The approach incorporated in this report is
motivated by uncertainty surface method be-
cause it is the only method which does not make
use of information originating from the cross-
correlation map of experimental images. Instead
it employs synthetic image generation to repro-
duce the experimental conditions. The approach
incorporated for a-priori error estimation here
is based on generation of synthetic images with
similar image attributes as experimental image.
The similarity of the synthetic images to ex-
perimental images can be tested by comparing
estimated error of synthetic and experimental
images with each other. This is followed by
estimating the error as a function of the domi-
nant error sources. This approach will help in
predicting errors before doing the experiment.
2 Objective
The main objectives of the assignment is to pre-
dict error for a PIV experiment before perform-
ing the experiment. Sub goals to fulfill the main
objective are mentioned as follows:
• Generate synthetic images with similar im-
age characteristics as experimental images.
• Validation of measured error of the vector
field estimated from the synthetic images
to those from the experimental images.
• a-priori error estimation of synthetic images
as a function of the relevant experimental
parameters.
Page 3 of 20
5. 3 Methodology
In this section the approach used to perform the
above mentioned objectives will be discussed in
detail.
3.1 Paticle image diameter & image
density estimation
To reproduce experimental images using syn-
thetic image generation, the particle image di-
ameter and image density needs to be estimated.
The approach of Scot O. Warner11 was used.
Yes
Figure 1: Flow Chart for calculating image density
3.1.1 Image density
When particle density is low it can be approx-
imated by counting particles. However when
image density is high it is impractical and time
consuming to count the particles. For the pur-
pose of better understanding, NI will be repre-
sented in particles per pixel (ppp) by dividing
NI from the number of pixel in an interrogation
window. The method used to estimate particle
density is adapted from the method discussed
by Scott O Warner and Barton L Smith10. The
relationship between the parameters was mod-
eled using a power-law function. Recently it has
been shown that the image particle density can
be estimated from the equation shown below:
NI ⇡ 48.2
t(RP )1.33
(d⌧ )1.99( ¯IP )0.84(A)1.35
(1)
where NI is image density, RP is auto-correlation
peak height, d⌧ is particle image diameter, ¯IP
is average peak intensity and A is the inter-
rogation window size. In principle, the image
density can be approximated if the e↵ect of all
the other parameters are quantified. Fig 1 shows
the flowchart used for creating a coding script
to estimate image density.
3.1.2 Auto-correlation peak
In PIV, the height of the auto-correlation peak
(Rh) is directly proportional to particle image
diameter (d⌧ ), image density (NI), average peak
intensity of particles (IP ) and interrogation win-
dow size (A).
Rh = f(d⌧ , NI, IP , A) (2)
An auto-correlation map for an interrogation
area can be determined using a frequency based
correlation by applying the Wiener-Khinchin
theorem8. The auto-correlation is computed as:
R = Re[FFT 1
(FFT⇤
(IA)FFT(IA))] (3)
where IA is the interrogation area, FFT is Fast
Fourier Transform , FFT* is complex conjugate
Page 4 of 20
6. Yes
Figure 2: Flow Chart for calculating particle diam-
eter
of FFT and Re is the real part of the complex
number. In order to use Eq. 3 the interrogation
window must be of size 2n where n is an integer.
3.1.3 Particle image diameter
The particle image diameter (d⌧ ) is the diameter
(in pixel units) of the particle as it appears in the
image and is directly proportional to the width
of the correlation peak. The correlation peak’s
width is calculated using the e 2 width of which
is calculated from the standard deviation of the
Gaussian intensity distribution. The particle
image diameter is then given as:
d⌧ ⇡ 2
p
2 (4)
In order to find the standard deviation ( ) from
the auto-correlation peak, 3 point Gaussian fit is
applied. Fig 2 shows the flowchart used for cre-
ating a coding script to estimate particle image
diameter.
3.1.4 Average particle intensity
As the name suggests, average particle intensity
is the average value of the intensity of all the
individual tracer particle over the entire image.
It is calculated by first locating particles using a
function that locates local maxima in the image.
Pixel intensity is an average of light intensity
falling on a pixel, thus particle intensity is maxi-
mum and nearest to the true maximum intensity
when a particle is centered on a pixel. In order to
find a pixel centred particle, the standard devia-
tion of the intensity of the four adjacent pixel is
calculated. If the calculated standard deviation
is less than a threshold value, it is considered as
a pixel centered particle and contributes to the
average intensity calculation. One thing to be
noticed is that the estimation of average particle
intensity is based only on local maxima and not
on the average intensity of the actual individual
particle.
3.2 Laser sheet thickness
The thickness of the PIV laser sheet can be in-
ferred from the auto-correlation from both the
cameras dewarped images. The images from the
same pulse exposure are used from both the cam-
eras. The correlation peak is smeared out by
perspective distortion due to particles contribut-
ing from di↵erent depths throughout the light
sheet. Discrete cross-correlation is performed
Page 5 of 20
7. with a large window size of 256⇥256 or 512⇥512
px2. A larger window size is used in this case to
increase the signal to noise ratio of the correla-
tion peak. B.Wieneke13 showed that the laser
sheet thickness can be estimated from the cam-
era angle and the correlation peak width. Eq.
5 shows the equation of correlation peak when
using dewarped images.
wc = Zo
1
tan(↵1)
+
1
tan(↵2)
(5)
where Zo is the light sheet thickness, wc is
the correlation peak width, ↵1 and ↵2 are the
viewing angle of cameras 1 and 2 for instance
relative to the x axis when the cameras are placed
horizontally along the x axis. This equation gives
the thickness of light sheet in pixels. For a typical
Stereo-PIV experiments with measured in-plane
displacement of 5-10 pixels, laser sheet thickness
is generally at least twice as thick to measure
z direction component of the same order. Thus
for a typical PIV experiment one can expect
a correlation peak width in the order of 10-20
pixels13.
3.3 Synthetic image generator
A synthetic images generator gives full control
in the hand of the user to change image charac-
teristics for parameters such as image dimension,
particle image peak intensity, mean particle di-
ameter, image density, laser sheet thickness, out-
of-plane displacement, in-plane displacement,
displacement gradient, background noise, etc.
The synthetic image used here is based on the
algorithm discussed by M. Ra↵el, C. E. Willert
and J. Kompenhans8. The particle are randomly
distributed onto the pixel array. The individual
particle in synthetic images are described by a
Gaussian intensity profile as shown in Eq. 6.
I(x, y) = Ioexp
✓
(x xo)2 (y yo)2
(1/8)d2
⌧
◆
(6)
where (xo, yo) is the center of particle image
and randomly distributed throughout the image.
In the above equation, particle image diameter
(d⌧ ) is defined as the e 2 intensity value of the
Gaussian bell shape which contains 95% of the
scattered light. For a light sheet centered at Z =
0 with Gaussian intensity profile for Io is given
as,
Io(Z) = q · exp
Z2
(1/8) Z2
o
!
(7)
where Zo is the laser sheet thickness also mea-
sured at e 2 intensity waist point of its Gaussian
profile.
Hence for generating particle image, parti-
cle position (X1, Y1, Z1) are randomly generated
within the specified laser sheet intensity pro-
file. The peak intensity (Io(Z1)) is calculated
using Eq. 7 for Gaussian intensity profile. When
all these inputs are fed to Eq. 6, it gives the
intensity captured by each pixel. To simulate
displacement, another image is generated with
same process as discussed above but with an
o↵set of the previous image particle location de-
pending on the x, y and z displacement specified
by the user.
Background noise in the synthetic image is
added as a product of uniform distribution of
random numbers and by the user defined noise
intensity. The (white) noise added in an image
pixel is uncorrelated with its neighbours and
with its counterpart exposure.
Correlated noise in the image was simulated
by using a brown noise image and super impos-
ing the same spatial pattern on both synthetic
images. Brown noise is signal noise created by
random motion. However, It has more energy
Page 6 of 20
8. at lower spatial frequencies. Power spectrum
(Sf )for brown noise is given as:
Sf = (u2
+ v2
)2 (8)
where u and v are set of frequencies in first and
second dimension respectively and is spectral
distribution. For brown noise, is considered
as -2. All the synthetic images created in this
report have image dimension of 500⇥500 px2.
3.4 PIV Processing
The PIV processing software used in this report
is PIVview (PIVTEC, G¨ottingen, Germany).
Throughout the report same PIV processing
steps are used. No filters were used in the im-
age pre-processing. A window size of 32⇥32 px2
with 50% overlap is used. The Multigrid Inter-
rogation Method is used having initial sampling
window size of 128⇥128 px2 and going down
to final window size of 32⇥32 px2. Multigrid
algorithm uses a pyramid approach by starting
o↵ with large interrogation window and refines
to smaller window size after every pass. Also
on the final multigrid pass, image deformation
is used to enhance the spatial resolution of the
image. This is done by sub-pixel image shifting
based on B-spline interpolation of order 3.
3.5 Error estimation
Identifying the source of error in PIV images is a
complicated task because it depends on various
factors. There are three forms of errors gener-
ally encountered in PIV. These are the outliers,
random errors and the bias errors6.
Outliers are the incorrect displacement esti-
mate because a random noise peak dominate the
true displacement peak. This occurs when there
are insu cient number of particles in the image
pair, high noise level, strong velocity gradients
and strong three-dimensional flow motions. Out-
liers usually appear randomly both in direction
and in amplitude. The amplitude of this error
is generally larger than one pixel and are easy
to detect.
Random error is the deviation of the par-
ticle displacements from their mean. It corre-
sponds to the precision of the estimation of the
peak location. Random error is given by Eq. 9
✏rand =
v
u
u
t
NX
i=1
di dmean
N
!
(9)
The random error are di↵erent for each mea-
surement and are associated with factors related
to image characteristics. The identification of
random error is bit challenging as typically it
lies in a range between 0.03 and 0.1 pixels9.
Bias error is the di↵erence between the mea-
sured displacement and true displacement. Bias
error (✏bias) is given by Eq. 10
✏bias = dmeas dtrue (10)
For PIV measurement in the empty test section,
the true displacement can directly be calculated
from the wind tunnel velocity if there is no out-of-
plane component. For a light sheet perpendicular
to the main flow the camera angles has to be
considered in calculating the true displacement.
Eq 11 can be used to calculate true displacement.
dtrue = dWT M
y
z0
(11)
where dWT is displacement calculated from wind
tunnel velocity and separation time interval. y
and z0 are the y-z co-ordinate of the camera
with respect to the image. Since the image is
having a longitudinal displacement gradient be-
cause of bias error, only the first row of averaged
measured velocity is computed for true displace-
ment.
Page 7 of 20
9. 4 Experimental Setup
In this section the experimental setup of the
two test cases used in this report is discussed.
The experiments were conducted at DNW’s large
low-speed facility (LLF).
4.1 Test Case I
In test case I, a PIV experiment was performed
in the open jet configuration of the wind tunnel.
The wind tunnel speed was kept constant at 20
m/sec. The light sheet was oriented parallel
to the tunnel flow. PIV images were acquired
using a pco.2000 14-bit CCD camera. The cam-
eras were mounted on a tower as shown in Fig
3. The camera positions are shown in Table
1. The flow was seeded using di-ethyl-hexyl-
Figure 3: Experimental test setup for test case I
Camera x-axis y-axis z-axis
Camera 1 102 mm 6391 mm 5048 mm
Camera 2 88 mm 6302 mm -5170 mm
Table 1: Camera position for test case I
sebacate (DEHS). To obtain converged statistics,
N = 100 images were acquired for various time
separations ranging between 60µs to 70µs. Mag-
nification factor of 6.5 mm/pixels was used.
4.2 Test Case II
In test case II, a PIV experiment was performed
in the 8m⇥6m closed test section. The wind
tunnel speed was kept constant at 47 m/sec.
The light sheet was perpendicular to wind tun-
nel velocity. PIV images were acquired using
a pco.2000 14-bit CCD camera. The cameras
were mounted on traverse unit along x-axis as
shown in Fig 4. The camera positions are shown
in Table 2. The laser sheet is illuminated in
Figure 4: Experimental test setup for test case II
Camera x-axis y-axis z-axis
Camera 1 4426 mm -570 mm -2790 mm
Camera 2 4429 mm -3555 mm -2788 mm
Table 2: Camera position for test case II
z-direction. The flow was seeded using di-ethyl-
hexyl-sebacate (DEHS). To obtain converged
statistics, N = 15 images were acquired for vari-
ous time separations ranging between 8 µs and
25 µs. Magnification factor of 7.4 mm/pixels
was used.
5 Results
In this section validation of image characteristic
of synthetic image to the experimental image is
done. This is followed by a-priori error estima-
tion for the dominant error sources.
Page 8 of 20
10. 5.1 Attributes of synthetic image
generator
In this section validation of estimated image
density is done by synthetic images with known
actual image density. This shows how well does
the image density estimation method works for
various input parameters as discussed hereafter.
5.1.1 Image density estimation
Synthetic images were generated with densities
ranging from 0.01 ppp to 0.09 ppp in steps of
0.01 ppp while keeping particle image diameter
and particle image peak intensity constant at 3
pixels and 1000 counts respectively. No out-of-
plane displacement and no noise was introduced
in the images. The image density was calculated
for interrogation window size of 32 ⇥ 32 px2.
From Fig 5a, it is evident that estimated density
shows good approximation until image density
of 0.06 ppp. After this point the model starts to
underestimate the image density.
Fig 5b shows a limiting case when particle
diameter is considered as 1 pixels. Similar to
previous case synthetic images were generated
with densities ranging from 0.01 ppp to 0.09 ppp
in steps of 0.01 ppp while keeping other parame-
ters constant. In this case estimated density is
over predicted from the particle density of 0.02
ppp onwards. This occurs because density is in-
directly proportional to particle image diameter
and average image intensity. In this case, there
is no overlap of tracer particles and the tracer
particles are small leading to an increase in the
estimated image density.
5.1.2 E↵ect of particle image diameter
The particle image diameter also a↵ects the pre-
diction of image density as shown in Fig 5c.
Synthetic images were generated with varying
particle image diameter from 1 pixel to 5 pixels
in steps of 0.5 pixel, with constant particle den-
sity of 0.1 ppp and no noise. The estimate are
within 10% of the actual density for the particle
diameter in the range of 1 pixel to 3.5 pixels.
5.1.3 E↵ect of noise
To study the e↵ect of noise on seeding density es-
timation, simulated images were generated with
particle image density and diameter of 0.05 ppp
and 3 pixels respectively. The random noise level
was varied between 10 and 40 %. Fig 5d shows
noise level have a significant impact on the par-
ticle image density. It leads to an increase in
estimation of density in an almost linear fashion.
5.1.4 Application to experimental images
An experimental PIV image with inhomogeneous
seeding density is considered (see Fig 6a). When
this image is processed for estimating particle
image density and particle image diameter with
32⇥32 px2 window size, the results shows that
the density estimate captures the seeding vari-
ation in experimental PIV images. Fig 6b and
Fig 6c shows estimation of particle diameter and
image density respectively. In both the images
there is significant range in the variation of par-
ticle image diameter and image density. The
estimated particle image diameter and image
density is taken as the mean value over all the
interrogation windows.
5.2 Laser sheet thickness estimation
In order to determine the laser sheet thickness,
one needs to determine the width of the correla-
tion peak as already discussed in Section 3.2. Fig
7 shows a typical example of the auto-correlation
Page 9 of 20
11. Actual Particle Density (ppp)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
EstimatedParticleDensity(ppp)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Window Size = 32 x 32
Actual Particle Density
Estimated Particle Density
(a) Estimated Particle Density vs. Actual Particle
Density for d⌧ = 3 pixels
Actual Particle Density (ppp)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
EstimatedParticleDensity(ppp)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
dτ = 1 px, Window Size = 32 x 32
Actual Particle Density
Estimated Particle Density
(b) Estimated Particle Density vs. Actual Particle
Density for d⌧ = 1 pixel
Particle Diameter (pix)
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
EstimatedParticleDensity(ppp)
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Window Size = 32 x 32
Actual Particle Density
Estimated Particle Density
(c) Estimated Particle Density vs. Particle Image
Diameter for NI = 0.1ppp
Noise (%)
0 5 10 15 20 25 30 35 40
EstimatedParticleDensity(ppp)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Window Size = 32 x 32
Actual Particle Density
Estimated Particle Density
(d) Estimated Particle Density vs. Noise for NI =
0.05ppp
Figure 5: Estimation of particle image diameter and particle image density
Figure 7: Cross-correlation peak width for laser
sheet estimation
peak. The correlation width is smeared in only
x-direction because of the horizontal positioning
of both the cameras.
5.3 Experimental vs. Synthetic images
In this section a comparative study of estimated
error level is done for the experimental images
and the synthetic images generated for both the
test cases.
Page 10 of 20
12. (a) Raw Image
(b) Particle image diameter es-
timation (WS = 322
)
(c) Particle image density esti-
mation (WS = 322
)
Figure 6: Estimation of particle image diameter and particle image density for image with inhomogeneous
seeding density
5.3.1 Test Case I
Stereo-PIV images for test case I is discussed
in this section. The experimental setup for this
test is described in Section 4.1.
E↵ect of Noise on error Synthetic Images
Figure 8: Error vs. Noise for displacement of 6.6
pixels
were generated with similar properties as esti-
mated from experimental image (d⌧ = 3px, NI
= 0.1 ppp). The image noise level was varied
from 0 to 20% to estimate the noise present in
the vector field from the experimental images.
From Fig 8 it can be inferred that random er-
ror increases as image noise level increases and
bias error remains almost constant with noise
level. A similar investigation was done for the
same experiment but at a di↵erent in-plane dis-
placement of 7.7 pixels. This was done to see
whether a similar noise level is present in all the
the experimental images.
Figure 9: Error vs. Noise for displacement of 7.7
pixels
Fig 8 and Fig 9 shows that measurement er-
Page 11 of 20
13. Correlation plane obtained at 174,556 Dx=-2.190 Dy=0.022 SNR=159.8 Corr=78.6% Date: vr nov 6 2015 Time: 13:58:04
(a) Experimental image
Correlation plane obtained at 185,154 Dx=7.590 Dy=0.005 SNR=86.2 Corr=66.6% Date: vr nov 6 2015 Time: 13:58:37
(b) Synthetic image with 15% noise level
Figure 10: Raw Images for Test Case I
Figure 11: Error vs. Displacement for 0% noise
level
Figure 12: Error vs. Displacement for 15% noise
level
ror the PIV processing of synthetic images with
15% noise level is similar to experimental im-
age’s measurement error. Fig 10a and Fig 10b
shows a 32⇥32 px2 window of an experimental
image and a synthetic image with 15% noise
level respectively. Henceforth all the graphs will
be having a noise level of 15% for Test Case I.
E↵ect of Displacement on error To deter-
mine the similarity of the image characteristic
of synthetic image compared to experimental
images, a validation study of measurement
error as a function of in-plane displacement is
done for both the images. Fig 11 and Fig 12
shows random errors and systematic errors with
respect to displacement for noise level at 0%
and 15% respectively. Fig 12 shows a better
agreement of the error in synthetic images
compared to the error level in experimental
images. It can also be noted that random error
is less sensitive to noise compared to bias error
for di↵erent displacement.
5.3.2 Test Case II
After the results of test case I, a more complex
test with out-of-plane component was considered.
The out-of-plane displacement was also included
in synthetic image generation. The experimental
setup for this test is described in Section 4.2.
E↵ect of Noise on error Similar to test case I,
synthetic images similar to experimental images
were generated with 3.25 pixel mean diameter,
particle image density of 0.13 ppp (estimated
from experimental image) and with 10% out of
plane component. Noise level was varied from 0
Page 12 of 20
14. Figure 13: Error as a function of noise level for
Test Case II
to 30% to predict the expected noise level. Fig 13
shows that random error increases as noise level
increases. PIV processing of synthetic images
with a noise level of approximately 24% showed
the same random error as the experimental re-
sults. However, the correlation strength of the
results from synthetic images was much lower
compared to experimental image. At a noise
level of 18%, the experimental and synthetic re-
sults showed a similar cross-correlation strength.
Henceforth, synthetic images with 18% noise will
be used for further investigation.
E↵ect of Displacement on error Fig 14 shows
the experimental random error and bias error
with respect to the displacement. An unexpected
behaviour was noticed for both experimental er-
rors. They shows a sudden change in random
and bias error when the displacement is between
2 - 3 pixels. Synthetic images were generated
with similar properties for di↵erent displacement
as discussed above with 0% and 18% noise level.
There is little similarity in the random error of
synthetic image PIV results compared to the
magnitude of error of the experimental results.
The results from the synthetic images shows the
expected increase of error with an increase in
out-of-plane displacement. Further investigation
was done to find out what might be the reason.
Additional possible processing of the vector fields
was formulated to remove the outliers from the
results by giving strict boundary condition of
±5 to remove any error value greater than the
prescribed value. Fig 16b shows the result with
a confidence interval of 95%. This means that
there is probability of 0.95 that the estimated
error lies within that region. The magnitude
of error decreased but the characteristic of the
curve remained same meaning the strange nature
is not due to outliers in the image. In order to
Figure 14: Error as a function of displacement for
experimental image
Figure 15: Error as a function of displacement for
di↵erent configurations
make sure whether the strange behaviour was
due to wind tunnel operating condition or exper-
imental setup, same analysis was done for the
left camera in the experiment shown in Fig 16a.
However it did not show similar characteristics
Page 13 of 20
15. (a) For left camera (b) For right camera
Figure 16: Random Error as a function of displacement with/without standard deviation filter
Figure 17: Correlated noise in correlation plane in
experimental image for 32⇥32 interoga-
tion window
for the random error. The experimental images
from di↵erent polar set was also compared to
see the behaviour of measurement error as a
function of in-plane displacement. Polar 9 (P9)
is the data set with wind tunnel speed at 0.14
Mach. Polar 11 (P11) is the data set with the
aforementioned wind tunnel speed but with a
new camera (upgraded version of previous cam-
era used for polar 9) showed similar behaviour.
Fig 15 shows the graph with these polars and
synthetic images.
A hypothesis was considered that the unex-
pected error characteristics may be due to corre-
lated noise present in the experimental images.
It was noticed that the seeding fog patter present
in the images was stationary for all the images
of a data set. Fig 17 shows that in the interroga-
tion window at the edge of the seeding fog there
is a band of noise in the center of the correlation
plane. This e↵ect was present on all the seeding
fog edges. Thus it was considered as correlated
noise.
Spatial Power Spectral Density (PSD): Spa-
tial analysis was used to further investigatethe
characteristics of the noise in experimental im-
ages. A straightforward approach to calculate
PSD is by taking the magnitude squared of its
Fourier transform of the image. For reference,
Fig 18a was plotted to show PSD of 3 synthetic
images namely, images with only 10 particle, im-
age with a 5% noise level and image having 10
particle with 5% noise level. The image with
10 particle i.e. only signal shows that its PSD
decreases after 102 1/pixel in the spatial domain.
For the other two plots they almost remain con-
stant with increase in spatial frequency which
shows the trend of white noise in PSD added in
the synthetic images.
Fig 18b shows the PSD of experimental image
and synthetic image over spatial frequency. It
is evident from the figure that the experimen-
tal image have more signal in the low frequency
component which cannot be seen in the synthetic
image with noise level of 18%. This served as a
Page 14 of 20
16. Spatial Frequency
100
101
102
103
-35
-30
-25
-20
-15
-10
-5
0
5
10
PowerSpectralDensity[db]
10 particles & no noise
no particles & 5% noise
10 particles & 5% noise
(a) Reference Plot
Spatial Frequency
100
101
102
103
PowerSpectraldensity[db]
-60
-50
-40
-30
-20
-10
0
10
Experimental Image
SI with 18% Noise
SI with 18% Noise & Brown Noise
(b) Experimental and Synthetic Images
Figure 18: Power spectral density as a function of spatial frequency
Figure 20: Synthetic image with 19% noise and
brown noise
Figure 21: Random Error as a function of displace-
ment
motivation to include low frequency component
in the synthetic images. The slope of PSD for ex-
perimental image is approximately -30db/decade
which is similar to the slope of brown noise. Thus
the same spatial pattern of brown noise is added
in both the exposures to simulate correlated
noise. A new synthetic image containing brown
noise is shown in Fig 20. The image looks similar
to experimental images (see Fig 6a) because now
low frequency noise is also added. As expected
PSD for the new synthetic image have low fre-
quency component and is indeed more similar
to experimental image.
The correlation peak for the experimental im-
age and synthetic image with brown noise shows
a band of correlated noise at the edges of the
seeding fog which cannot be seen in synthetic
image without brown noise as shown in Fig 19.
To validate it the unexpected behaviour of PIV
random error with respect to the displacement
was related to correlated noise, synthetic image
with and without brown noise were processed
(see Fig 16b). Fig 21 shows that there is an
increase in random noise of synthetic image with
brown noise. Also there is a small bump in the
random error in the displacement range of 2 to
3 pixel but the trend of increasing noise with an
increase in out-of-plane displacement was similar.
6 Numerical Assessment
The a-priori error estimation of PIV data was
analysed via synthetic image generation at dif-
ferent conditions. The e↵ect of 4 major error
Page 15 of 20
17. (a) Experimental image (b) SI with brown noise (c) SI with no brown noise
Figure 19: Correlation peak for di↵erent images
(a) Particle image diameter (b) Image seeding density
(c) Displacement gradient (d) Out-of-plane motion
Figure 22: RMS error prediction for Test Case I
sources typically in PIV measurement was anal-
ysed. The input parameters are with respect to
the image characteristic of test case I.
6.1 Particle image diameter
In the current simulation, images were gener-
ated with particle image density 0.1 ppp with
no out-of-plane displacement. The in-plane dis-
placement was constant at 7.5 pixels for all the
images. Particle image diameter was varied be-
tween 0.5 - 5 pixels. The RMS error decreases
as particle image diameter increases till 2 pixels
and after this point error starts to increase again.
This trend is expected because peak locking dom-
inates the measurement error for d⌧ 1 pixel.
And for d⌧ >1 random error starts to dominate
which was investigated by Westerweel12.
6.2 Seeding density
Simulated images were generated with particle
image diameter 3 pixels with no out-of-plane dis-
placement. The in-plane displacement was kept
Page 16 of 20
18. at 7.5 pixels. Particle image density was varied
between 0.02ppp to 0.2ppp. Fig 22b shows RMS
error decreases for increasing seeding density.
This is because at low seeding density there are
less particle pairs in each interrogation window
which makes it di cult to determine the true
displacement peak. With increase in seeding
density, the number of particle pair increases
and the true displacement peak is determined
with higher precision. It can also be seen that
after 0.15 ppp adding more seeding leads to a
minimal decrease to the magnitude of error.
6.3 Displacement gradient
A displacement gradient ranging from 0.02 to
0.2 pixels per pixel is considered in this case. Fig
22c shows that RMS error increases by almost
factor of 2 until 0.18 pixels/pixel. The high dis-
placement gradient decreases the quality of the
matching of paired particle images between the
interrogation window of the two images, yielding
a cross-correlation peak that broadens in the
shear direction. The WIDIM algorithm applies
in-plane deformation and compensates the gra-
dient e↵ect. As a result, the measurement error
is reduced by one order of magnitude5.
6.4 Out-of-plane motion
Measurement error due to out-of-plane motion
can be one of the main source of errors7. The
current simulation considers a uniform in-plane
displacement of 7.5 pixels and uniform out of
plane displacement ranging from 0 to 0.35 Z.
Fig 22d shows an exponential increase in RMS er-
ror with the out of plane displacement which was
also investigated by Nobach and Bodenschatz7.
This happens because of loss of particle pairs in
interrogation area of the two exposures.
7 Discussion
In this section the limitations of the methods
used through out the report are discussed.
Synthetic image generator: With higher seed-
ing density in simulated images (see Fig 5a) there
is an overlapping of the tracer particle. In this
case the average particle intensity is overesti-
mated because the intensities of the particle are
summed together. This in turn leads to under-
estimation of true image density because it is in-
directly proportional to average image intensity
(Eq. 1) . Also if window size is more than 32⇥32
px2 is chosen the method tends to over predict
the image density. A high level of background
noise has a significant impact on the prediction
of image diameter and density as shown in Fig
5d. This is expected because the model starts to
consider noise also as tracer particle and hence
with increasing noise level density also increases
significantly. Thus it can be inferred that the
method breaks down in case of high level of noise.
Hence it is always better to cross check the syn-
thetic images and experimental image for image
characteristic anomalies.
However, if the image with inhomogeneous
seeding density is having a large number of inter-
rogation windows (more than 1000 interrogation
windows) it helps in better estimation because
of averaging of the parameters. It is not feasible
to pre-process the experimental image before es-
timating image density and diameter as it may
lead to loss of the true nature of the image.
Experimental vs. Synthetic image: Error
from the PIV processing of synthetic images
was calculated and compared to the magnitude
of error in experimental results with respect to
noise level and in-plane displacement. Fig 8
Page 17 of 20
19. shows the e↵ect of error as a function of noise
level. Random error increases with added noise.
Bias error on the other hand does not depends
on increment of noise level (see Fig 8 and Fig 9)
because bias error usually comes from the peak
locking e↵ect which is less dependent of noise.
If more noise is added to the images in order
to make similar error level inPIV processing
then the signal of synthetic image gets lost in
the noise and the cross correlation strength
decreases. Hence throughout the report more
emphasis was give on cross-correlation strength
in selecting the noise level. In Fig 11 and Fig 12
typical peak locking sinusoidal behaviour of bias
error cannot be seen because the particle image
diameter is larger than the optimum diameter
of 2 pixels. However no explanation in literature
can be found yet for the periodic pattern.
The error estimated by PIV processing of syn-
thetic images is likely underestimated due to
fact that a lot of assumptions are considered
while generating them such as all the particles
in the synthetic images have Gaussian intensity
distribution which may not be the case with
experimental images. One further di↵erence be-
tween experimental and the synthetic images is
that the pixel fill factor (ratio between sensitive
area and total area of the pixel) is less than 1 in
the experimental images, while it is equal to 1
in the synthetic ones. Implementing a pixel fill
factor smaller than one in the synthetic images
is not a trivial task. Thus an error curve for
synthetic image will never perfectly fit with that
of the experimental image.
PIV processing of synthetic image with and
without brown noise as shown in Fig 21 showed
that fog density simulated by brown noise in-
creases the error level by 0.005 px compared to
synthetic image without brown noise and error
level of the results increases as the intensity of
brown noise increased. This is expected because
it starts to hinder the cross co-relation of the two
exposures which leads to increase in error. In
Fig 15, Polar 9 and 11 shows similar unexpected
behaviour at displacement of 2 pixels. Synthetic
images on the other hand showed the result what
one would expect from theoretical point of view.
No meaningful reason was found for the sudden
increase in the error of experimental image for
small in-plane displacement (see Fug 16b). This
means the source of error is something else which
was not covered in this report. Also, the con-
fidence interval in test case II was determined
from 14 images which gives limited power of esti-
mation. Thus it was not pursued further in this
report.
a-priori error estimation: Monte Carlo simu-
lation was used to predict the a-priori error es-
timation. Only random error was considered in
this process as it is the dominant error compared
to bias error. Fig 22a, Fig 22b, Fig 22c and Fig
22d shows the error propagation curve with re-
spect to particle image diameter, image density,
displacement gradient and out-of-plane motion
respectively. The plots generates confers with
the results shown by M. Ra↵el & co-authors8
and A. Sciacchitano1 in their respective research.
It was noticed that out-of-plane motion could
dominate the error for stereo PIV. Similar plots
were generate for test case II image properties
(d⌧ = 3.25 pixels, NI = 0.13 ppp) which showed
the same trend of the curve as shown by test
case I. Only anomaly in test case II is that the
error estimation is higher by 0.02 pixels due to
10% out-of-plane component(see Fig 22d).
Page 18 of 20
20. 8 Conclusion
In this report, a-priori uncertainty quantification
approach is used to determine the expected error
as a function of selected parameters namely parti-
cle image diameter, image density, displacement
gradient and out-of-plane motion for unifor flow
case in empty test section. Firstly, the image
properties were derived from the experimental
image to create similar PIV simulated images
which is generally expected from the experimen-
tal setup at DNW’s LLF wind tunnel. Particle
image diameter and density was estimated by
correlation-based density method developed by
Scot O. Warner11. Laser sheet thickness was es-
timated by the method discussed by Wieneke13.
Displacement component in x, y and z direction
was calculated from PIV processing the exper-
imental images. Background image noise was
also determined to find the noise level present
in the experimental images. From these inputs
similar synthetic images were generated. Using
Monte Carlo simulation, random error of the four
dominant parameters were studied. Both the
test cases showed similar trend of the random
error curve for respective parameters.
The work in this report is for uniform flow in
an empty test section. Similar analysis can be
done for more complex flow problems such as
vortex to make the method more robust. Also an
e↵ort is required to make better synthetic image
generator which a is more realistic representation
of the experimental images maybe by considering
only half of the randomly distributed particles
to have Gaussian distribution instead of all of
them as taken in this case.
In order to find out the sudden increase in the
random error for the right camera of test case
II, similar experiment with same experimental
setup can be carried out for lesser time separa-
tion between the two exposure of the camera to
determine the random error for lower displace-
ment. This would shed some light in determine
whether the increase seen in Fig 16b was just by
chance or there is a trend in the random error
for lower displacement.
The main conclusion which can be drawn from
this report is that synthetic images still needs
further development and cannot be used as a
stand alone method to generate simulated images
which can be expected in experimental test cases.
However analysis with synthetic image helped
in quantifying few of the characteristics present
in the experimental images.
References
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