Theoretical and experimental analysis of wavelengthmultiplexed PTV
R Wang and A J Moore
School of Engineering and Physical Sciences, Heriot-Watt University,
Edinburgh EH14 4AS, UK
[email protected]
Abstract. Time-resolved measurements of 3D velocity are required for characterizing
complex flows. Particle tracking velocimetry (PTV) tracks entrained-particle positions
through an image sequence, giving the 3D velocity throughout a volume of fluid. Multicoloured particles can enable an increased seeding density to be achieved than with
single colour particles, thereby increasing the measurement spatial resolution.
Numerical simulation and mathematical modelling were developed to study the
improvement achieved with wavelength-multiplexed PTV and to estimate the optimal
seeding density. It is shown that the measurement spatial resolution could be
improved by a factor of 10 with 10-colored particles for particles of diameter 3 pixels. A
proof-of-principle experiment was performed using multi-colour particles, where 3D
displacement components in a fluid were measured.
1. Introduction
The determination of the local flow velocity and the velocity distribution in complex flows is a
fundamental task in the field of fluid dynamics. Particle image velocimetry (PIV) and particle
tracking velocimetry (PTV) are well-established full-field optical techniques that visualize the
flow of neutrally-buoyant seeding particles that are sufficiently small to faithfully follow the
flow. PIV generally determines spatially averaged velocity vectors for uniformly spaced
interrogation windows in each image[1],[2],[3], so the number of particles must be sufficient for
statistical analysis, and uniform scattering from particles and smooth fluid boundaries are
required. 3D PIV is currently limited to planar measurement. In PTV, the position of
individual particles is determined, from which their velocity and trajectory can be calculated.
PTV provides long 3D trajectories of individual particles. The identification and tracking of
large numbers of particles in a three-dimensional (3D) volume of fluid through multiple
images has become computationally viable since the mid-1980s[4],[5].
Photogrammetric 3D PTV requires correspondences between particles recorded in the
multiple images to be established. Homologous particles are located by projecting the
epipolar line that passes through a particle in one view into the other available camera views:
the intersection of the projected lines defines the particle position in 3D space. Algorithms
and techniques that decrease the number of ambiguous correspondences are actively
sought in order to increase the spatial resolution of PTV measurements. In traditional PTV, to
identify uniquely the corresponding particle within each view and avoid ambiguities, the
seeding density may be reduced and (or) extra cameras may be added, thereby reducing the
spatial resolution of the measurement and (or) increasing the optical access requirement as
well as equipment cost.
In this paper, we describe an extension to PTV, namely wavelength-multiplexed PTV,
which reduces the number of ambiguous particle correspondences and enables volume 3D
velocity measurements to be made at higher seeding densities with only two cameras. Also,
a mathematical model and numerical simulation are used to determine the improvement in
spatial resolution achieved by adding more colours. The model is validated by experiments
using particles that can be optically distinguished by their colours. Finally, the applicability of
the technique is described, and conclusions drawn on the optimum seeding conditions for a
given experiment.
2. Theoretical simulation
Epipolar geometry is a photogrammetric approach to determine homologous points in a set
of images recorded from two or more directions. Any visible object point must lie on the
epipolar line, which is projected from the image of the particle out through the perspective
centre in to the object space. The distance of the particle from the perspective centre can be
determined by using more views. Thus, to determine the corresponding image of a particle in
the second view, the particle is required to be identified on the projected epipolar line. If only
one particle is observed on an epipolar line, the particle can be easily identified. As the
seeding density increases, two or more particles might be imaged on the epipolar line,
leading to ambiguity. However, in wavelength-multiplexed PTV, the particles can still be
discriminated by their colours. The increase in the probability of successful particle
identification was quantified by a mathematical model and a numerical simulation.
For the purposes of simulation, it is assumed that spherical particles are dispersed
randomly within the fluid volume. Particles of different colours can be distinguished according
to their hues, and the proportions of all colours are equal. If two or more particles of the
same hue lie on an epipolar line, is regarded as failure in the particle identification for that
particular hue. A particle is determined to be on an epipolar line if any part of it crosses that
line. Since the shape and size are assumed to be equal, the centre positions of particles
determine if they intersect with the epipolar line. Therefore the search area for particle
centres either side of the epipolar line is determined by the diameter of the particles.
In the mathematical modelling, the probability of successful particle identification at a
given seeding density of the flow is determined by mathematical calculations. If np particles
lie on a line, each of which could be any one of nc hues, then the total number of possible
n
1
)
hue combinations is given by (Cn
. Successful combinations occur when no more
than one particle on the line is in the hue under consideration. Therefore the probability of an
epipolar line being analysed successfully is given by:
p
c
p( Success ) 
Cn1 p (Cn1c 1 )
( n p 1)
(Cn1c )
 (Cn1c 1 )
np
np
(1)
Equation (1) was used to plot the probability of a success against the seeding density,
where the seeding density, S, is defined by:
S
np
(2)
d pL
where dp is the particle diameter and L is the length of the epipolar line. In order to
compare theoretical and experimental results more readily, a normalised seeding density
was defined:
Sn  S
 d p 2
4

 npd p
4
L
(3)
The probability of success in particle identification from the mathematical analysis is
plotted in Figure 1(a), which shows that the probability of success falls as the normalised
seeding density increases, because the epipolar line potentially intersects more particles.
Also, it is shown that the probability of success increases as the number of hues, nc,
increases, because more particles can be uniquely identified on the epipolar line. To express
in a dimensionless way, an effective seeding density, Se, is defined, that is proportional to
the number of successful measurement points.
S e  S n pSuccess 
(4)
From figure 1(b), it is shown that the effective seeding density increases with number of
hues used. For each additional hue introduced, the optimal effective seeding density occurs
at an increasingly higher seeding density.
In the numerical simulation, particles with random position were generated within the test
window. A threshold was chosen to control the normalised seeding density, and then
subdivided to present multi colours with equal proportion. At each seeding density, 10,000
searching regions of data were generated, and the number of times particles of a particular
hue could be successfully identified was recorded to determine a numerical estimate of the
probability of success. The results of the numerical simulation are also shown in Figure 1,
and they show good agreement with the mathematical model. Note that the simulation for
particles of a single colour gives a gradual reduction in the probability of success due to the
Gaussian distribution of number of particles, while the mathematical model for single
coloured particles shows a rapid transition between one particle and two particles.
0.035
100
1 colour
2 colours
3 colours
4 colours
5 colours
0.03
e
80
Effective seeding density, S
Probability of success, p(Success) (%)
90
70
60
50
40
30
1 colour
2 colours
3 colours
4 colours
5 colours
20
10
0
0.025
0.02
0.015
0.01
0.005
0
0
0.005
0.01
0.015 0.02 0.025 0.03 0.035
Normalised seeding density, S n
(a)
0.04
0.045
0.05
0
0.005
0.01
0.015 0.02 0.025 0.03 0.035
Normalised seeding density, S n
0.04
0.045
0.05
(b)
Figure 1 (a) Probability of success, p(Success) and (b) effective seeding density, Se,
plotted against normalised seeding density, for a particle diameter of three pixels. The
solid lines represent the mathematical model and the points the numerical simulation.
3. Experimental verification
A proof-of-principle experiment was performed to demonstrate volume 3D PTV
measurements using the two-coloured particles[6]. Furthermore, experiments were carried out
to verify the mathematical model and numerical simulation. Quantum dots with a CdSe core
and a ZnS shell, that had been encapsulated into PMMA particles were used (supplied by
Evident Technology). Five quantum dot emission wavelengths were selected (550, 580, 592,
606 and 632 nm), each with an emission bandwidth of ±10 nm. A UV lamp emitting at 366
nm was used as the excitation source. The emission spectra of the different particles are
shown in figure 2(a). The particles were fully mixed in water at the bottom of a cylindrical
container, of diameter 8.5 cm. Figure 2(b) shows a typical image captured by the digital
camera with a long-pass filter in place with the five coloured particles, which can be isolated
by their hues.
1000
550nm
580nm
592nm
606nm
632nm
900
Normalized intensity (a.u.)
800
700
600
500
400
300
200
100
0
500
520
540
560
580
600
620
Wavelength (nm)
640
660
680
700
(a)
(b)
100
100
90
90
Probability of success, p(Success) (%)
Probability of success, p(Success) (%)
Figure 2 (a) Emission spectra of the 5 PMMA particles used; (b) Typical experimental
images of wavelength-multiplexed particles recorded with UV illumination with optical
long-pass filter.
At each normalised seeding density, all the vertical and horizontal lines in the image were
processed as the epipolar line and analyzed for finding unique particle in a particular hue.
The experimental results are shown in figure 3(a), where the lines are still the results of the
mathematical model from figure 1(a) for comparison. Reasonable agreement is achieved
with some scatter about the modeled behaviour.
80
70
60
50
40
30
1 colour
2 colours
3 colours
4 colours
5 colours
20
10
0
0
0.005
0.01
0.015 0.02 0.025 0.03 0.035
Normalised seeding density, S n
(a)
80
70
60
50
40
1:0
1:1
1:2
1:3
1:4
30
20
10
0.04
0.045
0.05
0
0
0.005
0.01
0.015 0.02 0.025 0.03 0.035
Normalised seeding density, S n
0.04
0.045
0.05
(b)
Figure 3 Probability of success, p(Success): (a) is for 5 colour experimental images
and (b) is for 2 colour experiment. The solid lines represent the mathematical model
and the points the experimental data.
4. Discussions
The results of the mathematical model and numerical simulation were found to be in good
agreement. The experimental results achieved reasonable agreement with the theoretical
results although the scatter around the trend is quite large. Errors are introduced due to nonuniform distribution of particles and non-uniform particle size in the experiment.
In order to reduce the effects of different sizes between particles, a 2–colour PTV
experiment was repeated using only red and green particles, which were found to be of the
closest size and brightness. This experiment started with a small proportion of red particles
and then red particles were added to change the proportion between the two colours. In this
way, images were scanned and processed as the 5-coloured images. The results by using
two-coloured particles are shown in figure 3(b), which shows similar trends as in figure 3(a).
Comparing to the results of multi-coloured particles, it is found that adjusting the proportion of
the particles has the similar effect as adding or removing colours. Furthermore, the much
smaller scatter indicates that the large scatter can be improved by using particles with more
uniform particle size.
5. Conclusions
Wavelength-multiplexed PTV can be used to increase the particle seeding density with a
given number of views, thereby increasing the measurement spatial resolution. Alternatively,
the technique can be used to achieve a given seeding density but with fewer cameras,
thereby reducing the optical access requirements for the experimental arrangement. For a
certain number of colours, the probability of successful particle identification drops as the
seeding density rises. However, for a certain seeding density, more colours enable higher
probability of success, e.g. at a normalised seeding density of 0.05, the probability of
success with single colour is only 2%, while it is as high as 70% with 5–coloured particles,
which is increased by 34 times. The theoretical effective seeding density by using 3 pixel
particles with up to 10 colours is plotted in figure 4, in which it is shown the optimal seeding
density increases with the number of colours, e.g. the optimal normalized seeding density
with one colour is about 0.01, while it is more than 0.14 with 10 colours, which is increased
by 13 times.
0.08
Optimal Se
1 colour
2 colours
3 colours
4 colours
...
Effective seeding density, S
e
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Normalised seeding density, S n
0.35
0.4
Figure 4 Theoretical results with 1-10 colours particles. Points are optimal effective
seeding density with 3 different particle sizes and up to 10 colours.
The size of particles affects the probability of success in particle identification, which
decreases when using larger particles. Therefore, an optimal seeding density can be chosen
according to the particle diameter and colours available.
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[4]
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[6]
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