Stereoscopic Particle Image Velocimetry using
telecentric lenses
yx
T. Fournel
y
y
y
, S. Coudert , C. Fournier , and C. Ducottet
Laboratoire Traitement du Signal et Instrumentation, UMR CNRS 5516 Universite J. Monnet 23, rue du Dr P. Michelon, 42023 Saint-Etienne cedex 2, France
y
Telecentric imaging is presented in the general frame of Digital
Stereoscopic Particle Image Velocimetry (DSPIV). A 2D-3D mapping approach is
combined with telecentric imaging in order to obtain a user-friendly DSPIV technique.
This technique is tested on translation tests then in a round free jet.
Abstract.
Stereoscopic Particle Image Velocimetry, Telecentric lens, AÆne model,
Mapping function
Keywords:
PACS numbers: 06.30.G, 07.05.P
Submitted to: Meas. Sci. Technol.
x
To whom correspondence should be addressed (fournel@univ-st-etienne.fr)
2
SPIV using telecentric lenses
1. Introduction
Thanks to digital techniques, Particle Image Velocimetry allows the user-friendly
measurement of the velocity eld of a uid ow in a laser sheet. In order to get the three
components of the displacement between two exposures, video-cameras are positioned in
stereoscopic conguration. Two congurations of imaging system have been successfully
developed for DSPIV (Digital Stereoscopic Particle Image Velocimetry) : the translation
method (Solo et al 1997, Lecerf et al 1999) and the angular displacement method
(Willert et al 1997, Oord 1997) depending on whether the optical axes are at right
angle to the object plane or not. From a theoretical point of view, the angular
displacement method presents a higher sensitivity whereas the maximum possible
viewing angle in the translation method is limited by the o-axis aberrations and
the decrease of the modulation transfer function toward the edges of the eld-ofview. As conventional lenses are commonly used (Prasad 2000), perspective projections
induce image deformations due to the oblique orientation of the object plane. Such
deformations resulting from spatially varying magnication are increased when videocameras are focused on the object plane using Scheimpug condition (Hinsch et al
1993, Prasad and Jensen 1995). Then, magnication at a given location in the object
plane diers from left to right images. An approach consists in dewarping images or
vectors before reconstructing the three components of the displacement eld (Oord
1997). Another approach (Solo et al 1997) makes use of a mapping function F between
a 3D location in the object plane and the corresponding 2D location in the left
(respectively the right) image plane. To rst order, the displacement is related to
the displacement on a given image plane by
= rF ( ) (1)
In camera coordinate system, the jacobian matrix rF at location , also termed
the magnication matrix, is given (Solo et al 1997) by
!
1
0
tg (v )
rF ( ) = M (z) 0 1 tg( )
(2)
v
where v and v are the angles of the line of sight with respect to the optical axis
z. For simplicity, equation (2) is expressed as the optical axis of the imaging system is
at right angle to the object plane.
A telecentric imaging system can be used to avoid such deformations (Hinsch 1995,
Konrath and Schroder 2000, 2002, Fournel et al 2000). Indeed a telecentric lens can
perform a parallel projection in the direction of the optical axis. Thus, the angle v
and v are null and magnication M is unvarying. Consequently no perspective eect
arises. In (Konrath and Schroder 2000) telecentric lenses are used to investigate an air
pipe ow. As magnication is unvarying, a dewarping technique (dewarping of vectors)
is just performed to compensate optical distortions before reconstruction.
In order to make Stereoscopic PIV more and more user-friendly, a 2D-3D mapping
approach is here combined with telecentric imaging. The mapping function, given by the
X
x
X
x
x
X
X
X
X
3
sum of the geometrical image and distortions (section 3), is derived from section 2 where
telecentric designs are discussed in the frame of SPIV. The displacement equation is then
deduced. Section 4 is dedicated to experimental results. Measurement uncertainties
estimated by shifting a particle-seeded block of resin are rst discussed. Afterwards
SPIV measurements carried out in a round free jet are compared to experimental results
given in literature.
SPIV using telecentric lenses
2. Imaging with telecentric lenses
2.1. Perfect imaging
By positioning an aperture stop in the back focal plane of a lens L1 , the principal rays
of light (i.e. the rays passing through the aperture stop) are parallel to the optical axis
in object side. Furthermore, despite defocus blurring, the coordinates of the image on
the sensor stay unvarying as the object point moves in depth. Such a system whose
the entrance pupil is at innity, is said to be telecentric in object side. Obviously the
maximum size of the eld of view is limited by the diameter of the lens L1 . A second
lens L2 is used to adjust the magnication.
Aperture stop
(F1’=F2)
L1
L2
Image
plane
x
F2’
F1
f1
z
f2
Telescopic system. Principal rays (dashed line) are parallel to the optical
axis on both object and image sides.
Figure 1.
The lens L2 can be located one focal length f2 behind the aperture stop in order
to have telecentricity in image side (gure 1). Thus, principal rays are parallel to the
optical axis in both object and image sides. Such an afocal system, said telescopic, has
an unvarying magnication given by the ratio :
f
(3)
M= 2
f1
A more compact system is obtained by positioning the lens L2 in the aperture
plane (gure 2). So the image conjugate of a given (out-of-axis) point is the intersection
between the principal ray and the ray that should pass (without aperture stop) by the
center of the rst lens. However this last ray changes as the object point moves in depth.
4
More exactly magnication depends on the working distance z = zo according to the
following relationship
M (zo ) = f1 f2 =( f12 + f1 :f2 zo :f2 )
(4)
An object-sided telecentric system is usually designed to work at a distance z0 equal
to one focal length f1 . The magnication is then also given by equation (3).
SPIV using telecentric lenses
L1
Image
plane
x
Aperture
stop L2
F2’
z
F1
z0=f1
f1
f2
System telecentric on object side only. Such a system is more compact than
the telescopic one. Principal rays (dashed lines) are no longer parallel to the optical
axis on image side.
Figure 2.
In both cases, the focuses F1 and F2 0 are conjugate. By working at a distance equal
to one focal length f1 , the sensor is then located at the back focal plane of the second
lens L2 and the image cone of light is determined by the aperture of the lens L2 focused
at innity. Consequently the aperture number of such systems is the nominal f-number
f2 # of the lens L2 . So, the depth of eld Æz can be deduced from (Adrian 1991) :
2
Æz = 4(1 + M 1 )f2 #
(5)
where is the wavelength of the laser sheet.
2.2. Scheimpug condition
In conventional angular SPIV, the sensor is tilted according to the Scheimpug condition
(Hinsch et al 1993, Prasad and Jensen 1995) in order to be focused on the laser sheet
and to work with high aperture. The Scheimpug condition is given by
tan = M
(6)
tan where is the viewing angle and the Scheimpug angle (in the horizontal plane).
This condition is still valid for a thick lens system as for a telescopic lens system (Konrath
and Schroder 2000). The working distance being xed to f1 , it can be easily proved for
both telescopic and single-sided congurations by tracing the "ray" coming from the
5
SPIV using telecentric lenses
α
θ
L1
Image
plane
L2
F1
F2’
f1
Figure 3.
f2
Scheimpug condition for telecentric lenses
focus F1 (the dashed line in gure 3). It comes out the lens L1 parallel to the optical
axis then passes through the focus F2 0 .
One can notice from equation (4) that an object-sided telecentric lens system has
a varying magnication when tilting the sensor according to the Scheimpug condition.
For a 1/2 inch video-camera, the relative magnication can reach 4% for = 45o.
Consequently to provide in-focus images at unvarying magnication leads to increase
the aperture number f2#. In return the light energy that passes the aperture stop
decreases.
On the contrary, the magnication of a telescopic lens system remains unvarying in
Scheimpug condition because it is telecentric in image side (principal rays are parallel to
the optical axis in image side). Its eld of view decreases with a factor cos (whereas the
resolution increases with a factor 1=cos) stretching the image with a factor cos=cos.
2.3. Distortions
As described in section 2.1, a telecentric lens is a double lens system. So distortions
are generated by both lenses L1 and L2 . A slight angle Æ 1 (respectively Æ 2) exists
between the actual principal ray and the ideal one in object side (respectively in image
side) as illustrated in gure 4.
Distortions Æx along x-axis is the sum of distortions introduced by each lens :
Æx = B 0 B1 + B1 B12
From equation (3), this last expression can be approximated by
Æx ' f2 (Æ 2 Æ 1 )
(7)
when viewing at right angle.
Distortions can be fully compensated when imaging with a unit-magnication
telescopic lens by using a lens L2 identical to the lens L1 (4-f1 set-up). Otherwise,
6
SPIV using telecentric lenses
L1
B
Aperture stop
(F1’=F2)
δψ1
Image
plane
x
L2
δψ2
f1
f1
Figure 4.
f2
z
B12
B1
B’
f2
Distortions of telecentric lenses
distortions have to be taken into account. For object-sided telecentric lenses, the angle
Æ 2 can be neglected in comparison to Æ 1 because the lens L2 located nearby the
aperture works in Gauss conditions. The value of Æ 1 is usually about 0:1Æ.
3. 2D-3D mapping approach
3.1. Video-camera model
From section 2, telescopic systems (in Scheimpug condition or not) and compact
telecentric ones without tilt have at once an unvarying magnication M and principal
rays parallel to the optical axis in object side (v = v = 0). Compared with the
expression (2), the magnication matrix of such a system becomes unvarying. To take
account for the oblique orientation of the video-camera c = 1; 2 with respect to the object
plane, the rotation from the object coordinate system XY Z to the camera coordinate
system F20(c) x(c) y(c)z(c) is introduced. Thus, the magnication matrix is expressed by
0
rF (c) = M (c) @
(c) r(c) r(c) 1
r11
12 13
(c) r(c) r(c) A
r21
22 23
(8)
where rij(c) (i = 1; 2 and j = 1; 2; 3) are the coeÆcients of the rotation matrix.
Such a system performs a parallel projection by scaling in F2 0(c) x(c) y(c) the two
rst coordinates of an object point given by its coordinates in XY Z : (c) =
rF (c) + ! where !(c) is the parallel projection of the origin . This aÆne
transformation of coordinates is in practice aected by distortions Æ (c) at the image
point x(c) so that the 2D-3D mapping function is expressed by
(c) = rF (c) +
(c)
(9)
! +Æ
For a standard sensor format, distortions can be approximated by Æ (c) =
a1 r2 (x(c) ; y (c))t (Tsai 1987). They increase when the distance r from the image center
(i.e. the intersection between the optical axis and the image plane, here F2 0(c) ) increases.
X
X
x
x
(c)
x
x
X
x
(c)
x
x
7
They are generally about few pixels in the corner of the image. The image grid
corresponding to the mesh grid dened in the object plane for eld measurements has
to be corrected. The preliminary stage of calibration includes the determination of the
magnication matrices and distortions of the left and right video-cameras.
SPIV using telecentric lenses
3.2. Displacement equation
The relationship between the three-dimensional displacement eld and the
displacement eld (c) computed in the image plane can be approximated at the rst
order by dierentiating equation (9) :
g(c) ' rF (c) (10)
where g(c) is the image eld obtained by subtracting the distortion eld dÆ (c)
from the image displacement eld (c) .
At each position , the three unknowns X , Y , Z can be deduced from
equation (10) expressed for the left and right telecentric video-cameras (c = 1; 2) by
solving the overdetermined linear system :
0 g (1) 1 0
1
(1) r11 (1) M (1) r12 (1) M (1) r13 (1) 0
x
1
M
B
C
B
C
(1)
X
B
C
B
(1)
C
(1)
(1)
(1)
(1)
(1)
g
B
y CC = BB M r21 M r22 M r23 CC BB Y CC
B
(11)
B
B
@
A
(2) r11 (2) M (2) r12 (2) M (2) r13 (2) C
gx(2) C
B
C
B
C
M
@ (2) A @
A Z
(2)
(2)
(2)
(2)
(2)
(2)
g
M
r
M
r
M
r
21
22
23
y
X
x
x
X
x
x
x
X
4. Experimental results
The stereoscopic set-up used to test the proposed technique has two video-cameras
positioned horizontally in symmetrical angular conguration with the viewing angles
. Each video-camera is equipped with a compact telecentric lens (Melles & Griot
59LGJ423-LGC535 : f1 = 230mm, M = 0:15) and with a 640 480 pixel 1/2"
CCD sensor. Pincushion distortions aect images. The magnitude of distortions
obtained after calibration are less than 3 pixels in the corner of the image area in both
experiments hereafter reported. They are compensated during the image grid denition.
On each image plane, displacement (about 3 pixels) is computed by cross-correlation
with the discrete oset technique introduced by Westerweel et al (1997) and with the
parabolic three-point estimator. The 3D displacement linearly reconstructed according
to equation (11) is slightly distorted by 0:02 pixel per pixel (i.e. for a displacement of
one pixel) in the corner of the vector eld. So, distortions have been neglected in the
reconstruction of the 3D displacement eld.
4.1. Translation tests
A target was accurately shifted in order to study the uncertainties of displacement
measurements. The target made in transparent resin (74x92x8 mm3 ) and seeded with
8
SPIV using telecentric lenses
silica crystals (Lawson and Wu 1997) was positioned in a laser sheet plane of 2 mm
thick. It was step-by-step shifted by means of a remote control XY Z translation stage
(resolution < 1m). Displacements were so generated with respect to the coordinate
axes and the rst diagonal. The view angle was set to = 30Æ. The size of the eld is
38x25 mm2 . The eld reconstructed at each step has 266 displacement vectors.
RMS deviation (µm)
20
16
12
8
4
0
-273
-195
-117
-39
39
117
195
273
∆ Z (µm)
RMS deviations X (triangles), Y (squares) and Z (diamonds) for
out-of-plane displacements of a particle-seeded resin lighted with a laser sheet
Figure 5.
In all the cases, measurements present a bias lower than 3%. Concerning the
RMS (Root Mean Square) deviations of the components, they are quite constant over
the range of displacement as it is shown on gure 5. This behavior agrees with the
theoretical values :
"X = "=(M cos ); "Y = "=M; "Z = "=(M sin )
(12)
where " denotes the measurement uncertainty on image displacement coordinates.
These expressions are derived from
0
1
0
(1) + x(2) )=cos 1
(
x
X
B
C 1 B
C
B
C
(13)
@ Y C
A = 2M B
@ (y (1) + y (2) )
A
(1)
(2)
Z
(x x )=sin
extracted from (11) by rough estimating the experimental set-up with the parameter
values : M (i) = M , r11 (i) = cos , r13 (1) = r13 (2) = sin , r12 (i) = 0 = r21 (i) = r23 (i) and
r22 (i) = 1 for i = 1; 2.
The ratio between the RMS deviations Z and X extends from 1.69 to 1.78
which is close to the theoretical value 1= tan ' 1:73 deduced from equation (12). The
ratio between the RMS deviations X and Y is also close to the theoretical value
1= cos ' 1:15.
9
SPIV using telecentric lenses
4.2. Measurements in a round free jet
The stereoscopic system was used to investigate a round free jet in a water-lled tank
(440x450x900 mm3 ). The Reynolds number based on the nozzle hole diameter of 5
mm is 21000. The ow was seeded with 15 m particles. It was illuminated with a
30 mJ=pulse Nd:Yag laser from the top of the test facilities with a pulse separation
of 1.5 ms. In order to minimize aberration eects, the video-cameras were positioned
with a normal incidence regarding to the interface (Prasad and Jensen 1995). The view
angle was set to = 45Æ. The stereoscopic system pointed out to the axis of the jet,
just beyond the developing region where the shear ow is turbulent and consequently
three-dimensional. Thus, each line of left and right images is centered with respect to
the jet axis and corresponds to a given value of X=d in the 28-35 range.
1
0.9
0.8
<U>/Uo
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.02
0.04
0.06
0.08
η
0.1
0.12
0.14
Non-dimensional mean axial velocity hU i=U0 of the round free jet v.s. the
non-dimensional radius (Reynolds number of 21000, X=d in the 28-35 range) : t of
all the measured proles (continuous line) and theoretical prole (dashed line)
Figure 6.
As displacement is no longer constant regarding to translation tests (section 4.1),
the misalignment between the light sheet coordinate system and the target coordinate
system XY Z is accounted for using grid correction techniques (Coudert and Schon
2001). A translation of 1.8 mm along Z-axis and rotations of 1:3Æ and 0:5Æ around
respectively X- and Y-axes have been detected. 1000 image pairs were recorded on both
video-cameras giving after 3D reconstruction 1000 vector maps of 37 27 nodes.These
instantaneous velocity vectors were averaged and rendered non-dimensional in order to
10
calculate the symmetrical self-similar mean proles hU i=U0 , hV i=U0 and hW i=U0 where
U0 is the velocity along the jet axis. In the cylindrical coordinate system centered on
the jet axis, the theoretical axial prole is given by
1
hU i =
p
(14)
U0 (1 + S2 1 2 )2
where S is the spreading rate, = Y=(X X0 ) the non-dimensional radius and X0
the virtual origin (Pope 2000).
SPIV using telecentric lenses
2
0.05
<V>/Uo
0.04
0.03
0.02
0.01
0
-0.01
0
0.02
0.04
0.06
0.08
η
0.1
0.12
0.14
0
0.02
0.04
0.06
0.08
η
0.1
0.12
0.14
0.05
<W>/Uo
0.04
0.03
0.02
0.01
0
-0.01
Figure 7. Non-dimensional mean out-of-axis velocities hV i=U0 , hW i=U0 of the round
free jet v.s. the non-dimensional radius (Reynolds number of 21000, X=d in the
28-35 range) : ts of all the measured proles (continuous line) and theoretical proles
(dashed line)
The velocity U0 and the spreading rate S = 0:084 were extracted from Wygnansky
and Fiedler (1969). The light sheet assumed to be vertical was assessed to be 5 mm
behind the jet axis. The radius separating the measurement nodes from the jet axis
is in the range from 0.033 in the middle of the view eld to 0.113 at the borders.
Figure 6 shows that the measured and theoretical mean U-proles well overlap. The
non-dimensional RMS deviation of the U-component is 0.0066. The measured V- and
W-proles reported on gure 7 are in good agreement with the theoretical ones. The
non-dimensional RMS deviations of the V- and W-components are respectively 0.0018
and 0.0035. One can notice that the V-component is null and the W-component maximal
11
at the minimal value of . When the radius exceeds 0.08, both V- and W-components
decrease to zero.
SPIV using telecentric lenses
5. Conclusion
Telecentric lenses which form images by parallel projection allow to overcome diÆculties
lied to the use of entocentric (conventional) lenses in stereoscopic PIV. In comparison
to the classical SPIV methods, the (angular) telecentric method can support a high
viewing angle contrary to the translation method and does not present a systematic
image deformation as the angular displacement method does. In addition, when imaging
through a liquid-air interface at right angle, if necessary by using a liquid prism (Prasad
and Jensen 1995), the smear in the image plane produced by the light scattered from a
particle remains circular away from the lens axes. In return, the limitation of telecentric
lenses is the maximum size of the eld-of-view which is restricted by the diameter of
the lens entry. In practice most telecentric lenses cover less than 15 cm. From the two
telecentric congurations : the object-sided and the telescopic congurations, only the
telescopic one forms images by parallel projection in Scheimpug condition.
Combining the aÆne video-camera model with the 2D-3D mapping approach makes
appear distortion compensation as an additive correction of both mesh points and
displacement eld. The ability of this combination has been demonstrated on translation
tests and on measurements in a free jet. When the correction of the displacement
eld is negligible, such a method leads to a linear reconstruction using the constant
magnication matrices. SPIV measurements can so be achieved within 0.1 pixel at
standard magnication on a 1/2" video-sensor as the telecentricity is about 0:1Æ or less.
In this frame, the technique can be successfully implemented for in-line controls of ows.
References
Adrian R. J. 1991 Particle-imaging techniques for experimental uid mechanics Annual Review of Fluid
Mechanics 3 261-304
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Fournel T., Coudert S., and Riou L. 2000 Stereoscopic 2D3C DPIV with telecentric lenses : calibration
and rst results Proc. Euromech 411
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Fluid Mechanics
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stereoscopic approach Experiments in Fluids 33 703-708
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techniques Measurement Science and Technology, 8 1455-1464
SPIV using telecentric lenses
12
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application to an isotropic turbulent ow Experiments in Fluids 26 107-115
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image velocimetry Measurement Science and Technology 8 1441-1454
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323-344
Van Oord J. 1997 The design of a stereoscopic DPIV system Report MEAH-161 Delft the Netherlands:
Delft University of Technology
Westerweel J., Dabiri D. and Gharib M. 1997 The eect of a discrete window oset on the accuracy of
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Acknowledgements
This work has been performed under the EUROPIV 2 project. EUROPIV 2 (A Joint
Program to Improve PIV Performance for Industry and Research) is a collaboration
between LML URA CNRS 1441, DASSAULT AVIATION, DASA, ITAP, CIRA, DLR,
ISL, NLR, ONERA, DNW and the universities of Delft, Madrid, Oldenburg, Rome,
Rouen (CORIA URA CNRS 230), St Etienne (TSI UMR CNRS 5516), Zaragoza.
The project is managed by LML URA CNRS 1441 and is funded by the CEC under
the IMT initiative (CONTRACT N: GRD1-1999-10835).
The authors thank Pr K. Hinsch and Dr H. Royer for their helpful remarks.
SPIV using telecentric lenses
13
Figure captions
Telescopic system. Principal rays (dashed line) are parallel to the optical
axis on both object and image sides.
Figure 1.
System telecentric on object side only. Such a system is more compact than
the telescopic one. Principal rays (dashed lines) are no longer parallel to the optical
axis on image side.
Figure 2.
Figure 3.
Scheimpug condition for telecentric lenses
Figure 4.
Distortions of telecentric lenses
RMS deviations X (triangles), Y (squares) and Z (diamonds) for
out-of-plane displacements of a particle-seeded resin lighted with a laser sheet
Figure 5.
Non-dimensional mean axial velocity hU i=U0 of the round free jet v.s. the
non-dimensional radius (Reynolds number of 21000, X=d in the 28-35 range) : t of
all the measured proles (continuous line) and theoretical prole (dashed line)
Figure 6.
Non-dimensional mean out-of-axis velocities hV i=U0 , hW i=U0 of the round
free jet v.s. the non-dimensional radius (Reynolds number of 21000, X=d in the
28-35 range) : ts of all the measured proles (continuous line) and theoretical proles
(dashed line)
Figure 7.