4th International Symposium on Particle Image Velocimetry
Göttingen, Germany, September 17-19, 2001
PIV’01 Paper 1032
Corrections for misalignment between the laser sheet plane and
the calibration plane: measurement in a turbulent round free jet
using stereoscopic PIV with telecentric lenses
S. Coudert, C. Fournier, N. Bochard, T. Fournel, J-P. Schon
Abstract
The stereoscopic PIV system used consists of the so-called angular configuration with telecentric lenses. Constant
magnification of telecentric lenses help to align the different components of the setup. Using large lens, the amount
of light that reach the CCD, is the same than using standard lenses. The stereoscopic PIV system uses a 3Dcalibration scheme. Despite, a slight misalignment between calibration plane and laser sheet plane always occurs,
even if the best is done in practice to align both planes. Corrections for this misalignment are described. These
corrections are based on image mapping algorithm, in order to compute 3D-vectors with homologous 2D-vectors
(i.e. from the same physical origin). The misalignment is reconstructed, so its magnitude is well known for each
experiment. This stereoscopic PIV is used on a turbulent round free jet. Results show good agreement with the
literature experiments.
1
Introduction
A stereoscopic PIV setup utilizes simultaneous viewing with two cameras to reconstruct the three component (3C)
of the velocity in a planar (2D) cross section of the observed flow.
The stereoscopic PIV system used consists of the so-called angular configuration with telecentric lenses. The
angular configuration is preferred to the translation one, as the angle of triangulation is higher (up to the optimal
45 ° angle). For the angular configuration with standard lenses, the image magnification is no longer constant; i.e.
the images have a perspective distortion. The use of telecentric lenses removes this problem of perspective
distortion. So both viewing areas overlap exactly. With the telecentric lenses used, there is no loss of light compared
to standard lenses.
When using a stereoscopic PIV system, a slight misalignment between calibration plane and laser sheet plane
always occurs, even if the best is done in practice to align both planes. It was pointed out by Prasad (2000) that
errors due to a misalignment are inherent to these methods, and that the problem has been neglected so far. In our
facilities, the misalignment is in the order of 1 mm. If the stereoscopic PIV system doesn't take into account this
misalignment, the measured 3D vectors are computed (during the stereoscopic reconstruction) using 2 vectors from
two different physical zones of the flow. The correction of the misalignment is done by using image mapping
method.
Previous works on our stereoscopic PIV system has been done: Riou et al. (1999) describe the 3D calibration
scheme with standard lenses and Fournel et al. (2000) introduce telecentric lenses. Besides, misalignment
considerations and corrections are described in Coudert & Schon (2001) for angular stereoscopic PIV system with
Scheimpflug conditions.
In this paper, the misalignment between the calibration grid and the light sheet, and also the use of 2D3C angular
stereoscopic PIV with telecentric lenses on a turbulent round jet is considered. Considerations and corrections for
misalignment between the laser sheet plane and the calibration plane lay in section 2. Section 3 describes the setup
and experiments. Results and outlooks are presented in section 4. Section 5 contains the final conclusions.
2
Misalignment considerations
Misalignment
Considering a misalignment, which can be locally considered as a uniform translation in the out-of-plane direction,
as shown in Figure 1, two principal errors can be laid out:
S. Coudert, C. Fournier, N. Bochard, T. Fournel, J-P. Schon, laboratory ’Traitement du Signal et Instrumentation’, LTSI, Saint-Etienne, France
Correspondence to:
Dr. S. Coudert, laboratoire Traitement du Signal et Instrumentation (LTSI), Université Jean-Monnet Saint-Etienne,
23 rue du Docteur Paul Michelon, 42023 Saint Etienne cedex 2, France, E-mail: coudert@univ-st-etienne.fr
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PIV’01 Paper 1032
The first error relates to the vector origin that is computed (e.g. Figure 1). For example, in the case of a 90°
stereoscopic angle, this error is at least twice the misalignment distance, and it corresponds to an offset in the
order of 1 mm (i.e. 10 pixels) between ‘corresponding’ interrogation regions. Such large errors cannot be
neglected.
The second error concerns the vector length. After the first correction, this error is usually negligible, and it
completely vanishes in a paraxial approximation where the back-projection ray is parallel to the optical axis.
For uniform displacements, no corrections are needed to get good results. Indeed, as every vector in a uniform
vector field have the same magnitude, stereoscopic reconstruction works well even if 2D-vectors are not
homologous vectors (i.e. from the same physical origin).
Misalignment corrections
The misalignment is measurable using the mapping algorithm, as outlined by Willert (1997). The mapping method
refers to image mapping method. It consists of the back-projection of the recorded image to the object plane. From a
recorded image, a new image is created in an other plane of space, such as the object plane (e.g. Coudert
& Westerweel 2000). On mapped images, an object in the object plane is seen at the same pixel on both left and
right mapped images. If a misalignment exist, an object in the laser sheet appears no more at the same pixel on the
mapped image. Consequently, the mapping can measure the misalignment between the laser sheet and the object
plane. It is easy to determine a possible misalignment by recording simultaneous a single-exposure PIV image on
each camera. Those two images are mapped on the object plane, using the mapping algorithm, and then crosscorrelated using standard PIV interrogation. The resulting 2D-vector field gives at least a qualitative idea of the
misalignment. Using this field of 2D vector to shift the windows in both mapped images permit to compute vectors
of the same physical zone (e.g. Figure 2/laser sheet point). In these conditions, 2D-vectors from the same physical
zone are used to compute a 3D-vector.
The back-projection is realized using the calibration matrix of each camera. Both calibration matrixes are
determined during the 3D-calibration scheme. A calibration matrix connects the space coordinates (X,Y,Z) to the
image coordinates (x,y) (e.g. Riou et al. 1999). Back-projection of the recorded image to object plane refers to
projection to the plane Z=0. In this paper, calibration matrixes are referred to as MC1 and MC2 for left camera and
right camera respectively, and the position in object plane as Z=0.
Misalignment reconstruction
A misalignment along the Z-axis is simulated to understand the possibility of using the mapping method to solve a
misalignment. To measure the value of the misalignment, displacements along the out of plane axis are
reconstructed at each node of the cross-correlation vector field. The cross-correlation vector Ux is used to compute
local out of plane misalignment W (e.g. Figure 2) using the stereoscopic geometry so that:
Ux
f ( M n + 1)
f ( M n + 1)
with
and
cos θ
sinθ
W =H
H=
B=
(1)
2B − U x
Mn
Mn
Geometric parameters (i.e. aiming angles θ, focal lengths f and nominal magnifications Mn) are optimized by
applying a non-linear least-mean-square routine for the computation of calibration matrix (i.e. during the 3Dcalibration scheme).
This yields to a field of 3D vectors that represents the misalignment of the laser sheet with respect to the object
plane. An example of this misalignment is represented in Figure 7.
origin error
laser sheet point
laset sheet plane
reconstructed misalignment
object point
object plane Z=0
cross-correlation vector Ux
left camera ray
right camera ray
Figure 1. Origin error: this error is relatively large
considering fluid mechanics. In this configuration, the
origin error is 2 time higher than the misalignment
amount along the out of plane axis.
Figure 2. Local reconstructed misalignment using
equation (1).
Correction of the misalignment error using mapping
This correction is processed on special images. It uses two PIV mapped images recorded at the same instant with
both cameras.
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PIV’01 Paper 1032
At each origin node of the common grid, these two special images are cross-correlated using a standard PIV
processing pipeline. Each vector Ux length, converted to world units, gives the origin error magnitude due to
misalignment. The resulting vector field (averaged and smoothed) is used as a correction vector field (e.g. Figure 3),
in order to get the same physical vector origin (e.g. Figure 2/laser sheet point).
For one camera, the common grid is modified taking into account the corrections. Each common origin is
corrected by the vector of the correcting vector field e.g. Figure 3. At this stage, one camera has an irregular grid,
and the other have the regular common grid e.g. Figure 4. Of course, these corrections can be reported on both
cameras, by modifying the origin by half the vector of the correcting vector field.
For each camera, its grid is projected to the image, using its own calibration matrix. This gives an irregular grid,
where the PIV pipeline works e.g. Figure 4.
PIV C2 im
MC1 & Z=0
PIV C2 im
image mapping
MC2 & Z=0
image mapping
mapped image C1
mapped image C2
corrected grid
MC1
common grid
MC2
PIV cross-correlation
common grid
projection
projection
irregular grid C1
irregular grid C2
correction vector field
origin correction
Figure 4. Origins of the 3D-vectors in the object plane
are projected to the image planes.
corrected grid
Figure 3. Correction of the 3D-vector origins: using
the common grid computed in the previous step, the
correction vector field is computed. Then it is used to
correct the origin error due to the misalignment.
Computation of the 3D displacement vector
The computation of the 3D vector is represented in Figure 5. As the PIV pipeline is done on the nodes of the
irregular grid for both cameras, it includes corrections of the origin vector due to misalignment between the object
plane and the laser sheet. The algorithm consists of the following steps:
computation of 2D-vectors on recorded image: using the irregular grid as windows center, the 2D-vectors are
computed on the recorded image. Windows shifting algorithm is used (e.g. Westerweel et al. 1997).
reconstruction of 3D-vector: the 3D-vector is triangulated from the two homologous 2D-vectors of the left and
right cameras, using the stereoscopic reconstruction based on calibration matrixes (e.g. Riou et al. 1999).
irregular grid C1
MC1
SPIV C1 im
SPIV C2 im
irregular grid C2
PIV pipe line
PIV pipe line
vector field
vector field
stereoscopic reconstruction
MC2
3D vector field
Figure 5. Computation of the 3D vector: it includes correction of the origin vector due to misalignment between the
object plane and the laser sheet.
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PIV’01 Paper 1032
Accuracy gain
The accuracy gain depends on the laser sheet misalignment magnitude, which can be measured using the mapping
algorithm (e.g. section Misalignment reconstruction). For actual experiments, the accuracy gain should be between 2
and 1 order of magnitude in theory. Previous tests showed that the accuracy benefit is one order of magnitude in
practice (e.g. Coudert & Schon 2001).
3
Setup and experiments
Stereoscopic PIV system with telecentric lenses
The stereoscopic PIV system developed in the laboratory ’Traitement du Signal et Instrumentation’ (LTSI) is based
on telecentric lenses (e.g. Fournel et al. 2000). In the angular configuration, the system allows high angle of view
(up to 45 degrees). Such lenses-performs a parallel projection. So, both camera areas overlap exactly, i.e. no loss of
pixel areas for vector computation. Working in Scheimpflug conditions (e.g. Konrath & Schröder 2000) allow to
increase the amount of light that reach the CCD.
The 2D3C stereoscopic PIV setup is composed of two 648×484 pixel CCD cameras (PULNiX TM-6705AN)
equipped with telecentric lenses (Melles-Griot Invaritar (R) base lens 59 LGC 535 and attachment lens 59 LGJ 423)
that are mounted on a flat plate. The opening angle for the cameras was 90°, and symmetric with respect to the lightsheet plane normal. Using commercial telecentric lenses, this amount of light is the same compared to standard
lenses (i.e. 24 mm Nikon), as it have a large first lens (i.e. base lens 115 mm in diameter).
For the 3D-calibration scheme, the calibration target, which consists of dots on a rectangular grid, is printed out
on transparent slide with a resolution of 600 dpi. It is glued to a white flat plate mounted on a 3D-translation stage.
The precision of the remote translation stage is 1 µm for Z-axis. The calibration matrixes are determined for both
cameras from 5 pairs of recorded images of the calibration target. The geometric parameters are optimized by
applying a non-linear least-mean-square routine to the dot positions of the calibration target. With this procedure, it
was found that the actual viewing angles for the left camera and the right camera are 45.3 ° and 45.2 ° respectively.
Turbulent round jet facility
The turbulent round jet facility is represented in Figure 6. The facility is composed of:
a tank: filled of water, size 450x450x900 mm3, flat Plexiglas wall, free surface on top face at constant level.
a round hole: placed in the middle of tank bottom face, size 5 mm in diameter.
a laser sheet plane: double pulse laser (30 mJ per pulse) and optics.
a window: placed on the free surface in order stabilize the inner surface for the laser sheet.
an L arm: hold the calibration target used for 3D-calibration scheme, it is mounted on the remote translation
stage and removed when calibration is over.
Dot pattern for 3D
calibration scheme
Cameras with
telecentric lenses
Origin of the jet
Figure 6: Test facilities: the turbulent round free jet in the water tank, the 3D calibration arm in the middle of the
tank and the stereoscopic PIV system on the left hand side.
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PIV’01 Paper 1032
a huge translation stage: it carries the stereoscopic cameras mounted the flat plate, so the stereoscopic system
can move up and down along the jet.
a flow meter: to measure the flow rate.
The laser sheet plane is aligned regarding to the calibration plane at Z=0. This particular calibration plane is the
so-called the object plane. Despite, a slight misalignment always exists. This misalignment is measurable using a
mapping method (e.g. section 2).
Experiments
The stereoscopic system is moved along the jet using the huge vertical stage. At each step, several experiments are
realized: images are recorded for different Reynolds numbers. Using the misalignment corrections, re-calibrations in
the laser plane are not necessary when moving the stereoscopic system along the jet. Only the computation of the
reconstruction of misalignment is necessary, in order to compute a 3D vector with 2D vectors from the same
physical zone.
For each experiment, 1000 images pairs are recorded on each camera. The reconstruction of the misalignment of
the laser sheet plane regarding to the object plane is computed. For each camera, irregular grid is generated from this
misalignment plane. Then, 2000 2D-vectors maps are computed. After the stereoscopic reconstruction, 1000 3Dvector maps are ready to be compiled through statistics. Each map has 266 vectors.
4
Results and outlooks
Misalignment
The reconstruction of misalignment is represented on Figure 7. The reconstruction of the laser sheet plane regarding
to the object plane gives a misalignment of 1.8 mm along Z-axis and rotations of 0.7 ° and 1.3 degrees around
respectively X and Y axes.
As the position of the laser sheet plane regarding to the SPIV system may vary as the system is moved,
reconstruction of misalignment is done for each experiment. The system doesn't need to be re-calibrated close to the
laser sheet plane.
Z
X
Y
4
2
Z
0
-2
-4
-10
-10
0
0
Y
Object plane
X
Reconstructed laser sheet
10
10
Laser sheet plane
Figure 7: Reconstruction of the laser sheet misalignment: the laser sheet (gray level surface) is reconstructed locally
using mapping and equation (1). Then, an optimal plane (black mesh) for this reconstructed laser sheet is computed.
This laser sheet plane (black mesh) is always misalign regarding to the calibration plane (white mesh). Using
mapping method, the misalignment is measured. Then, corrections for calculation of the 3D vector are integrated in
the stereoscopic reconstruction.
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PIV’01 Paper 1032
Turbulent round jet
From Pope (2000), the theoretical profile for axial velocity in the round jet is:
f(η) = (1+aη2)-2
with
a = (√2-1) S-2
and
η = r (x-xo)-1
(2)
where η is the non-dimensional radius and S the spreading rate. The experimental value of S is around 0.1 and
the turbulence intensity is around 25 % in axial direction. From Hussein et al. (1994), magnitudes of mean square
fluctuations for the 3 components are respectively <u²> = 0.076, <v²> = 0.047 and <w²> = 0.049 on the jet axis.
In our experiment, the axial mean velocity profile is plotted in non-dimensional form in Figure 8. This
experimental configuration refers to axial position x/do = 30, Reynolds number Re = 21 000 and local Reynolds
number Reo = 12 000. The S value is 0.96 and the turbulence 21 %. Magnitudes of mean square fluctuations are
respectively 0.065, 0.045 and 0.045. The magnitudes of the measured values are little less than those from Hussein
et al. (1994).
Next step is to process data in order to compute velocity correlations and length scales.
1
0.9
0.8
U/Uo
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.05
η
0.1
0.15
Figure 8: Axial velocity profile: the non-dimensional velocity U/Uo (266 dots) is plotted against the nondimensional radius η of the jet for xo/d = 30. The black curve represent the theoretical profile with S = 0.96.
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Conclusions
Using telecentric lenses compare to standard lenses has real advantages when setting up the system (i.e. same
magnification on whole viewing areas and exact overlapping) and when processing the 2D-vectors (constant
magnification, no more perspective distortion). Contrary to what is believed, the same amount of light reaches the
CCD using such lenses compared to standard lenses.
The misalignment between the laser sheet plane and the object plane is corrected by using algorithm based on
only one image mapping per experiment. Using these corrections for misalignment, homologous vectors (i.e. from
the same physical origin) are used for stereoscopic reconstruction.
Results of our experiments are close to literature results on the turbulent round free jet.
Acknowledgement
This work has been performed under the EUROPIV 2 project (CE project n° GRD1-1999-10835).
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PIV’01 Paper 1032
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