Rouen, 29-31 May 2000
EUROMECH 411
STEREOSCOPIC 2D3C DPIV WITH TELECENTRIC
LENSES : CALIBRATION AND FIRST RESULTS
T. Fournel, S. Coudert and L. Riou
Laboratoire Traitement du Signal et Instrumentation
UMR CNRS 5516 – Université J. Monnet
23, rue du Dr P. Michelon, 42023 Saint-Etienne cedex 2, France
SHORT ABSTRACT
In this paper, the use of telecentric lenses for Stereoscopic PIV is
proposed. The reliability of a tilt telecentric set-up is discussed and the
calibration of a telecentric lens is described. A tilt telecentric system is
carried out on a 3D test pattern.
THE APPROACH
The two most common set-ups in Stereoscopic Particle Image Velocimetry are the tilt and the
parallel set-ups (Gauthier, 1988). In the parallel set-up, the axis of the first lens is parallel to the
second one and perpendicular to the light sheet. So for each camera, magnification is constant
and there is a whole overlapping of the object field (a part of the median plane of the sheet) by
shifting the sensors (Arroyo, 1991), (Lecerf, 1999).
But in a symmetric arrangement, the relative error ε between the error on the transversal
component Z and the error on the in-plane component Y is given (to within the factor 1/ 2 ) by
(Lawson, 1997) :
ε=
2
tgβright − tgβleft
where βleft and βright are the angle of the chief rays corresponding to the considered point in the
object plane. In practice, as angle is bounded to about 15° (conventional lenses are made for
working correctly at a maximum angle around 8°), the relative error can not be more reduced.
In the tilt set-up (Grant, 1991), the angle between the lens axis can be freely adjusted. In particular
angle can be set to a high value or to an optimal value regarding to light scattering by particles.
But the laser sheet is not parallel to the image plane. The tilt of the sensor according to the
Scheimpflug condition allows to be focused (Hinsch, 1995). It implies that magnification is not
constant and the image on the sensor corresponds to a trapezoidal field in the object plane. These
trapezoid have an opposite direction of compression if the two cameras are located at the same
side of the laser sheet. The overlapping is partial and a direct image processing has to take it into
account (direct correlation or zero-padding on trapezoidal mesh) (Lourenço, 1998). A whole
overlapping is obtained by placing cameras in a symmetric arrangement on both sides of the laser
sheet (magnification is non constant but identical for each camera). Nevertheless the relative error
ε still depends on the view angles of the mesh point considered in the object plane.
In fact, performing a parallel projection rather than a central projection of the object plane
according two different directions ensures a constant magnification for each recording system
(Fig.1).
Telecentric lenses perform a parallel projection. For a symmetric arrangement of two cameras
equiped with telecentric lenses working respectively under the angles –β and +β, the overlapping
is complete and the relative error ε becomes independent of the mesh point in the object plane :
ε =
1
tgβ
1
Rouen, 29-31 May 2000
EUROMECH 411
As a tilt set-up, the angle between the axis of the lenses can be freely adjusted.
Z
Principal ray
O
Y
βleft β
S=F’
F1
E=H’
F2’
y
E
L2 z
L1
v
Image
S sensor
u
Q
Figure 1 – Standard lens with Scheimpflug condition (at the left) and telecentric lens (at the right)
IMPLEMENTATION
The previous symmetric telecentric tilt set-up is actually implemented as described below. Particle
images with a single exposure per frame are recorded on each camera at two successive times.
For each camera, numerical images are divided into square cells (corresponding to rectangular
cells in the object plane) and a 2D velocity field is computed by local (direct) cross-correlation and
(parabolic) interpolation of the correlation peak. Every measured displacement AB is obtained
from the two previous velocity maps and from the respective calibration matrix M and N of the
cameras. The point A corresponding to a given cell centre A1 in the left map is computed by
assuming that the point A is in the object plane (Z=0) and with the help of the left matrix M. Then
the corresponding vector A2B2 on the right map, lied to the subpixel projection A2 = N.A of the
mesh point A, is obtained by interpolation with the surrounding vectors. Having the two projections
A1B1 and A2B2 of the displacement AB, the end point B is given by the intersection of the
projection lines passing respectively trough B1 and B2 (Riou, 1999).
THE CALIBRATION STEP
The thin model of a telecentric lens
The determination of the calibration matrix M or N assumes a camera model. Telecentric lens
(telecentric in the object space) can be described as a centred system compound of two
convergent thin lenses, the second lens L2 being located at the image focus F1’ of the first one L1
(Fig. 2). In this system, the centre E of the lens L1 is identical to the image principal point H’ and
the centre S of the lens L2 is identical to the image focus F’. Therefore the focus length f is equal
to that of the first lens f 1. Moreover, this system is thick (H’H = f 1²/f 2).
Because of the diaphragm close to the second lens L2, the principal ray is not the ray passing
through the point E but the ray passing through the point S. But the point S is also the image
focus, so the principal ray is parallel to the optical axis before entering in the centred system (Fig.
2). As the intersection of the principal ray with the sensor plane gives the image of any object point
in the depth of field (this image point is the centre of the image spot), the image point is
independent of the object distance and magnification g remains constant.
Generally such systems are made to work at a distance AE equal to the focus length f 1 of the first
lens L1. Thus, the object for the second lens is at the infinity so that the image is in the focus plane
2
Rouen, 29-31 May 2000
EUROMECH 411
of the lens L2 (f 2 fixing the magnification value) Telecentric lens only scales down by the
magnification factor g = -f 2/f 1 the coordinates x and y (in the camera coordinate system Sxyz) of
any object point (Fig. 1).
Figure 2 – Thin model of a telecentric lens
The calibration matrix
The coordinates x and y arise from the change of the world coordinate system OXYZ to the
camera coordinate system Sxyz (a rotation R and a translation) :
æ xö
ç ÷
è yø
æ Xö
ç ÷ æx ö
= R23 ⋅ ç Y ÷ + ç O ÷
ç ÷ è yO ø
è Zø
where R23 is the under-matrix compound of the first two lines of the rotation matrix R and where
(xO,yO) are the coordinates (in the camera coordinate system Sxy) of the origin O of the world
coordinate system.
Moreover according to the previous section, telecentric lens acts as a scale transformation of
coordinates (x,y). Thus the coordinates (u,v) expressed in pixel unit in the image coordinate
system Quv (Fig. 1) are given by :
æ u − uO ö
ç
÷
è v − vOø
æ Xö
ç ÷
= K 22 ⋅ R23 ⋅ ç Y÷
ç ÷
è Zø
where K22 is the scale transformation matrix and where (uO,v O) are the coordinates of the point O
in the image coordinate system Quv. Hence, the calibration matrix that can be written as
M = K22.R23 = (mij), is a 2x3 matrix. The six coefficients mij can be assessed by solving the system
:
æ
ç
ç Xn
ç0
ç
è
M
Yn
Zn
0
0
M
ö
÷
÷
X n Yn Z n ÷
÷
ø
0
0
0
æ m11 ö
ç
÷
ç m12 ÷
çm ÷
13 ÷
⋅ç
ç m 21 ÷
ç
÷
m
ç 22 ÷
ç
÷
è m 23 ø
=
M
æ
ö
ç
÷
−
u
v
O÷
ç n
çv − v ÷
n
O
ç
÷
M
è
ø
where (Xn,Yn,Zn) are the well-known coordinates in the world coordinate system OXYZ of the nth
reference point of the calibration pattern and where (un,v n) are their corresponding coordinates in
the image coordinate system Quv.
TESTS
Experimental conditions
As (Lawson, 1997) the test object is a transparent resin plate seeded with particles. The particles
which are silica crystals having a mean diameter of 80 µm, are randomly distributed in the volume
3
of the plate (74x92x8 mm ). This 3D rigid pattern was lighted with a laser sheet. The laser was a
300 mW continuous laser (λ=512 nm) and the thickness of the sheet was 2 mm. Two 648Hx484V
½” cameras equiped with Melles&Griot telecentric lenses (aperture number 11, base lens focused
at the infinity) were located in an horizontal plane (YZ) with an angle β = +/- 30° regarding to the
normal of the plate and at a working distance of about 200 mm (Fig. 3). The size of the field is 38
mm x 25 mm. The plate was shifted by a motorised XYZ translation stage (resolution 0.1 µm).
3
Rouen, 29-31 May 2000
EUROMECH 411
Figure 3 – View of the tilt telecentric set-up working on the test resin plate
Results
Measured displacem ent (m m )
The tilt telecentric system was calibrated to perform displacement measurements as described
previously. Series of particle images were recorded on each camera for different locations of the
3D test pattern, according to the axis X, Y and Z and according to the first diagonal line (X=Y=Z).
The generated displacements are in the range of +/- 0.6 mm (i.e. 60% of the thickness of the light
sheet). Every velocity maps contains a number of vectors in the order of 100. The bias induced by
the measurement system, assessed by the magnitude of the measurements obtained for a null
displacement, is in the order of 10 µm for axial displacements (with a mean value lower than 5
µm).
Zm = 1 , 0 1 7 * Z
0,6
R
0,4
2
= 0,99
0,2
0
-0,6
-0 ,4
-0 ,2- 0 , 2 0 ,0
0 ,2
0,4
0,6
Y
-0,4
Z
-0,6
A c t u a l d is p la c e m e n t ( m m )
Figure 4 – Measured displacement vs actual displacement according to the Z axis
Figure 4 shows the ability of the tilt telecentric system to measure the third Z component of the
displacement. The root mean square σZ (rms), about 17 µm, does not rise up with the
displacement (the “rms” of the rms values is about ten times lower than the mean). This fact is not
due to the large scattering of the light in the resine plate which could reduce the effect of the Z
bias whereas displacement exceeds 30% of the sheet thickness. Indeed, the same behavior of the
rms σZ , still about 17 µm, is obtained in similar conditions with a 2D test pattern (random dots
printed on a paper and lighted with a white light).
For any tested direction, the rms ratio σZ/σY is close to the theoretical relative error
ε = 1/tg(30°) = 3 (Fig. 5). This is still checked on the 2D test pattern.
σZ/σY
X
1,694
Y
1,778
Z
1,725
XYZ
1,743
Figure 5 – Values of the ratio σZ/σY for different directions of displacement
st
The deviation of the measurements for a displacement according to the 1 diagonal line is given
on figure 6. High at low displacement, the relative dispersion passes down 4% for a relative
displacement greater than 20 % of the sheet thickness.
4
Rouen, 29-31 May 2000
EUROMECH 411
Relative deviation (% )
12%
10%
8%
6%
4%
2%
0%
-0,6
-0,4
-0,2
0,0
0,2
0,4
0,6
A ctual dis placem ent (m m )
Figure 6 – Ratio between the deviation norm and the norm
st
of the displacement according to the 1 diagonal line (X=Y=Z)
CONCLUSION
Though telecentricity has been known for long in optics (Born, 1965), until now it has not been
largely exploited in computational vision. In this paper, the ability to measure the three
components of a particle displacement by a tilt telecentric set-up is demonstrated on an
experimental test. The experimental results obtained on reliability are in good agreement with
theory. The implementation can probably be improved (for example by matching with the help of
an experimental map obtained when the fields of the camera are overlapped in the calibration
step). The tilt telecentric set-up has now to be tested on an actual flow.
REFERENCES
Gauthier V. & Riethmuller M.L. (1988), Application of PIDV to complex flows : Measurement of
the third component, Von Karmann Lecture Series, Vol. 1988-06
Arroyo M.P. & Greated C.A. (1991), Stereoscopic Particle Image Velocimetry, Meas. Sc. Tech.,
Vol. 2, p 1181 – 1186
Lecerf A., Renou B., Allano D., Boukhalfa A. & Trinite M. (1999), Stereoscopic PIV : Validation
and application to an isotropic turbulent flow, Exp. In Fluids, Vol. 26, p 107 – 115
Lawson N.J. & Wu J. (1997), Three-dimensionnal particle image velocimetry : Error analysis of
stereoscopic techniques, Meas. Sc. Tech., Vol. 8, p 894 – 900
Grant I., Zhao Y., Tan Y. & Stewart J.N. (1991), Three component flow mapping : experiences in
stereoscopic PIV and holographic velocimetry, Laser Anemometry Advances and Applications, Eds
Dybbs & Ghorashi, New-York:ASME, Vol. 1, p 365 – 371
Hinsch K.D. (1995), Three-dimensionnal particle velocimetry, Meas. Sc. Tech., Vol. 6, p 742 – 753
Lourenço l.m., Alkislar M.B. & Sen R. (1998), Measurement of velocity field spectra by means of
PIV, The 9th Int. Symp. On Applications of Laser Tech. To Fluid Mech., Vol. 6, p 742 – 753
Riou L., Fayolle J. & Fournel T. (1998), PIV measurements using multiple cameras : The
calibration method, The 8th Int. Symp. On Flow Visu., p 95-1 – 95-11
Lawson N.J. & Wu J. (1997), Three-dimensionnal particle image velocimetry : Experimental error
analysis of a digital angular stereoscopic system, Meas. Sc. Tech., Vol. 8, p 1455 – 1464
Born C. & Wolf E. (1965), Principle of optics, London:Pergamon
5