3D Velocity measurements by Optical Flow PIV based on a simple and accurate camera
calibration
**A. Rambert, **F. Lusseyran, **P. Gougat, *G. Quénot,
*CLIPS IMAG, BP 53 , 38041 Grenoble, **LIMSI CNRS, BP 133, 91403 Orsay, France
Abstract
The aim of this work is to present first results for 3D velocity field measurements obtained by Optical
Flow PIV coupled with an original technique of camera calibration . The calibration method search the
parameters of an optimal pinhole camera model using only a single image of a dot grid target, the known camera
focal length and sensor pixel size. After the target marks have been located, an optimal projective transform is
calculated by a least square method between the target plane coordinates and the image plane coordinates . The
camera model is recovered via a canonical decomposition of the linear part of the projective transform and an
iterative search for the location where the optical axis intersects the image plane.
The interaction between a flow and a cavity is a test configuration useful to study the unsteady flows
and the development of the 3D instabilities. It also represents a configuration well adapted to studies relating to
transport (aerodynamics of vehicles) and environment (air renewal in a cavity).
Flow issued from a boundary layer and a cavity interaction was characterised by PIV based on Optical
Flow measurements performed inside and out of the cavity for two perpendicular planes which showed that the
flow is three dimensional. Therefore we choose this configuration to perform 3D velocity measurements by
stereoscopic PIV.
The results presented in this paper were obtained for low velocities, close to the laminar turbulent
transition, for a given geometry of cavity characterised by a shape factor H/L=0.5, where H is the height of the
cavity and L its length. The Reynolds number based on the height of the cavity and the mean velocity is about
4000.
1
INTRODUCTION
Particle Image Velocimetry based on the optical flow computation was proven to be a valuable method
for quantitative two dimensional flow structure evaluation [Quénot et al., 1998]. Using two or more cameras it is
also possible to obtain the out of plane velocity component.
In order to accurately combine the 2D apparent velocity fields from the different viewpoints into a
single 3D velocity field, an accurate camera model giving the correspondence between an (x,y,z) object location
and a (i,j) image location is required for each viewpoint. This correspondence is usually modelled by functions
of various forms (typically polynomial) whose parameters are evaluated from the analysis of several images
(typically 3 or 5) of a dot grid target placed at different locations, parallel to the PIV light sheet and at various
distances from it [Riou et al., 1998]. The main drawback of such methods is the need to take several images
while placing the target grid very accurate at the desired locations. Moreover, the result may be unstable relative
to non parallel and not evenly spaced target locations.
The proposed calibration method is based on a search for the parameters of an optimal pinhole camera
model using only a single image of a dot grid target and known camera focal length [Quénot et al., 2001].
The search is carried out in three steps:
(i) Extraction of the locations of the grid markers
(ii) Search for an optimal (x,y) to (i,j) projective transform
(iii) Search for the camera location using the optimal projective transform and the characteristics of the
camera (focal length and pixel size). This step which is the most difficult and the main contribution of this paper
is achived via a canonical decomposition of the linear part of the extracted projective transform. The projective
transorm is defined as:
(i,j)=((ax+by+c)/(gx+hy+1), (dx+ey+f)/(gx+hy+1))
Least square minimisation is used in the second step to find the coefficients (a,b,c,d,e,f,g,h) of the
optimal projective transform.
In this paper will present briefly the camera model and the method used to obtain the camera location
and orientation. The result of such a calibration is used to compute a 3D velocity field inside the cavity.
Experimental set-up
The experimental set up is presented on Figure 1. The characteristic dimensions for the cavity are H=50
mm, L=100 mm and l=300 mm. The Reynolds number based on the mean velocity and the cavity height is
about 4000.
U00
z
H
l
x
y
L
2
Figure 1: Experimental set-up
The laser sheet was obtained from an Argon laser and spectacle smoke was using as seeding.
Measurements were performed using two digital Pulnix cameras ’’0’’ and ’’1’’ (768 x484 pixels). The size of
the pixels is 10.6 x 13.6 µm and the focal length of 25 mm. The camera are located in the (xy) plane at 377 mm
from the target origin and form an angle of 12 ° between them.
The time between two illuminations is 10 ms and the duration of one illumination is 3ms.
Results
Local intensity histogram normalisation, thresholding connected component labelling and finally
gravity centre extraction are used in the first step to achieve a subpixel marker localisation.
The target image on the camera ’’0’’ and the marker localisation are presented on Figure 2 and Figure
3. For the (x,y,0) to (u,v) projective transform function the RMS error between the extracted target locations and
their prediction is below 0.5 pixels.
Figure 2 : Target image on the camera ’’0’’
Figure 3 Extraction of the localisation of the target markers (camera ’’0’’)
The global camera location and orientation is searched for as (Figure 4):
3
Figure 4: Camera model (combination of camera translations and rotations)
(i)
(ii)
(iii)
a translation in the target plane by a (X0, Y0) vector followed by
a rotation of an angle ϕ around an axis of rotation located inside the target plane and making an angle θ
with the x axis and
a translation along the optical axis z by an angle ψ
After calculation, the parameters of the image centre are u0=272.7, v0 = 262.0 pixels.
The camera model described above is used in order to accurately combine the 2D apparent velocity
fields inside the cavity computed by Optical Flow (Figures 5a et 5b) from the different viewpoints into a single
3D velocity field (Figure 5c ). A pure red and a pure blue correspond to a velocity of +/- 1 mm by step time.
Figure 5a: 2D velocity field rectified camera ‘’0’’
Figure 5b: 2D velocity field rectified camera ‘’1’’
4
Figure 5c : 3D velocity field at instant t
On figure 6 we present two 3D velocity fields for different times which emphasises the three
dimensional and unsteady character of the flow for such a configuration.
t + 264 ms
t + 666 ms
Figure 6: 3D velocity field for different times
In order to compare the results obtained by this method we represented the instantaneous velocity
components Ux and Uy profiles as a function of Y for different abscises X The indices ’0’ and ’1’ denote the
velocity field calculated with images from the camera ’0’ and ’1’, respectively , ’3D’ corresponding to the three
dimensional calculation. The coordinates system is represented on Figure 7.
Y (mm)
0
X (mm)
Figure 7: The coordinates system used to show velocity profiles inside the cavity
The horizontal Ux and vertical Uy velocity components variation with Y are presented on Figure 8a
and 8b for different abscises.
5
X=15
1.5
ux0
UX
0.5
-20
-10
-0.5 0
UX1
10
20
UX3D
-1.5
Y
X=21
1.5
UX
1
ux0
0.5
UX1
UX3D
0
-20
-10
-0.5 0
10
20
Y
X=28
1.5
UX
1
ux0
0.5
UX1
UX3D
0
-20
-10
-0.5 0
10
20
Y
X=33
UX
1.5
1
UX0
0.5
UX1
UX3D
0
-20
-10
-0.5 0
10
20
Y
Figure 8a: Ux variation with Y calculated for the two cameras ’0’ and ’1’
6
UY
X=15
-20
-10
0.8
0.6
0.4
0.2
0
-0.2 0
uy0
UY1
Uy3D
10
20
Y
X=21
0.6
uy0
UY
0.4
UY1
0.2
UY3D
0
-20
-10
0
10
20
Y
UY
X=28
-20
-10
0.6
0.4
0.2
0
-0.2 0
-0.4
UY0
UY1
10
20
UY3D
Y
UY
X=33
-20
-10
0.6
0.4
0.2
0
-0.2 0
-0.4
UY0
UY1
10
20
UY3D
Y
Figure 8b: Uy variation with Y calculated for the two cameras ’0’ and ’1’
One can notice the very good agreement between the calculations based on the images from the both
cameras and the three dimensional calculation.
7
Conclusion
We have presented first results of 3D velocity fields measurements based on the stereoscopic PIV
coupled with a new method for camera calibration using only an image of a plane target and the intrinsic camera
parameters (focal length and pixel size). The method was evaluated using a pair of digital cameras looking at a
target dot grid. The angular location of the camera appeared to be at least as accurate as what it was possible to
measure on our experimental set-up. The main accuracy check was done using the locations of the recovered
camera optical centres . The distance between the optical centres and the target origin was found with an
accuracy better then 2 mm (about 0.5 % for a distance of 377 mm). The distance between the optical centres was
found also with an accuracy better than 2 mm. The angular accuracy is evaluated via the error on the distance
between the optical centres related to their distance from the origin (about 0.5 %).
Analysis of the velocity components profiles obtained for different images from the two cameras show a
very good agreement and can valid this new method of calibration and 3D velocity field calulation.
Future work concerns the validation of theses measurements for different velocities and different
configurations. Using a third camera will perform more accurate 3D velocity measurements.
References
Quénot G.M., Pakleza J., Kowalewski T: Particle Image Velocimetry with Optical Flow, Experiments in Fluids ,
vol 25, No. 3, pp 177-189, 1998
Quénot G., Rambert A., Lusseyran F., Gougat P.: Simple and accurate PIV camera calibration using a single
target image and camera focal lenght, 4th International Symposium on Particle Image Velocimetry , Gottingen,
Germany, September 17-19, 2001
Riou L., Fayolle J and Fournel T.: PIV Measurements using multiple cameras: the calibration method, 8th
International Symposium on Flow Visualisation , Sorentto, Italy, 1-4 September, 1998
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