Comparison of interpolation techniques for 2D and 3D velocimetry
Laurent DAVID, Aurélie ESNAULT, Damien CALLUAUD
Laboratoire d’Etudes Aérodynamiques
SP2MI, Téléport 2, Boulevard Marie et Pierre Curie
86960 Futuroscope cedex, France;
Laurent.David@univ-poitiers.fr
Abstract
Some difficulties appear when the researcher want to compare some information located in different positions
and calculated by several measurements techniques. Even so to simplify the algorithms for 3D PIV
measurements, the choice to calculate velocity vectors on specific location from classic 2D PIV vector fields is
often prefered than to determine directly velocity vectors on an unstructured grid mesh. As it could introduce
some errors, data interpolation seems then necessary. This preliminary work consists to assess four interpolation
methods : the bicubic and the thin shell splines, a least squares technique and a weighting function in relation
with the distance between the vectors. Irregularly and regularly spaced data have been used for the interpolation.
Different vector selection are compared and the influence of this step is estimated on 2D and 3D PIV
measurements (figure1).
Figure 1 : Scatterplot and probability function for an instantaneous velocity vector field and an interpolated field.
Introduction
During measurements in the flows, the interpolation techniques to evaluate velocity vectors in specific locations
are often employed. On the one hand, these techniques can be used via random vector distributions to calculate
velocity vectors on a regular grid. Scalars characterizing the flows are then deduced and the velocity vectors
resulting from measurement techniques like PIV, PTV or PSV can be compared at the same locations. On the
other hand, the interpolation techniques are also used, via regularly distributed velocity vectors, to determine
velocities on an irregular grid. For 3D PIV measurements, these points result from projecting a regular grid of
the object reference in the image references. Some authors compared methods to interpolate random distributed
vectors on nodes of a regular mesh. Robinson and Rockwell (1993), Spedding and Rignot (1993) carried out
comparisons between the techniques Adaptative Gauss Window and Spline Thin Shell onto theoretical and
experimental velocity fields. Imaichi and Ohmi (1983), David and Texier (1995) also studied these interpolation
effects with the methods of least squares, of weighting according to the distances and by Krigging from random
velocity fields. In order to determine the most suitable algorithms to carry out interpolations on a regular grid or
on an unstructured grid, a comparative and systematic study between four interpolation techniques (Spline Thin
Shell, weighting by the square of the distance, minimization by least squares, Bicubic Spline) is presented in this
paper. Using a Columnar vortex, 2D simulated and 3D simulated and projected by real matrices of projection
obtained during measurements by 3D-PIV, the interpolation effects of the velocity fields are assessed. In
particular, the selection of the points used for the interpolation, the capacity to respect, as far as possible, the
instantaneous or mean values of the velocity vectors, the faithful restitution of the velocity gradients are elements
particularly highlighted.
Interpolation Techniques
Four interpolation techniques are compared; they are all well defined in the literature and we present for each
only the main characteristics.
•
The inverse distance weighting methods (called here AGW)
This method consists in calculating velocity vector for a location by summing vectors of the selected points
origin, weighted by a function W, specific to each point origin. The function W is related to the distance between
the points origin (with the number of N) and the point of the interpolation Agui and Jimenez(1987). One can thus
write the estimator in the following way :
k=N
Um =
∑
wk ( x,
k =1
k=N
y ).U k
(1)
∑ wk ( x, y)
k =1
The function W can take various expressions. In the case of the interpolation weighted by the inverse of the
distances, the function W is written : wk=dk-2, with the distance d k = ( xm − xk ) 2 + ( ym − yk ) 2 and
k=N
∑ wk
= 1.
k =1
In the case of the interpolation by Gaussian window (Adaptive Window Gauss), W takes the
−
dk
2
σ2
(σ being the width of the window of selection). The first of these two expressions has been
form wk ( x, y ) = e
computed in this comparison.
• The Thin Shell Spline (noticed STS)
The principle of the method rests on the search of N functions G and N coefficients λ such that the required
velocity value Um could be calculated in a location (x,y) by U m ( x, y ) =
k=N
∑ λk Gk ( x, y )
(2)
k =1
The problem consists to find functions G suitable. It was proven that a class of named Thin-Shell Splines
function minimized a global function proportionnal to the Laplacian of the velocity fields. The following Um
function satisfied this condition :
U m (t ) =
k=N
∑ λk Gk ( M k , M ) + α1x + α 2 y + β
k =1
(3)
with
k=N
∑ λk
k=N
∑ λk M k = 0 and M(x,y), Mk(xk,yk), and Gk(Mk,M)=|dk2|log|dk2|
=0,
k =1
(4)
k =1
L.Paihua Montès (1978) studied the resolution of the system which rises from this formulation and calculated the
first derivative in the following way:
k=N
∂vi
= α1 + 2 ∑ λ2, k d k , x log d k 2 + 1
(5)
∂x
k =1
{
(
)}
{
(
)}
k=N
∂ui
= α 2 + 2 ∑ λ1, k d k , y log d k 2 + 1
∂x
k =1
(6)
It is noticed that the interest of this method holds in the fact that calculation of interpolation takes into account
the velocity derivative.
• The least squares minimization (noticed MC in the paper)
The method of interpolation by least Squares is based on a development of Taylor carried out in an origin point
starting from the sought value in a location. To determine the Um velocity in a point M, Imaichi and Ohmi (1983)
wrote the development in a point K where the velocity components are known:
∂U m
∂U m
Uk = Um +
( xk − x m ) +
( yk − ym )
(7)
∂x
∂y
(The same development is applied for V) .
Then, Um and Vm are estimated such as the quantities S and T are minimal:
S=
k=N
∑ [U k − U m −
k =1
T=
k=N
(8)
∂V
∂Vm
( xk − xm ) − m ( y k − y m )]2
∂y
∂x
(9)
∑ [Vk − UVm −
k =1
•
∂U m
∂U m
( xk − xm ) −
( yk − ym )]2
∂y
∂x
The Bicubic Spline (noticed BC)
The Bicubic Spline interpolation (from Numerical Recipes in C) is carried out by two successive interpolations
according to direction X then Y. The velocity vectors are defined in nodes of a Cartesian grid Ui = U(xi), for i=1
to N and each value U in x location is computed by the following formulation according to a direction X then Y :
U = AU j + BU j +1 + CU ' ' j + DU ' ' j +1
(10)
x j +1 − x
x − xj
, C = 1 ( A3 − A ) ( x j +1 − x j ) 2 and D = 1 ( B 3 − B ) ( x j +1 − x j ) 2
6
6
If the expression (10) is derivated, we obtain the expression of U’i:
with A =
x j +1 − x j
,B =
x j +1 − x j
dU U j +1 − U j 3 A2 − 1
3 B2 − 1
( x j +1 − x j )U ' ' j +
=
−
( x j +1 − x j )U ' ' j +1
6
dx
x j +1 − x j
6
and the second derivative :
U ' ' = AU ' ' j + BU ' ' j +1
(11)
(12)
(13)
Now to evaluate the second derivative supposing known, the fundamental idea of the Bicubic Spline second
interpolation is to seek there such as the first derivative is continuous. This imposes that the equation (12) has the
same value in xj and xj+1 what after arrangement gives:
x j − x j −1
6
U " j −1 +
x j +1 − x j −1
3
U '' j +
x j +1 − x j
6
U ' ' j +1 =
U j +1 − U j
x j +1 − x j
−
U j − U j −1
x j − x j −1
(14)
It’s a system of N-2 equations and by using two boundary conditions (y ’’1 or y’’N or the two, equal ones to zero
or y’’1 and y’’N calculated from (12) by imposing the boundary conditions with the first derived ), the N
unknown U can then be calculated.
Principle of the comparison
In order to suppress the uncertainties of experimental data, a Columnar vortex is simulated Robinson and
Rockwell (1993). This swirl has a tangential velocity component
vt =
( )


Γ
 1 − exp  − r


Vs * Rs * r 
Rs

2
(15)



and a vertical velocity component,
  r 2 
V0
− exp −   
v=
  Rs  
Vs


(16)
with the parameters :
Γ : circulation,
Vs : velocity,
Rs : vortex radius,
V0 : vertical external velocity.
To study the influence of the interpolation step of vectors from a random distribution onto a regular grid, three
2D vortices (without vertical velocity) are calculated (figure2 - table1) with different sizes and amplitudes,
randomly and repetitively, to obtain 100 fields of velocity vectors. Then they are interpolated on nodes of a
regular grid by different interpolation methods : minimization by least squares (noted MC), weighting close to
the Adaptative Gauss Window (AGW) and an approximation by a Spline (Spline Thin Shell).
In the same way, to evaluate the influence of the interpolation step in a data processing sequence by 3D-PIV,
three 3D vortices (Table 1) are simulated and projected to the nodes of a regular grid mesh in two image
references (right-hand side and left) using projection parameters resulting from an in-situ calibration Calluaud
and David (2002). These regular fields of velocity vectors are interpolated onto an unstructured mesh and are
combined by backward projection in the object reference (figure 3). Thus the influence of the preceding
interpolation techniques and the influence of approximation by Bicubic Spline are evaluated as a whole in the
data processing sequence.
0.3
e
m
ro
n
0.2
0
0.1
2
m)
m
(
X
4
0
6
0
2
8
4
6
8
Y (mm)
10
10
Figure 2 : Velocity vector magnitude (mm/s ) for the three simulated vortices.
Vortex
V0
RS
VS
Γ
2D
reference
0
2
5
5
2D
plain
0
4
5
5
2D
tip
0
2
5
10
3D
reference
15
12
10
30 000
Table1 : Parameters for the simulated vortex
3D
plain
15
18
10
30 000
3D
tip
30
12
10
30 000
Three modes of vector selection were compared: the four or eight closest vectors or a minimum of one vector per
quadrant. The interpolated velocity fields are compared using the calculation of the difference of the Euclidian
norm between the real and simulated velocity on the complete field. The relative errors on a velocity profile
passing by the center of the vortex is determined too and some random velocity vector fields are examined in
details to understand the errors on some vectors and the behavior near the vortex center.
MG
MD
INTERPOLATION
INTERPOLATION
M-1G
M-1D
Figure 3 : Schematic representation of the steps to estimate the influence of the interpolation
on 3D PIV measurements with the backward projection.
Results
Three aspects are observed closely in the results :
- the best point selection according to the interpolation techniques;
- the faithful restitution of the instantaneous velocity fields,
- the influence of the starting distribution of the vectors on the interpolation techniques.
At first, the optimum modes of selections of the points for each interpolation type were given starting from the
random fields of vectors and the fields uniformly distributed. The methods splines, in general, require a
homogeneous distribution of the vectors without hole of information. The approximation by STS is correctly
conditioned when the data are not aligned and that the first three points forms an equilateral triangle. The choice
of the four closest points remains the mode of the selection most favourable to this method. The method Bicubic
Spline, only tested from a regular grid mesh, takes account of all the points to determine the second derivative
employed by the approximation. For the method by weighting or of quadratic minimization, the number of
points and their locations strongly influence the accuracy of the method. If the criteria of reproducibility is
preferable, the choice of the four or the height closest points is better for technique AGW whereas for the
method of least squares the height closest points give the most satisfactory results. If we seek to reduce the
average error obtained from 100 velocity fields, the choice of the four closest points is essential in the areas of
strong velocity gradients (figure 4). If the center of the swirl is examined, the two methods behave in the same
way i.e the height points closest then a vector by quadrant.
1,0
0,8
0,6
0,4
0,2
0,0
-0,2
-0,4
-0,6
-0,8
-1,0
-1,2
-1,4
-1,6
-1,8
-2,0
R
el
at
iv
e
er
ro
r
AGWP4ermoyU
AGWP4ermaxU
AGWP4erminU
4 nearest points
R
e
l
a
t
i
v
e
e
r
r
o
-2
0
2
4
6
8
10
12
14
16
1,0
0,8
0,6
0,4
0,2
0,0
-0,2
-0,4
-0,6
-0,8
-1,0
-1,2
-1,4
-1,6
-1,8
-2,0
AGWP8ermoyU
AGWP8ermaxU
AGWP8erminU
8 nearest points
-2
0
2
4
R
el
at
iv
e
er
ro
r
MCP4ermoyU
MCP4ermaxU
MCP4erminU
4 nearest vectors
-2
0
2
R
el
at
iv
e
er
ro
r
4
6
8
8
10
12
14
16
AGWQ4erminU
1 vector by
quadra nt
-2
0
2
4
Distance to the origin
Distance to the origin
1,0
0,8
0,6
0,4
0,2
0,0
-0,2
-0,4
-0,6
-0,8
-1,0
-1,2
-1,4
-1,6
-1,8
-2,0
6
AGWQ4ermoyU
AGWQ4ermaxU
1,0
0,8
0,6
Rel
0,4
ativ
e 0,2
erro 0,0
r
-0,2
-0,4
-0,6
-0,8
-1,0
-1,2
-1,4
-1,6
-1,8
-2,0
10
12
Distance to the origin
14
16
1,0
0,8
0,6 8 nearest points
0,4
0,2
0,0
-0,2
-0,4
-0,6
-0,8
-1,0
-1,2
-1,4
-1,6
-1,8
-2,0
-2
0
2
MCP8ermoyU
MCP8ermaxU
MCP8erminU
4
6
8
10
Distance to the origin
6
8
10
12
14
16
Distance to th e origi n
12
14
16
1,0
0,8
0,6
Rel 0,4
ativ
e 0,2
err 0,0
or
-0,2
-0,4
-0,6
-0,8
-1,0
-1,2
-1,4
-1,6
-1,8
-2,0
MCQ4ermoyU
MCQ4ermaxU
MCQ4erminU
1 vector by quadrant
-2
0
2
4
6
8
10
12
14
16
Distance to the origin
Figure 4 : Velocity profiles for the U component and the three modes of vector selections (reference vortex).
For the study of the three two-dimensional swirls, we have conserved every mode of selection but we comment
only the best mode of points selection for each technique. The norms of velocity amplitude difference show that,
from random velocity fields, the method by least squares gives the best results and the more reproductible. In
regard to the velocity profiles taken on a diagonal, it seems similar in zones of low gradient since the method
AGW better represents the areas of strong velocity gradients. In this place, the minimization by least square
tends to smooth the vectors and the velocity field is then underestimated. Near the vortex center, the least square
method gives in appearance better results than with the AGW weighting function. The location of the center is
correctly positioned. Technique STS does not appear regular enough to be used without a validation by filtering.
In general, the results are very accurate but for some points this spline introduces too significant errors (> 200%).
Reference vortex
Tip vortex
Plain vortex
Figure 5 : Norm of the velocity amplitude difference for the three 2D vortices.
For the study of the three-dimensional vortices, some behavior modifications appeared during vector
interpolation from a regular grid onto an unstructured grid mesh. First, because the origin points are better
conditioned than those calculated, the methods Spline (Bicubic or STS) give very good results on the values of
the velocity difference norm and the relative errors. This is bound, in particular, to the fact that these methods
take into account the first or second derivatives and that the latter are correctly estimated on this type of grid.
Methods AGW and MC, though, give satisfactory results with nevertheless some differences according to the
components (figure 6). The AGW weighting function introduces between 5 to 10% of errors like for the
interpolation on the 2D cases. On the other hand, the other techniques seem to be improved. To check that these
results are not related to a smoothing during of the passage in the stereoscopic method of backward projection,
complementary calculations were carried out from a regular grid onto a unstructured grid mesh. The obtained
results confirm the improvement of the accuracy on the vectors interpolated from a regular vector distribution.
Referenc e v ortex
Norm W
Referenc e v ortex
N
o
r
m
m
m
/
s
N
or
m
m
m
/s
7.5
7.0
6.5
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
6.0
Norm U
5.5
N
o
r
m
5.0
4.5
4.0
3.5
m
m
/
s
3.0
2.5
2.0
1.5
1.0
0.5
AGW P8
BS
MC P8
STS P4
N
n
or
m
Norm V
8.0
7.5
7.0
6.5
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
M
m
/s
AGW P8
BS
MC P8
STS P4
AGW P8
BS
13
12
11
10
9
8
7
6
5
4
3
2
1
0
MC P8
STS P4
Norm U+V+W
AGW P8
BS
MC P8
STS P4
Figure 6 : Norm of the difference for the 3 components and the amplitude of the velocity
from a regular grid mesh to a unstructured grid mesh .
Other tests have been made with the bicubic spline interpolation. The introduction of errors (10, 20, 30%) on one
or two modules of the velocity vectors for just one of the vectors fields (left or right velocity projection) used for
the reconstruction by backward projection, makes it possible to highlight the propagation of this error (figure 7)
on the calculation. The velocity errors appear clearly around the place where the vector has been changed and the
norm of the relative error reveal some privilegated direction of the error.
Conclusion
In conclusion, the interpolation techniques are to be employed with prudence. Since a random
distribution, the methods involving minimization by least squares and AGW give reliable results respectively for
low and strong velocity gradients. For interpolations using vectors distributed regularly on a structured grid
mesh, the Spline methods and in second the approximation by least squares seems to be more accurate for the
calculation of the third component. The improvement results mainly to the fact that those techniques give better
estimations from structured grid mesh and not to the employment of the backward projection However these
results on simulated fields are to be tempered on realistic velocity vectors resulting from experiments and sullied
with spurious vectors. An other aspect showed in the figure 1 is the scatter plot and the histogram of the
displacement of the vector for the component U. The interpolated vectors from a regular grid mesh have a
representation modified after the calculation and it is clear that these variations should have some influence for
the determination of the 3D vectors.
References
J.C. Agui, J. Jimenez, 1987 : On the performance of particle tracking, Journal of Fluid Mechanics, vol 185, 447468.
D. Calluaud, L. David, 2002 : Backward projection algorithm and stereoscopic particle image velocimetry
measurements of the flow around a square section cylinder. 11th International Symposium on Applications of
Laser Techniques to Fluid Mechanic, Lisbon.
L. David, A. Texier, 1995 : Influence des techniques d’interpolation sur la précision des champs des vitesses
instantanées caractérisant un écoulement. 6ième Congrès de Visualisation et de Traitement d’images en mécanique
des fluides, St-Etienne (France).
K. Imaichi, K. Ohmi. 1983 : Numerical processing of flow visualization pictures. Measurement of two
dimensional vortex flow. Journal of Fluid Mechanics, vol 86, 283-311.
WH. Press, SA. Teukolsky, WT. Vetterling, BP. Flannery, 1995 : Numerical Recipes in C, Cambridge Press
University.
L. Paihuas Montès, 1978 : Quelques méthodes numériques pour le calcul des fonctions splines à une ou plusieurs
variables. Thèse de Doctorat de l’Institut Polytechnique de Grenoble.
O. Robinson, D Rockwell, 1993 : Construction of 3D images of flow structures via particule tracking
techniques. Experiments in Fluids 14, 257-270.
G.R. Spedding, E.J.M. Rignot, 1993 : Performance analysis and application of grid interpolation techniques for
fluid flows. Experiments in Fluids 15, 417-430.
Figure 7 : Bicubic Spline Interpolation without and with introduction of magnitude errors in one of the image
references, Reconstructed 3D velocity fields for these two interpolations in the object reference.