INSTITUTE OF PHYSICS PUBLISHING
MEASUREMENT SCIENCE AND TECHNOLOGY
Meas. Sci. Technol. 13 (2002) 1058–1071
PII: S0957-0233(02)31785-5
Some considerations on the accuracy and
frequency response of some derivative
filters applied to particle image
velocimetry vector fields
J M Foucaut and M Stanislas
Ecole Centrale de Lille, LML URA 1441, BP 48, Cité Scientifique,
F59651 Villeneuve d’Ascq Cedex, France
E-mail: jean-marc.foucaut@ec-lille.fr
Received 11 December 2001, in final form and accepted for publication
27 May 2002
Published 20 June 2002
Online at stacks.iop.org/MST/13/1058
Abstract
This paper concerns the computation of derivatives from particle image
velocimetry (PIV) velocity fields with the goal of obtaining the vorticity
component normal to the plane. A variety of derivative schemes are
characterized by their transfer function, taking into account the truncation
and noise amplification. The PIV measurement noise is supposed to be a
white one in the Fourier space. A spectral approach is used in order to
choose the best filter for turbulent flows. The derivative spectra are
discussed. An application is presented on a real turbulent flow with two
interrogation window sizes and different derivative schemes. The most
significant schemes are also applied to a velocity field containing a single
vortex. A comparison of the maximum of vorticity obtained with each
scheme and through a least-square fit with an Oseen vortex, allows us to
quantify the effect of the band pass filter and to select the best scheme.
Keywords: PIV, derivative computation, vorticity, transfer function, accuracy
Nomenclature
a, ai
b
c
E11
Enoise
F11
i, j , l
k
klc
khc
n
r
R
RO
T
Tn
Tr
coefficients of derivative schemes
coefficients of compact derivative schemes
coefficients of compact derivative schemes
velocity spectrum
noise level in Fourier space
derivative spectrum
index
wavenumber
low cutoff wavenumber
high cutoff wavenumber
order of derivative filter
radial coordinate
non-dimensional cylindrical coordinate
radius of an Oseen vortex
transfer function of derivative filter
transfer function of derivative filter, noise effect
0957-0233/02/071058+14$30.00
© 2002 IOP Publishing Ltd
u
û
u∗
uθ
v
x, y, z
X
α
β, χ
ε
x, y
σu
ω
ωc
ω0
Printed in the UK
transfer function of derivative filter,
truncation effect
velocity component along x
RMS velocity fluctuation
characteristic velocity of an Oseen vortex
tangential velocity component
velocity component along y
Cartesian coordinates
interrogation window size
coefficient of the truncation term
parameters of compact difference schemes
noise amplification
spatial resolution
circulation of an Oseen vortex
noise level
vorticity component normal to the PIV plane
estimation of ω by means of a filter
vorticity at the centre of an Oseen vortex
1058
The accuracy and frequency response of some derivative filters applied to PIV vector fields
1. Introduction
Particle image velocimetry (PIV) is a well established
measurement technique which allows us to obtain large
amounts of data on all kinds of flows. The theory of PIV
is now quite extensive (Westerweel 1997) and clearly shows
one of the main characteristics of this technique: it provide
a discretization of the velocity field with a limited spatial
resolution. Nevertheless, through its spatial character and the
large number of samples that can be recorded nowadays, PIV
allows a lot of new insights into many flows and leads scientists
to adapt progressively the processing tools used in CFD. In
this development, care should be taken due to the fact that the
PIV data are fairly different from CFD data. In PIV the spatial
resolution is limited toward the large scales by the length of the
field and toward the small scales by the interrogation window
size. Moreover, the data carry a measurement noise inherent to
the method. This noise is introduced during recording (optical
distortion, light sheet non-homogeneity, transfer function of
CCD, particle characteristics, speckle) (Allano et al 1998)
and during the data processing (peak fitting algorithm, image
interpolation, peak deformation) (Westerweel et al 1997).
Thus the CFD tools cannot be transposed directly and care
should be taken to check their accuracy and reliability. This
problem is addressed in the present paper for one specific tool:
the computation of derivatives of the field, illustrated by the
estimation of the vorticity. Derivatives of the velocity field are
important characteristics in fluid dynamics as they appear in
the Navier–Stokes equations and in important operators such
as the divergence, the vorticity or the stress tensor. These
operators are extensively used to understand the physics of
the flow under study. These checks are particularly important
for turbulence, where the bandwidth of the measurement
technique with respect to these operators is fundamental for
the interpretation of the results. The PIV spatial bandwidth
being limited by nature, a guideline is thus needed to select
the best suited estimator of these operators and information
must be provided on the spectral response and accuracy of the
selected filter. In the present paper, the interest of such an
approach will be illustrated with the vorticity operator applied
to two types of flow for which it is of particular interest: a
turbulent flow (a boundary layer in the present case) and an
isolated vortex.
The use of such derivative operators implies a certain
continuity and smoothness of the measured velocity field
which can be put in default by the presence of too many
outliers. This implies a preliminary validation and eventual
interpolation of the vector field. This problem was addressed
by several authors, most recently by Foucaut et al (2000). To
obtain derivatives, the literature proposes several techniques.
The first one, which is probably the one most used, is based
on discrete differential operators applied to neighbouring grid
points. Lele (1992), Lourenco and Krothapalli (1995), Kim
and Lee (1996), Raffel et al (1998) and Luff et al (1999)
have used several schemes of this kind which will be discussed
later. Even if a differential approach behaves like a filter, it is
evident that, using this technique, the derivative estimate can
be disturbed by the measurement noise. Following Lourenco
and Krothapalli (2000), it can be assumed that the transfer
function of PIV is mainly due to the windowing effect. In the
present paper, the measurement noise will be supposed to be
white. A second technique to estimate derivatives is based on
the fit of an analytic function to the discrete velocity field. The
derivative is then obtained by an analytical derivation of the
fitted function. This method is proposed by Fouras and Soria
(1998) by means of a χ 2 fit of a second-order polynomial and
by Abrahamson and Lonnes (1995) who use a least-square
approach. In that case, the local velocity is fitted to a model
consisting of uniform translation, rigid rotation, a point source
and plane shear. The advantage of such methods is to reduce
the effect of the noise. The drawback is that a filtering of the
data is performed when the fit is applied. A recent paper of
Ruan et al (2001) propose a third method to obtain directly
the vorticity from the particle images. They use a pattern
matching technique to determine the particle displacement due
to rotation between both exposures.
In this paper, only the differential technique is studied.
In the following section the different schemes found in the
literature will be presented. Methods to optimize the choice
of derivative filter are then proposed in the case of turbulent
flows and for an isolated vortical structure.
2. Derivative computation method
For the derivative computation, a compromise has to be made
between the order of the filter, the number of points used for the
derivation, the frequency response and the noise amplification.
Generally, for classical finite difference schemes, the order
increases with the number of points used for the computation.
The frequency response increases when the truncation error
decreases, which is directly linked to the order of the filter. On
a regular grid, the first derivative at a point j , with a centred
scheme of order n, can be estimated by the following equation:
∂u 1
=
ai (uj +i − uj −i )
∂x j ax i=1,n/2
σu
x i−1 ∂ i u αi
+ε
(1)
+
i
i! ∂x j
x
i=n+1,∞
where x is the grid spacing around point j . A centred scheme
implies that the order
of a and αi are
is even. The values given by a = 2 i=1,n/2 i ai and αi = a2 l=1,n/2 al k i =
i
l=1,n/2 al k
l=1,n/2 al k. The values of ai are obtained by
means of limited expansions of the velocity. The second term
on the right-hand side is the truncation error and the last term
is the noise error estimated here by a probabilistic quadratic
approach (Neuilly 1998). σu is the measurement noise level
and ε is the noise amplification coefficient obtained by
ε2 =
4 2
ai .
a 2 i=1,n/2
(2)
Lourenco and Krothapalli (1995) have considered that σu can
be estimated by the uncertainty in the velocity measurement.
This error is random and is characterized in that case by
its RMS value. It will be seen that the noise amplification
coefficient ε increases with the order of the scheme.
In the present study, four standard filters based on centred
difference schemes (equation (1)) were first studied with
second, fourth, sixth and eighth order of accuracy. The values
1059
J M Foucaut and M Stanislas
Table 1. Detailed characteristics of centred difference derivative
filters.
Order
a
a1
2
4
6
8
2
12
60
840
1
0
8
−1
45
−9
672 −168
a2
a3
a4
0 0
0 0
1 0
32 −3
αn+1
ε
1
4
36
576
0.71
0.95
1.08
1.17
of a, ai , αn+1 and ε are given in table 1. As expected, the
truncation error decreases when the order increases, but ε
increases and is even higher than 1 for the sixth- and eighthorder filters.
Luff et al (1999) compared three schemes: the second- and
fourth-order centred difference and the circulation computed
over eight velocity points, following the method of Reuss et al
published in 1989. This method gives directly the vorticity
while the differential ones provide the gradient field:
 v −v

ui,j +1 −ui,j −1
i+1,j
i−1,j
−
2x
2y



1
 1 vi+1,j +1 −vi−1,j +1
ui+1,j +1 −ui+1,j −1 
ω(i, j ) =  + 2
−
 . (3)
2x
2y
4



v
−v
u
−u
+ i+1,j −12xi−1,j −1 − i−1,j +12yi−1,j −1
As can be seen in equation (3) the method of Reuss et al
corresponds to a linear combination of second-order centred
difference filters computed at the point of interest and its
neighbours. Luff et al used synthetic images computed
from an Oseen vortex and experimental ones obtained by
a photographic recording technique and an auto-correlation
analysis method. They estimated the error on the vorticity
field with and without noise added. Without noise, they
obtained average errors of 4.3, 2.8 and 0.004%, respectively,
for the eight-point circulation second-order and fourth-order
schemes. These errors correspond to a truncation effect. With
noise, they obtained the opposite behaviour from 9.2 to 18.9%
due to the noise amplification coefficient which increases.
They discussed the effect of the interrogation window size.
They have also tested the effect of a 3 × 3 Gaussian smoothing
filter to avoid the noise amplification. In that case, they
obtained a global error of the order of 3.8% whatever the filter
used.
Three compact difference schemes were also studied here.
These schemes are presented by Lele (1992) and Kim and Lee
(1996). They are implicit and need a matrix inversion. They
are given by
βuj −2 + χuj −1 + uj + χuj +1 + βuj +2
uj +3 − uj −3
uj +2 − uj −2
uj +1 − uj −1
=c
+b
+a
(4)
6x
4x
2x
where β, χ, a, b and c are five parameters which are optimized
to obtain the selected order. By the use of Taylor series
coefficients of various orders, five relations between these
parameters can be solved to cancel the high-order terms. The
form of equation (4) allows us to obtain derivative filters up to
the tenth order.
If c = 0 is imposed, one step of optimization leads to
a sixth-order filter family which is given by the following
1060
parameters:
3χ − 1
,
a = 29 (8 − 3χ),
12
57χ − 17
b=
,
c = 0.
18
β=
(5)
As can be seen equations (4) and (5) show that only four
velocity values are necessary and β, a and b depend only on χ .
In this case, for each value of χ the truncation terms up to sixth
order are cancelled. The choice of β = 0 (χ = 1/3) leads
to a sixth-order scheme which necessitates only a tridiagonal
matrix resolution. An optimization of χ at a value of 4/9
allows us to find an eighth-order derivative filter.
The parameter c can also be optimized. In that case, a
eighth-order family is obtained with the following parameters:
8χ − 3
,
20
568χ − 183
,
b=
150
β=
12 − 7χ
,
6
9χ − 4
c=
.
50
a=
(6)
Here, β, a, b and c depend on χ, six velocity values and a
pentadiagonal resolution are necessary. An optimum value of
χ = 0.5 can also be found which enables us to get a tenth-order
filter. The values of the coefficients of the different compact
schemes studied here are presented in table 2.
The derivative estimation presents the same types of errors
as previously (see equation (1)):
∂u σu
x i−1 ∂ i u = uj +
αi
+ε
.
(7)
i
∂x j
i!
∂x
x
j
i=n+1,∞
The compact difference schemes look interesting because ε
is smaller for the tenth-order filter than for the fourth-order
centred scheme, together with a very small truncation error.
For the lowest frequency the noise can become the
principal source of error. This is why Lourenco and Krothapalli
(1995) recommended using the Richardson extrapolation
which takes the values far from the measurement point into
account, in order to improve the accuracy by minimizing the
noise amplification. In the present contribution, seven filters
based on Richardson extrapolation were tested. The principle
of this method is to take a linear combination of secondorder centred difference schemes computed with different
data spacing x, 2x, 4x and 8x. The coefficients
of the combination are adapted in order to minimize the
truncation error or the noise error, depending on the objective.
The Richardson extrapolation is equivalent to the following
equation (Lourenco and Krothapalli 1995):
uj +i − uj −i
∂u 1 =
ai
∂x
a
2ix
j
+
i=1,2,4,8
i=n+1,∞
αi
x i−1 ∂ i u
σu
+ε
.
(n + 1)! ∂x i
x
(8)
The term (uj +i − uj −i )/2ix is a second-order centred
difference scheme with a grid spacing of ix. The coefficients
ai are optimized to increase the order by use of Taylor
expansions or to minimize ε by a least-square method.
Four schemes has been derived with an optimization of the
The accuracy and frequency response of some derivative filters applied to PIV vector fields
Table 2. Detailed characteristics of compact difference derivative filters.
Order
a
b
c
χ
β
αn+1
ε
6
8
10
1.5555
1.4814
1.4166
0.1111
0.4630
0.6733
0
0
0.01
0.3333
0.4444
0.5
0
0.0278
0.05
4
16
144
1
0.9
0.84
truncation error (for second, fourth, sixth and eighth orders)
and three with a minimization of the noise coefficient ε (for
second, fourth and sixth orders). The characteristic coefficients
are presented in table 3. The second- and fourth-order filters
obtained with truncation error minimization are exactly the
same as the corresponding centred difference schemes. The
sixth and eighth order with the truncation error minimization
also give ε larger than 1. The second-, fourth- and sixth-order
filters with a minimization of ε show a truncation error αn+1
(see equation (1)) about 10 times higher than the two other
methods.
The least-square filter proposed by Raffel et al (1998)
is of second order and shows a smaller noise amplification
coefficient than the classical centred schemes:
−2uj −2 − uj −1 + uj +1 + 2uj +2
∂u =
∂x j
10x
+ 3.4
x 2 ∂ 3 u
σu
.
+ 0.316
(3)! ∂x 3
x
(9)
It is obtained from a noise minimization comparable to the
Richardson method computed on the four first neighbouring
points.
Raffel et al (1998) have studied the effect of overlapping
higher than 50% on the computation of vorticity. This effect
was discussed in Carlier (2001) as far as the velocity is
concerned. The conclusion concerning the cutoff frequency
for the vorticity is the same as for the velocity. The PIV
results present a low and a high cutoff frequency. The low
one is due to the field size. The high one depends only on
the interrogation window size whatever the overlapping is.
However, it is clear that PIV samples the turbulence with a
limited range of frequencies. The effect of each derivative
filter has thus to be discussed by comparison of the derivative
power spectra.
Raffel et al (1998) describe the sources of error in
differential estimation. These sources are: the measurement
uncertainty on velocity vectors, the over-sampled data when an
overlapping higher than 50% is used, the interrogation window
size which limits the spatial resolution, the truncation of the
curvature of the particle motion due to the use of only two
pulses in PIV. The main measurement error which can affect
the derivative is the measurement noise. In the spectral domain
its effect is coupled with the under-sampling and with the
windowing effects. The curvature error can be minimized by
decreasing the PIV laser pulse delay but, of course, with a loss
in accuracy (Raffel et al 1998).
3. Turbulent flows
The first case for which derivative filters are characterized is a
turbulent flow. The optimization is based on a spectral analysis
and on the transfer function of each filter.
3.1. Frequency response of PIV
For turbulent flows, the PIV spectral response is characterized
from experimental PIV records. The study is based on a
series of experiments which allowed us to build a model of
the PIV spectrum and to characterize the PIV measurement
noise. One of these experiments has been carried out in the
configuration shown in figure 1 (Carlier 2001). The flow is
a turbulent boundary layer obtained in a specific wind tunnel.
The velocity was measured in a plane parallel to the wall. The
external velocity was 3 m s−1 , corresponding to a Reynolds
number of Rθ = 7800. The camera was a Kodak DCS 460,
with a CCD of 3072 × 2048 pixels, the size of which is 9 µm
by 9 µm. A 100 mm Nikkor lens was used with an f -stop
of 2.8. The magnification was 0.173. These measurements
were made at 100 wall units (12 mm). At this wall distance,
the mean velocity is about 2 m s−1 . A total of 200 fields were
recorded. The PIV delay was optimized in order to obtain a
dynamic range of about 15 pixels (400 µs). Such a delay leads
to a turbulence intensity of 2.2 pixels. Cross-correlation with
symmetrical discrete local shift of both windows was used,
in order to measure the velocity with a second-order accuracy.
The overlapping was 50%. The analysis was made with 24×24
window size (zero padding into 32 × 32 FFT windows) which
gives about 40 000 velocity vectors in the map. Such a flow
configuration allows us to use two homogeneity directions
to compute a spectrum with a better convergence (this was
checked in the present case). For each vector line, the spectrum
is computed along x by means of a FFT algorithm. The spectra
are then averaged along z and over the 200 fields to obtain
convergence. Figures 2 presents the comparison of the PIV
spectrum with hot wire anemometry (HWA) results. A Taylor
hypothesis (x = U t) is made on the HWA spectrum to convert
them into spatial ones to allow comparison.
The PIV spectrum is in agreement with HWA up to
ks = 500 rad m−1 . The spectrum model, which is also
presented in figure 1, is given by
1/2
1/2
E11P I V = (E11H W A + Enoise )2
sin k X/2
k X/2
2
(10)
where Enoise is the white noise level which is optimized to
fit the PIV spectrum. The value of Enoise varies with the
inverse of the window size. The cardinal sine function is
the Fourier transform of a gate function which represents the
windowing effect (X is the interrogation window size and k
is the wavenumber). The model of equation (10) is in good
agreement with the PIV spectrum. The value of Enoise is
of 2.6 × 10−7 m3 s−2 (0.29 pix3 ) for the above experiment.
The sinc function introduces a cutoff wavenumber given by
kc = 2.8/X. This cutoff wavenumber is of 2200 rad m−1 ,
which is larger than ks . The difference is due to the noise level
which is high for that window size.
1061
J M Foucaut and M Stanislas
Table 3. Detailed characteristics of Richardson extrapolation derivative filters.
Order
a
ai+1
2
4
6
8
2∗
4∗
6∗
1
3
45
2 835
65
1 239
41 895
1
0
4
−1
64
−20
4096 −1344
1
0
272
1 036
714 −14 063
ai+2
∗
indicates a noise minimization.
ai+4
ai+8
αn+1
ε
0
0
1
84
0
0
41 356
0
0
0
−1
−64
−69
13 888
1
4
64
4096
63.03
214.5
3156
0.71
0.95
1.18
1.35
0.088
0.334
0.425
1.E-02
1.E-04
3
2
E11 (m /s )
1.E-03
1.E-05
HWA spectrum
24 x 24
Equation (10)
1.E-06
1.E-07
kc
kmin
1.E-08
1
10
100
500
1000
10000
k (rad/m)
Figure 2. Power spectra of velocity along x, 24 × 24 interrogation
windows.
1.E-02
An optimization of the interrogation window based on
a signal-to-noise ratio of 1 at the cutoff wavenumber is also
done. This optimization leads to an interrogation window size
of 44 pixels in the present case. Figure 3 shows the power
spectrum obtained with this window size, compared to the
model of equation (10). The PIV spectrum is now in agreement
with HWA up to kc = 1200 rad m−1 . The lower significant
wavenumber kmin = 40 rad m−1 is deduced from the length
of the velocity field. By this new analysis, the value of Enoise
decreases to 1.4 × 10−7 m3 s−2 (0.16 pix3 ). This study allows
us to give an estimation of the standard deviation of the noise
which was of the order of 1% of the dynamic range in this
case. In section 3.3 of the present paper, the spectra of first
derivatives computed from both analyses will be presented.
3.2. Transfer function of derivative
The spatial character of PIV allows us, after derivative
computation, to assess one vorticity component. This section
concerns the spatial derivative computation in order to obtain
the gradient field. The discrete derivative operator needs
to make a compromise between the truncation error, which
decreases when the order of the filter increases, and the noise
amplification which, in general, increases with this order. The
truncation error affects particularly the higher frequencies and
the noise, which is white, can disturb all frequencies. The
best way to obtain a representation which contains all this
information is to use the transfer function of the derivative
filter.
1062
1.E-03
1.E-04
E11 (m 3 /s 2)
Figure 1. PIV experimental set-up.
1.E-05
HWA spectrum
1.E-06
44 x 44
Equation (10)
1.E-07
kmin
kc
1.E-08
1
10
100
1000
10000
k (rad/m)
Figure 3. Power spectra of velocity along x, 44 × 44 interrogation
window, frequency optimization.
The fourth-order centred difference scheme is used as an
example to illustrate how the transfer function can be obtained:
uj −2 − 8uj −1 + 8uj +1 − uj +2
∂u =
12x
∂x j
+4
x 4 ∂ 5 u
σu
+ 0.95
.
5
(5)! ∂x
x
(11)
In this equation only the first term of the truncation error is
kept.
The frequency response of this filter can be obtained by
several methods. First, the Dirac response can be computed.
The input signal is a step function whose derivative gives a
Dirac pulse. Due to the discrete character of the signal, the
step has a width x and its derivative is a quasi -Dirac function
which is 1/x in width. Figure 4 shows the input signal and
the response of the fourth-order derivative filter (equation (11))
The accuracy and frequency response of some derivative filters applied to PIV vector fields
800
4th order centered difference
Cardinal sin wave
Input signal x 200
700
600
Step response
500
400
300
200
100
0
0.01
-100
0.02
0.03
0.04
0.05
0.06
-200
-300
x
Figure 5. Comparison of transfer functions of the fourth-order
centred difference derivative filter.
Figure 4. Step response of the fourth-order centred difference
derivative filter compared to a sinc function.
compared to the response of a centred scheme of infinite order
(which is, in fact, a sinc function). The Fourier transform of
the step response of the derivative operator, multiplied by the
Fourier transform of the digitization gate function, gives the
transfer function of the derivative filter.
Another method to obtain the transfer function is to use
directly a Fourier mode: u = û exp(ikx) in the derivative
scheme (Lele 1992), where k is the spatial wavenumber and û
can be the turbulence intensity at the considered wavenumber.
The transfer function is then directly obtained by
Tr =
8 sin(kx) − sin(2kx)
.
6kx
(12)
The comparison between both methods is presented in figure 5,
which shows perfect agreement. This figure also shows an
estimation of the transfer function given by the truncation error
x i−1 ∂ i u
(∂u/∂x)est
i=n+1,∞ αi i! ∂x i
=1−
Tr =
(∂u/∂x)true
(∂u/∂x)true
(kx)4
(kx)i−1
=1−4
(13)
=1−
αi
i!
5!
i=n+1,∞
where the index ‘est’ means estimated. The variable u is
replaced here by the Fourier mode as in equation (12). Figure 5
shows that if only one term of the truncation error ((n + 1)th
order of the expansion) is considered (last term of (13)), the
transfer function is the same only for the lowest wavenumbers.
If three terms of the truncation error are kept, the transfer
function obtained is then close to that coming from the direct
method.
This result shows that, if the noise is not taken into account,
a derivative scheme is a low pass filter and that it is also
possible to deduce its transfer function from the truncation
error. The same approach can be proposed to take into account
the measurement noise. If the random error is considered as
a white noise, the transfer function of the noise effect can be
expressed as
Tn = 1−
εσu x
σu 1
=1−ε
û kx
(∂u/∂x)true
(14)
which shows that the noise effect is comparable to a high pass
filter. This kind of behaviour was also observed by Lourenco
Figure 6. Transfer functions of the fourth-order centred difference
derivative filter.
and Krothapalli (1995). That expression shows the fact that,
due to the noise, when the frequency goes to zero, the error on
the derivative can become very important. This error varies
also with the inverse of the turbulence intensity. Figure 6
presents an example, for the fourth-order centred difference
scheme (equation (11)) of the transfer functions of:
• the low pass filter due to the truncation,
• the high pass filter due to the noise,
• the band pass filter deduced from both: T = T n T r.
The local fluctuation rate û and the noise σu are chosen
respectively at 40% (local fluctuation of the turbulent field)
and 1% corresponding sensibly to the case of section 2.
The above method has been generalized for the different
schemes presented in section 2. Four filters based on centred
difference schemes were first studied with second, fourth,
sixth and eighth order. The values of a, ai , αn+1 and ε (see
equation (1)) are given in table 1. As expected, the truncation
error decreases when the order increases, but ε is higher than 1
for the sixth- and eighth-order filters.
The complete transfer functions of the four filters based
on centred differences are presented in figure 7. The low pass
filter transfer function (without noise) can be generalized by
the following equation:
i=1,n/2 2ai sin(ikx)
Tr =
.
(15)
akx
1063
J M Foucaut and M Stanislas
Figure 7. Transfer functions of centred difference derivative filters.
The bandwidth and the lower cutoff frequency increase
with the order of the filter. A recapitulation of the cutoff
frequencies of each filter studied will be given later. Figure 7
shows also an estimation of the transfer function in the
case of the eight-point circulation method presented by Luff
et al (1999). As this filter gives directly the vorticity, the
computation of the transfer function was difficult and needed
some hypothesis: the fluctuations are considered isotropic
(u = û exp(ikx) exp(iky)) and the resolution along x and y
is the same (x = y). The most important hypothesis
is that v = 0 everywhere in the field. Consequently, the
vorticity
becomes identical to a single derivative: ω =
1
∂v/∂x
−
∂u/∂y = − 21 ∂u/∂y. With this last hypothesis
2
the filter becomes
ui,j +1 − ui,j −1
∂u 1
1 ui+1,j +1 − ui+1,j −1
=
+
∂y j
2y
2
2
2
2 3
ui−1,j +1 − ui−1,j −1
σu
x ∂ u
+
+ 0.433
+
. (16)
2
(3)! ∂x 3
y
The transfer function is then
Tr =
sin(kx)
(1 + cos(kx)).
2kx
(17)
For the three compact schemes, the equation for the low pass
transfer function is
Tr =
1
c sin(3kx)
3
+ 21 b sin(2kx) + a sin(kx)
.
(1 + 2β cos(2kx) + 2χ cos(kx))kx
(18)
As can be seen, if β and χ are equal to zero, the explicit centred
difference scheme is recovered. The compact difference
schemes are shown in figure 8 compared to the eighth-order
centred difference one. The compact schemes present a
bandwidth much wider than the centred one.
The most significant transfer functions based on
Richardson extrapolation are presented in figure 9. They
are compared to the tenth-order compact difference scheme.
The noise-minimized Richardson extrapolation filters present
a bandwidth smaller than the tenth-order compact filter. The
Richardson extrapolation low and high cutoff frequencies are
about ten times smaller than for the compact scheme. This
decrease of the low cutoff frequency can be very interesting
in the case of non-turbulent flow. In fact, the minimization
of ε described by Lourenco and Krothapalli (1995) as a
minimization of the noise leads us to move the bandwidth
1064
Figure 8. Transfer functions of compact difference derivative filters.
Figure 9. Transfer functions of Richardson extrapolation derivative
filters.
toward the lower frequencies. The consequence is not really
a minimization of noise but a strong filtering of the frequency
higher than kx = 0.175. Figure 9 also shows the least-square
transfer function which is very comparable to the fourth-order
Richardson extrapolation.
A test of noise minimization, based on compact difference
schemes, was made. From equations (5) and (6) χ was
optimized in order to minimize the noise. The obtained filter
characteristics are not very significant. The bandwidth is
only slightly increased for the lower but also for the higher
wavenumbers due to the denominator of equation (18).
In summary, table 4 presents the low and high cutoff
frequencies, respectively kcl and kch in each case.
The compact difference schemes present a very large
bandwidth, reaching 2.79 which is very close to the ideal
value of π . The low cutoff frequencies are, in that case, of
the order of 0.07. The lowest cutoff wavenumber is obtained
with the second-order Richardson extrapolation filter with a
value of 0.0075. However the high cutoff wavenumber of this
filter is only 0.175. The second-order centred difference filter
shows a cutoff wavenumber of 1.39, which is comparable to
that of PIV. Attention will be focused on this filter in the test
presented in section 4.
3.3. Power spectra of derivative
∂u
In the present section, power spectra of the derivative ∂x
are
computed from both velocity fields whose spectra are shown
in figures 2 and 3. A classical method to obtain the power
spectrum of a derivative is to take the power spectrum of the
The accuracy and frequency response of some derivative filters applied to PIV vector fields
Figure 10. Power spectra of white noise derivative with and without
windowing effects.
Table 4. Frequency characteristics of derivative filters.
noise minimization.
∗
indicates a
Scheme
Order
kcl x
kch x
Centred differences
2
4
6
8
6
8
10
6∗
8∗
2
4
6
8
2∗
4∗
6∗
2∗
2
0.0606
0.0811
0.0924
0.0996
0.0854
0.0768
0.0717
0.0733
0.0682
0.0606
0.0811
0.1007
0.1152
0.0075
0.0285
0.0363
0.0270
0.0370
1.392
1.923
2.163
2.305
2.562
2.718
2.794
2.849
2.911
1.391
1.923
2.050
2.079
0.175
0.814
1.098
0.760
0.897
Compact differences
Richardson extrapolation
Least squares
Eight-point circulation
velocity and to multiply it by k 2 in the Fourier domain. Using
the model of equation (10), a test can be made with no velocity
in such a way that the spectrum is only a white noise convoluted
by a squared sinc function. In that case, the derivative spectrum
F11 , which should be continuous with a slope of 2 for the
derivative of a white noise, gives, due to the sinc function, a
squared sine wave:
F11 = E11 /k 2 = Enoise sin2 (kx).
(19)
Figure 10 shows a comparison between the continuous
theoretical spectrum with a slope of 2 and the model of
equation (19). The noise level Enoise is taken equal to 1. The
model follows the slope of 2 up to k0 x = 0.39 as for the
velocity power spectrum (with a confidence interval of 95%).
For a reduction of −3 dB, a cutoff frequency can be found
which is exactly the same as for the velocity power spectra
kc x = 1.4. This result gives the limit of the PIV method not
only for the velocity measurement but also for the derivative
computation .
Figure 11 compares the power spectra of derivatives
computed from the most significant schemes detailed in
Figure 11. Power spectra of derivative of velocity along x, 24 × 24
interrogation windows.
section 3, applied to the data of figure 2 obtained from
24 × 24 interrogation windows. The spectra deduced from
the PIV and HWA (obtained by multiplying E11 by k 2 ) are
also given in this figure. The schemes compared are the
tenth- and sixth-order compact differences, the second-order
centred differences, the least-square schemes and the secondand fourth-order Richardson extrapolation schemes. The last
two are used with a noise minimization. As can be seen,
above k1 = 500 rad m−1 , the direct computation of the PIV
spectrum shows a very strong amplification of the noise, which
leads to an unrealistic evolution of the spectrum close to the
cutoff wavenumber. The compact difference scheme spectra
are close to the direct computation due to the fact that they
present a very weak filtering. The best filters seems to be
the fourth-order Richardson extrapolation and the least-square
filters whose spectra are superimposed in this figure. The
second-order Richardson extrapolation shows a very strong
filtering effect which leads the spectrum to leave the HWA
one at a wavenumber of about 100 rad m−1 . The value of
k1 x is of the order of 0.31, which should be the best cutoff
frequency. This value is between the cutoff wavenumbers of
the fourth-order Richardson extrapolation (0.81, see table 4)
and of the second-order Richardson extrapolation (0.175 in
table 4). Concerning the low wavenumbers, the value of
kmin x is here of the order of 0.05. Table 4 indicates that the
Richardson extrapolation filters are the best suited to resolve
such low wavenumbers.
Figure 12 presents the same comparison as in figure 11 but
with the optimized interrogation windows of 44×44 pixels. In
this figure, the derivative power spectra deduced from the PIV
velocity spectrum are close to HWA up to k = 700 rad m−1 .
The compact difference schemes present the same behaviour.
The fourth-order Richardson extrapolation and the least-square
filter are always superimposed and show a strong filtering
effect. The best filter is the second-order centred difference
scheme which presents the same cutoff wavenumber as PIV.
The low cutoff wavenumber of this filter is of the order of 0.06
which is not very far from the value of kmin x = 0.05. A
derivative computation by means of the sixth-order compact
difference scheme was also tested on the velocity data filtered
at the PIV cutoff wavenumber. The filter used was a cutoff one.
The result, also plotted in figure 12, shows that this approach
does not improve the derivative computation.
1065
J M Foucaut and M Stanislas
Figure 12. Power spectra of derivative of velocity along x, 44 × 44
interrogation windows.
Finally, the derivation in physical space is equivalent to
multiplying the velocity spectra by k 2 in the Fourier space.
If the velocity field is noisy (the noise appearing mostly in
the high frequencies), this noise is strongly amplified. The
best filter is the one with a cutoff frequency located where
the noise becomes predominant. In that case, the slope of
the filter cancels the noise amplification. It is clear that, for
the derivative computation, the best way is to use a secondorder centred difference scheme on a velocity field computed
from an optimized window size analysis. If the optimization
is not possible, it is necessary to look for the higher significant
wavenumber and to choose the filter which present a high cutoff
as close as possible to this wavenumber.
3.4. Instantaneous fields
From the experiments discussed in section 3.1, a region has
been extracted and is shown in figures 13(a) and (b) analysed
respectively with 24 × 24 and 44 × 44 interrogation windows.
It is clear that in figure 13(a) the spatial resolution is better
and the noise does not appear visibly. However, if the vorticity
is computed the noise is amplified and becomes apparent. To
compute the vorticity, a second-order centred difference filter
was applied to the velocity fields of figures 13(a) and (b). The
results are presented respectively in figures 13(c) and (d). As
can be seen, the vorticity of figure 13(c) is very affected by
the noise. The result of figure 13(d) is smoother and the noise
seems less perceptible. Moreover, the cutoff wavenumber of
the second-order centred difference, in the case of a 24 × 24
interrogation window size, is 1100 rad m−1 . This value is
twice the wavenumber of about 500 rad m−1 observed in
figure 2. This explains the high level of noise observed in
figure 13(c). Figure 13(e) presents the vorticity computed
from the field of figure 13(a) with a fourth-order Richardson
extrapolation scheme. The filtering effect of this scheme
allows us to reduce the noise, according to the observation
deduced from figure 11. The cutoff wavenumber of the fourthorder Richardson extrapolation is of 650 rad m−1 with the same
window size. The second-order centred difference applied to a
44 × 44 window size has a cutoff wavenumber of 610 rad m−1 .
Finally the useful bandwidth is not increased by the use of
a 24 × 24 window size as far as the vorticity is concerned.
Figure 13(f) corresponds to the vorticity computed with a sixthorder compact on filtered data from figure 13(b). The filter is
1066
a cutoff with the same cutoff frequency as PIV (presented in
figure 12). The result of figure 13(f) is comparable to the
one of figure 13(d). Only the vorticity levels are slightly
different, probably due to the shape of the spectra close to
the cutoff frequency (figure 12) and to the difference of slope
of the filter. The second-order centred filter applied on the
field with an optimized interrogation window size seems to
give the best results on the instantaneous fields and on the
spectra. Theoretically, the spatial resolution being optimized,
the accuracy on the vorticity is then the best obtainable.
If the results shown in figure 13 allow us clearly to
characterize the effect of the noise and spatial resolution on
the global shape of the vorticity field, they do not allow any
conclusions about the absolute value of the vorticity itself. This
would be possible with a known velocity field, for example
using synthetic images.
4. Vortex structure
Beside turbulent flows, where vorticity is a parameter of prime
interest, PIV is often used to study isolated vortices such as
those encountered in the wakes of airplanes or helicopter rotors,
for example. In that case, vorticity, which is characteristic of
the strength of the vortical structure, is also a relevant parameter
to be extracted from the instantaneous velocity maps. This is
why the filters presented in the previous sections were also
applied to a PIV image provided by DLR Göttingen. This
image shows a single vortex recorded in the wake of an airplane
in the DNW wind tunnel. This flow field is close to that of an
Oseen vortex. In the present contribution, a method is proposed
to optimize the choice of the filter and different schemes are
tested.
4.1. Best choice of a derivative filter for an Oseen vortex
As was shown in the previous section, the optimal choice of a
derivative filter in a turbulent flow is linked to the frequency
response of the PIV record. In the case of an Oseen vortex, the
choice should depend a priori on the vortex characteristics, the
spatial resolution and the measurement noise. The proposed
method of optimization is based on the fact that the vortex is
similar to an Oseen one. The limited expansion of the Oseen
vortex tangential velocity is given by
b2
u∗
−b0 R 2
) = u∗ b0 R − 0 R 3 + · · ·
uθ = (1 − e
R
2
n
b
− (−1)n 0 R 2n−1 + · · ·
(20)
n!
where R = RrO is the non-dimensional radial coordinate. RO
is the Oseen vortex radius taken at the maximum tangential
velocity uθmax . The velocity u∗ is given by u∗ = 2πRO ,
where is the circulation of the vortex. In the case of an
Oseen vortex, the parameter b0 is equal to 1.256. In fact, the
velocity u∗ can be well estimated by u∗ = 1.4uθmax . Given
equation (20), if the vortex is exactly centred on the mesh, the
vorticity at the centre is exactly ω0 = uR∗ Ob0 . Using equations (1)
and (20), an estimation of the derivatives by a differential
method can be obtained. For example, with the fourth-order
The accuracy and frequency response of some derivative filters applied to PIV vector fields
e)
f)
Figure 13. Velocity field: (a) and (b) analysis by 24 × 24 and 44 × 44 interrogation windows. Vorticity field: (c) and (d) computed by
second-order centred difference schemes with, respectively, 24 × 24 and 44 × 44 interrogation windows, (e) computed from fourth-order
Richardson extrapolation with 24 × 24 interrogation windows and (f) sixth-order compact difference on filtered data from 44 × 44
interrogation window analysis.
1067
J M Foucaut and M Stanislas
10
∆ω o /ω o
1
0.1
2nd order Richardson
4th order Richardson
Least squares
2nd order centered dif.
6th order compact Dif.
10th order compact Dif.
8 points circulation
0.01
0
5
6
10
15
20
25
30
35
40
Ro/∆x
Figure 14. Error estimation on vorticity of an Oseen vortex as a
function of the radius for different derivative filters.
centred difference (equation (11)) this estimation is



∂uθ 
=
u

∗
+
∂x j


(−2x )−8(−x )+8(x )−(2x ) b0
RO 12x
1!
2
3
3
3
3

x ) +8(x ) −(2x ) b0

+ (−2x ) −8(−
3
2!
RO
12x

5
5
5
5 b3
.
(−2x ) −8(−x ) +8(x ) −(2x ) 0

+
·
·
·
5
3!
RO 12x

44x b03
u∗ b0
= RO + 0 − R5 3! + · · ·
O
(21)
The error on the vorticity value at the centre of the vortex can
then be estimated as
n/2
ε σu R O
x n b0
ω0 − ω c
ω0
+√
=
= αn+1
RO (n/2 + 1)!
ω0
ω0
2 u∗ b0 x
(22)
where ωc is the vorticity computed with a differential estimator.
The parameter αn+1 , ε and σu are the same as in equation (1).
The parameter n corresponds to the order of the filter used.
Only the most significant term of the truncation is kept.
In the case of the vortex studied in the present paper,
the value of u∗ is of the order of 13. The noise level in
the centre of the vortex is high due to a lack of seeding in
the PIV images. It is estimated to be of the order
of 0.5
pixel. Figure 14 shows ω0 /ω0 = f RO /x for some
of the derivative filters presented in section 2. For each
filter, this curve present a minimum. If RO /x is small,
the truncation error, given by the first term on the righthand side of equation (22), is preponderant. In that case the
compact difference scheme is the best. With this noise level
the minimum value to compute vorticity to a good accuracy
is RO /x = 1.5. If RO /x is large the noise error is then
preponderant and the second-order Richardson extrapolation
is the best filter. For the considered vortex the value of RO /x
is about 6, which shows that the best filters are the fourth-order
Richardson extrapolation and the least-square scheme. These
schemes present errors of the order of 9%. The noise level
can be the most difficult parameter to estimate. Figure 15
shows the influence of that level on equation (22) in the case
of a fourth-order Richardson extrapolation and a tenth-order
compact scheme. The comparison is done for noise levels
of 0.5, 0.8 and 1.2. As shown in this figure, the choice of the
filter is not very sensitive to the noise level.
1068
Figure 15. Same as figure 14, influence of the noise level.
4.2. Instantaneous fields
In order to validate the previous optimal choice of derivative
filters, different schemes were tested on the field containing
the vortex provided by DLR (figure 16(a)). The velocity field
is computed with a multi-pass algorithm with a three-point
Gaussian peak fitting. A fit of a viscous vortex using the leastsquare method allows us to obtain the value of the maximum
of vorticity. This value, which is of the order of 0.168 with
a regression coefficient R 2 = 0.986, gives a good idea of the
expected result. Figures 16(b) and (c) present the vorticity field
computed respectively from the second-order and fourth-order
Richardson extrapolation, 16(d) the least-square, 16(e) the
second-order centred difference, 16(f) the sixth-order compact
difference schemes, 16(g) the same scheme applied on filtered
data (the filter was a cutoff at the same frequency as PIV)
and 16(h) the eight-point circulation. In figure 16(b) the peak
of vorticity is much wider than in the other figures. The
maximum values of the vorticity are given in table 5 for each
filter tested. These values go from 0.069 for the Richardson
extrapolation scheme, which presents a strong filtering effect,
to 0.192 for the sixth-order compact difference scheme which
present a large bandwidth. The values of vorticity closer to
the expected one are given by the fourth-order Richardson
extrapolation and by the least-square filters and are of 0.164
and 0.159. These values correspond to errors of the order of
4%, which is smaller than the estimation given in figure 14.
This is probably due to the fact that the model is based on a
vortex centred on the computation mesh, which is the less
favourable case. As can be seen in figures 16(c) and (d)
the vorticity given by these filters seems not too affected by
the noise. The results given by the filters which present a
larger cutoff wavenumber are more noisy and overestimate
the vorticity peak. The results obtained with the secondorder centred difference and the eight-point circulation are in
agreement with the Luff et al (1999) comparison. In their
paper, the error reduces by a few per cent when going from the
second-order centred difference to the eight-point circulation
scheme. In the present case, the error on the maximum vorticity
decreases sensibly from 6 to 5%. This error is located in
the centre of the vortex and corresponds to the maximum
error on the vorticity field. The average error on the field
is probably smaller than 2%. Looking back at figure 9 allows
The accuracy and frequency response of some derivative filters applied to PIV vector fields
200
200
10 pix
0.0 25
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g)
-200
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x (pix)
100
-200
200
h)
-200
-100
0
100
200
x (pix)
Figure 16. (a) Velocity field vortex 32 × 32 interrogation windows, vorticity field (b), (c), (d), (e) and (f) computed, respectively, by the
second- and fourth-order Richardson extrapolation, second-order least-square, second-order centred difference, sixth-order compact
difference schemes, (g) sixth-order compact difference from filtered data and eight-point circulation.
1069
J M Foucaut and M Stanislas
Table 5. Maximum of the vorticity deduced from different filters.
Scheme
Vorticity
Second-order Richardson extrapolation
Fourth-order Richardson extrapolation
Least-square
Second-order centred differences
Sixth-order compact differences
Sixth-order compact difference filtered data
Eight-point circulation
0.067
0.164
0.159
0.178
0.192
0.187
0.177
us to conclude that the significant wavenumber of the vortex is
probably of the order of kc x = 0.8, which corresponds to the
cutoff frequency of the optimal derivative filter. A least-square
fit of the velocity field has also been tried in order to compute
the vorticity from an analytic derivative. The fit was made
on 9, 13 and 21 points with a bi-quadratic polynome following
the method of Fouras and Soria (1998). The obtained vorticity
values are, respectively, 0.187, 0.171 and 0.157. The best
result is obtained for a fit with 13 points in that case. It is not
possible to generalize this approach. The method proposed by
Abrahamson and Lonnes (1995), which consists of fitting with
a model representative of the flow, has also been tested and
gives exactly the same result as the one from Fouras and Soria.
The proposed optimization allows us to select the best
filter for a vortex which is close to an Oseen one. The method
is based on a limited expansion of the analytic expression of
the velocity field. It is possible to use the same approach with
another expression for the vortex flow field or for another kind
of flow for which an analytical model exists.
5. Conclusions
In the present study, a characterization of different derivative
schemes with the aim of computing the vorticity component
normal to the plane was performed. These schemes are already
used in the post-processing of LES or DNS computation. They
have to be adapted to the case of PIV, due to the limited
spatial resolution and the measurement noise. The present
characterization is based on two different applications: a
turbulent flow and a single vortex.
In the case of a turbulent flow, the characterization is
based on the transfer function of each derivative scheme.
Ideally, these schemes behave like low pass filters. The cutoff
frequencies of these filters depend on the order and thus on
the truncation error. With measurement noise, a statistical
quadratic approach shows that the noise effect makes them
behave as band pass filters, the low cutoff frequency of which
increases with the noise level and the noise amplification factor.
This low cutoff frequency increases also when the turbulence
intensity decreases.
Several schemes have been characterized. The transfer
functions of all of them were determined. A spectral analysis
shows that a second-order centred difference scheme is enough
to obtain a good accuracy of the derivative. This result is
linked to the fact that PIV presents a cutoff frequency, due
to the windowing effect, which is the same as the secondorder centred difference scheme. An analysis of the derivative
spectrum shows that the first derivative of the velocity field
(and thus the vorticity) have the same cutoff frequency as the
1070
PIV velocity itself. It is not possible to obtain a derivative
response with a good accuracy above this frequency. Another
problem is the measurement noise. A derivative computation
in the spectral domain is equivalent to a multiplication by k 2 .
If the high frequency signal is noisy, the derivative effect is to
drastically amplify the noise. A solution is then to filter the high
frequencies before computing derivatives or to use a derivative
scheme which presents the same high cutoff frequency as the
PIV. If the interrogation window size is optimized following the
method cited in section 3.1, the best filter is the second-order
centred difference scheme. In that case the frequency response
is optimized but the uncertainty on the effective vorticity level
is still to be quantified. The best way is probably to make
use of synthetic images computed from DNS (see Lecordier
et al (2001)). In this way the exact vorticity will be known a
priori and this will allow us to validate definitely the derivative
scheme choice.
The case of a single vortex has also been studied. In that
case, a new method of optimization was proposed based on the
limited expansion of an Oseen vortex. This method allows us
to choose the filter which gives the best accuracy. The error for
each filter can be estimated and is slightly overestimated. It
is influenced by the local noise which remains a parameter
difficult to determine. The results obtained on the studied
vortex are in agreement with the ones of Luff et al (1999).
The optimization method can be applied to other kinds of flow
if an analytic model is available. If no model or spectrum are
available, the second-order centred difference scheme should
be used as its transfer function is comparable to the PIV one.
The method of the χ 2 fit proposed by Fouras and Soria
(1998) seems very interesting because the filtering effect of this
fit limits the noise influence. To obtain the transfer function of
this method presents real difficulties. The fit is equivalent to
a low pass filter whose cutoff frequency depends on the order
of the polynome used and the noise attenuation depends on
the number of points used for the fit. Fouras and Soria used
a bi-quadratic polynome (six parameters to optimize) fitted
on 9, 13 or 21 points. This method presents a cost in term of
computation time much more important than the differential
one. The improvement is not evident.
A recent paper by Ruan et al (2001) proposes a method
to obtain directly the vorticity from the PIV images. They
use a method of matched patterns to determine the particle
rotation between both exposures. As a perspective, it could be
interesting to compare this method to the present one.
Acknowledgments
This work has been performed under the EUROPIV2 project.
EUROPIV2 (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 and the universities of Delft,
Madrid, Oldenburg, Rome, Rouen (CORIA URA CNRS 230),
St Etienne (TSI URA CNRS 842) and Zaragoza.
The project is managed by LML URA CNRS 1441 and
is funded by the European Union within the 5th framework
(Contract no G4RD-CT-2000-00190).
The accuracy and frequency response of some derivative filters applied to PIV vector fields
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