RESEARCH ARTICLE
A simplified, flow-based calibration method for stereoscopic PIV
Silvano Grizzi • Francisco Pereira •
Fabio Di Felice
Received: 14 November 2008 / Revised: 5 June 2009 / Accepted: 15 September 2009 / Published online: 7 October 2009
� Springer-Verlag 2009
Abstract An improvement of the process of stereo-PIV
calibration is presented. The central feature of the proposed
technique is that the calibration of the stereoscopic system
is based upon the measurement of a calibrated flow. This is
achieved through an initial two-dimensional calibration of
the measurement plane using a single target point, followed
by a perspective and laser sheet thickness optimization that
makes use of the measurement of a known reference flow,
e.g., a uniform flow. This technique results in planar
domain, three-component (2D–3C) measurements with a
simpler calibration phase, which delivers uncompromised
accuracy, compensates for the mechanical misalignment
and eliminates the errors deriving from the classical target
plate dot identification. The technique is particularly well
suited for towing tanks and other large facilities, and is
applied to the vortex system shed from an underwater
bridge sail.
1 Introduction
Stereo particle image velocimetry (stereo-PIV or SPIV)
extends the capabilities of the original planar, two-com-
ponent implementation of PIV (see e.g., Arroyo and
Greated 1991; Willert and Gharib 1991; Prasad 2000).
Using principles of stereoscopic vision, SPIV reconstructs
the three-component velocity field in a plane using the
information from two apparent velocity fields captured
from two distinct viewpoints. Through equations calibrated
to account for the geometrical and optical characteristics of
the stereoscopic system, this reconstruction process links
the image information to the physical space.
Different calibration techniques have been devised to
operate this transformation. Geometric reconstruction
combines the measured two-dimensional velocity fields
based on the knowledge of the geometrical parameters of
the stereoscopic system (Prasad and Adrian 1993). Those
parameters can be difficult to determine precisely and can
be easily affected by measurement uncertainty, especially
for non-trivial stereoscopic configurations. Two-dimen-
sional calibration uses geometrical parameters to perform
the reconstruction, in a similar way to the geometric one.
However, it differs from this latter in that the image dis-
tortion field is derived from a calibration process in image
space rather than from the geometric parameters. Three-
dimensional (3D) calibration techniques are more com-
monly used than two-dimensional ones (Soloff et al. 1997).
The main advantage of three-dimensional calibration is that
geometric knowledge of the stereoscopic system is not
required. The optical as well as geometrical distortions are
corrected through mapping functions, which are calibrated
using a number of parallel planes near the imaging loca-
tion. The common three-dimensional calibration technique
uses a target plate, which normally consists of a regular
Cartesian grid of markers and is placed ideally in the plane
of the light sheet (Lawson and Wu 1997). Unfortunately,
misalignment errors of the calibration target with the laser
plane, as well as errors linked to the positioning of the
target at different locations along its normal axis, are ele-
ments that play an important role on the quality of the flow
computation. Willert (1997) introduced a correction
scheme, later enhanced by Wieneke (2005), which operates
a cross-correlation between stereoscopic particle images to
generate, and consequently try to minimize, the so-called
disparity maps. This additional information is used to
S. Grizzi � F. Pereira (&) � F. Di Felice
INSEAN, Italian Ship Model Basin, Rome 00128, Italy
e-mail: [email protected]
123
Exp Fluids (2010) 48:473–486
DOI 10.1007/s00348-009-0750-2
correct the misalignment errors between the target and the
laser light sheet.
The aim of the present work is to introduce a 3D-class
calibration method, short-named ‘‘flow-calibration’’, where
the mapping matrix is defined upon a drastically simplified
target-type calibration phase and upon the measurement of
a calibrated, reference flow field. Through this new strat-
egy, the method corrects in one streamlined process, for the
optical and geometrical distortions while intrinsically
eliminating target misalignments.
2 Flow-calibration
When performing a calibration designed upon the Soloff
et al. (1997)’s approach, the calibration phase is usually
accomplished by the determination of polynomial mapping
functions between the object and image planes. This type
of calibration calculates the polynomial coefficients
through least-square minimization applied to a set of three-
dimensional points given by various target positions and to
their corresponding two-dimensional images. Usually, the
calibration points are taken from a dot-type target placed at
different parallel planes around the center of the laser
sheet.
The approach proposed here establishes the correspon-
dence between the camera and object planes based on the
two-dimensional information provided by a standard yet
greatly simplified target, as in the Soloff et al. (1997)’s
approach. However, the mapping functions are here eval-
uated with the help of an a priori known uniform flow field,
i.e., W0(x, y, z, t) = constant. In fact, the knowledge of
such a flow, as recorded on both camera planes, embeds the
necessary information about magnification non-uniformity,
optical distortions as well as misalignment errors of the
system. Moreover, this calibration is not affected by target
displacement errors, since moving the target across the
light sheet is no longer required. The definition of the
mapping functions using the uniform flow input gives
the capability to reconstruct the 2D–3C flow field and to
simultaneously correct for misalignment errors.
The present technique divides the calibration procedure
into two steps, each with different accuracy orders and
requirements:
(1) The first step is the planar calibration part of the
process. The physical plane, where the measurements
are performed, is illuminated by the actual laser light
sheet and mapped on the camera image plane. The
calibration can be performed using a classical two-
dimensional dotted target or, as proposed in this work,
by placing a point-like object (for instance, a pin) in
the measurement plane and at specific points with the
help of a traversing mechanism. This latter approach
greatly simplifies the process since it actually
removes the dot target as the defining plane of the
measurement domain. The normal process requires
that one places the solid plate target in that plane by
physically aligning it with the light sheet using the
side mirrors and controlling that the incident and
reflected light sheet overlap in a satisfactorily manner.
This process is essentially subjective and depends on
the ability of the operator. Furthermore, the user is
required to perform this alignment manually, unless
costly positioning devices are available. The user is
also strongly exposed to the risk of laser radiation.
The method proposed here removes these constraints.
The operator, from a remote location and through the
live image, controls the traverse and moves the pin tip
to the center of the light sheet thickness, where light
is scattered with maximum intensity. This positioning
is typically done with an accuracy below 0.1 mm in
physical space and one pixel in image space. Note
that, with this procedure, the light sheet plane with
z = 0 is literally and accurately pinpointed, that is, no
assumption ‘‘z equal 0’’ is made as in the standard
target procedure. Although misalignment errors
remain, these are in any case smaller and remain so
in a consistent manner from calibration to calibration.
Therefore, the pin-based process requires little or
none alignment caution with respect to a standard
plate-based calibration. The only requirement is on
the correct alignment of the traverse system respec-
tive to the flow coordinate system, which is self-
evident. In fact, the traversing system is an additional
element between the PIV system to be calibrated and
the flow geometry. Therefore, one must take the
necessary caution to make sure that this element is as
much as possible part of this flow geometry.
(2) The second step is needed to acquire information
about the out-of-plane displacement and is related to
the local magnification factor and local distortion. In
this phase, a high accuracy is required and is here
obtained using the particle displacement generated
by a uniform flow. The particle displacement is here
determined at sub-pixel level from the correlation
map. The spatial accuracy and spatial resolution are
therefore far higher than in the case where a traverse
is used to simulate the out-of-plane displacement. The
only strict requirement is on the quality of the
uniform flow, which needs to be accurately generated.
Therefore, the new technique is particularly suited for
towing tanks, where it is trivial to create such a
uniform flow, by simply towing a stereoscopic-PIV
probe in a virtually stagnant flow and at a precise
translation speed. Assuming that the plane z = 0 is
474 Exp Fluids (2010) 48:473–486
123
the center plane of the laser sheet, the reference
acquisition is made at a constant speed W0 with a
given time delay Dt between two consecutive frames,
such that the measurement domain defined by the
laser volume lies between z ¼ �W0 � Dt=2 and
z ¼ þW0 � Dt=2. Changing the speed W0 or the time
delay Dt allows one to refine the information inside
the measurement domain. To provide a complete
description of the laser sheet thickness, Dt can be
varied in small steps as long as the correlation is
valid. Equivalently, the velocity W0 can be varied to
achieve the same result.
Following the Soloff et al. (1997)’s approach, the 2D–
3C field reconstruction is based on a first order approxi-
mation of the displacement:
DX � rFðxÞ � Dx ð1Þ
where x is the particle three-dimensional vector with
coordinates (x, y, z), Dx is the displacement vector of the
particle and DX the displacement vector of its image. For a
given camera, rF(x) is the gradient matrix defined by
ðrFÞi;j ¼oFi
oxj¼ Fi;j
where Fi;j is the so-called gradient tensor of the mapping
function F, with i = 1,2 being the image coordinate index,
and j = 1,2,3 being the space coordinate index. For a
stereoscopic system using two cameras, the complete
mapping system is described by
DXð1Þ1
DXð1Þ2
DXð2Þ1
DXð2Þ2
0BBB@
1CCCA ¼
Fð1Þ1;1 F
ð1Þ1;2 F
ð1Þ1;3
Fð1Þ2;1 F
ð1Þ2;2 F
ð1Þ2;3
Fð2Þ1;1 F
ð2Þ1;2 F
ð2Þ1;3
Fð2Þ2;1 F
ð2Þ2;2 F
ð2Þ2;3
0BBBB@
1CCCCA
Dx1
Dx2
Dx3
0@
1A ð2Þ
rF(x) is also referred to as the magnification matrix. The
equations of the system above are in practice independent
of each other, since errors are always present in real
systems. In this case, the system is over-determined, for
which a least-square minimization is the recommended
approach to solve the system. The mapping functions F
that link the physical space to the image space are chosen,
because of their ease of implementation, to be of the
polynomial type and can be expressed in the general form:
X ¼ FðxÞ ¼Xlþmþn
k¼0
akxlymzn ð3Þ
where the real space object coordinates (x, y, z) of x and
the image coordinates (X, Y) of X are known input data,
ak are the lþ mþ n vectors of the polynomial coeffi-
cients and l, m and n are the polynomial exponents.
Hence, ak is a four-component vector, since there are two
mapping polynomials per camera, one for each image
coordinate.
For our flow-calibration procedure, a quadratic form on
x and y and linear on z is chosen for the function F.
However, it must be understood that there is no limitation
in extending the technique to higher orders, with a mapping
function similar to that of Soloff et al. (1997)’s. In any
case, as outlined in Soloff et al. (1997), this choice is
purely subjective and is driven by the actual level of optical
distortions. The case presented here is the worst case sce-
nario to test the new approach, since it is built on the
minimal set of input information, as explained hereafter.
Besides, it will allow us to perform a comparison with the
calibration based on the dual-plane target, which is by
definition linear on z. The mapping function is thus
expressed as
FðxÞ ¼ a0 þ a1xþ a2yþ a3zþ a4x2 þ a5y2 þ a6xy
þ a7xzþ a8yz ð4Þ
We assume z = 0 on the reference plane, which in practice
corresponds to the center plane of the laser light sheet.
With this trivial condition, Eq. 4 provides a relation where
six coefficient vectors remain: a0, a1, a2, a4, a5 and a6,
which can be evaluated using six physical points in space.
A dotted calibration target plate can be used; however, the
simplified procedure proposed here makes use of a single
target point, specifically a pin, conveniently moved to six
different points of the measurement plane. As noted before,
this relieves the user of the effort normally required to align
accurately a standard target plate.
The second stage of the calibration procedure requires a
constant displacement in physical space, which is in our
case generated by a uniform flow. A coordinate change is
necessary to introduce the velocity information from the
constant field: x ¼ x0 þ Dx with Dx ¼ U0 � Dt, where x0
is the point coordinate on the reference plane and
D0ðDx0;Dy0;Dz0Þ is the displacement vector of the uni-
form field. The displacement can be expressed as a gradi-
ent, as per Eq. 1:
DX ¼ ða1 þ 2a4xþ a6yþ a7zÞDx0 þ ða2 þ 2a5yþ a6x
þ a8zÞDy0 þ ða3 þ a7xþ a8yÞDz0 ð5Þ
DX is a four-component vector, as per Eq. 2. Equation 5 is
written for a generic uniform flow. If the flow is coplanar
with the light sheet, the procedure based on an actual
uniform flow cannot be applied since the system of equa-
tions cannot be solved. In fact, the Dz0 term is canceled,
and the coefficients a3, a7, a8, cannot be determined.
However, this difficulty can be easily overcome using the
traversing system to move the laser plane along the cross-
flow axis with the fluid at rest, thus creating an artificial
flow along that axis. In that particular situation or if the
Exp Fluids (2010) 48:473–486 475
123
actual fluid flow is orthogonal to the laser light sheet, Dx0
and Dy0 are equal to 0 and Dz0 ¼ cst, in which case only
three coefficients are to be determined. Those two cases are
the preferred practical situations in towing tank applica-
tions, since they are the most common.
The first derivative of Eq. 4 with respect to the z-coor-
dinate thus expresses simply as
DX ¼ ða3 þ a7xþ a8yÞDz0 ð6Þ
There are two such equations per camera, one for each
image coordinate. Equations 4 and 6 are sufficient to fully
determine the whole set of mapping functions, hence the
magnification matrix rF(x). The 2D–3C field reconstruc-
tion can then be fulfilled by inverting Eq. 2, or more
robustly as suggested by Soloff et al. (1997), by performing
a least-square minimization.
In general, a stereoscopic PIV calibration target is a
precisely machined plate with a Cartesian grid of dots
printed on it, uniformly distributed along two normal axes
with a predefined spacing, and with some sort of distinctive
mark (also known as the fiducial mark) that defines a ref-
erence physical point. The target is attached to a traversing
system, if multiplane calibration is to be performed, and is
subsequently placed in the measurement plane illuminated
by the laser light sheet. The accuracy with which this
positioning is performed relies mostly on the user’s judg-
ment and on some basic control elements such as the light
reflected from a mirror, thus introducing a rotation/trans-
lation bias that is generally corrected a posteriori using a
variety of methods (e.g., Wieneke 2005). Once in place, the
target plate is moved accurately by its supporting traversing
mechanism to the various calibration planes. Although this
general procedure is the standard approach in most stereo-
scopic applications, it is known to be of low practicality in
some instances where the measurement area is of difficult
reach or when it is away from the control station. Di Felice
and Pereira (2008) report the cases of PIV operation in a
towing tank and in a large cavitation tunnel, both illustrat-
ing the difficulties of calibrating a stereoscopic PIV system
in environments where invasive structural elements or
extended distances can easily impair the calibration process.
To overcome these difficulties, the calibration procedure
described here makes use of a single target point that is
imaged at the required six locations needed to perform the
first stage of the flow-calibration process. The single target
point is in our case a simple pin, which is placed across the
light sheet, such that the light reflection over the pin tip is
imaged and recorded. The operation is performed from the
control station and uses the direct imaging of the pin to
achieve precise adjustment. For small-scale experiments,
the PIV system (laser light sheet comprised) is fixed
whereas the pin can be positioned at six locations, using a
dedicated traversing system.
However, the preferred approach for larger scale situ-
ations consists in having one or more pins fixed at known
locations in the physical coordinate system and in trans-
lating the PIV system, and the laser light sheet altogether,
to the measurement planes by relative displacement from
the pins’ location. In this family of applications, the tar-
geted measurement area is generally scaled accordingly,
and spatial resolution can become a limiting issue. To
guarantee a high spatial resolution in these situations, the
only practical approach consists in measuring smaller
areas, translating the measurement system to adequate
positions and finally patching together the individual areas
to build up the final larger measurement domain. To that
purpose, the PIV probe is mounted on its own traversing
system and is translated in steps such that each patch
overlaps the adjacent ones to some amount. Di Felice and
Pereira (2008) provide a case of a stereoscopic PIV
experiment performed in a large towing tank where an
area of about one square meter is built out of 18 smaller
areas patched together with an overlap of 25%. Thereby,
the spatial resolution could be kept as low as 2.5 mm, for
a final number of velocity vectors in the 105 range. This
sort of resolution would not have been possible if the
whole area had been imaged at once, except with
exceptionally high resolution sensors that are not yet
commonly available. A patching technique becomes very
effective if the reference initial patch is precisely located
with respect to the experiment geometry. This can be
easily achieved with one or more pin-type markers
attached to that geometry. From the initial patch location,
a simple in-plane translation is sufficient to move to the
next patch position, and so forth until the measurement
plane is fully scanned.
In the following sections, the results obtained with the
new calibration procedure are compared to those obtained
with the standard three-dimensional calibration based on
target plate. A practical application on an underwater
vehicle (UWV) is also presented.
3 Experimental setup
The stereoscopic PIV system used here is the underwater
probe described by Pereira et al. (2003). It is equipped with
two 2,048 9 2,048 pixels CCD cameras and a 200 mJ
Nd:YAG laser. The system is highly modular and can be
assembled into different configurations to accommodate
the experimental constraints. Figure 1 shows the module
system and the configuration chosen for our experiment.
Hollow glass spheres with a mean diameter of 30 lm are
used to seed the flow. The targeted investigation area is
about one square meter, composed of a series of patches
with a dimension of 250 9 300 mm.
476 Exp Fluids (2010) 48:473–486
123
The flow-calibration uses mapping functions that are
quadratic for the in-plane components, and linear along z,
as outlined in the previous section, see Eq. 4. For com-
parison, a standard 3D calibration is performed using a
500 9 500 mm2 single-plane calibration target plate
placed at two parallel positions along the flow axis
(z = 0 mm and z = 2.5 mm). This case is equivalent to a
calibration using a two-plane target plate, which is also
commonly used in most stereoscopic PIV applications. A
second instance of this standard 3D calibration is also
performed using the same single-plane target plate, but
placed at thirteen parallel positions along the z-axis as for a
multiplane analysis, with a spacing of 2 mm. In both cases,
the mapping polynomial is expressed with a quadratic
dependency on x and y and is linear with z, in order to be
comparable with the flow-calibration. This polynomial will
hereafter be referred to as ‘‘221-polynomial’’.
Figure 2 describes the experimental setup, the coordi-
nate system and photographic overviews of the setup
components. The stereo-PIV probe is supported by its
own three-axis translation traverse, which sits on the rails
of the tank carriage. The necessary caution is put in order
to insure that the translation system is itself perfectly
aligned with the tank carriage, and consequently with the
flow coordinate system. For the target calibration, the
plate is placed at a depth of 160 cm in water and requires
a diver to have it fixed in place under the towing tank
carriage.
To generate the uniform flow required to perform the
second stage of the flow-calibration, the stereo-PIV probe
is simply towed through the water tank by the towing
carriage. The speed W0 of the carriage, and therefore of the
probe itself, is computer-controlled and maintained within
0.1% of the nominal value. Note that the towing tank is a
very large pool of still water, with a width of 13.5 m, a
length of 470 m and a depth of 6.5 m, see Fig. 2a. After
each tow, a rest period of about 20 min is usually allowed
so that the large scale convection flow of the tank, if any, is
dampened and the flow turbulence levels become negligi-
ble. Flow dampeners are also available at the bottom and
sidewalls of the tank to accelerate this process. The
undisturbed flow is obviously measured without the
Fig. 1 Underwater stereoscopic
PIV probe: a modules and
b experimental configuration
Exp Fluids (2010) 48:473–486 477
123
underwater vehicle. A photograph of a calibration/align-
ment pin is shown in Fig. 3.
The performance of the calibration procedure proposed
here is evaluated against the traditional approach. The
choice of the number and position of the calibration points
is first discussed. In a second time, we compare the per-
formance of the proposed method versus the standard 3D
calibration approach.
Fig. 2 Experimental setup:
a schematics of the towing tank;
b the setup with the coordinate
system and corresponding
velocity components; c view of
the probe on its traversing
system; d laser light sheet in
operation in wake of UWV
478 Exp Fluids (2010) 48:473–486
123
4 Results and discussion
Flow-calibration is performed through the evaluation of
140 velocity fields retrieved from the uniform flow
measurement. The non-dimensional velocity mean fields
(U/W0, V/W0, W/W0) and the normalized RMS fields are
calculated.
The influence of the spatial distribution of the calibra-
tion points on the velocity estimation is investigated, see
Figs. 4 and 5. Six different distributions are considered:
full-target plate with more than 185 dots, nine points
placed uniformly along the two axes of the measurement
plane, seven points placed on a I-type and a H-type patterns
and finally the minimum set of six points required to
determine the mapping coefficients. In the latter case, two
situations are considered: six points placed on the borders
of the field, and six points placed on a cruciform configu-
ration, with one point located in a corner. Note that the full-
target plate analysis is based on dot images, whereas the
other analyses use the pin image.
Figure 4 shows the U/W0 velocity distribution. A
maximum deviation of about 3% is observed in all cases,
the expected value being 0 since U is the cross-flow
component. The W/W0 velocity field is shown in Fig. 5,
and the expected value is unity. The choice of the spatial
distribution of the target points has a relatively small
influence on the way the error is distributed. A full-target
plate with more than 185 dots has a seemingly more
uniform distribution of errors with respect to a calibration
based on a reduced number of points. In fact, this latter
causes a 1% increase for a total relative accuracy of 2%,
while the full-target calibration is mostly below 1%. This
is observed on all components (V is not shown). The
results are summarized in Table 1, where we report the
mean and root mean square values of the normal and
cross-flow terms, normalized by the towing velocity
nominal value W0. The cross-flow nominal velocities are
U0 = V0 = 0.
Overall, the polynomial behavior seems to favor the new
and simpler calibration strategy. This result derives from
the fact that the full-target calibration, and consequently
any traditional 3D calibration, is prone to various sources
of error, aside from the known misalignment issue. The
accuracy with which the dot centers are determined is
undermined to some extent by their manufacturing char-
acteristics, in particular their millimeter-range size, and by
the incident light distribution. Both factors introduce errors
that are then part of the polynomial optimization process
needed to solve the over-determined system of equations.
In contrast, even a minimal six-point calibration, which
uses a pin with a diameter in the submillimeter range, can
produce identical if not more accurate results than a full-
target calibration. In particular, errors on the borders and
especially in the corners are comparable or even lower with
a cruciform-type configuration, as outlined in Table 1.
Furthermore, this latter arrangement offers an exact ana-
lytical solution to the problem of six equations and six
unknowns. For instance, the system for coordinate X on a
given camera would be
X1
X2
X3
X4
X5
X6
26666664
37777775¼
1 x1 y1 x21 y2
1 x1y1
1 x2 y2 x22 y2
2 x2y2
1 x3 y3 x23 y2
3 x3y3
1 x4 y4 x24 y2
4 x4y4
1 x5 y5 x25 y2
5 x5y5
1 x6 y6 x26 y2
6 x6y6
26666664
37777775�
a0
a1
a2
a4
a5
a6
26666664
37777775
ð7Þ
The other configurations lead to over-determined
systems that require optimization algorithms to be solved
based on least-square minimization (Levenberg 1944;
Marquardt 1963). Optimization on as few as seven or
nine points are prone to convergence difficulties, and the
errors listed in Table 1 confirm this observation for the
nine-point and seven-point configurations. With 185 points,
as in the full-target configuration, the optimization process
is more robust and provides better results, though at the
same level as the flow-calibration.
The important result is that the flow input is the central
player in the flow-calibration process since it is enough,
with as few as six physical points, to reach a satisfactory
level of accuracy comparable with a full-target calibration.
More pin locations would certainly increase the accuracy,
but to a very limited extent.
It is further understood that a quadratic mapping func-
tion as in Eq. 4 may not be adequate for a system with
stronger distortions, caused either by additional optical
windows or by low quality lenses. A higher-order mapping
function is required in that case. However, if a linear
dependence is still forced along the z-axis, the procedure
proposed here can be applied as it is. A cubic mapping
function for x and y would consist of 16 coefficients, 10 of
which would be determined by the pin positions and 6 by
Fig. 3 Photograph of the alignment pin on underwater body
Exp Fluids (2010) 48:473–486 479
123
the flow information. A higher order on z would simply
require further calibration velocities W0.
The choice of the calibration points for the cruciform-
type calibration is based on geometrical considerations:
• The center point is an obvious candidate as origin of the
paraboloid defined by the function a0 þ a1xþ a2yþa4x2 þ a5y2 in Eq. 4;
• The four cardinal points placed around the center dot
are also obvious candidates to define the orientation of
the minor and major axes of the paraboloid given
earlier;
• The sixth point placed in a corner plays the role of an
anchor point that helps define the perspective effects,
which are embedded in Eq. 4 in the bilinear function
a6xy.
Based on these considerations, we proceed with the
cruciform-type pattern for the comparison work between
the new flow-calibration procedure and the standard
approaches.
Figure 6 shows the accuracy error measured on the
mean value of the out-of-plane component W normalized
by the actual free-stream velocity W0. On the same figure,
we report the statistical distribution of the mean W/W0
value for each type of calibration. All velocities are nor-
malized by the out-of-plane nominal velocity W0. The
comparison is done between the flow-calibration, the dual-
Fig. 4 Effect of number and position of calibration points on the normalized mean U-velocity field. Black dot markers represent the location of
the reference points
480 Exp Fluids (2010) 48:473–486
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plane and multiplane calibrations. Note that the ‘‘221’’
term indicates that the polynomial is quadratic in x and y,
and linear in z. The dual-plane calibration provides a higher
degree of uniformity (low 3r) but this comes with an
important bias on the mean value (about 1%). The multi-
plane calibration displays better accuracy (about 0.1%) but
Fig. 5 Effect of number and position of calibration points on the normalized mean W-velocity field. Black dot markers represent the location of
the reference points
Table 1 Effect of number of target points and arrangement on the mean and RMS values
|Wmean-W0|/W0 Wrms/W0 |Umean-U0|/W0 Urms/W0 |Vmean-V0|/W0 Vrms/W0
Full target 0.0004 0.0078 0.0008 0.0093 0.0003 0.0041
Six points 0.0011 0.0125 0.0008 0.0127 0.0015 0.0045
Seven points I 0.0011 0.0121 0.0010 0.0124 0.0003 0.0042
Seven points H 0.0007 0.0098 0.0023 0.0104 0.0007 0.0047
Nine points 0.0005 0.0094 0.0004 0.0101 0.0001 0.0044
Cruciform 0.0012 0.0081 0.0002 0.0094 0.0003 0.0041
Expected cross-flow values are: U0 = V0 = 0
Exp Fluids (2010) 48:473–486 481
123
large 3r. Instead, the flow-calibration provides high accu-
racy (\0.2%) and narrow 3r. These results extend also to
the U and V maps, not shown here.
From a practical standpoint, the flow-calibration
greatly simplifies the calibration operation. This is a key
point when the calibration needs to be done in a non-
trivial experimental environment, where the simplest
operations require significant effort on the operator side.
Moreover, alignment problems typical of a target plate
calibration are here canceled in an intrinsic manner. The
application described hereafter clearly demonstrates these
advantages.
Fig. 6 Mean out-of-plane flow field W/W0: a standard dual-plane, b multiplane calibration, c flow-calibration, d ensemble statistical
distributions; values are normalized
482 Exp Fluids (2010) 48:473–486
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5 Underwater vehicle flow
The flow-calibration is applied to the flow field in the wake
of an underwater vehicle (UWV), more specifically to the
twin-vortex system generated by the bridge sail. The sail
behaves like a wing having a large thickness and low
aspect ratio. Because the UWV is moving at 0� drift angle,
the pressure distribution created by the body geometry
generates a swirling cross-flow characterized by two vor-
tices of opposite rotation. This flow is illustrated in Fig. 7.
The W/W0 plot shows a velocity deficit in the wake of
the UWV sail, as expected. In this application, W0 is the
operating upstream velocity, and is 68% higher than the
calibration velocity.
Figure 8 shows the non-dimensional velocity W/W0
calculated using a standard three-dimensional calibration,
made following the best possible practices in the field, and
the one obtained with a six-point flow-calibration. No
alignment correction has been applied to the 3D calibra-
tion, purposely to show the difference with the flow-based
calibration on the issue of alignment compensation. Most
importantly, the alignment errors, caused by the inaccurate
positioning of the target plate on the plane of the light
sheet, are here fully accounted for by the new calibration
procedure. The bottom plot of Fig. 8 represents the dif-
ference between the two axial velocity plots. A clear linear
gradient causing a 0.6% difference between the top-right
corner and the bottom-left corner of the field of view can be
observed. Using simple trigonometric geometry, this cor-
responds to a misalignment of 2.5� between the target
plane and the light sheet along the diagonal of the mea-
surement domain. This is consistent with the misalignment
error obtained from the disparity map analysis (Wieneke
2005) applied to the target plate data, which leads to about
2�. This explicitly demonstrates that the target alignment,
done following the best possible practices in the field, is the
actual source of the error. In the case of this specific
application, this sort of error cannot be monitored nor
controlled because of the experimental environment (scale,
mechanical errors, access, etc.). The flow-calibration pro-
vides a simple means to minimize this error.
Moreover, this application is a case where the area to be
measured is very large, i.e., in the order of one square
meter or more. One needs to perform measurements on
smaller areas to reach an acceptable spatial resolution, as
outlined in a previous section. These smaller domains are
then patched together, with a predefined overlap, to form
the full vector flow field. The initial patch is simply
Fig. 7 Underwater vehicle
flow: a tip vortex couple
generated by the bridge sail,
b cross-flow velocity field,
c longitudinal velocity field;
values are normalized
Exp Fluids (2010) 48:473–486 483
123
centered on a reference pin attached to the hull, see Fig. 3,
that is the PIV probe is positioned such that the image of
the pin is centered on the image. The remaining patches of
the area of interest are then reached in physical space with
the traversing system, relative to that reference position by
known dimensional steps in all three axes.
Unfortunately, errors tend to be important on the edges
of the single measurement domain or patch, especially
when working with low f-numbers, mirrors and high per-
spective angles as in our case. During the assemblage of
the different patches, which is normally done through
averaging, these errors create visible discontinuities in the
overlap zones. Using a standard 3D calibration method,
further compensated with a misalignment correction based
on disparity maps, strong discontinuities are found as
shown in Fig. 9a. Applying the flow-based calibration with
a six-point cruciform configuration, these discontinuities in
the overlap region are neatly reduced, as shown on the
right-hand plot of Fig. 9. This demonstrates clearly that
this novel flow-based method accounts and corrects for a
wider spectrum of imaging parameters with respect to the
standard approach, thanks to the highly detailed and
accurate reference flow input.
6 Conclusion
The flow-calibration method is a simple and efficient way
to perform the calibration of stereoscopic PIV systems. It
does not require the standard target plate, thus canceling
Fig. 8 Normalized velocity
maps: a full-target calibration,
b six-point flow-calibration,
c difference between the two
maps; values are normalized
484 Exp Fluids (2010) 48:473–486
123
the need for the usually difficult operation of target
placement in, and alignment with, the plane of the laser
light sheet.
This calibration approach follows a very different
strategy with respect to the standard three-dimensional
methods. In fact, those use three-dimensional targets to
map the space around the laser plane without necessarily
increasing the accuracy on the out-of-plane mapping.
The flow-calibration implemented here uses a mono-
dimensional target, i.e., a simple pin, and achieves high
accuracy mapping functions from the reference uniform
flow, with a sub-pixel precision proper to the cross-
correlation standards. This accuracy depends upon the
magnification factor, which is normally a fixed parame-
ter, and upon the particle pixel displacements, which can
be easily adjusted by simply varying the uniform flow
velocity or the time delay in the PIV pairs. Different
pixel displacements can be used to reach a volumetric
and refined mapping of the light sheet, through high
order functions. The polynomial mapping functions, built
upon these information, allow to compensate accurately
for most of the optical distortions as well as for the
target alignment errors that would normally incur in the
standard calibration process.
The flow-based calibration is particularly suited for
towing tank applications and other large facility applica-
tions, where creating a uniform flow is generally straight-
forward. The method allows a reduction of the test setup
time. For a typical towing tank experiment on a hydrody-
namic model, we suggest the following four-step procedure
to setup the calibration and the alignment of the mea-
surement plane:
(1) Coarse alignment in air of the SPIV cameras, through
adjustment of the mirror (if any) and of the
Scheimpflug angle for each camera;
(2) Immersion of the SPIV system in water and fine
adjustment of the focus and Scheimpflug angles,
based on the direct observation of actual seeding
particles;
(3) Recording of the reference uniform flow, by towing
the carriage with the SPIV system through the tank at
a preset constant speed;
(4) Immersion of the hydrodynamic model in water and
alignment of the measurement plane with the model
coordinate system. This is done using the reference
pins mounted on the model. The six points necessary
to perform the flow-based calibration are then imaged
and recorded.
Acknowledgments The work was sponsored by the INSEAN 2007-
2009 Research Program funded by Ministero dei Trasporti and by
the European Network of Excellence ‘‘Hydrotesting Alliance (HTA)’’.
Dr G. Aloisio and Dr. M. Falchi are gratefully acknowledged for
their contribution to the experiments.
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