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: f.pereira@insean.it

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

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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

123

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

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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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