9TH INTERNATIONAL SYMPOSIUM ON PARTICLE IMAGE VELOCIMETRY – PIV’11
Kobe, Japan, July 21-23, 2011

Single-View Volumetric PIV using High-resolution Scanning andLeast Squares Matching
1

1

1

D. Hess , Ch. Brücker , J. Kitzhofer , T. Nonn
12Institute of Mechanics and Fluid Dynamics, University of Freiberg, Germany{christoph.bruecker,david.hess,jens.kitzhofer}@imfd.tu-freiberg.de
2
Dantec Dynamics A/S, Skovlunde, Denmark
thomas.nonn@dantecdynamics.com

ABSTRACTThis paper shows a measurement technique for a fully time
resolved 3D-3C velocity field using a scanning light sheet anda single high-speed camera. The acquired particle images of
one scanning cycle are stacked together to span a voxel
volume as a single camera reconstruction technique. Thus, theparticles get an ellipsoidal shape in scanning direction. The
influence of the ellipsoidal particles is discussed and thevolume is analysed by using a 3D least-squares-matching(LSM) algorithm. The measurement technique is applied on avortex ring to examine its instabilities and break up behaviour.

conditions are well known. Thus, it is possible to evaluate thedata maintained from the experiments.

2. Experimental Setup
The illumination is generated by a high-speed pulse Nd:YlfLASER (30mJ per pulse @ 1kHz) and light sheet optics. After
passing the optics, each pulse is reflected on a facet of thepolygon at different angles. This leads to an angular separationof the light sheets (Figure 1).

1. Introduction
During the end of the last decade 3D-3C measurement
techniques became more and more important. Thus, highspeed cameras offer the possibility to analyse complex threedimensional flow structures in full spatial and temporalsolution. The data generation can be performed by several
reconstruction techniques, i.e. holographic or tomographic
reconstruction methods. Those are evaluated by using threedimensional particle image velocimetry (3D PIV) [1,2] orparticle tracking velocimetry (3D PTV) [3]. In all techniques acomplex multi-camera-setup is needed to avoid ghost particles
in the particle reconstruction. Those multi camera setups
require multiple optical accesses for the cameras and complex
calibration methods. The large processing overhead in multi
camera approaches is overcome in single camera approaches.
Willert showed in 1992 a possibility to generate 3D data,using the defocusing technique based on a single camera
equipped with a three-hole aperture [4]. However, themaximum particle density is limited by overlapping particles
in the measured volume.
Different techniques with scanning light sheets alreadyshowed their potentials, for instance to enlarge the particle
density. The light sheet scans the volume and divides it into
separate parts. A single light sheet can be analysed withclassical PIV to evaluate the velocity distribution in eachplane. The result is a 3D-2C velocity field. The 3rd velocity
component can be obtained by using the continuity equation[5] or out of plane correlation [6]. High scanning frequencies
freeze the observed flow in time as Thurow et al. (2010) [7]demonstrated. He used a high-speed-LASER and camera
system to scan a jet flow within 100µs, but number of LASER
cycles is limited per run. Thus, it is not yet possible to repeat
one scan fast enough to acquire double images.
Here a camera with a telecentric lens system is used together
with a scanning light sheet to acquire the volume of interest.
In this case the observed flow is a vortex ring. The vortex ring
was chosen due to its complex 3D-flow structure, especiallyafter becoming unstable. Whereas the stationary flow

Figure 1: Overview of the experimental set up with thescanner in the upper part, the camera and the waterbasin with a visualization of the vortex ring.
A cylindrical lens parallelizes the angular separation. Because
the lens also focuses the thickness of the light sheet, a second
cylindrical lens is needed to compensate this effect. The
second lens is located between the light sheet optics and thepolygon mirror. In order to position the polygon in time to adefined angular position, the polygon needs to be driven by astepper motor. The whole process is controlled by TTL trigger
signal generated by a sequencer. Generally, there exist two
different operating modes of the polygon scanner [8]. In thiscase scans of complete volume are finished, before a newscanning cycle starts. A high-speed camera with a telecentriclens system is mounted perpendicular to light sheets. The
setup allows separating the volume of interest into maximum
200 single slices with a time between two following slices of
0.1 ms. For the smaller illuminated volume the particle density
can be increased dramatically. The frequency between twocomplete volume-scans is 40 Hz to 16 kHz, depending on thenumber of light sheets used.

vortex core. This was modeled by Feng et al. (2009) andexperimentally observed by Allen et al. (2002).

To elongate the measurement time and eliminate the travelling
speed of the vortex ring relative to the camera, the complete
water basin can be moved with the same travelling speed anddirection of the vortex ring. The vortex is generated in tube
(diameter 30 mm) with a chamfered end of 20°. A piston,driven by a linear motor, asides the water from the tube, andforms the vortex ring. Controlling the acceleration, travelling
distance and speed of the piston, the characteristics of thevortex ring can be influenced. Therefore, the Reynolds
number of the vortex ring can be varied. Table 1 shows theparameters used to generate the vortex ring. Furthermore, thesettings of the polygon scanner are given.
Parameter
Vortex Ring
DiameterTraveling speed min
Re minScannerNumber of faccets
Number of light sheets
Volume scanning frequency
Δt light sheet
Frame rate within volume
Max rpm of Polygon

Value

Unit

0.03
0.06
1800

M
m/s
/

40
100
80
0.1
10
1600

Hz
Ms
kHz
Rpm

Table 1: Important experimental settings used forscanner, camera, vortex ring and LASER
The results of the experiments are up to 140 time steps of 100
images, one image for each scanned plane, with a size of
752_x_752 pix². The pixels in each plane scale 0.0834
mm/pix, because of the telecentric imaging technique. The
images from each time step can be stacked easily together tospan a voxel volume. Without the use of the telecentric lenssystem, the different scale factors of the images must be
normalized.

Figure 2: Visualization of the vortex ring withstRhodamine at a stable and an unstable stage (1 row)
the red lines indicate the plane for the reconstructed top
ndviews (2 row).

4. Data evaluation
The 3D voxel representation of the particle distribution in thescanned volume can be reconstructed from the imaged planes,
if the light sheets are overlapping. A typical shift of the planes
is 0.53 mm. With a light sheet thickness of about 2.4 mm thedepth of the reconstructed volume is about 55 mm. Figure 3
illustrates the overlapping light sheet in one image. Moreover,each 2nd plane is shifted vertically about 100 pixels. In order toreceive a better visibility and responded to the huge overlap,
every 1st plane horizontally of about 10 pixels, due.

3. Visualization and scanner evaluation
Visualizations of the vortex ring have been done tocharacterize the vortex generation and the cmera settings.
Because the visualization was not for quantitative analysis,
200 light sheets were used for one complete scan.
For the visualizations Rhodamine was used as contrast fluid.
Figure 2 shows the result of the visualizations. The vortex ringis displayed at two different time steps. The left part shows astable state and the right part an unstable state. The first row
shows an image of the measured planes, whereas the second
row displays a reconstructed top view of the images. We must
admit that the local length of the 2nd cylindrical lens of thescanner is not big enough. Thus, it was not possible to observe
the complete vortex ring, which has cut at one side. The
reconstructed view can be achieved by commercial software.
For the purpose of our studies ImageVis3D, a program for Xray computed tomography images was used. The program uses
the different slices to put the complete vortex ring together
and computes different perspectives. In the stable state of thevortex ring, the top view shows concentric circles, as expectedfrom a laminar ring vortex. Thus, in later stages the circles
become irregular and start to form azimuthal waves around the

Figure 3: Arrangement of the light sheets. The images
are overlaid in a staggered arrangement to illustrate theoverlapping planes.
The picture is set together from the light sheet calibrations
pictures, where its positions in the observed volume aredetermined. The arranged picture is necessary to evaluate if
the scanner and the triggering of the cameras is working
properly. Incase it is not working, the distance between twolight sheets would alter.
The particles in one plane of an image have a Gaussian
distribution, for the stacked images this also needed indirection of the reconstructed depth. Hence, a small shift ofthe measured planes is necessary and the planes should
overleap. If one particle, in a sequence of light sheets, isstaring to be illuminated, the grey value intensity of theparticle is slender, because only parts of the particle are within
the light sheet (Figure 4, see Plane 1). In the following Planes
the grey value enlarges, because more parts of the particle arelighted (Figure 4, see Plane n). When the particle moves out of
the light shees, the grey values decreases, due to theincomplete illumination (Figure 4, see last plane).

part of the voxel representations of the particles are kind of
ghost particles in single-view imaging, which appear
systematic in viewing direction in front and behind of theactual object. On the other hand, compared to standard
reconstruction methods such as MART and SART, there is no
generation of ghost particles by multiple line-of-sightcombinations. Due to the thin light sheets there is shadowing
effect on the reconstruction of neighboring particles inviewing direction.

Figure 4: Scheme of the intensity distribution of a single
particle in the different scanned planes.
Hence, the particle is displayed in different planes; it gets anellipsoidal shape instead of a spherical one. An example of theacquired images (x-y planes) is shown in Figure 5.

Figure 6: reconstructed top view from the 100 measured
planes of one scanning cycle after transformation of theimage stack into a voxel volume with equally sized voxel
cuboids

5. Error Influence of the ellipsoidal particles
In order to evaluate the influence of the ellipsoidal
representation of the particles in the voxel space on the LSM
results, two synthetic voxel volumes are generated. One voxel
volume is filled with spherical and the other one withellipsoidal particles. The centre of the particles is the same forboth particle shapes. For the second time step, a Hill-type
vortex displaces the particles. The exact solution allows us toinvestigate the influence of different representations ofparticles in the voxel space.

Figure 5: Example of an Image of the recorded scan in
the center plane of the vortex ring (red line indicates thereconstructed plane of Figure 6).
As the scaling in x–y-direction is 62.72 x 62.72 mm² on 752 x
752 pix² is different to the scaling in reconstructed depth, an
expansion factor is needed to correct that influence. Therefore,
every plane is stacked multiple times (herein the expansion
factor is 5) behind each other to achieve cuboidal voxel
representations. A cutout plane from the reconstructed volume
representing a top view from the center of the vortex ring isshown in Figure 6. The resulting image format in this plane is752 pix by 500 planes.
The reconstruction of ellipsoidal shapes of particle
representations in the voxel space is well known in holography
techniques [12]. For further processing of the data, usually thecenter of mass of the ellipsoid is determined and theellipsoidal shape is replaced by a spherical blob with aGaussian grey level distribution [13]. The ellipsoidal extended

The Hill-type vortex represents an exact solution of theNavier-Stokes Equation for the inner spherical region as well
as for the outer part in form of a spherical vortex ring. The
velocity field in cylindrical coordinates for r<R in the innerpart reads:

Here UR is the radial and UΦ the circumferential velocity
component with U0 as the translational overlayed velocity. For
the outer part of the Hill-type vortex UR and U0 are describedas follows:Figure 7: Two time steps of one x-z-slice extracted fromthe simulated voxel volume using ellipsoidal particles
The simulated volume has the size of cuboid with 301 x 301 x
301 voxel elements. It has a typical mean particle
displacement in order of two voxel sizes. The sphericalparticles are represented as a spherical Gaussian grey value
distribution and fill a volume of 3 x 3 x 3 voxel elements.
Besides that ellipsoidal particles are expanded five times in ydirection with an ellipsoidal Gaussian distribution. Two
extracted slices in the x-z-plane are shown in Figure 7 todemonstrate the ellipsoidal extension of the particle images inz-direction.

The two time steps of the spherical and ellipsoidal particle
representations in the voxel volumes are analysed by LSM. As
shown by Tokumaru et al. (1995), LSM is also suited fordiscrete particle image data. Thus, 3D LSM is used to evaluate
the data. The method applies deformation and rotation to
cuboids in an iterative fashion using a least squares adjustment
procedure to determine the affine transformation between agrey cuboid at a time t1 and a time t2. The resulting affine
transformation contains 12 parameters for each translation,
rotation and shear. Conservation of mass for incompressible
fluids is predefined and therefore a much faster convergenceof the iterative process is ensured (for details of the 3D LSM
implementation see Westfeld et al. 2010).
3D LSM is applied to a cuboid size of 25 x 25 x 25 voxel³
within an enlarged search region and the cuboid region is
sampled with a spacing of 10 voxels in every direction (15.000
vectors). Figure 8 shows a comparison of the velocity field bymeans of color-coded plots of the velocity magnitude andisosurfaces of the vertical velocity component. The plots
compare the simulated velocity field (a), the 3D LSM result
for spherical particles (b) and the 3D LSM results for
ellipsoidal particles (c).
The measurement uncertainty introduced by the ellipsoidal
particles on the results obtained by 3D LSM is defined as theabsolute value of the difference vector between the results
obtained for ellipsoidal and spherical particles (disparity
vector). The RMS value of the disparity is about 0.15 voxel iny-direction and about 0.015 voxel in the other dimensions.
Only two vectors out of 15.000 (< 0.013%) have a disparity
larger than 1 voxel in direction of the long axis of the ellipsoid(z-direction). As a result of the higher measurement
uncertainty in z-direction, the isosurface of the sphericalparticles appears slightly sheared in z-direction, see Figure 8.
Nevertheless, the resulting error has only a small influence on
the overall impression of the flow. Furthermore, the error
decreases when the expansion ratio of the ellipsoidal particles
is lowered. This can be achieved by using the Gaussian
intensity distribution in the overlapping planes together withappropriate weighting functions for the expansion of thevolume. Hence, it can be said that the method of light sheet
stacking is a meaningful way to achieve a full 3D voxel
volume with only one high-speed camera and single view
access to the region of interest. Nevertheless, the conditions oflight-sheet thickness, overlapping and frame rate are chosen
appropriate for the given flow conditions.

6. Results

Figure 8: Velocity field results of the simulated voxel
volumes with z-velocity component of (0,25Uabs/Norm)shown as an isosurface, a) prescribed velocity field b)
3D LSM with spherical particles c) 3D LSM withellipsoidal particles

The result of stacked planes is a complete voxel volume withthe dimensions of 752 x 752 x 600 voxel³. A typical cuboid
size for the 3D LSM is 65 x 65 x 65 voxel³ with a shift of 20
voxels. The velocity map with 20700 vectors has a grid of 30
x 30 x 23, the gradient tensor of each grid point are also a
result of the 3D LSM algorithm. Figure 9 shows two
successive time steps of the measured results within the periodof vortex ring transition. The relatively large shift is only for
reducing the processing time of the LSM results for the large
number of voxel reconstruction in the time-series of the flow.
For a single time step the total processing time of 3D LSM isabout 30 min. on a 8-core Intel 17 860 processor. This will belargely reduced by using the Cuda computing on graphic
boards, see Nonn (2010).

a)

The figures show the raw data after 3D LSM with no postprocessing such as smoothing or filtering. Figure 9a indicatesthe surfaces of constant velocity magnitude and the vector
field to illustrate the reconstruction quality of the quasicylindrical flow structure. Furthermore, Figure 9b and 9c
provides some details of the evolving inner flow structure of
the hydrodynamic instabilities of the vortex. The red
isosurface indicates areas of a constant y-velocity component.
From these surfaces one can see a formation of pockets
staggered in azimuthal direction with a certain wavelength.
The blue isosurface shows the unstable vortex core computedfrom the gradient tensors as a surface of constant magnitude ofvorticity. The waviness is clearly seen as a 3D wave structure
overlaid on the previous rather concentric vortex shape.
Furthermore, wave-like disturbance is in horizontal andvertical direction. Hence, the results are well comparable tothe qualitative results of the visualization study using thescanning light-sheet method.

7 Conclusionsb)c)This work shows the potential of scanning light-sheet methods
in combination with voxel-based reconstruction and 3D leastsquares matching for volumetric PIV. The strength of thismethod in applications with only single-view access can easily
be seen from the quality of the results when proper conditionsof high-speed imaging and light-sheet scanning are ensured
for the given flow problem. Due to the 70% overlap of thelight-sheet planes, particles are reconstructed also with aGaussian intensity distribution in scanning direction (the same
particle is seen at least in 3 successive planes). This can beused to improve the measurement uncertainty introduced by
the ellipsoidal shape of particle reconstruction, similar as it isdone in in-line holographic PIV. The results deliver a detailed
evolutionary process of an unstable vortex ring. In comparison
to the qualitative results of LIF-scanning, the quantitative 3D
results can be used for further processing of the vorticity field,
the Lamb-vector field and its divergence. The simulations withsynthetic particle fields showed a minor influence of theellipsoidal shape of the reconstructed particles with an RMS
value of approx. 0.15 voxel disparity. With the optics used andthe 10000fps camera the maximum measurable speed islimited to about 0.2 m/s in our configuration with a volume ofinterest of approx. 63 x 63 x 53 mm3 at a mean particle
displacement of 10 voxel. The in-plane particle density is
about 0.03 ppp at a seeding density of 2 particles/mm³.
Though, with faster cameras, the maximum measurable
velocity will rise. In addition, it was shown that bymechanically subtraction of the mean velocity by using a
traverse system the measureable Reynolds-number as well as
the feature tracking period rise proportionally.

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results show the formation of azimuthal instabilities witha wavelength of 120 voxel sizes (0.85cm). Flow direction
is from bottom to top.

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