15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010

Estimation of 3D deformation and rotation rate tensor from volumetric particle
data via 3D least squares matching
J. Kitzhofer1, P. Westfeld2, O. Pust3, T. Nonn3, H. G. Maas2 and C. Brücker1
1: Institute of Mechanics and Fluid Dynamics, University of Freiberg, Germany, jens.kitzhofer@imfd.tu-freiberg.de
2: Institute of Photogrammetrie, University of Dresden, Germany
3: Dantec Dynamics A/S, Skovlunde, Denmark

Abstract This article discusses the estimation of the deformation rate and rotation rate tensor from results
obtained by 3D Least Squares Matching (LSM), which is applied to volumetric voxel spaces for the
calculation of 3D velocity fields. It is shown that LSM already yields the velocity gradient matrix forcomputation of the deformation and rotation rate tensor. Vorticity, shear and strain are calculated without
applying central differences schemes. It is shown that LSM is well suited for the extraction of thedeformation parameters in cases of small to medium deformations. The accuracy and constraints will bepointed out by analyzing the coefficients of the tensors. Results are presented for simulated and experimental
datasets.

1. Introduction
In the last decades several investigations have been performed to experimentally describe themotion of fluid. The fundamental theorem by Helmholtz says that every infinitesimal motion of a
fluid element can be decomposed in translation, rotation and deformation.
Initially, the measurement technique Particle Image Velocimetry (PIV) was used for 2D
investigations yielding a planar field with translational velocities in the Eulerian frame [1]. Thus,
only one part of the fundamental theorem could be described in 2D. Developments like Scanning
PIV [2], Holographic PIV [3] and Tomographic PIV [4] expanded the description of fluid motion to
3D. Here, correlation based techniques, like 3D cross correlation, are frequently applied in the postprocessing on gray value voxel spaces to extract the zero order translational velocity components
neglecting the higher order terms of rotation and deformation. The assumption is that the flow fieldis smooth and not significantly influenced by rotational or shear displacements, thus yielding thezero-order translational displacement field with an additional uncertainty in measurement due toneglecting the higher-order terms. Reduction of the measurement uncertainty can be achieved bywindow deformation techniques [5]. The higher-order motion terms are then estimated by finite
difference schemes of the velocity field information on discrete grids (indirectly by consideration of
the translational velocities of neighbouring elements). The assumption is that the higher order fluid
motion of an element is only affected by the translational velocity components of the neighbouring
elements.
Another approach for estimation of velocity fields is Least Squares Matching (LSM) [6]. LSM has
already been successfully used in the analysis of 2D particle images. Nevertheless, cross correlation
techniques outperformed LSM because of the shorter processing times. In contrast to correlation
based techniques, Least Squares Matching (LSM) shifts, rotates and stretches a fluid area. For thispurpose, the least squares matching algorithm iteratively compares gray value information of aninterrogation area in the first time step with the gray value information in the second time step. Thisis an iterative least squares procedure applying an affine transformation on the interrogation areas.
In 2D this results in six transformation parameters and in 3D this results in twelve transformation
parameters for each interrogation area.
So, the advantage of LSM is that while calculating the zero order translational velocities, the first-1-

15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010

order terms of motion are considered increasing the accuracy of the velocity field. Moreover, theaffine transformation includes parameters like rotation, shear and strain of the interrogation area.
This paper shows how to extract these parameters from the affine transformation and to transformthem in the description of fluid motion.
For this purpose, this article first discusses the necessary principles (fundamental theorem of fluid
motion, Least Squares Matching) and the connection of both principals to calculate the deformation
rate and rotation rate tensor via the results obtained by LSM. A parametric study is performed on asingle volumetric cube to show the accuracy and the constraints of the estimation of deformation
and rotation rate. A simulated Hill type vortex shows the strength of the method for identifying
regions in the flow of high fluid mechanical interest. At the end we will show results for thebenchmark experiment of a vortex ring.

2. Principles
This section summarizes the basic principles for understanding the estimation of the rotation anddeformation rate tensor via the dataset obtained by 3D Least squares matching. For this purpose, thefundamental theorem for fluid motion is shortly described. The equivalent description in terms oftransforming one system (basic cube) into another system (deformed cube) as obtained by 3D LSM
is discussed and both mathematical descriptions are compared showing that 3D LSM yields thevelocity gradient matrix and consequently the rotation and deformation rate tensor.

Fig 1. Definition of fluid motion: E represents the fluid element at two successive time steps; P is a point
inside the infinitesimal fluid element; the velocity of P (uP) is the sum of the translational velocity u0
and the deformation velocity udef

2.1 Fundamental theorem of fluid motion
The fundamental theorem by Helmholtz says that the velocity at a point x+dx inside an infinitesimal
fluid element E at x can be decomposed in a constant translational velocity u0 (x) of the fluid
element and in a velocity udef (dx) resulting from the deformation of the fluid element as a functionof the distance dx from the center of the fluid element as shown in figure 1. Mathematically, this is
written:u ( x + d x, t ) = u 0 ( x, t ) + u def (d x) (1)The velocity at dx in the infinitesimal volume element resulting from deformation is written in-2-

15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010

terms of the velocity gradient matrix V:
⎡ ∂u
⎢ ∂x
⎢
∂v
u def (d x) = ⎢
⎢ ∂x
⎢ ∂w
⎢
⎢⎣ ∂x

∂u
∂y
∂v
∂y
∂w
∂y

∂u ⎤
∂z ⎥⎥
∂v ⎥
d x = V ⋅ d x (2)
∂z ⎥
∂w ⎥
⎥
∂z ⎥⎦

It is generally that the Jacobian of the velocity can be decomposed in a symmetric tensor D and an
asymmetric tensor R, where R describes the rotation rate tensor and D the deformation rate tensor.
V = R + D (3)The deformation rate tensor describes strain εij, i=j and shear εij, i≠j of the infinitesimal volume
element resulting from velocity gradients in the inner of the volume.
⎛
∂u
⎜
∂x
⎜
⎜ 1 ⎛ ∂v ∂u ⎞D = ⎜ ⎜⎜ + ⎟⎟⎜ 2 ⎝ ∂x ∂y ⎠⎜ 1 ⎛ ∂u ∂w ⎞⎜⎜ ⎜ +⎟⎝ 2 ⎝ ∂z ∂x ⎠1 ⎛ ∂v ∂u ⎞⎜ + ⎟2 ⎜⎝ ∂x ∂y ⎟⎠∂v∂y1 ⎛ ∂w ∂v ⎞⎜+ ⎟2 ⎜⎝ ∂y ∂z ⎟⎠1 ⎛ ∂u ∂w ⎞ ⎞⎜ +⎟⎟2 ⎝ ∂z ∂x ⎠ ⎟⎡ε xx
1 ⎛ ∂w ∂v ⎞ ⎟ ⎢⎜+ ⎟ ⎟ = ε yx
2 ⎜⎝ ∂y ∂z ⎟⎠ ⎟ ⎢⎢⎟ ⎣ε zx∂w⎟⎟∂z⎠

ε xy ε xz ⎤⎥ε yy ε yz ⎥ (4)ε zy ε zz ⎥⎦The rotation rate tensor describes the vorticity ω in each spatial direction x, y, z.
⎡
1 ⎛ ∂v ∂u ⎞ 1 ⎛ ∂u ∂w ⎞ ⎤0− ⎜⎜ − ⎟⎟⎟⎥⎜ −⎢2 ⎝ ∂x ∂y ⎠ 2 ⎝ ∂z ∂x ⎠ ⎥⎢⎛ 0⎢ 1 ⎛ ∂v ∂u ⎞1 ⎛ ∂w ∂v ⎞⎥ ⎜R = ⎢ ⎜⎜ − ⎟⎟0− ⎜⎜− ⎟⎟⎥ = ⎜ ω z
x
y
y
2
2
∂
∂
∂
∂z ⎠⎥ ⎜
⎝
⎠
⎝
⎢
− ωy
⎥ ⎝
⎢ 1 ⎛ ∂u ∂w ⎞ 1 ⎛ ∂w ∂v ⎞0⎥⎢− 2 ⎜ ∂z − ∂x ⎟ 2 ⎜⎜ ∂y − ∂z ⎟⎟
⎠
⎝
⎝
⎠
⎦
⎣

− ωz
0

ωx

ωy ⎞
⎟
− ω x ⎟ (5)0 ⎟⎠

It is obvious that calculation of the deformation rate and rotation rate tensor for fully describing
fluid motion needs the evaluation of the velocity gradient matrix V.
2.2 3D Least Squares Matching (3D LSM)
The basic principle of Least Squares Matching is the evaluation of a transformation, whichtransforms the state of a system into another state of the same system. For this purpose the states ofthe system are compared via an iterative least squares procedure. In the application of 3D LSM tovolumetric particle data as obtained by Tomo PIV, the system is an interrogation volume (cuboid)and the state of the system is the gray value representation of particles. The transformation isevaluated in an iterative least square adjustment procedure, where the gray values in state one aretransformed into the gray values in state two. A detailed description can be seen in [7]. The
geometrical model is an affine transformation for each cuboid E transforming all positions at state-3-

15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010

one in the inner of the cuboid into the positions at state 2.
x 2 = a 0 + a1 ⋅ dx1 + a 2 ⋅ dy1 + a 3 ⋅ dz1
y 2 = b0 + b1 ⋅ dx1 + b2 ⋅ dy1 + b3 ⋅ dz1(6)z 2 = c0 + c1 ⋅ dx1 + c 2 ⋅ dy1 + c3 ⋅ dz1 E

The index 1 means the state of the volume element at time step 1 and the index 2 the state of thevolume element at time step 2. The parameters a0, b0 and c0 are the constant translational
displacements valid for each point in the infinitesimal cuboid E. In the following the translational
displacements in the cuboid E at x will be t0(x) = (a0, b0, c0)T directly yielding the translational
velocity of all points in the cuboid E at x.
u 0 ( x) =

t0
(7)
∆t

The deformation of the cuboid is described by the transformation matrix T:⎡a1 a2T = ⎢⎢ b1 b2⎢⎣ c1 c2

⎡ ∂x2
⎢ ∂x
a3 ⎤
⎢ 1
∂y
⎥
b3 ⎥ = ⎢ 2⎢ ∂xc3 ⎥⎦ E ⎢ ∂z 1
⎢ 2⎢⎣ ∂x1

∂x2
∂y1
∂y2
∂y1
∂z2
∂y1

∂x2 ⎤
∂z1 ⎥⎥
∂y2 ⎥
(8)
∂z1 ⎥
∂z 2 ⎥⎥∂z1 ⎥⎦ E

2.3 Velocity gradient matrix
The velocity gradient matrix is necessary for the computation of the deformation and rotation rate
tensor. The result from LSM is the transformation matrix und the displacement vector. So, for agiven point P1 = (dx ,dy,dz)T in the inner of the cuboid E the position of the point P2 is calculated
via the transformation matrix assuming that the translational displacement is zero:P2 = T ⋅ P1 (9)The velocity of the point due to the deformation is calculated by:

⎡1

0
1

⎢⎣ 0

0

⎢
P − P T ⋅ P1 − P1 (T − I )u def (d x) = 2 1 ==⋅ P1 , I = ⎢ 0
∆t
∆t
∆t

0⎤
0 ⎥⎥ (10)1 ⎥⎦So, by comparison of (2) and (10) it is clear that the velocity gradient matrix V of a volume element
E and as a consequence the rotation rate R and deformation rate tensor D is derived by subtracting
the identity tensor I from the transformation matrix T and by following division by the separation
time ∆t.
V =(T − I )
∆t
-4-

(11)

15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010

3. Numerical assessment
In this section we present numerical results for the estimation of the rotation rate and deformation
rate tensor obtained by LSM. The accuracy of the estimation is analysed via synthetic generated
voxel spaces. For this purpose a parametrical study is performed by analysing the deformation of awell defined cube. The interpretation of fluid mechanical aspects is shown at the end via asimulated Hill type vortex.
3.1 Cube deformation
The following parametrical study extracts influences on the estimation of the rotation rate anddeformation rate tensor. Parameters are number of particles, noise and strength and number of
deformation. The synthetic set-up is shown in figure 2. A cubic element (60x60x60 voxel) is
generated with the coordinate system in the center of gravity. A part of the element (40x40x40) isthen filled with homogenous sized particles (3x3x3 voxel elements) at random positions having a
Gaussian intensity distribution. The positions of the particles are chosen with sub pixel accuracy
and represent the original cube without deformations. The particle positions are then deformed via
defined shearings and strains and following rotations. The shifted particle positions represent the
deformed cube. The undeformed and deformed cubes are then analysed via LSM to calculate thetransformation matrix and in the following processing step the velocity gradient matrix. The
computation of the LSM parameter is performed by an adapted code given by Thomas Nonn from
Dantec Dynamics.

Fig 2. Numerical set-up for the parametrical study of the cube deformation

As LSM is an iterative least squares procedure, it is sensitive to starting values. The analysis herein
is performed without the knowledge of starting values resulting in a higher number of necessary
iteration steps to calculate the transformation matrix within predefined boundaries. To make theresult independent of the number of iteration steps, the iterative procedure is stopped after 200
iterations, which is far beyond necessarily about 10 iterations for small deformations [10].
In the following the value dev is defined as the relative accuracy of the measured deformations,where σcalc is the calculated value (ωi, εij) and σ0 is the initial value (ωi0, εij0):dev =σ calc − σ 0⋅ 100%
σ0

Figure 3 on the left shows the accuracy of the absolute value of rotation as a function of particle per
pixel (ppp). Particle per pixel is defined by generated number of particle positions divided by the-5-15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010

size of one side of the cube (herein 40x40 pix²). The deformation consists of rotations around eachaxis and strain and shear in two planes representing a cube with low magnitude in deformation but
large number of different deformations. At low particle densities the error is very high up to 80 %converging to 10 % within a slightly increasing ppp. Common 2D PIV images have a number
particle density of 0.1... 0.2 ppp. This is much higher than the observed border at 0.005 ppp in
figure 1. This is in agreement with [8]. They used a 2D LSM approach for Particle Tracking
Velocimetry, which measures velocities at lower number particle density. Nevertheless, thefollowing investigations will be performed at 0.1 ppp, because the aim of this paper is theapplication of LSM to Tomo PIV. But to point out, LSM delivers a constant accuracy of 10 %beyond 0.005 ppp.

Fig 3. Accuracy in measurement;
left: as a function of ppp (deformation: ω = φ = κ = 5°; dxx = 1.1, dyy = 0.9, dzz = 1; dxy = 0.1, dxz = 0.1,
eyz= 0);right: as function of noise (deformation: ω = 12; φ = κ = 10°; dxx = 1, dyy = 1, dzz = 1; dxy = 0, dxz = 0,eyz= 0)Figure 3 on the right shows the deviation as a function of noise. Noise is included in a way that thegray value of each voxel position is superimposed by a randomly standard deviation. The maximum
standard deviation is defined between 0 and 255 meaning noise 0 is a standard deviation of 0 andnoise 1 is a standard deviation of 255. It is clearly recognizable that dev increases to more than 10% from a noise of 0.2. This that LSM is applicable, if the noise does not increase 20 % of thehighest magnitude in the images, herein 255. The reason probably is that the shapes of the particles
start to smear with the surrounding.
The relative deviation in absolute rotational strength as a function of increasing rotational strength(increasing deformation) shows figure 4. The solid line shows the deviation for ωy ,ωz = 0° and thedashed line shows the deviation for ωy ,ωz =15°. The solid line remains nearly constant up to arotation in x-direction of about 15°. The deviation is pretty low (< 3%). Between 15° and 20° thedeviation dramatically increases being beyond 10% at about 22°. If rotations around the other axes
are included, two characteristics can be concluded (dashed line). The deviation for small rotations isincreased (≈ 7%) and the sudden increase of the deviation is shifted to lower values of ωx. This
problem is well known in literature and results from the assumption of the affine transformation.
The transformation is linear, which is valid for small deformations. But when the absolute
deformation is increased, the transformation becomes non-linear.

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15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010

Fig 4. Accuracy in measurement as a function of ωx(deformations: φ = κ = 0° -solid- /15° -dashed-; dxx = 1 dyy =1, dzz = 1; dxy = 0, dxz = 0, dyz= 0)

Figure 5 shows the influences on the accuracy on shear and strain. In both classes of deformation
trends are visible. On the one hand the deviation again dramatically increases, when a certain
deformation is exceeded. On the other hand the deviation increases for lower deformation, if
multiple deformations are included. Nevertheless, the deviation of strain and shear is in the order ofmagnitude of about 1% remaining nearly constant also in the case if multiple deformations areincluded.

Fig 5. Accuracy in measurement;
left: as a function of shear in xy (deformations: ω = φ = κ = 0° ; dxx = 1 dyy =1, dzz = 1; dxz = 0 -solid/0.3 -dashed-, dyz= 0 -solid- /0.3 -dashed-);right: as a function of strain in x (deformations: ω = φ = κ = 0° ; dxx = 1 dyy =1, dzz = 1; dxz = 0 -solid/0.3 -dashed-, dyz= 0 -solid- /0.3 -dashed-)

The parametrical study of the cube deformation shows that LSM can handle multiple deformations
up to certain strength. The accuracy is in the order of 1% if only one class of deformation is
included. If multiple classes of deformation are included, the accuracy decreases to about 10 %.
Nevertheless, strong deformations are not capable of the LSM algorithm. The reason is probably inthe assumption of an affine transformation.

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15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010

3.2 Hill type vortex
In the numerical assessment we use the hill type vortex ring for validation of the extraction of thedeformation and rotation rate tensor from the LSM results. The hill type vortex is characterized byspherical streamlines, a donut like vortex core and two stagnation points (upstream anddownstream). The exact solution of the Navier Stokes equations for the inner part (r<R) is written in
cylindrical coordinates:⎛r2 ⎞U R = −1.5 ⋅ U 0 ⋅ cos(Φ ) ⋅ ⎜⎜1 − 2 ⎟⎟⎝ R ⎠⎛ 2r 2 ⎞U Φ = 1.5 ⋅ U 0 ⋅ sin(Φ ) ⋅ ⎜⎜1 − 2 ⎟⎟R ⎠⎝UR is the radial velocity, UΦ is the circumferential velocity and U0 is the transport velocity of thevortex. For the outer part of the vortex (r>R) the solution is written:⎛ R3 ⎞U R = U 0 ⋅ cos(Φ ) ⋅ ⎜⎜1 − 3 ⎟⎟
r ⎠
⎝
⎛
R3 ⎞
U Φ = 1.5 ⋅ U 0 ⋅ sin(Φ ) ⋅ ⎜⎜1 − 3 ⎟⎟⎝ 2r ⎠As the LSM algorithm is defined in Cartesian coordinates, the velocity field is transformed fromcylindrical coordinates into Cartesian coordinates. The size of the generated velocity field is
101x101x101 voxel. The Cartesian grid is then filled with 1000 homogenous sized particles (3x3x3
voxel) with a Gaussian intensity distribution, meaning that 9% of the whole volume is filled withparticles. The sub-pixel displacement of the particles due to the velocity field is then estimated witha Runge-Kutta scheme. Afterwards random noise with standard deviation of 1.5 is superimposed.
The chosen conditions (random particle positions and noise) are a good approximation for realexperimental conditions. The size of the analyzed cuboids is 15x15x15 voxel with an overlap of
75%. This results in 10648 translational displacements and transformation tensors.
The visualization of some results shows figure 6. The figure shows the velocity vectors in thesymmetry plane x-z, the 3D streamlines starting above the upstream stagnation point, andisosurfaces of the vorticity magnitude (gray), the positive (orange) and the negative (blue) strain.
Moreover, some numerical results of the deformation rate and rotation rate tensor are included. The
generated streamlines clearly follow the exact solution of the hill type cortex. The streamlines forma spherical shape. The velocity vectors in the symmetry plane identify the velocity distribution of ahill type vortex. No outliers are recognizable. The isosurfaces of vorticity and strain excluded fromthe transformation matrix derived by LSM identify the main features of the hill type vortex without
any further post processing steps. The upper and lower stagnation points are highlighted. The
volume elements in the upper stagnation point are compressed in z-direction and expanded in x- andy-direction (εzz<0, εxx>0, εyy>0). In the lower stagnation point the scene is inverted. The isosurfaceof the vorticity magnitude identifies the donut like vortex core. The numerical result for the rotation
rate tensor in the symmetry plane x-z at the vortex core for x<0 and x>0 is given. The prefixes ofthe entries for vorticity ωy agree with the physical interpretation.

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15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 20100 ⎤⎡ 0 .3 0⎢
D = ⎢ 0 0 .3
0 ⎥⎥
⎢⎣ 0
0 − 0.6⎥⎦⎡ 0 0 − 0 .7 ⎤R yz , x > 0 = ⎢⎢ 0 0
0 ⎥⎥⎢⎣0.7 00 ⎥⎦0 0.7 ⎤
⎡ 0⎢
R yz , x < 0 = ⎢ 0
0 0 ⎥⎥⎣⎢− 0.7 0 0 ⎥⎦00⎤⎡− 0.3⎢D=⎢ 0− 0.3 0 ⎥⎥
0
0.6⎦⎥
⎣⎢ 0

Fig 6. Visualization of some results obtained by LSM applied to the simulated hill type vortex ring;
isosurface of vorticity magnitude (blue) identifies the donut like vortex core, isosurface of positive(red) and negative (blue) strain rates identify stagnation points; 3D streamlines show a sphericalshape and the velocity vectors in the symmetry plane y-z characterize the velocity field of the hill
type vortex

The numerical results show that the estimation of the rotation and deformation rate tensor via LSM
delivers correct results without an exhausting post processing via central derivatives. The cube
deformation showed an accuracy of about 1 % for small and single deformations and 10 % for small
and multiple deformations. The Hill type vortex showed that physical aspects can simply be
excluded from LSM.

4. Experiment
The experimental set-up is shown in figure 7. The laser beam of a continuous Argon-Ion laser
Coherent Innova 70 (2 W) passes an optical lens system to adjust the desired thickness of the lightsheet. The rotating mirror drum reflects the laser beam into the direction of the observed volume
and generates successively 10 parallel light sheet planes with a thickness of 10 mm. The studied
flow is a vortex ring travelling in an octagonal glass tank filled with water. The vortex is generated
at the exit of a piston tube with a diameter of 50 mm. The neutrally buoyant seeding particles (100
microns) are injected into the center of the vortex generator. The particle images are recorded with athree camera system consisting of digital high speed cameras Photron APX RS with a resolution of1024 x 1024 Pixel² and an angular displacement of 45°, -45° and 90°. The cameras are equippedwith telecentric lenses at f-number 16 resulting in a parallel projection for the observed volume. The
side of the octagon opposite to the entrance side for the laser is covered with a light absorbing matthat reduces stray reflections. This is also valid for the faces opposite the cameras, thus giving aperfectly black background. The experiments are performed with a recording rate of 1000 frames/s.
Using 10 scanning planes results in a separation time of 10 ms for each subsequent illumination of
one scan plane. The image size of the cameras and the light sheet thickness define the measured
volume, resulting in about 90 x 90 x 10 mm³ for one light sheet.

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Lisbon, Portugal, 05-08 July, 2010

Fig 7 Experimental set-up consisting of three high speed cameras APX RS equipped with telecentric lenses

The volumetric reconstruction as well as LSM is performed by Dantec Dynamics Software. The
volumetric reconstruction bases on a multiple projective transformation of each camera view into
the depth layer of the observed volume and a minstore algebraic reconstruction technique. For
further information on the volumetric reconstruction see [10]. In total a regular grid of
278x1112x944 voxel with gray value information is generated for each time step. The velocity field
as well as the parameters of the affine transformation is calculated as presented in [7]. Tosummarize, the LSM algorithm is performed on 25³ cuboids resulting in 25³ velocity vectors andtransformation matrices. The result is visualized in figure 8. The vortex ring is clearly recognizable.
Beside the velocity vectors the isosurface of vorticity magnitude is included. To point out, thevorticity magnitude is not calculated via central difference schemes, but via the parameters of theaffine transformation.

Fig 8 Vortex ring: the velocity vectors have a uniform length; the blue isosurface represents the vorticity
magnitude showing the vortex core

5. Conclusion
The article discussed the estimation of the rotation and deformation rate tensor via the affine
parameters resulting from LSM. It is shown that the calculation is straight forward without using
any central difference schemes. The accuracy is estimated via the deformation of a well defined
cube. The result is that LSM is valid for small multiple deformations and for medium single
deformations. The accuracy is about 1% for single deformations and under 10 % for multiple- 10 -

15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010

deformations. It seems that LSM is not sensible to high deformation rate, but sensible to highdeformations. The extraction of the parameters is performed on a simulated as well as on an
experimental vortex ring, both showing that the physical character of the vortex ring is described by
the parameters of the affine transformation. Future work has to be performed in cases of large
deformations. Here, the linear affine transformation needs to be replaced by a non linear model.

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International Symposium on Appliocations of Laser Techniques to Fluid Mechnaics, 5-8 July 2010,
Lisbon[8] Ishikawa M, Murai Y, Wada A, Iguchi M, Okamoto K, Yamamoto F (2000) A novel algorithm
for particle tracking velocimetry using the velocity gradient tensor. Exp. In Fluids 29[9] Kitzhofer J, Brücker C, Pust O (2009) Tomo PTV using 3D Scanning Illumination andtelecentric Imaging. Proceedings of the 8th International Symposium on Particle Image
Velocimetry, Melbourne, Australia[10] Maas HG, Westfeld P, Putze T., Botkjaer N, Kitzhofer J, Brücker C (2009) Photogrammetrictechniques in multi-camera tomographic PIV. Proceedings of the 8th International Symposium on
Particle Image Velocimetry, Melbourne, Australia- 11 -