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

3-D least squares matching for volumetricvelocimetry data processing
Patrick Westfeld1, Hans-Gerd Maas1,
Oliver Pust2,
Jens Kitzhofer3, Christoph Brücker3
1: Institute of Photogrammetry and Remote Sensing, Technische Universität Dresden, Germany,{patrick.westfeld,hans-gerd.maas}@tu-dresden.de
2: Dantec Dynamics A/S, Skovlunde, Denmark, oliver.pust@dantecdynamic.com
3: Insitute of Mechanics and Fluid Dynamics, Technische Universität Bergakademie Freiberg, Germany,{kitzhofer,bruecker}@imfd.tu-freiberg.de

Abstract The paper presents a three-dimensional least squares matching approach applied to time-resolved
volumetric particle image velocimetry data to determine 3-D velocity fields. It will introduce the functionaland stochastic model of the tracking algorithm which includes the geometric concept, the parameterization of
the transformation, the derivation of information on precision, determinability and reliability of thetransformation parameters as well as the integration of additional constraints. The implementation of 3-D
least squares matching has been validated with both simulated and real data. The accuracy potential as wellas the reliability will be documented comparatively for all data sets. The parameterization for different flows
and the percentage of significant parameters in accepted trajectories will be analyzed and a comparison
between 3-D cross correlation and 3-D least squares matching is given.

1. Introduction
Photogrammetric 3-D motion analysis is a well-established field of close-range photogrammetry
and allows the extraction of geometric information from images with high precision and reliability.
In this context, least squares techniques such as least squares matching depict powerful, flexible andwidely used tools for the computation of velocity vectors from 2-D or 3-D image sequences.
Volumetric PIV generates a tomographic reconstruction of a particle constellation from a limited
number of synchronized camera views by applying a feasible reconstruction algorithm like MART
(Herman & Lent, 1976) or MinART (Maas et al., 2009). The advantage of PIV is the insensitivity tohigh seeding densities. This kind of tomographic reconstruction facilitates the implementation of a
volume-based particle tracking technique rather than a discrete PTV (particle tracking velocimetry)approach, which may be prone to ambiguities at high seeding densities (Papantoniou & Dracos,
1989).

Fig. 1

Volumetric Data Processing (Willneff, 2003).

Volume-based tracking techniques deliver dense flow velocity field information by dividing time-1-

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

resolved volumetric velocimetry data into cuboids of a predefined size and tracking these cuboids
through the reconstructed sequence of voxel spaces. Herein, 3-D cross correlation (3-D CC, Keane& Adrian, 1992) is a rather straightforward technique to determine 3-D displacement vectors
between cuboids of two consecutive epochs with sub-voxel precision by calculating a 3-D crosscorrelation coefficient field and fitting a Gaussian function into it to obtain sub-voxel accuracy. An
implementation of this approach is quite simple but limited to the determination of the shifts in eachdirection only. Thus, cuboids with significant deformations will not be tracked reliably. Scarano(2002) partially improves this by iterative image deformation methods, but at much larger
computational efforts. Here, 3-D least squares matching (3-D LSM) offers the advantage of being
adaptive to cuboid deformation, rotation and shear, which makes it a rather interesting alternative to
3-D CC.

2. Principle
3-D least squares matching (3-D LSM) is a volumetric matching technique (Maas et al., 1994). In
analogy to 2-D least squares matching (Ackermann, 1984; Förstner, 1984; Grün, 1985), 3-D LSM
utilizes an iterative geometric and radiometric transformation between two (or more) consecutive
cuboids in a way that the sum of the squares of the gray value differences between the voxels of the
cuboids reaches a minimum. The geometric transformation is a 3-D affine transformation with 12
parameters, the optional radiometric adjustment may use a 2-parameters linear correction term (Fig.
2).

Fig 2.

3-D affine transformation of a cuboid. From left to right: translation, scale, rotation, shear.

The essential advantage of these additional non-translational cuboid transformation parameters isthe adaptivity to changes in scale, rotation or shear. If not considered, they may hamper thedetermination of velocity vectors or systematically deteriorate the quality or results.
Compared to the 3-D cross correlation (3-D CC) method, cuboid tracking formulated as a leastsquares problem has the following implications:•

The geometric transformation includes additional scale, shear and rotation parameters.
These parameters allow adapting to linear deformations of cuboids.

•

The mathematical model can be extended by additional information or constraints. Whenapplied to liquid flow data, an incompressibility constraint has to be introduced to force thevolume of a cuboid to remain constant during the iterative transformation.

•

Assuming a correct mathematical model and sufficient contrast within the cuboids, 3-D
LSM can achieve very high sub-voxel precision down to 1/100 of a voxel.

•

3-D LSM converges in a few iterations and may reduce the computational effort.
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15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010•

The observation equations are non-linear. The linearized equations requires initial values,
usually obtained by pre-knowledge or by hierarchically applying the technique on a
resolution pyramid.

•

The adjustment has a high redundancy, allowing for general acceptance decisions and thedetection and elimination of outliers.

•

Accuracy estimates of the unknown parameters can be delivered.

3. Mathematical Model
3.1 Functional Model
Template cuboid g1 and search cuboid g2, taken from consecutive volume data sets 1 and 2,
provide gray value observations for the adjustment at each position (x,y,z)1 resp. (x,y,z)2. The
geometric and radiometric relations between those cuboids can be formulated as
g1 ( x, y, z ) − e( x, y, z ) = r0 + r1 ⋅ g 2 ( x, y, z )(1)whereas r0 and r1 model changes in brightness and contrast, and e considers a small noise fraction.
The geometric 3-D affine transformation model is given byx2 = a0 + a1 x1 + a2 y1 + a3 z1
y2 = b0 + b1 x1 + b2 y1 + b3 z1(2)

z2 = c0 + c1 x1 + c2 y1 + c3 z1
with (a0,b0,c0) being shifts, (a1,b2,c3) scales, and (a2,a3,b1,b3,c1,c2) rotations and shears. Within aGauss-Markov Model (GMM), the parameters can be estimated by minimizing the sum of thesquares of the gray value differences between g1 and g2.
Suppose approximate values are available for all introduced parameters of the cuboid g2 at its initial
position g20, Eq. 1 can be linearized as

g1 ( x, y, z ) − e( x, y, z ) = g 20 ( x, y, z ) +∂g 20 ( x, y, z )∂g 0 ( x, y, z )∂g 0 ( x, y, z )dx + 2dy + 2
dz
∂x
∂y
∂z

(3)

with
dx = (∂x ∂pi ) ⋅ dpi , dy = (∂y ∂pi ) ⋅ dpi , dz = (∂z ∂pi ) ⋅ dpi , pi ∈ (a j , b j , c j r0 , r1 ), j = {0,1,2,3}(4)

With the gray value gradients in three coordinate directions
gx =∂g 20 ( x, y, z )∂g 0 ( x, y, z )∂g 0 ( x, y, z ), gy = 2, gz = 2∂x∂y∂z

this leads to a linearized observation equation

-3-

(5)

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

Φ( pi ) = g1 ( x, y, z ) − e( x, y, z ) = g 20 ( x, y, z ) + g x da0 + g x x20 da1 + g x y20 da2 + g x z20 da3+ g y db0 + g y x20 db1 + g y y20 db2 + g y z20 db3+ g z dc0 + g z x20 dc1 + g z y20 dc2 + g z z20 dc3(6)+ r0 + r1 g 20 ( x, y, z )Like all non-linear least squares approaches, the estimation of the unknowns takes place in an
iterative way. They are updated after each iteration until a preset iteration criteria has been reached.
In general, the residual amounts are non-integer values and the gray values in the search cuboid willneed to be determined by interpolation.
For the special case of tracking particles in liquids, a cuboid, while changing its shape due todeformations, will have a constant volume. Consequentially, a volume constraint was implemented
to consider the incompressibility of the liquid. The incompressibility constraint is formulated asa1 ⋅ b2 ⋅ c3 = 1(7)

and results in one additional linearized equation for the GMM:
∂Ψ ( pi0 )∂Ψ ( pi0 )∂ Ψ( pi0 )Ψ ( pi ) =dc3 = 0
db2 +
da1 +
∂c3
∂b2
∂a1

(8)

3.2 Stochastic Model

The stochastic model describes the variances and co-variances of the observations. In the present
case, homogeneous data are available and all gray value observations are introduced with same apriori accuracy specifications in terms of equal variances resp. equal stochastic weights.

⎡ s02 s12
⎢
P =⎢
n ,n
⎢
⎢
⎢⎣

s02 s22⎤⎥⎥= I⎥ n ,nO⎥
s02 sn2 ⎥⎦(9)where s0 is the a-priori standard deviation of unit weight and si is the a-priori standard deviation of
the observation k, k={1,2,...,n}. Due to independent and homogeneous observations, the weightmatrix of the observations P will often be an identity matrix I (s0 = si ≡ 1).
The incompressibility constraint (Eq. 7) is introduced with highest priority by setting its a-priori
standard deviation to 'zero' (sΨ ≡ 0, PΨ ≡ 0).

3.3 Gauss-Markov Model
Each voxel i in a cuboid produces one observation equation Φi. For each consecutive pair of
cuboids, c = 1 additional equation has to be set up, which considers the incompressibility constraint
Ψ . The complete equation system is solved in an extended Gauss-Markov Model (conditional least-4-

15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010squares adjustment) to estimate u unknown transformation parameters by n gray value observations
per cuboid:
v = A xˆ − ln ,1n ,u u ,1

n ,1

(10)

w = Β xˆc ,1c ,u u ,1Assuming normal distributed errors, the following conditions for the residuals v resp. theinconsistencies w resultsv T Pv + 2k (Bxˆ + w ) a min,which leads to the extended normal system of equations solved for the vector of unknowns:⎡ xˆ ⎤ ⎡ AT PA BT ⎤ ⎡ AT Pl ⎤⎥⎢⎥⎢k ⎥ = ⎢− PΨ−1 ⎦ ⎣ w ⎦⎣ ⎦ ⎣ B

(11)

Herein, the Jacobian matrix A consists of the derivatives of Φ(pi) (Eq. 6) with respect to theunknowns and describes the functional relation between the parameters. In analogous way to theobservation equations, B consists of the derivatives of Ψ(pi) (Eq. 8).
⎡ ∂∂Φa 1⎢ ∂Φ02A = ⎢ ∂a0⎢ Mn ,u⎢ ∂Φ n⎣⎢ ∂a0

[

B= 0

c ,u

∂Φ1
∂a1
∂Φ 2
∂a1

∂Ψ1
∂a1

∂Φ1
∂a2
∂Φ 2
∂a2

∂Φ1
∂a3
∂Φ 2
∂a3

∂Φ1
∂b0
∂Φ 2
∂b0

∂Φ1
∂b1
∂Φ 2
∂b1

∂Φ1
∂b2
∂Φ 2
∂b2

∂Φ1
∂b3
∂Φ 2
∂b3

∂Φ1
∂c0
∂Φ 2
∂c0

∂Φ1
∂c1
∂Φ 2
∂c1

∂Φ1
∂c2
∂Φ 2
∂c2

∂Φ1
∂c3
∂Φ 2
∂c3

∂Φ1
∂r0
∂Φ 2
∂r0

M

M

M

M

M

M

M

M

M

M

M

M

∂Φ1
∂a1

∂Φ n
∂a2

∂Φ n
∂a3

∂Φ n
∂b0

∂Φ n
∂b1

∂Φ n
∂b2

∂Φ n
∂b3

∂Φ n
∂c0

∂Φ n
∂c1

∂Φ n
∂c2

∂Φ n
∂c3

∂Φ n
∂r0

0 0 0 0∂Ψ1
∂b2

0 0 0 0

∂Ψ1
∂c3

0 0

⎤
⎥
⎥
M ⎥
∂Φ n ⎥
⎥
∂r1 ⎦
∂Φ1
∂r1
∂Φ 2
∂r1

(12)

]

The vector l is denoted as the vector of the reduced observations:⎡ g1,1 − g 2,1 ⎤⎢g − g ⎥1, 2
2, 2 ⎥
l=⎢
n
⎢
⎥
M
⎢
⎥
⎣ g1, n − g 2, n ⎦(13)

The vector of Lagrangian multipliers k are not normally of interest. The vector of the unknowns
contains the estimated parameters:xˆ = [a0
u

a1 a2

a3 b0

b1 b2

b3

c0

c1 c2

-5-

c3

r0

r1 ]

T

(14)

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

4. Internal Quality Measures
As a least squares adjustment method, 3-D least squares matching (3-D LSM) delivers informationon the precision, determinability and reliability of the 12 affine transformation parameters. This
includes the standard deviation of each of the parameters as well as the correlation between
parameters.
The a-posteriori standard deviation of unit weight is given bysˆ0 =vT Pvalid v
r(15)

with an redundancy r = n − u + c.
The a-posteriori standard deviation of an estimated parameter i is given bysˆi = sˆ0 qii

(16)

(

)

−1

where qii is the ith diagonal element of the cofactor matrix Q = AT PA .
Furthermore, the level of significance can be calculated for each parameter in each cuboid. To
decide whether a transformation parameter is significant or not, the one-dimensional Student test
function can be calculated for each introduced parameter pi . The parameters are significant, if
ti =

pi
>q
sˆi

(17)

is fulfilled. The quantile q can be obtained by the inverse of the Student's t inverse cumulative
distribution function using the degrees of freedom r and a given probability P that the parameters
are significant. Non-significant parameters may be excluded from the estimation process (and set tozero) in order to improve the strength of the solution. Assuming at least one non-significant
parameter in the introduced set of parameters pi , the least squares adjustment is repeated until all
used parameters are significant.

5. Some Results
5.1 Simulated Data Set
A simulated data set was generated by composing two consecutive voxel spaces 1 resp. 2 with avolume of (101 voxel)3. These volumetric reconstructions contain the flow of a vortex ring. The
second state of the vortex was achieved by a defined translation and rotation of the first one.
Further, some random noise was added to the gray values of both voxel spaces.
The velocity field information were determined comparatively by 3-D cross correlation (3-D CC)
and 3-D least squares matching (3-D LSM). A regular grid of 173 voxel cuboids was defined into
1. For each grid position, the corresponding cuboids were determined either by calculation the 3-Dcross correlation coefficients or by calculating the 3-D LSM affine transformation parameters. All-6-15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010

tracking results were accepted. Neither a threshold for the cross correlation coefficient nor 3-D
LSM outlier criteria were applied.
For 3-D CC, the template cuboid g1 was shifted through the search volume 2. At each position thenormalized cross correlation coefficients between the gray values of the template g1 and thecorresponding search cuboid g2 were calculated. A Gaussian function was fitted into the computed
3-D cross correlation coefficient field to obtain sub-voxel accuracy. The position with the highest
coefficient represents the location of g1 in 2. For 3-D LSM, the 12 parameters of the 3-D affine
transformation were computed as described in Sec. 3. Parameters were excluded from thetransformation if they turned out insignificant in the significance test (Sec. 4).

Vector lengths

Nominal/actual variations

3-D CC

3-D LSMFig 3.

Cross sections of color-coded vector lengths (left) and nominal/actual variations (right) ofa simulated vortex ring achieved by 3-D CC (top) and 3-D LSM (bottom).

The results can be seen in Fig. 3 and Tab. 1. At first the cross sections of the resulting velocity
vector fields looks similar for 3-D CC and 3-D LSM. Analyzing the variations between the nominal
and actual particle positions show the great advantage of 3-D LSM. In contrast to 3-D CC, thetrajectories were determined more precisely and reliability. Mainly in the center of the vortex, an
improved precision by factor 2-6 can be achieved, which proves the relevance of determining not
only translation parameters in cuboid tracking.

3-D CC
3-D LSMσ∆x1.1182e-001
1.7027e-002

σ∆y1.1566e-001
1.6948e-002

σ∆z1.6526e-001
4.3520e-002

Tab. 1 Standard deviations of actual/nominal variations in x, y and z (in voxel).

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

5.2 Real Data Set
The real data set originates from the following experimental setup: A vortex ring in a water tank is
illuminated by a 3-D laser beam device. A rotating mirror generates parallel light sheet planes witha thickness of 10 mm. This volume of about (10×10×1) cm3 is recorded by a system of foursynchronized high speed cameras (1024×1024 pixel, 1000 fps) equipped with telecentric lenses(Fig. 4 and 6). Neutrally buoyant seeding particles are injected into the center of a vortex generator.
See (Kitzhofer et al., 2009) for detailed specifications of this experimental setup.

Fig. 4. Experimental setup with telecentric lenses(Kitzhofer et al., 2009).
X

Y
Z

Fig. 5. Voxel space of the volumetric reconstruction
at one epoch (Westfeld & Maas, 2010).

Fig. 6. Vortex ring imaged with fourcameras at one epoch (Maas etal., 2009).

The volumetric reconstruction (Fig. 5) results from improved reconstruction techniques presented in(Maas et al., 2009) and (Westfeld & Maas, 2010). The process bases on a multiple projective
transformation of each camera view into each depth layer of a the voxel representation of the object
space and a MinART (minstore algebraic reconstruction technique) to fill-up the voxel space withgray value information.
The velocity field information can then be obtained by applying the presented 3-D LSM to the timeresolved voxel space representation. Again, a regular grid of 253 cuboids was defined into thevolumetric reconstruction of (278×1112×944) voxel3 and the 12 affine parameter were determined.
The initial values for all parameters were computed automatically by applying 3-D LSM
successively on a 2-level pyramid of the object space. If adequate initial values are available, 3-D
LSM converges in a few iterations (Fig. 7 and 8). Non-significant parameters were excluded.
Trajectories which do not fulfill the criteria of an outlier detection scheme were removed:•

Affine transformation parameter standard deviation: The results of cuboids with standard
deviations exceeding a preset threshold were deleted.

•

Convergence behavior: Cuboids with a diverging or oscillating solution were rejected.
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15th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 05-08 July, 2010•

Vector length: Translation vectors exceeding a preset threshold were eliminated.

•

Neighborhood correlation: The differences of the translation vector components between
neighboring cuboids were analyzed. Vectors with deviations from their neighborhood
exceeding a preset limit were eliminated.

Fig. 7. Template cuboid g1 and search cuboids Fig. 8. Convergence behavior of all 12 affine
g2 through seven iterations.
transformation parameters.
The following figures visualize the 3-D LSM results. Fig. 9 shows the cross sections of color-coded
velocity in voxel space. Fig. 10 shows the direction of the velocity vectors for one half of the vortex
ring in frontal view.

Fig. 9. Cross sections of color-coded velocity Fig. 10. Cross sections of color-coded velocity
in voxel space (Maas et al., 2009).
in voxel space (Maas et al., 2009).
The standard deviation of unit weight was 2.5 gray values, avaraged over all accepted cuboids.
Tab. 2 shows the average standard deviations of the 12 affine transformation parameters. As one
can see, the internal precision of the cuboid translation parameters is in the order of 1/100 of a voxel.
However, one has to consider that these internal precision figures are only realistic if the assumed
functional and stochastic model is correct (3-D affine transformation and least squares adjustment
assuming Gaussian error distribution). Furthermore, Tab. 2 gives an overview on the percentage ofsignificant 3-D affine transformation parameters over all accepted cuboids. As the cuboidtranslation parameters (a0,b0,c0) were not excluded as a rule in the significance tests, they all have
100% here. The scale parameters (a1,b2,c3), constrained by the incompressibility condition, were-9-

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

only significant in relatively few cuboids, while the rotation and shear parameters (a2,a3,b1,b3,c1,c2)
were significant especially in the center of the vortex (Fig. 11). In total, about 20% of the cuboids
showed at least one significant non-translation parameter, proving the adequateness of the 3-D LSM
approach. Further, the gained 3-D LSM non-translation parameters can be used to estimate the 3-D
deformation tensor as well as the 3-D rotational tensor directly (Kitzhofer et al., 2010).

σi = [vx]Sig = [%]

a00.0132100

b00.0105
100

c0
0.009
100

σi = [vx]Sig = [%]

a1
2.4e-3
1.95

b2
1.8e-32.68

c3
1.6e-33.75

σi = [vx]Sig. = [%]a2
2.4e-3
4.42

a3
2.3e-3
10.47

b1
1.7e-36.40

b3
1.8e-310.77

c1
1.6e-36.07

c2
1.6e-3
12.61

Tab. 2 Average standard deviations of transformation parameters and percentage of significant
parameters in accepted trajectories.

Fig.11. Velocity vector display with vectors belonging to cuboids with at least one significant nontranslation 3-D affine transformation parameter coded in green (Maas et al., 2009).

6. Outlook
The paper presented a volume based tracking technique to compute Eulerian 3-D velocity field
information from time-resolved voxel space representations. Mainly due to its adaptivity to cuboid
deformation and rotation, 3-D least squares matching (3-D LSM) forms a rather interesting- 10 -

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

alternative to conventional 3-D cross correlation. It could be shown that an increase in precision andreliability can be achieved, if scale, rotation and shear information were considered in the tracking
process. Moreover, these parameters enable the determination of a shear tensor for eachinterrogation cube.
Future work will concentrate on extension of the linear transformation model in 3-D LSM byintroducing higher order polynomials. The resolution of the velocity field may also be improved by
identifying individual particles in voxel space and tracking those particles, using the results of thevolume-based tracking as good approximation.

References
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Förstner, W., 1984. Quality assessment of object location and point transfer using digital image
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Grün, A., 1985. Adaptive least squares correlation - a powerful image matching technique. In:
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Scarano, F., 2002. Iterative image deformation methods in PIV. Measurement Science andTechnology 13:R1-R19, IOP Publishing Ltd.
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Reconstruction of Dense Flow Marker Fields. Accepted for publication in the Proceedings of theCommission V, WG V/1 Symposium 2010, 21 – 24 June 2010, Newcastle.
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