Reconstruction Techniques for Tomographic PIV (Tomo-PIV) of a Turbulent Boundary Layer

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(Tomo-PIV) of a Turbulent Boundary Layer

Callum Atkinson, Dillon-Gibbons Craig, Sophie Herpin, Julio Soria

To cite this version:

Callum Atkinson, Dillon-Gibbons Craig, Sophie Herpin, Julio Soria. Reconstruction Techniques for

Tomographic PIV (Tomo-PIV) of a Turbulent Boundary Layer. 14th Int Symp on Applications of

Laser Techniques to Fluid Mechanics, Jul 2008, Lisbon, Portugal. �hal-03123179�

Reconstruction Techniques for Tomographic PIV (Tomo-PIV) of a Turbulent Boundary Layer

Callum H. Atkinson

, Craig J. Dillon-Gibbons

, Sophie Herpin

, Julio Soria

1: Laboratory for Turbulence Research in Aerospace and Combustion, Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Australia, Callum,[email protected]

Abstract To date Tomo-PIV has involved the use of the multiplicative algebraic reconstruction technique (MART) where the intensity of each 3D voxel is iteratively corrected to satisfy one recorded projection, or pixel intensity, at a time. This results in reconstruction times of multiple hours for each velocity field, explaining why Tomo-PIV has typically been limited to data sets of 1400×130×700 voxels and 500 realizations (Elsinga et al., 2007). In this paper we present an alternative reconstruction algorithm based on a multiplicative line-of-sight estimation and a single iteration of the simultaneous algebraic reconstruction technique (Andersen and Kak, 1984), which we shall call MLOS-SART. Monte Carlo simulations of reconstructions are presented for both the MLOS-SART and MART algorithms. Application of MLOS- SART and MART to a turbulent boundary layer at Re

= 2200 using a 2 camera stereo system are discussed.

Reconstruction techniques are compared with existing Stereo-PIV data (Herpin et al., 2007a,b).

1. Introduction

Tomographic particle image velocimetry (Tomo-PIV) is a recently introduced tool for three- component three-dimensional (3C-3D) velocity field measurements (Elsinga et al., 2006), in which multiple instantaneous views of a seeded flow field are used to estimate the three-dimensional distribution of particles in the flow. 3D cross-correlated is then used to determine the particle displacements and the corresponding 3C-3D velocity vectors. The requirement for instantaneous imaging and the limited number of cameras and optical access points that are commonly found in fluid measurement facilities, limits the types of tomographic techniques that may be applied in fluid flow measurements (Michael and Yang, 1991). Algebraic reconstruction techniques (Herman and Lent, 1976) are well suited to handling the limited views and arbitrary viewing angles, but are computationally demanding and have hence not been commonly employed in fields such a medical diagnostics (Mueller and Yagel, 1999), where unlimited and uniformly spaced viewing angles have favored faster Fourier based techniques.

To date Tomo-PIV has involved the use of the multiplicative algebraic reconstruction technique

(MART) (Hain et al., 2007; Elsinga et al., 2007; Schröder et al., 2007) and has typically been

limited to volumes of 1400×130×700 voxels and only 500 realizations or velocity fields per data

set. This stems from both the computational demand and storage required to perform each

reconstruction. Typical reconstructions times for a single reconstructed volume pair of

730×730×184 voxels, based on 5 MART iterations, have been quoted at approximately 1 hour per

volume object (Scarano et al., 2006). For 16-bit intensity fields this corresponds to reconstructed

volume file sizes on the order of 200 MB. Based on this it would be expected that for larger

volumes on the order of 4000×2600×500 voxels, corresponding to the use of PCO4000 (4008×2672

pixel) cameras, approximately 50 hours would be required per reconstruction with a resulting file

size of approximately 10 GB per volume object or 20 GB per volume pair. This imposes a

significant restriction on the applications of Tomo-PIV, especially in study of turbulence where

thousand of realizations are typically required. If this technique is to develop into a common tool

for complex 3D flow topology investigations then significant improvements in reconstruction times are required.

In this paper we present an alternative reconstruction technique based on an initial projection line- of-sight multiplication step that drastically reduces the number of voxels that have to be considered in subsequent iterative intensity corrections. The combination of this multiplicative line-of-sight (MLOS) with a single iteration of the simultaneous algebraic reconstruction technique (SART), referred to as MLOS-SART, was found to produce particle reconstructions on par with the MART technique for approximately 30 times less processing time than the MART algorithm. System memory requirements were also significantly reduced, as were the volume output file sizes.

Applications of both the MLOS-SART and MART algorithm to a turbulent boundary layer are presented. A Stereo-PIV setup is used to enable the comparison of these results with Stereo-PIV measurement.

2. Tomographic reconstruction

In Tomo-PIV projections of an illuminated particle seeded volume are recorded using multiply cameras or CCD arrays. The intensity recorded by each pixel P

on the array represents the integration of intensity along the pixel’s line-of-sight s

through the volume:

P

=

I x,y,z ( ) ^ds

∫

⁽¹⁾

where I(x,y,z) represents the intensity source fuction of the illuminated particle field. A schematic of this can be seen in figure 1. The aim of tomographic reconstruction is therefore to invert equation 1 and determine the intensity distribution and hence the particle locations, within the volume.

Fig. 1 Schematic of multi-camera algebraic reconstruction technique. Filled voxels represent particle locations required to satisfy the filled pixels in each CCD or camera projection

Medical and astronomical imaging typically makes use of Fourier reconstruction techniques involving convolution. Such techniques are highly efficient, but generally require a large number of uniformly spaced projections (Michael and Yang, 1991). Such projections may easily be acquired using a translating sensor and a stationary target, as is done in computed axial tomography or CAT scans. This luxury is not however available in unsteady fluid flows. Instead simultaneous recordings

CCD 1

CCD 2

CCD 3 Pixel line-of-sight

Voxels

Pixels y x

z

using multiple cameras at finite viewing positions will be required. In most experimental flow facilities optical access is also typically limited, meaning that it will rarely be possible to obtain the projection spacing and recording numbers that are typically required by such techniques.

Algebraic techniques such as the multiplicative algebraic reconstruction technique (MART) (Gordon et al., 1970) which operate in the spatial domain are much better suited to the limited view reconstructions that will commonly be encountered in fluid flow measurements. Such techniques involve the division of a volume into a grid of discrete voxel elements (see figure 1). This enables equation 1 to be expressed in terms of a weighting matrix W

that represents the contribution of each voxel to each pixel, such that the integral along a pixel’s line-of-sight can be expresses as:

P

= W

I

j

∑ ⁽²⁾

where I

represents the intensity of each voxel. The intensity value of each voxel is iteratively corrected until the difference between the projected pixel intensity P

and the recorded pixel intensity is minimized. This iterative correction forms the basis of all algebraic reconstruction techniques. The evaluation of a number of different implementations for Tomo-PIV applications showed that the commonly used MART algorithm provided significantly better particle field reconstructions for less iteration than most of the alternatives (Atkinson and Soria, 2007). For this reason the MART algorithm will be considered as the present state of the art in Tomo-PIV.

2.1 Multiplicative algebraic reconstruction technique (MART)

The MART techniques involves a multiplicative correction to the voxel intensity in each iteration k, based on the ratio of the recorded P

to the projected ( Σ

W

I

)

pixel intensity:

I

= I

P

W

I

∑





 





 

µW_ij

(3)

where µ is a relaxation parameter typically chosen between 0 and 2 (Gordon et al., 1970). The intensity of each voxel is corrected to match the intensity of one pixel at a time, until every pixel has been considered and the iteration is completed. This requires m × n operations per camera per iteration and a weighting matrix of the same size, where m is the number of pixels and n the number of voxels. For a 1280×1024 pixel camera and a volume of 1000×1000×200 voxels this would involve 2.6×10

operations per camera per iteration and corresponding weighting matrix file size on the order of 500 TB. Typically the weighting matrix will be pre-calculated based on geometric calibration of the camera arrangement, a process that can require considerable processing time and storage space. Fortunately a given pixel will see only a small portion of the volume, which enables the matrix and number of operations to be reduced via a sub-grid method such as that discussed in Atkinson and Soria, 2007. This sub-grid approach restricts the reconstruction to considering only those voxel within a sub-grid along the pixel’s line-of-sight, reducing the number of operation to m

× nw × nz, where nw is the number of voxels in a plane of the sub-grid and nz is the number of

voxels along the line-of-sight direction. For the case above the reconstruction process is reduced to

2.4×10

operations per camera per iteration with a weighting matrix file size of approximately 4.8

GB. This sub-grid MART implementation will be used from this point on.

3. The MLOS-SART technique

In PIV a typical experimental setup aims to capture around 10 particle pairs in each cross- correlation region, with particle sizes of approximately 2-3 pixels or voxels in diameter. For a 32

voxel interrogation region containing 10 particles approximately 10×3

or 270/32768 voxels should have intensity values higher than the background. In other words approximately 99% of the voxel in each window will have negligible intensity and will play not part in the reconstruction process. If these non-zero voxel can be predetermined then algebraic reconstruction can be limited to roughly 1% of the voxel intensity field. By neglecting these zero-intensity particles it is theoretical possible that a 100× improvement in reconstruction time could be achieved. The following technique takes advantage of this concept via an initial intensity distribution estimation described below and preprocessing of each camera image to set the background to zero intensity.

3.1 Multiplied line-of-sight (MLOS) estimation

A multiplied line-of-sight approach attempts to isolate expected non-zero voxels by using the calibration mapping to consider the intensity of each pixel that is viewing a given voxel. If we consider one of the black voxels in figure 1, we can trace its projection to corresponding pixels in each camera via their lines-of-sight. In order for the intensity of this black voxel to be greater than zero, the intensity of each of the corresponding pixels must also be greater than zero. If each of these pixel intensities is greater than zero then their product is likewise greater than zero. This enables the use of the multiplication of corresponding pixel intensities to rapidly determine whether or not a voxel will have a negligible intensity. Corresponding pixels can be found directly from the calibration of each camera. For instance if calibration is performed using the multidimensional polynomial fit (Soloff et al., 1997), then the polynomial equations for each camera that can readily be solved to determine where any voxel in the volume (x, y, z) maps to in each camera plane (X, Y):

X = a

+ a

x + a

y + a

z + a

x

+ a

xy + a

y

+ a

xz + a

yz + a

z

+ a

x

+ a

x

y + a

xy

+a

y

+ a

x

z + a

xyz + a

y

z + a

xz

+ a

yz

Y = b

+ b

x + b

y + b

z + b

x

+ b

xy + b

y

+ b

xz + b

yz + b

z

+ b

x

+ b

x

y + b

xy

+b

y

+ b

x

z + b

xyz + b

y

z + b

xz

+ b

yz

(4)

Fig. 2 Schematic of the multiplied line-of-sight (MLOS) approach to determining non-zero voxels

200 0 0 0 0

0 0 0 0 2400

0 0 0 1500 0

0 10

0 50

0

80 0

20 0

0 30

0

Multipled intensities Voxels

Lines-of-sight

CCD 2 CCD 1

Pixel intensities

Once the corresponding camera coordinates have been determined for a given voxel, the associated pixel intensities can be determined by either a nearest neighbor or interpolation approach. The pixel intensity from each camera can then be multiplied to provide an initial estimate of that intensity in that voxel. An illustration of this method is provided in figure 2. If this is done for each voxel the result will be a volume populated by a series of zero intensity voxels, where no particle can be present, and a smaller set of non-zero voxels. These non-zero voxels represent points where lines- of-sight of non-zero pixels intersect and where particles maybe present. It is important to note that this method does not account for the possibility of ghost particles.

Ghost particles in this sense are defined as regions of non-zero intensity that do not correspond to actual particle locations. They are essentially reconstruction artifacts that result from there being multiple possible voxel intensity distribution that can satisfy a given set of projections. Without prior knowledge of the true particle locations it is not possible to distinguish between true and ghost particles. Monte Carlo simulations show that ghost particles are typically of a lower intensity than true particles (Elsinga et al., 2006; Atkinson and Soria, 2007), which fortunately minimizing their effect on the cross-correlation peak. This lower intensity is tied to the nature of algebraic reconstruction techniques. When multiple possible particle locations exist the corresponding pixel intensities are divided between these locations, such that the projection matches the recorded image.

The consequence of this is that the intensity in the image must be split between multiple locations, resulting in less intense ghost particles. This is a desirable effect when it comes to cross-correlation of the reconstructed volumes. Unfortunately the MLOS approach alone provides no such intensity attenuation.

3.2 Simultaneous algebraic reconstruction technique (SART)

In order to reduce the relative intensity of ghost particles an iterative voxel correction is still required. The advantage of the initial MLOS step is that the intensity corrections now only involve a small percentage of the total voxels in the reconstruction volume, requiring fewer operations and less system memory. The simultaneous voxel correction provided by the SART algorithm ( Andersen and Kak, 1984) is favored over MART in this case owing to the arrangement of data that results from the MLOS approach. Since only a small number of voxel will need to be considered, it is no longer necessary to pre-calculate the large weighting matrix W

. In fact the weighting can now be calculated on the fly, based only on the intersection of a non-zero voxel’s projection to the image plane with the nearest pixels. An entire projection image can then be rapidly determined by summing the influence of one non-zero voxel at a time and mapping this to corresponding pixels, rather than concentrating on one pixel at a time and considering every voxel that falls on this pixels line of sight, as done in MART. Consider the SART algorithm below:

I

= I

+ λ

P

− W

I

n=1

∑

W

n=1

∑





 

 





 

 

i

∑ ^W

W

i

∑ ⁽⁵⁾

where Σ W

I

represents the projection of all voxels to a given pixel or in other words the intensity

of the projected image at pixel i. The weighting influence of all voxels on a given pixel Σ W

can

easily be obtained in the same manner as the projected image by assuming each non-zero voxel has

an intensity of unity. P

- Σ W

I

is simply the difference between the recorded and projected image, and λ a relaxation parameter typically taken to be less than unity. The correction for each voxel j can then be determined based on a sub-grid of pixels in the vicinity of the voxel’s projection.

4. Reconstruction simulations

In order to access the reconstruction accuracy of this two step MLOS-SART techniques a series of 3D particle distributions and corresponding projection images were simulated. A 100×100×100 voxel volume was used with 3 linear projections taken in a common plane at angles of -30, 0 and +30 deg around the y-axis, similar to the arrangement in figure 1. The projections simulated for each camera consisted of a 150×100 pixels, with 300 spherical particles of 3 voxel diameter Gaussian intensity randomly located within the volume. 10 particles fields and corresponding projections were generated. Reconstructions were performed using both the sub-grid MART algorithm and the MLOS-SART approach, with parameters shown in table 1. In the case of MLOS- SART no initial intensity is assumed, however the results of the MLOS step were used as an initial intensity distribution for the SART algorithm. In all cases considered in this paper a single SART iteration was found to be sufficient.

Table 1 Simulation reconstruction parameters

Parameter MART MLOS-SART

Initial intensity 1 MLOS

Relaxation parameter 1 1

Iterations 5 1

The overall reconstruction quality was evaluated by determining a correlation coefficient Q between the reconstructed intensity field I

and the generated Gaussian sphere intensity field I

:

Q =

I

I

∑

I

∑

^I

2

∑

⁽⁶⁾

The percentage of reconstructed ghost particles relative to true particles was examined via the use of a region merging techniques. This involved merging interconnecting non-zero intensity voxels into discrete region, which were then compared to the true particle locations.

Results from 10 random particle fields are shown in table 2 and an example of the original and reconstructed intensity fields can be seen in figure 3. While MART was seen to produce a slightly higher reconstruction coefficient, a significant reduction in processing time can be seen when the MLOS-SART approach is used. A slight reduction in the number of ghost particle was also observed.

Table 2 Reconstruction Results

Parameter MART MLOS-SART

Q (mean ± std) 0.772 ± 0.012 0.679 ± 0.013

Ghost particles (%) 68.31 65.07

Time volume pair (sec) 71.52 3.69

(a) (b)

(c)

Fig. 3 Iso-contours of Voxel intensity fields. (a) Generated Gaussian spherical particles, (b) MART

reconstruction, (c) MLOS-SART reconstruction

5. Application to a turbulent boundary layer

In order to compare the accuracy and processing time of the MART and MLOS-SART algorithms under proper experimental conditions, it was decided to attempt to run both algorithms on existing Stereo-PIV images of a turbulent boundary layer at Re

= 2200. A through discussion of these images is provided in Herpin et al., 2007a,b. A single stereo system, consisting of 2 PCO 4000 (4008×2672 pixel) CCD array cameras was used to view a wall-normal stream-wise section of the boundary layer at a nominal off-axis stereo angle of θ = 30 deg (see figure 4). This setup involved a lightsheet thickness of 0.8 mm, with a spatial resolution of 0.02 mm/pixel, enabling a slightly larger reconstruction volume of 80.16×53.44×1.28 mm or 4008×2672×64 voxels to be used.

Fig. 4 Schematic of Stereo camera arrangement

Tomographic reconstruction was performed using both the MART and MLOS-SART algorithms with the same parameters as shown in table 1. In both cases the images were preprocessed to remove the background intensity as required by the MLOS-SART algorithm. This was done by subtracting an average image from each camera, then thresholding each image to a uniform zero intensity background. Gaussian smoothing was also applied in conjunction with local intensity normalization. Cross-correlation was then performed using an in-house multipass 3D PIV program, using 64

voxel interrogation regions with no overlap. Results from Stereo-PIV analysis are shown for comparison.

(a) (b)

Fig. 5 Comparison of in-plane velocity vectors. Black vectors denote Stereo-PIV velocity fields in each case:

(a) MART and Stereo-PIV vectors, (b) MLOS-SART and Stereo-PIV vectors

1 2

x z y

Water-prism θ

Flow

Illuminated volume

Water-air interface

(a)

(b)

(c)

Fig. 6 Instantaneous velocity fields and contours of out-of-plane velocity: (a) Stereo-PIV, (b) MART, (c) MLOS-SART

The instantaneous velocity fields (figure 5) show little difference in the in-plane velocity

components obtained from each technique. A more significant departure appears to be present in the

out-of-plane velocity component, however this difference is typically less than a 0.5 voxel

displacement. The advantage of the MLOS-SART processing can be seen in the required

reconstruction time for a single volume pair. MART reconstruction of this 4008×2672×64 voxel

volume required 6 hrs and 46 mins, not including the time taken to pre-generate the weighting

matrix. In comparison MLOS-SART required only 13 minutes, indicating a reconstruction

approximately 30 times faster than the MART algorithm. Since only non-zero voxel are considered

it is possible to now store only the non-zero voxel intensity values, which based on previous estimates should results in output volume intensity files in the vicinity of 5% of the full voxel grid output file size. In this particular implementation of the reconstruction algorithms, the requirement to only maintain a record of non-zero voxel also reduced the algorithms memory requirement from 6 GB of RAM for MART down to only 525 MB for MLOS-SART.

6. Conclusion

A new algorithm for Tomo-PIV based on an initial multiplied line-of-sight and simultaneous algebraic reconstruction technique is described. Monte Carlo simulations show a similar reconstruction quality to the MART algorithm with a 30 times reduction in reconstruction time.

Tomographic reconstructions of turbulent boundary layer flow are performed, based on a standard 2-camera Stereo-PIV system. Preliminary results for both reconstruction algorithms show that the instantaneous velocity field displacement vary by less than 0.5 voxels from the Stereo-PIV results.

Results indicate a promising improvement in processing time, with reconstruction of a 4008×2672×64 voxel volume now requiring only 13 minutes of processing time per volume pair.

Acknowledgements

The support of Australian Research Council for this work is gratefully acknowledged. C.H.

Atkinson and C.J. Dillon-Gibbons were supported by an Australian Postgraduate Scholarship while undertaking this research.

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