'Shake The Box' - a 4D PTV algorithm: Accurate and ghostless ...

17th International Symposium on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2014

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‘Shake The Box’ - a 4D PTV algorithm: Accurate and ghostless reconstruction of Lagrangian tracks in densely seeded flows

Daniel Schanz*, Andreas Schröder, Sebastian Gesemann

German Aerospace Center (DLR), Inst. of Aerodynamics and Flow Techn., Dep. of Experimental Methods, Germany

* correspondent author: [email protected] Abstract The ‘Shake The Box’ (STB) PTV - method for the accurate reconstruction of particle tracks in densely seeded flows was first introduced in the scope of the 10th International Symposium on Particle Image Velocimetry (PIV13) in 2013 [Schanz 2013b]. At the time it was shown that the proposed method of incorporating the temporal information into the process of particle position reconstruction is able to improve results in various aspects compared to conventionally used techniques. However, only an experimental dataset was available to explore the abilities of the technique [Schanz 2013b]. The aim of the current work is to quantify the key features of the STB-method - and to identify the benefits and drawbacks. To this end, a scheme for the creation of synthetic tracks from input - vector volumes was created, allowing the calculation of physically (sufficiently) correct track systems at arbitrary seeding densities and time separations. Projections of such track systems on virtual cameras are used as input data for both STB, as well as an MLOS-SMART [Atkinson 2009] tomographic reconstruction algorithm. The results are compared with respect to positional accuracy of reconstructed particles, occurrence of ghost particles and reconstruction time. It is shown that, after a sufficient number of snapshots to allow for convergence of the algorithm, the STB-method is able to correctly identify nearly all particles in existence, even for high seeding density up to 0.125 particles per pixel (ppp). On the same time, the average position error stays very low (0.023 px for 0.125 ppp) – a gain of more than an order of magnitude compared to MLOS-SMART. The occurrence of ghost particles is almost completely suppressed (ghost level below 0.2 percent for 0.125 ppp). The efficient utilization of the temporal information allows a significant reduction in processing time, with STB being 4 to 6 times faster compared to MLOS-SMART. The introduction of noise to the particle images reduces the achievable accuracy for both STB and SMART. For high noise levels, STB shows positional errors up to 0.25 pixel (which can be improved to around 0.14 pixel using temporal fitting of the reconstructed trajectory). Still, the vast majority of particles is found and ghost levels stay very low also for these cases. As the temporal information plays such a dominant role in the STB method – both in the gained results, as well as in the reconstruction process itself – the method cannot be described as pure particle reconstruction, but rather as a method to reconstruct the full temporal development of tracer particles within their residence in the measurement volume. Therefore it is more appropriate to speak of a track reconstruction scheme - or 4D PTV.

1. Introduction The evaluation of time-resolved particle-based experimental investigations of three-dimensional flow fields typically falls into two different domains: on the one hand tomographic PIV (TOMO-PIV) [Elsinga 2006] which relies on a tomographic reconstruction of the particle distribution and a subsequent three-dimensional cross-correlation; on the other hand 3D PTV-methods (e.g. [Maas 1993]), which triangulate particle positions and try to find matching occurrences in successive time-steps in order to form Lagrangian tracks. Both methods have advantages and disadvantages: TOMO-PIV is very robust and allows the use of high seeding densities. The underlying Eulerian grid enables easy computation of derived values, such as vorticity from the large number of tracer particles. Yet, the occurrence of ghost particles tends to bias the velocity results (especially for high seeding densities), the cross-correlation introduces a window-size dependent spatial smoothing and the whole processing is computationally costly. 3D-PTV yields precise results at the location of the tracer particles without any spatial smoothing, while a temporal smoothing allows for better accuracy. Particle statistics of velocities and accelerations can give a different insight to flow


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conditions, compared to a regular grid. On the downside, the methods are only applicable to seeding densities that are around one order of magnitude lower, compared to TOMO-PIV, typically limiting PTV to the evaluation of mean values (as too few tracer particles are present within each instantaneous shot). The ‘Shake The Box’ – method [Schanz 2013b] introduces a novel evaluation approach: it relies on the principle that a particle whose trajectory is known for a certain number of time-steps should (a) not disappear within the measurement volume and (b) its position should be quite accurately predictable for the next time-step, based on a fit on the previous time-steps. Small errors introduced in the prediction can be corrected using image matching schemes, as introduced in [Wieneke 2013]. Using this principle, it is possible to use information that was gathered from past time-steps to help with the reconstruction of the current time-step, allowing greater accuracies, higher seeding densities and faster reconstruction times. It has been shown that the method is able to process densely seeded experimental data and to extract high-quality tracks in great number and with high track-lengths [Schanz 2013b]. However, a precise characterization of results is difficult using experimental data; therefore the current work focuses on the application of the STB-method to synthetic data with known ground truth. In the next chapter, the working principle of STB will be explained, followed by a description of the process of creating synthetic tracks. Subsequently, the parameters and results of the application of STB will be presented, a comparison to results gained by tomographic reconstruction and a study on the effects of image noise conclude the work. 2. Scheme of the STB- method

Conventional methods of evaluating three-dimensional particle-based measurements rely on an individual treatment of every single snapshot of the particle distribution and a subsequent treatment of the results for consecutive images (volume correlation for TOMO-PIV, partner search for 3D PTV). However, when sufficiently time-resolved data is present, the solutions to the reconstruction problem differ only by the movement of the particles from step to step. It is therefore possible to utilize results from previous time-steps in order to optimize the reconstruction of the current one. Under the assumption that the trajectories of (nearly) all particles within the system are known for a certain number of time-steps, the STB-method scheme for a single time-step is as follows:

1. Fit a function (polynomial) to the last n positions of every tracked particle. 2. Predict the position of the particle in the next time-step n+1 by evaluating the fitted

polynomial. 3. Shake the particles to their correct position and intensity, eliminating the error introduced by

the prediction - this step is realized using an image matching technique discussed later. 4. Find new particles entering the measurement zone using triangulation on the residual

images. 5. Shake all particles again to correct for residual errors. 6. Iterate steps 4 and 5, if necessary. 7. Add new tracks for all new particles that are identified within four consecutive time-steps. 8. Remove particles from track system either if leaving the volume or if intensity falls below

threshold.

The process of ‘shaking’ the particles was first described by Wieneke in the scope of the technique of ‘Iterative reconstruction of Volumetric Particle Distribution’ (IPR) [Wieneke 2013]. This method allows for the accurate reconstruction of particle positions in a single image with seeding densities up to 0.05 ppp by using an iterative process of triangulation and particle position refinement. Particles found by triangulation are moved (shaken) around the volume in small steps (0.1 px per


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direction of space), until a minimum of the local residual images (‘image matching’, including the use of a calibrated optical transfer function, OTF [Schanz 2013a]) is found. After this process, new particle candidates are triangulated from the (now sparser) residual images and again shaken into the optimal position. The steps 4 - 6 of STB basically represent the IPR-method with reduced number of iterations, as only few particles have to be considered. For more detailed information on the IPR-process, see [Wieneke 2013]. Using the described scheme, the algorithm is capable of working through long time-series, extending known tracks step-by-step and continually adding particles that enter the measurement zone to the list of tracked particles. As the major part of the system is already solved after the prediction, only small errors need to be corrected by shaking and by adding new particles. Therefore, the method is computationally very effective. However, the algorithm can only work efficiently if the majority of the present particles is known and can be predicted. It is therefore very important to initialize the track system in a way that allows STB to converge to the correct number of tracked particles. In the current study, this initialization is done in a relatively simple way: For the first four time-steps, the particle distribution is determined using the IPR-method with a high number of iterations. Within these four time-steps, particles that follow a reasonable trajectory (with deviations from a linear fit below a certain threshold) are identified. This initial track-finding is supported by a 3D vector field, gained by classical TOMO-PIV evaluation of the first four time-steps, which is used as a predictor for the search of fitting particles. For higher seeding-densities, the IPR-method produces significant numbers of ghost particles per time-step – yet, these are greatly reduced by the temporal condition of connected tracks within the four images. The particles identified within the initialization process are added to the list of tracked particles and their positions are extrapolated to the fifth time-step. From there on, the scheme depicted above is applied with a reduced number of iterations and no assistance by correlation results.

3. Creation to synthetic tracks

In order to quantitatively capture and compare the accuracy of different methods, a process to extract synthetic tracks from a known vector field was designed. For the first image, particles are randomly distributed in the selected domain; the velocity of the particles is calculated as the Gaussian weighted average of the eight neighboring velocity vectors of the source vector volume. In the next step, the particles are moved according to the determined velocity and the chosen time separation. Thus, smooth particle tracks following the source vectors can be created. To guarantee constant densities, particles leaving the measurement domain are reinserted at a suitable location of the volume border. To this end, all source velocity vectors located at the interfaces of the volume are examined. A list of such vectors with a component directed to the inside of the volume is created, weighted with the magnitude of this component. To determine the reinsertion point of a particle, first the general region is chosen randomly from the weighted list of interface vectors. Second, a position within a radius of 15 pixels around this point is determined randomly. As source vector volume, a result from an experimental dataset of the flow behind a series of periodic hills was chosen. This time-resolved TOMO-PIV dataset was already used to yield the first STB-Results given in [Schanz 2013b]. The experimental setup is explained in detail in this publication. Evaluations of the experiment are also presented on the 17th Lisbon conference by Schröder et. al [Schröder 2014]. TOMO-PIV processing of the images gained by six PCO Dimax / LaVision Imager Pro HS (4 MegaPixel each) yield vector data in a volume of 90x94x20 mm3. The voxel spaces used for tomographic reconstruction are of dimension 1951x2035x405 voxel. Cross correlation was performed using a final window size of 483 voxels with 75 % overlap, resulting in 163x170x34 vectors with a spacing of 12 voxels. From the vector results, a case offering high dynamic range was chosen. A subvolume of 1000x1000x400 was cut from the middle of the


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volume and the vector results originating from this volume were used as source vectors for the track creation. For the placement of reinserted particles, a border of 30 non-imaged voxels was left at each interface of the volume, so that an imaged volume of 940x940x340 voxel remained. An important prerequirement is that the source vector field is divergence free, as otherwise particles accumulate in sinks. To this purpose, a routine from LaVision Davis 8 was used to ensure divergence free input vector volumes.

Fig. 1: (Top) Vector slices from the divergence-free source vector volume used to create synthetic tracks. (Bottom) Tracks created for ppp = 0.01. Only 15 time-steps are shown. Flow in positive x-direction, color coded stream-wise velocity.


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As time separation, the original sampling rate was chosen, resulting in a mean particle displacement of around 6 px and a maximum displacement of around 11 px. Fig. 1 illustrates the vector volume used as source and shows tracks created for a low seeding density case of 0.01 ppp. Only 15 time-steps are shown in order to be able to discern single tracks. The particle positions determined from the track creation scheme are projected using parallel projection onto four virtual cameras (1200x1200 pixels each) in pyramidal configuration with a square basis and an angle of +– 30° in x- and y- direction. As weighting function (OTF), a two-dimensional Gaussian peak is used, which leads to particle diameters of 3-4 pixel. Seeding densities ranging from 0.01 ppp to 0.125 ppp (calculated in respect to the imaged volume – in this case ppp = Np/940/940, with Np: number of particles) were realized. Fig. 2 shows excerpts of a camera image for three seeding densities.

Fig. 2: Details from camera image for ppp = 0.01, ppp = 0.05 and ppp = 0.125 (from left to right). 4. Application of STB to synthetic data

The Shake The Box – Scheme was applied to the particle images originating from synthetic tracks. For initialization, the first four time-steps were treated with IPR as follows: the allowed triangulation error was set to ε = 0.5 px for all iterations; each triangulation iteration was followed by 8 shake-iterations; 6 triangulation iterations using all 4 available cameras were conducted, followed by 4 triangulation iterations using a subset of 3 cameras (thus 4 triangulations in total for every of these iterations, each with one camera missing; done to reduce the effects of overlapping particles). In the resulting particle distributions, 4-step tracks are identified and those are chosen that show a total deviation of less than 1.5 pixel in every direction of space from a linear fit. These tracks are moved to the list of tracked particles and the position prediction is performed. For all consecutive steps, the number of triangulations is reduced to 3 triangulations using all cameras and 2 triangulations using a subset of cameras. The number of shake-iterations remains unchanged. In the prediction step, a second order polynomial was fitted to the four last time-steps and extrapolated to the current one. New particles are added to the tracking system in a similar way as performed in the initialization (4-step tracks with less than 1.5 px deviation from linear fit). Time-series of 50 images were processed for each seeding density. Fig. 3 shows the temporal development of three parameters, describing the quality of the reconstruction. The reconstructed tracks are compared to the original ones by searching for reconstructed particles in a radius of 1 px around every real particle. The deviation of found real particles to the original position is determined and averaged.


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Fig. 3: Temporal development over 50 successive images of STB-runs for different seeding densities. The right side always displays a zoom into time-steps 40-50, where the algorithm has converged for all cases. (First row) Number of real particles not being detected (within 1px); (Second row) Development of the fraction of reconstructed ghost particles; (Third row) mean positional error for all reconstructed real particles

For lower seeding densities (<= 0.05 ppp), the track initialization is very effective in finding nearly all real particles: For 0.05 ppp only 2 percent of the real particles are not found by the initialization. When using higher seeding densities, this value quickly rises: For 0.075 ppp already 19 percent of the particles are not found, for 0.125 ppp the number climbs to 75 percent. The fraction of ghost particles (calculated by searching around all reconstructed particles in a radius of 1px within the source volumes – if no particle is found, the reconstructed one is counted as a ghost particle) is quite low for the first step (around 3 percent for 0.125 ppp) due to the small allowed triangulation error of ε = 0.5 px. For ppp <= 0.05 this fraction is very low (less than 0.1 percent of ghosts) even directly after the initialization, as four time steps are sufficient to eliminate nearly all ambiguities in such low seeding densities. The position error Δp for the reconstructed particles in the first time-steps is very low for ppp <= 0.05 (Δp = 0.002-0.004 px). High seeding densities show much higher


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errors (≈ 0.3 px for 0.125 ppp). The higher the seeing density, the longer the STB-scheme needs to converge. For 0.075 ppp, the system needs 3-4 time-steps to correct for the errors that were introduced during the initialization (not all particles could be found, the ones found show a positional error, as they are shifted by overlapping particle images and the occurrence of ghost particles). The 0.1 ppp case needs around 6-7 time-steps after initialization to converge to stable conditions, for 0.125 ppp this number rises to around 20-22. Especially for this case it can be seen that the initialization is still very compromised with less than 25 percent real particles identified and those with a high uncertainty of around 0.3 px. Therefore, a percentage of the four-step tracks found by the initialization will either be completely false or not predictable and are thrown out of the tracking process again – this explains the notable rise in ghost particles during the first images after the initialization. The particles are not following the correct track and are therefore seen as ghost particles. Step by step these are thrown out of the tracking system, until after around 25 steps they are gone. At the same time, the correctly tracked particles (though only a low fraction at first) help to reduce the complexity of the reconstruction problem and allow the detection of new correctly tracked particles. This process can be seen by the constant decline of the fraction of undetected particles. The accuracy of the detected particles is improved with the identification of more and more real particles, as less ghost particles can shift the particle and as more particle overlap conditions are resolved by correct positioning of the involved particles. Looking at the right column of plots, the converged results for the reconstructions of all seeding densities can be examined. It can be seen that the system stays at a pretty constant state, with only small variations, depending on instantaneous imaging conditions or small variations in particle density. The fraction of undetected particles falls below 0.5 percent, even for the 0.125 ppp case; the fraction of ghost particles is reduced even below 0.2 %. Particle peak accuracy is very high, with Δ < 0.005 px for ppp <= 0.075 and reaching only 0.025 px for ppp = 0.125. It has to be noted that for the 0.125 ppp case, the number of normal triangulation iterations was increased to 4, followed by 3 iterations with reduced camera numbers. Each triangulation is followed by 10 shake iterations. Using the iteration numbers applied to the other cases leads to a significantly slower convergence (around 60 images). Above 0.125 ppp, the algorithm fails to converge: The IPR reconstruction of the first images is highly erroneous in respect to found real particles, number of ghost particles and particle accuracy, such that the algorithm is not able to compensate for these errors in the following steps. For such high seeding densities, different approaches to the initialization have to be attempted - for instance taking the particle peaks from tomographic reconstruction (which is able to reconstruct the majority of the real particles in single images as shown in the next paragraph, albeit with a large amount of ghost particles and position error) for the first images. 5. Comparison to tomographic reconstruction

In order to compare the results to the technique commonly used for processing high-density three-dimensional data, tomographic reconstructions of the synthetic images were performed, using an MLOS-SMART algorithm [Atkinson 2009][Schanz 2013a]. As this technique does not utilize temporal information, only five volumes (time-steps 40-44, enabling a direct comparison to STB data in converged state) per seeding density were reconstructed. Following the MLOS initialization, five iterations of SMART with subsequent volume smoothing and contrast elevation were performed. In order to be able to compare position accuracy as well as the fraction of undetected and ghost particles, a 3D Gaussian peak finder from LaVision Davis 8 was used to identify particle positions within the reconstructed volume.


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Fig. 4: Comparison of results gained by tomographic reconstruction and subsequent particle peak identification to tracking results by STB. Values averaged over images 40-44 of the time series discussed in Fig. 3.

The accuracy determination and particle-/ghost-identification from the original track data was conducted analogous to the STB data. For both MLOS-SMART and STB, the results from steps 40-44 were averaged and are given in Fig. 4 and Table 1, respectively. Looking at the MLOS-SMART tomographic reconstruction, most real particles are correctly reconstructed (99 percent for low ppp, 92 percent for 0.125 ppp) with a positional error that rises from 0.13 px for 0.01 ppp to 0.31 px for 0.125 ppp. The fraction of the summed ghost particle intensity to the summed real particle intensity is low for ppp <= 0.025, but rises quickly with increasing ppp. For 0.125 ppp, the ghost particles contain more energy that the reconstructed real particles (fraction 1.3). As given in Table 1, the number of detected ghost particles surpasses the number of real particles by 0.075 ppp, however as the average intensity of a ghost particle is lower than for a real particle, the intensity fraction is around 0.5. For 0.125 ppp over 280.000 ghost particles are found – a factor of 2.5 to the real particle number of around 110.000. The peak accuracy results for MLOS-SMART are in good agreement to the values given for MART by Wieneke in [Wieneke 2013]. However for high ppp, both the number of correctly detected real particles as well as the number of ghost particles is significantly higher for the MLOS-SMART case, compared to MART in [Wieneke 2013]. As MART and SMART should produce comparable results, this difference is most likely explained by different thresholds used for the 3D peak detection: A higher value leads to a reduction of the detected ghost particles, but also reduces the number of correctly found real particles (those reconstructed with low intensity). Turning to the results gained by Shake The Box, it becomes obvious that the inclusion of the temporal information opens a door to achieve results of much higher quality. The number of correctly identified particles remains above 99.6 %, even for 0.125 ppp. The ghost particle problem is nearly completely resolved – for 0.125 ppp a ghost particle proportion of 0.13 % is determined. In particle numbers, this means that 143 ghost particles occurred for STB, while MLOS-SMART


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produced over 280.000. For lower seeding densities ghost levels of less than 0.05 % can be reached. The (nearly) absolute prevention of ghost particles, combined with the highly accurate image matching (shaking) process leads to a significant increase in achievable accuracy: For low seeding densities (<= 0.05 ppp), errors of around 0.003 px are found – a factor of over 50, compared to tomographic reconstruction. The error rises for higher seeding densities, but still remains below 0.025 px for 0.125 ppp, which is more than an order of magnitude lower compared to SMART (0.31 px) and a factor of five lower compared to SMART at 0.01 ppp (0.13 px). Wieneke already demonstrated the very high accuracy achievable by the image matching process used in IPR - however in case of single images, this hold only for very low seeding densities [Wieneke 2013]. Starting at 0.005 ppp, the error rises and reaches 0.1 px by around 0.03 ppp. For 0.1 ppp Wieneke finds average errors of around 0.6 px for single-image IPR. He writes: “Convergence starts to fail above 0.05 ppp when the solution is no longer unique”. This problem is solved by the inclusion of the temporal domain, as each snapshots provides a new view on the system (essentially adding a new system of cameras, as argued by Novara [Novara 2010]), leaving the whole spatio-temporal system only one solution – the real one – to converge to. The fact that the system is already close to the real solution after the prediction step allows for the low number of used triangulation iterations. Each of these is fast, as the residual images are sparse. This combined effect leads to the computational efficiency documented in Fig. 4. Reconstruction times for a single snapshot on an eight core Xeon server (2 * Xeon E5520 quad core CPUs, 24 GB Ram) are compared for the different seeding densities. It can be seen that STB is 4 to 6 times faster compared to MLOS-SMART. Computation times rise with ppp because of the increasing number of peaks detected on the images (leading to a more complex triangulation process) and the increase in tracked (and shook) particles. The unproportional rise for 0.125 ppp is caused by the increase of used triangulation iterations. For SMART the percentage on non-zero voxels increases, leading to rising computational effort. On larger evaluation machines (48 or 64 cores), the difference between the algorithms is expected to be smaller, as SMART with its large number of parameters (voxels) seems to parallelize better. However, it has to be kept in mind that the reconstructed voxel spaces still need to be processed further (3D cross correlation or particle peak fitting and partner search), while STB directly yields velocity (and acceleration) data.

ppp 0.125 0.1 0.075 0.05 0.025 0.01

Real particles 110868 88584 66439 44233 22060 8904

Undetected particles, SMART [%]

9387 (8.48 %)

3400 (3.83 %)

1119 (1.68 %)

405 (0.91 %)

135 (0.61 %)

47 (0.53 %)

Undetected particles, STB [%] 393 (0.35 %)

210 (0.23 %)

107 (0.16 %)

54 (0.12 %)

15 (0.06 %)

9 (0.10 %)

Ghost particles, SMART 282090 (254.8 %)

206990 (233.7 %)

121680 (183.1 %)

42229 (95.5%)

3721 (16.8 %)

134 (1.5%)

Ghost particles, STB 143 (0.129 %)

66 (0.076 %)

38 (0.057 %)

18 (0.041 %)

7 (0.030 %)

2 (0.023 %)

Avg. position error, SMART [px] 0.308 0.278 0.243 0.201 0.155 0.135

Avg. position error, STB [px] 0.0226 0.0104 0.0046 0.0033 0.0025 0.0023

Table 1: Comparison of tomographic reconstruction (MLOS-SMART) and subsequent particle peak identification to tracking results of STB. Values averaged over images 40-44 of the time series discussed in Fig. 4.


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6. Influence of image noise As shown in the previous chapters, the concept of particle position prediction, followed by a position refinement by image matching yields very good accuracy results for a wide range of seeding densities when looking at perfect imaging conditions. However, image noise will have an influence on several parts of the algorithm: The image matching process will not be able to find a perfect match for the particle position, as the peak information is altered by the noise. Secondly, the triangulation process for identifying new particles will be affected, as noise tends to shift 2D particle position identification. This will directly influence the triangulation error - therefore the allowed value ε has to be altered in order to find a sufficient number of particles. A higher value of ε will lead to a higher probability of ghost particle formation. In order to judge the effects of image noise, datasets have been created for three different noise levels at two different seeding densities. The noise was introduced after imaging the synthetic particle distribution on the four cameras by adding a randomized intensity to every pixel with a value taken from a normal distribution with variance σ derived from the average maximum peak intensity of a particle image Ip,avg. Images with noise levels corresponding to : σ = 0.03*Ip,avg, σ = 0.1*Ip,avg and σ = 0.2*Ip,avg were created for seeding densities of 0.01 ppp and 0.05 ppp. Fig. 5 shows exemplary excerpts of one camera image for the 0.01 ppp - case. The first two cases can be seen as representative for good to very good experimental circumstances (considering noise levels), while the σ = 0.2*Ip,avg – case is approaching experiments with poorly controlled conditions (sparse illumination, small tracer particles). Application of any kind of image preprocessing was intentionally omitted in order to not introduce further parameters. Time-series of 50 time-steps were created and reconstructed both by STB and by MLOS-SMART. Concerning STB, the allowed triangulation error was set ε = 0.85 for the two cases with lower noise and ε = 1.1 for the high-noise case. Results of the STB-track-reconstruction can be examined in Fig. 6 (left side). It can be seen that the convergence-time of the algorithm rises with the noise level. Especially the high-noise case with 0.05 ppp illustrates that the system has to work much harder in order to identify real particles and to get rid of ghost particle tracks. For this case, convergence is reached around 20 time-steps after the initialization, with a then constant ratio of undetected particles of around 0.065. The cases with lower noise level converge faster (3 and 7 time-steps after initialization, respectively). The 0.01 ppp case converges instantly for the lower noise levels, but needs around 3 iterations for the high seeding density. Looking at the mean displacement of the particles it is obvious that the very high accuracies seen in the perfect imaging case cannot be reached. For σ = 0.03*Ip,avg displacement errors of around 0.03 pixels are found. The error rises to 0.1 pixel and 0.24 pixel for the higher noise cases.

Fig. 5: Detail view from camera image for ppp = 0.01 for different levels of artificially added noise: σ = 0.03*Ip,avg, σ = 0.1*Ip,avg, σ = 0.2 * Ip,avg (from left to right). σ is the variance of the normal distribution used for the random noise generation, Ip,avg denotes the average maximum peak intensity of a particle image.


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0 10 20 30 40 500

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snapshot

0.01 ppp, σ = 0.30.01 ppp, σ = 0.10.01 ppp, σ = 0.20.05 ppp, σ = 0.030.05 ppp, σ = 0.10.05 ppp, σ = 0.2

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STB, 0.01 pppSTB, 0.05 pppSMART, 0.01 pppSMART, 0.05 ppp

0.03 0.1 0.20

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σ [Ip,avg]

Fig. 6: (Left): Temporal development over 50 successive images of STB-runs for seeding densities 0.01 ppp and 0.05 ppp for varied amounts of image noise. (Right): Comparison of results gained by tomographic reconstruction and subsequent particle peak identification to tracking results by STB for varied amounts of image noise. Values averaged over images 40-44 of the time series from the left hand side. As soon as the system is converged, the error is largely dependent on the noise level and less on the seeding density – clearly indicating the compromised accuracy of the image matching process as source for the position error. The ghost particle level stays very low for the two low-noise cases. For the high-noise case the ghost level rises. In this case, single noise peaks can be of approximately the same intensity as the particle images. These false particle peaks can lead to


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random ghost particle tracks if they match by chance in four consecutive images. The fraction of ghost particles is indeed higher for the lower seeding density, as less real particles are present to counterweigh the ghost particles. However, also for these cases the ghost level is low with a fraction of 0.025 and 0.06, respectively. Fig 6 (right side) compares the converged STB-results to reconstructions using MLOS-SMART and a subsequent particle peak identification. The fraction of undetected particles is largely similar between the two techniques. The mean positional error rises for both STB and SMART at approximately the same rate. As already discussed, STB does not reach the very high accuracy seen in the previous chapters, but always hold an accuracy advantage compared to tomographic reconstruction. SMART exhibits an error of around 0.39 pixel for the high-noise case, while STB shows around 0.24 pixel. The comparison of the total ghost particle intensity shows – as before – distinct advantages for the STB technique. Maximum values of a fraction of 0.025 for STB are opposed by values of around 2.2 for SMART. The SMART case shows the same pattern as recognized for STB: For high noise levels the ghost ratio is higher for lower seeding densities. It should be noted that no temporal fitting of the particle tracks gained by STB was conducted for the data shown in Fig. 6. A fit of a moving polynomial of length 15 to the track system enhances the accuracy for the two cases with higher noise (average errors of around 0.07 pixel for σ = 0.1*Ip,avg and 0.14 pixel for σ = 0.2*Ip,avg are found). 7. Conclusion

The 4D-PTV scheme, casually termed ‘Shake The Box’ (STB), was introduced in 2013 and at the time applied to an experimental dataset, showing profound advantages over conventionally used techniques to evaluate time-resolved three-dimensional data (tomographic PIV, 3D PTV) [Schanz 2013b]. For the work presented here, the method is characterized in terms of accuracy, completeness of the reconstruction, occurrence of ghost particles and computation time by means of synthetic data. Systems of synthetic tracks were derived from divergence-free source vector fields taken from a tomographic PIV experiment. Time-series of 50 snapshots each were created for different seeding densities ranging from 0.01 ppp to 0.125 ppp and projected onto four virtual cameras using parallel projection. Application of the STB-method to the gained image-series showed that for low to moderate seeding densities (<= 0.05 ppp) the algorithm needs very little convergence time, with nearly all particles being correctly identified by the track initialization. For higher seeding densities, the initialization (consisting of only four time-steps) introduces more and more errors that need to be corrected in subsequent steps. For 0.125 ppp, the algorithm needs around 25 time steps to converge to a stable situation. Above 0.125 ppp no convergence could be accomplished with the currently used initialization system. However, if convergence is achieved the results are highly accurate and complete: For ppp <= 0.075 ppp the average position error remains below 0.005 px, for 0.125 ppp a mean accuracy of 0.023 px is determined. The occurrence of ghost particles is effectively prevented by the system (less than 0.2 % ghost particles for 0.125 ppp), while nearly all true particles are correctly captured (0.35 % missing particles for 0.125 ppp). Comparing the results to tomographic reconstruction (MLOS-SMART), the benefits of incorporating temporal information in the reconstruction process are obvious: SMART suffers heavily under ghost particles for higher seeding densities; the accuracy of particle placement is more than an order of magnitude lower, compared to STB-results. It is feasible to extract track information from SMART-reconstructed volumes by searching particle peaks and connecting fitting partners [Schröder 2011]. This method considerably reduces the fraction of ghost particles [Elsinga 2013]. A subsequent temporal fit of the found trajectories would also improve peak position


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accuracy; however it is doubtful that results resembling those gained by STB can be realized. High quality tomographic reconstructions and velocity information have been shown to be achievable using advanced reconstruction methods that incorporate temporal information, like Motion tracking-enhanced MART (MTE) [Novara 2010] or Fluid trajectory correlation (FTC) [Lynch 2013] or combinations of both. Nevertheless the application of these techniques is computationally very costly. When introducing noise to the particle images, the realized accuracy is reduced. However, at noise levels of a well-conditioned experiment, positional error of under 0.1 pixels should be achievable if using a temporal fit on the particle trajectory. The ghost level remains very low also for images with high noise levels. For images with even worse conditions it is expected that the STB-method starts to break down at some point and that TOMO-PIV will be more suited to extract velocity information due to the robust character of the tomographic reconstruction and the correlation. The STB-method allows for the first time to extract highly accurate Lagrangian information from densely seeded flows. Such information is especially valuable for the validation and development of CFD-code and turbulence models, demanding precise measurements of derivation tensors. Turbulence research benefits greatly from the availability of accurate acceleration pdfs. 8. Outlook

While the ghost level is remarkably low, it is desirable to remove these last residues, as such particles might carry plainly wrong velocity data - potentially biasing velocity- and acceleration statistics. To this end, the algorithm will be modified to allow walking backwards in time. Ghost particles in STB are often particles that were not predicted correctly and have wandered off their position during the shake-iterations. Their place was taken by a new particle that starts a new track at the location the original particle should have been moved to. When walking backwards in time the two tracks - representing the same real particle - can be connected and the remaining ghost particle will be deleted. This method also enables a second processing of the first images that show reduced particle number and accuracies due to the initialization. After the second pass these will show the same accuracy as the ones in the converged state. Concerning the generated synthetic track data it is still to be determined to what extent the origin of the source vector field (3D cross correlation) has smoothed out the most extreme particle accelerations present in real data. This smoothing could potentially give an advantage to the STB-method compared to experimental data, as highly accelerated particles are the most difficult to predict. Variations of the velocity dynamic range of the synthetic data will be processed by STB in order to characterize maximum accelerations that still can be processed by the algorithm. First attempts have been made to apply the STB-idea to short time-series (even down to 2 pulses). For these cases of few measurement points per particle track, the focus is less on the prediction of particle positions (which at the moment is only possible if at least 5 time-steps are present, as the minimum track length is 4). Instead, IPR is applied to all time-steps, yielding particle distributions for every time-step. Within these, matching particles are identified using a predictor vector field gained from TOMO-PIV. Afterwards, only those particles found to be within a track are kept, a residual image is calculated and the process is iterated on this image. This method has shown good results on a four-pulse TOMO-system [Schröder 2013], opening a path to an accurate track-based evaluation of high-speed flows. Acknowledgements The work has been conducted in the scope of the DFG-project “Analyse turbulenter Grenzschichten mit Druckgradient bei großen Reynoldszahlen mit hochauflösenden Vielkameramessverfahren”. The experiment providing the data for the first experimental application of STB, as well as the source velocity field for the synthetic tracks, was funded by the European Research Council through


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the AFDAR project (grant no. 265695). The authors thank Bernhard Wieneke from LaVision for fruitful discussions and providing software to eliminate divergence from a given vector volume. References Atkinson C and Soria J (2009) An efficient simultaneous reconstruction technique for tomographic

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'Shake The Box' - a 4D PTV algorithm: Accurate and ghostless ...

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