# Time-resolved Three-dimensional

# Particle Tracking Velocimetry Algorithm

A typical time-resolved **three-dimensional particle tracking velocimetry (4D-PTV)** technique involves four recursive steps inspired by **Shake The Box (STB)**, as shown in Fig. 1.** **These steps are particle reconstruction, track initialization, prediction, and optimization.

**What is particle reconstruction?**

Particle reconstruction with triangulation is a process that converts multi-view two-dimensional (2D) particle images to particle positions in a 3D domain. However, the triangulation only works for sparse particle concentrations lower than 0.001 particles per pixel (ppp). The number of ghost particles drastically increases in triangulation for higher particle concentrations due to overlapping particles. Hence, Wieneke2 proposed IPR with an additional step to overcome the triangulation inaccuracy. In Iterative Particle Reconstruction (IPR), the triangulation is followed by an iterative optimization procedure that searches for the best particle position minimizing the intensity discrepancy between the original and the reprojected particle image.

**What is track initialization?**

The reconstructed 3D positions from IPR are thereafter fed into the initialization part (see** **Fig. 1). Track initialization is a process that builds tracklets from 3D particle positions. The idea of initializing a possible track in four frames, known as four frame best estimate (4BE), has been widely used in Lagrangian Particle Tracking (LPT) / PTV studies.

**What is prediction?**

After the tracklets of the first few frames are built, the prediction function then estimates positions of the next time step (tn+1) using polynomial or Wiener filter predictors. Prediction is the most tricky part of **3D particle tracking. **The whole particle tracking process fails If the prediction function fails to follow the flow behavior.

**What is optimization?**

The optimization takes part from the predicted positions until the optimal positions at tn+1 are found. The most straightforward optimization technique is using **Shak****e-The-Box (STB)** by a simple quadratic process called "Shaking". Generally, we try to minimize the discrepancy between the predicted position and recorded signals (camera images) and then pick the optimal solution.

During the prediction-optimization phase, particles continuously enter the domain. Those new entry trajectories must be fed into the tracked poll; otherwise, there would eventually be no tracks left since all tracked particles would have left the domain for a flow with a main advection. In complex flow motions or with high particle concentrations, some particles lose their trajectories due to the optimization failure. In this scenario, the lost particles are kept in the residual images, but their tracks will be removed from the tracked poll (see Fig. 1). It is vital to reconstruct those lost particles and build tracklets since an increasing number of lost tracks will lead to the divergence of the tracking algorithm.

### References:

D. Schanz, S. Gesemann, and A. Schroeder, “Shake-The-Box: Lagrangian parti- cle tracking at high particle image densities,”

*Exp. Fluids*57, 1–27 (2016).B. Wieneke, “Iterative reconstruction of volumetric particle distribution,”

*Meas. Sci. Technol*. 24, 024008 (2013).S. Tan, A. Salibindla, A. U. M. Masuk, and R. Ni, “Introducing OpenLPT: New method of removing ghost particles and high-concentration particle shadow tracking,”

*Exp. Fluids*61, 1–16 (2020).

**VIDEO****. 1.** In the 3rd Workshop and 1st Challenge on Data Assimilation & CFD Processing for PIV and Lagrangian Particle Tracking, we proposed a novel track initialization technique as a complementary part of 4D-PTV, based on local temporal and spatial coherency of neighbor trajectories.

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Cite as: Ali Rahimi Khojasteh*, *Yin Yang*, *Dominique Heitz*, and *Sylvain Laizet , "Lagrangian coherent track initialization", Physics of Fluids 33, 095113 (2021) https://doi.org/10.1063/5.0060644