# Particle Track Initialisation

## Build initial Single Particle Tracking (SPT)

## 1- Enhanced Track Initialisation

## (4BE-ETI)

**4BE-ETI** is improved version of four frame best estimate. All possibilities around the particle would be considered to be analysed. 4BE-ETI performs original 4BE for each candidate around the particle starting from the first frame. Similarly, the track is built if there is a unique pass. The same predictive function for each possible trajectory is implemented. ETI-4BE can be implemented in complex motions.

**Application:**

Single particle tracking (SPT) schemes

Trajectories with small number of cross intersections

high particle per pixel (ppp) densities

small displacement

**Source code: ****Python code**** **

**Related studies: **

https://doi.org/10.1088/1361-6501/ab0786

## 2- Four Frame Best Estimate

## (4BE)

**4BE** is the most common method to track particles in 2D and 3D trajectories. First track would be built by nearest neighbour technique from t1 to t2. 4BE predicts the next position by linear extrapolation function for further timesteps. Then it looks for nearest candidate around the predicted position among particles at the next timestep. The initial trajectory is tracked if there is unique solution in four following frames.

**Application:**

Single particle tracking (SPT) schemes

Trajectories with small number of cross intersections

low particle per pixel (ppp) densities

small displacement

**Source code: ****Python code**** **

**Related studies: **

https://doi.org/10.1088/1361-6501/ab0786

## 3- Nearest Neighbour

## (k-NN)

This trajectory technique is the easiest way to track a single particle. It takes nearest candidate at timestep tn+1 compared to the particle position at timestep tn. Maximum particle displacement should be taken into account in order to control the method from wrong trajectories. The **nearest neighbour **approach is fastest algorithm to build initial tracks which is highly recommended if particle positions are relatively far from each other.

**Application: **

simple single particle tracking (SPT) schemes

low particle per pixel (ppp) densities

smooth trajectory behaviours

small displacement

**Source code: ****MATLAB code**

**Related studies: **

https://doi.org/10.1093/bioinformatics/btu793

# 4- Lagrangian Coherent Track Initialization (LCTI)

The current initialization technique tries to find coherent tracklets in four frames. It should be noted that particles of a cluster are coherent if they spatially behave together over a finite time. A starting step is required in LCTI for the first time step t1, where there is no neighbor track information. It can be done by a classic four-frame scheme with a narrow threshold to index the most reliable tracks. It is assumed that a track is relatively reliable if it has comparable small velocity and acceleration standard deviations in four time steps to avoid false tracks. The standard deviation of the particle image intensity can also determine whether a possible track is reliable. In practice, the LCTI steps can be listed as the following algorithm:

**FIG. 2.** Schematic view of the LCTI algorithm when two possible four frame solutions exist. (a) LCTI four frame algorithm considering all possible neighbor candidates at t2 followed by linear predictions (blue dash line arrows). Candidate matching at time step t3 inside first search volume (blue circle r1). Second order prediction (red and black dash line arrows) to match possible candidates at t4 inside second search volume (gray circle r2). (b) Coherency check between two possible track matches and neighbor coherent motion.

Referring to the LCTI algorithm, we need to define the search volumes to index possible candidates at each time step. Clark et al. enhanced the probability of finding true tracks by applying adjustable anisotropic search volumes as a function of mean flow direction. Anisotropic means that if the mean flow (obtained from the predicted velocity) is dominant in one direction, the search volume in that direction is larger than in the other directions. Adjustable search volumes introduce local spatial motions (i.e., physics-based information) into four frame schemes, which can significantly tackle the high gradient threshold issues. On the other hand, using the adjustable search volume limits the number of possible candidates, avoiding non-coherent solutions by following the local spatial motion. The search volume in LCTI is based on the local maximum displacement map calculated from neighbor particles. Therefore, the first search volume is computed as a function of neighbor maximum displacements in each spatial direction between t1 and t2 as shown in Fig. 2(a). Then, every neighbor particle inside the search volume at t2 is a candidate. These candidates are in one of the following categories: the true position of the target particle at t2, the true position of other undetected tracks, and noise (i.e., false particle). Afterwards, a linear predictor [blue dash line arrows in Fig. 2(a)] between the target particle at t1 and the possible candidate at t2 is performed for every possible match. Similarly, the second search volume around the predicted position determines which particles are more likely to be in the true position at t3. A possible track is removed if there is no candidate inside the search volume. The pro- cess is repeated for the next time step with a higher order prediction function [red and black dash line arrows in Fig. 2(a)]. A unique four frame solution is expected for flows with low velocity and acceleration gradients or low particle concentrations. When more than one solution exists, LCTI selects the most coherent track to solve the ambiguities, as shown in Fig. 2(b). Otherwise, a particle can spatially meet a group of other particles with no coherency link between them. We recall here that coherent refers to a group of particles that are having the same Lagrangian behavior spatially and temporally.

**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

# Coherent track detection

Recently, Lagrangian Coherent Structures (LCS) have been applied in PIV/PTV experiments for flow structure analyses.17–20 However, previous studies have not yet combined the LCS extraction with the velocimetry algorithms and mostly focused on using LCS as a post-processing tool, to the best of our knowledge. Several methods have been proposed to identify LCS by looking for separatrix regions in time.13,21 Separatrices exist in boundaries (i.e., ridges) between different structures. A schematic view of the boundaries between vortices in a 2D isotropic homogeneous turbulent flow is shown in Fig. 3. Multi-clusters of particles spatially exist in the vicinity of the target particle (the dark blue particle in Fig. 3). All red and blue particles are neighbors of the target particle. However, the trajectories of each colored cluster temporally evolve in separated directions. LCS can be used to determine if a spatially neighbor particle is coherent or non-coherent over a temporal scale.

Suppose the flow is dominated by coherent structures such as in a 2D isotropic homogeneous turbulence illustrated in Fig. 3, the global LCS analysis can extract meaningful boundaries between structures. Difficulties in interpretation arise, however, when the flow carries 3D complex motions and numerous local structures. We, therefore, suggest local coherent structure extractions instead of a global calculation for which only coherent clusters and boundaries over neighbor trajectories are computed. Therefore, the complexity of the global LCS view is simplified into a small number of clusters around the target particle, such as in Fig. 3. In the local view, the curve or surface boundaries divide the local spatial area into discrete regions with different dynamic motions,22 and motions across these boundaries are negligible.23 Furthermore, the LCS boundaries can move, evolve, and vanish in spatial space as the flow pattern changes temporally.

**FIG. 3. **2D schematic of particle motions inside vortices. Each color belongs to a group of coherent clusters. The target dark blue particle with coherent neighbor particles is located in a clockwise vortex (blue cluster), while non-coherent particles belong to different clusters. The target particle is non-coherent with neighbor paticles in the red cluster.

In the local Lagrangian frame, separatrices can be obtained from Finite Time Lyapunov Exponent (FTLE) by measuring the amount of stretching between the target particle and its neighbor particles over finite time.24,25 Raben et al.25 showed that the normalized average error and normalized root-mean-squared (RMS) error of the FTLE map decreases with increased particle concentration. This trend is favorable because ongoing PIV/PTV experiments consistently succeed in achieving higher particle concentrations. Meanwhile, it is less likely to have ambiguities due to multi-possible solutions in low particle concentration cases. As a result, there is no critical need for the coherency check in low particle concentration cases.

A lower FTLE value means the neighboring particle is acting similarly, with no sign of separation with the target particle over the finite time. High values in the FTLE field show the existence of ridges that divide the local area into different clusters of coherent particles. With this formulation, it is possible to index a group of neighbor particles as coherent or non-coherent with the target particle. As we discussed in Sec. II B, the LCTI algorithm checks if the Lagrangian coherency is valid for each possible four frame tracks to avoid non-coherent reconstructions. Assuming two possible matches exist for the target particle [see Fig. 2(b)], we start by fitting a smooth curve over each known neighbor tracks to reduce the noisy reconstruction effect on the coherency detection. Then LCTI locally computes the FTLE map over the fitted tracks without considering two possible matches. If the FTLE map shows local separations, neighbor tracks in the same cluster with the target particle are classified as coherent neighbors.

As illustrated in Fig. 3, the local region is divided into two blue and red clusters. Only the neighboring tracks inside the blue cluster are coherent with the target particle. All neighbor tracks are coherent neighbors if no separation is detected. To quantify a threshold for the FTLE ridge detection, we assessed the FTLE map for the case of 2D homogeneous isotropic turbulence given by Direct Numerical Simulation (DNS). We found that values above the threshold 0.25 are optimal criteria to estimate the FTLE ridge positions. This threshold was in agreement with studies using global FTLE ridge calculation in the range of 50% 80% of the maximum FTLE value.26 There are valuable studies in ridge detection algorithms with extensive computation costs that can be employed instead of using a constant FTLE threshold (see, e.g., Shadden et al.23). After the coherent neighbor determination, LCTI checks the FTLE value for each possible match and neighbors. Finally, the most coherent match with the coherent neighbors will be indexed. This process continues iteratively until no track is found to be coherent with the tracked poll.

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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