# Lagrangian Coherent Track Initialisation (LCTI)

Ali Rahimi Khojasteh  Yin Yang<EMAIL_ADDRESS>Dominique Heitz
INRAE, OPAALE, 17 avenue de Cucillé, 35044, Rennes, France  Sylvain Laizet
Turbulence Simulation Group, Department of Aeronautics, Imperial College
London, Exhibition Road, London SW7 2AZ, United Kingdom

###### Abstract

Advances in time-resolved Particle Tracking Velocimetry (4D-PTV) techniques
have been consistently revealed more accurate Lagrangian particle motions. A
novel track initialisation technique as a complementary part of 4D-PTV, based
on local temporal and spatial coherency of neighbour trajectories, is
proposed. The proposed Lagrangian Coherent Track Initialisation (LCTI) applies
physics-based Finite Time Lyapunov Exponent (FTLE) to build four frame
coherent tracks. We locally determine the boundaries (i.e., ridges) of
Lagrangian Coherent Structures (LCS) among neighbour trajectories by using
FTLE to distinguish clusters of coherent motions. To evaluate the proposed
technique, we created an open-access synthetic Lagrangian and Eulerian dataset
of the wake downstream of a smooth cylinder at a Reynolds number equal to
$3900$ obtained from 3D Direct Numerical Simulation (DNS). The dataset is
available to the public. Performance of the proposed method based on three
characteristic parameters, temporal scale, particle concentration (i.e.,
density), and noise ratio, showed robust behaviour in finding true tracks
compared to the recent initialisation algorithms. Sensitivity of LCTI to the
number of untracked and wrong tracks are also discussed. We address the
capability of using the proposed method as a function of a 4D-PTV scheme in
the Lagrangian Particle Tracking challenge for a flow with high particle
densities. Finally, the LCTI behaviour was assessed in a real jet impingement
experiment. LCTI was found to be a reliable tracking tool in complex flow
motions, with a strength revealed for flows with high particle concentrations.

††preprint: AIP/123-QED

##  I Introduction

Recent developments in the Shake-The-Box (STB) approach (Schanz, Gesemann, and
Schröder, 2016) have led to a renewed attention on time-resolved Particle
Tracking Velocimetry (4D-PTV) for the study of turbulent flows. STB introduces
a fast and efficient tracking idea based on a particle position prediction
step followed by an image space optimisation scheme solved with "Shaking". STB
was first initiated and shaped after the Iterative Particle Reconstruction
(IPR) concept proposed by Wieneke (2012). A typical 4D-PTV technique comprises
four recursive steps inspired by STB, as shown in Fig. 1. These steps are
particle reconstruction, track initialisation, prediction, and optimisation.
Particle reconstruction is a process that converts with triangulation multi-
view 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, Wieneke (2012) proposed IPR with an additional step to
overcome the triangulation inaccuracy. In IPR, the triangulation is followed
by an iterative optimisation procedure that searches for the best particle
position minimising the intensity discrepancy between the original and the
reprojected particle image. The reconstructed 3D positions from IPR are
thereafter fed into the initialisation part (see Fig. 1). After the tracklets
of the first few frames are built, the prediction function then estimates
positions of the next time step ($t_{n+1}$) using polynomial or Wiener filter
predictors (Schanz, Gesemann, and Schröder, 2016; Tan  _et al._ , 2020). The
optimisation takes part from the predicted positions until the optimal
positions at $t_{n+1}$ are found. During the prediction-optimisation 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 optimisation 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. To this end, the
track initialisation attempts to build tracklets for three types of particles,
particles in the first four/five frames, new entries, and lost particles (see
Fig. 1).

Figure 1:  4D-PTV flowchart starting from particle reconstructions in four
frames using IPR Wieneke (2012). LCTI recursively add new entry and lost
tracks into the optimisation poll.

4D-PTV methods like STB and Kernelized Lagrangian Particle Tracking (KLPT,
Yang and Heitz (2021)) require an appropriate and reliable number of
initialised true tracks at every time step. Otherwise, the tracking process
fails to reconstruct trajectories for the majority of particles. Such a
failure illustrates the importance of implementing a robust multi-frame track
initialisation technique to prevent 4D-PTV divergence, particularly in dense
and complex situations. The idea of initialising a possible track in four
frames, known as four frame best estimate (4BE), has been widely used in
LPT/PTV studies (Malik, Dracos, and Papantoniou, 1993; Ouellette, Xu, and
Bodenschatz, 2006; Dou  _et al._ , 2018; Clark, MacHicoane, and Aliseda, 2019;
Machicoane  _et al._ , 2019). Four frame tracking methods with simple nearest
neighbour particle matching have been introduced for low density, and smooth
flow behaviours (Malik, Dracos, and Papantoniou, 1993). New studies recently
improved four frame track initialisation performance, including four-frame
best estimate (Ouellette, Xu, and Bodenschatz, 2006, 4BE-NNI method) by
looking for the nearest neighbours in sequential frames until a unique
solution is found or Enhanced Track Initialisation (Clark, MacHicoane, and
Aliseda, 2019, 4BE-ETI method) by looking for all track possibilities with an
adjustable search volume (see also Machicoane  _et al._ , 2019). Dou  _et al._
(2018) proposed initialising with two nearest candidates in a similar spirit,
then kept predicting and particle matching in the next two following time
steps (four frames in total) until a unique match is found. Cierpka, Lütke,
and Kähler (2013) have shown that the four frame methodology could be extended
to multi-frame tracking with the combination of neighbour possibility and
temporal prediction in sequential steps. The most straightforward prediction
function is the linear predictor that can be calculated from the position
difference of every two possible matches. The use of a linear predictor
improved the probability of finding true tracks as well as reducing the
computation time by having a targeted search volume. Some studies also suggest
applying the prior Particle Image Velocimetry (PIV) velocity field as a
predictor (Cardwell, Vlachos, and Thole, 2011; Schanz, Gesemann, and Schröder,
2016). Although this idea is applicable to 2D and 3D studies, extracting a 3D
PIV velocity field is expensive due to its spatial resolution, uncertainty,
and complex experimental issues.

For complex flow motions and experiments with high particle densities, a
further step for the track validity check is required to avoid false
detection. Guezennec  _et al._ (1994) originated this validity check as a
self-coherency algorithm with the concept of path coherence. Their technique
minimises a penalty function from all possibilities in five frames. In their
study, a possible track is a coherent path if it is a smooth trajectory in
position, velocity, acceleration, and rate of acceleration. This spatial and
temporal self-consistency penalty function only focuses on a single track
behaviour. Recent four frame based techniques also performed similar self-
consistency approaches. As an example, Dou  _et al._ (2018) checked if the
velocity differences between two frames exceed a certain threshold to validate
a possible track. With a comparative approach, Ouellette, Xu, and Bodenschatz
(2006) controlled the acceleration change instead of velocity. These post-
treatments rely on spatial and temporal filtering of the trajectories to avoid
false tracks. However, as complex flows often feature high gradient motions,
over-smoothing dynamics could lead to a quality degeneration of the
reconstructed field. The degeneration problem is that there is no direct
learning based on the physics of flow included in the classical four frame
schemes, which brings more challenges for complex and high-density flow
motions.

We argue that when a track can not be reconstructed successfully solely due to
ambiguities caused by overlapping and multi possibilities, it is always
beneficial to extract more information from its neighbourhood. If the
reconstructed tracks are available in the neighbourhood, we can use this
information to gain a better insight into the target particle’s potential
behaviour and eventually solve the ambiguity problematic associated with the
initialisation. As we are dealing with the fluid flow, assuming a constant
solid local rigid neighbourhood is naive and erroneous, it is essential to
consider a coherent neighbourhood where both the target particle and neighbour
particles share the coherent motion.

In this study, we propose a novel Lagrangian Coherent Track Initialisation
(LCTI) technique for finding tracks in four (or multi) frames that belong to
clusters of coherent motions. We apply Lagrangian Coherent Structures (Haller,
2015, LCS) to distinguish coherent and non-coherent neighbour trajectories.
The LCS, also known as the skeleton of flow, determines separatrix lines or
surfaces that divide flow structures into different coherent regions. We use
Finite Time Lyapunov Exponent (FTLE) that is the most common method in
quantifying these separatrices boundaries (Balasuriya, Kalampattel, and
Ouellette, 2020). The paper is organised as follows: we first present the LCTI
algorithm based on Lagrangian coherency in four frames in section II. We then
address in section III the concept of clusters of coherent particles, and how
to quantify the coherent trajectories. In the following section IV, we discuss
how to create synthetic datasets in the wake behind a smooth cylinder at a
Reynolds number equal to $3900$ (based on the cylinder diameter and freestream
velocity) because it is an effective tool to evaluate particle tracking
techniques (Lecordier and Westerweel, 2004) and has been utilised widely in
recent PTV studies (Schanz, Gesemann, and Schröder, 2016; Tan  _et al._ ,
2020; Patel  _et al._ , 2018; Clark, MacHicoane, and Aliseda, 2019). LCTI
robustness, accuracy and sensitivity analyses are also illustrated in terms of
particle concentration, temporal scale, and noise level. Finally, we
demonstrate the LCTI performance for the LPT challenge wall-bounded wake flow
behind a cylinder (Cfdforpiv.dlr.de, 2020) and a jet impingement experiment in
sections IV and V.

##  II Four Frame Coherent Track Initialisation

The current initialisation 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 $t_{1}$, where there is no neighbour 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,

1:Index possible candidates inside the search volume at $t_{2}$;

2:Two consecutive predictions and candidate matching for $t_{3}$ and $t_{4}$;

3:Index possible tracks from $t_{1}$ to $t_{4}$;

4:Coherency check with neighbour tracks for each possible track;

5:Index the most coherent track.

Algorithm 1 LCTI Figure 2:  Schematic view of the LCTI algorithm when two
possible four frame solutions exist. a) LCTI four frame algorithm considering
all possible neighbour candidates at $t_{2}$ followed by linear predictions
(blue dash line arrows). Candidate matching at time step $t_{3}$ inside first
search volume (blue circle). Second order prediction (red and black dash line
arrows) to match possible candidates at $t_{4}$ inside second search volume
(grey circle). b) Coherency check between two possible track matches and
neighbour coherent motion

Referring to the LCTI algorithm 1, we need to define the search volumes to
index possible candidates at each time step. Clark, MacHicoane, and Aliseda
(2019) 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 neighbour particles. Therefore, the first search volume is
computed as a function of neighbour maximum displacements in each spatial
direction between $t_{1}$ and $t_{2}$ as shown in Fig. 2a. Then, every
neighbour particle inside the search volume at $t_{2}$ is a candidate. These
candidates are in one of the following categories: the true position of the
target particle at $t_{2}$, the true position of other undetected tracks, and
noise (i.e., false particle). Afterwards, a linear predictor (blue dash line
arrows in Fig. 2a) between the target particle at $t_{1}$ and the possible
candidate at $t_{2}$ 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 $t_{3}$. A possible track is
removed if there is no candidate inside the search volume. The process is
repeated for the next time step with a higher order prediction function (red
and black dash line arrows in Fig. 2a). 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. 2b. 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 behaviour spatially and temporally. A
function is therefore required to determine coherent and non-coherent clusters
of particles locally. More details and principles on the coherent motion of
particles are discussed in Section III.

##  III Coherent Track Detection

Recently, Lagrangian Coherent Structures (LCS) have been applied in PIV/PTV
experiments for flow structure analyses (see, e.g., Krishna, Green, and
Mulleners (2018)). 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
(Haller, 2015; Hadjighasem  _et al._ , 2016). 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 neighbours of the target particle. However, the trajectories of
each coloured cluster temporally evolve in separated directions. LCS can be
used to determine if a spatially neighbour 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 neighbour 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,
curve or surface boundaries divide the local spatial area into discrete
regions with different dynamic motions (Samelson, 2012), and motions across
these boundaries are negligible (Shadden, Lekien, and Marsden, 2005).
Furthermore, the LCS boundaries can move, evolve, and vanish in spatial space
as the flow pattern changes temporally.

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 neighbour particles over finite time (Alligood, Sauer,
and Yorke, 1996; Raben, Ross, and Vlachos, 2014). Raben, Ross, and Vlachos
(2014) showed that the normalised average error and normalised root-mean-
squared (RMS) error of the FTLE map decreases with increased particle
concentration. This trend is favourable 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.

As discussed in section II, if a possible track is coherent with its neighbour
tracks, it will be indexed into the tracked poll. To compute FTLE, we assumed
that the local flow map is the particle trajectory in finite time from $t_{0}$
to $t_{0}+T$. The idea is to analyse the spatial displacements between the
target particle and its neighbours. The flow map of a single particle can be
formulated as

$\emptyset_{t_{0}}^{t_{0+T}}\left(x\right)\\-:\ x(t_{0})\ \rightarrow\
x(t_{0}+T)\ ,$ (1)

where $x(t_{0})$ is the starting position of the interval time $T$, and
$x(t_{0}+T)$ is the final position. Difference between the flow maps of the
target particle $\emptyset_{t_{0}}^{t}\left(x_{0}^{p}\right)$ and its
neighbour $\emptyset_{t_{0}}^{t}\left(x_{0}^{n}\right)$ would result in a
vector displacement as following,

${\delta
x(t)=\emptyset}_{t_{0}}^{t}\left(x_{0}^{p}\right)-\emptyset_{t_{0}}^{t}\left(x_{0}^{n}\right).$
(2)

This vector displacement contains transformation between the initial and final
positions of two particles. Equation 2 can be linearised by using the first
term of the Taylor series of $\emptyset_{t_{0}}^{t}\left(x_{0}^{n}\right)$
expanded about $x_{0}^{p}$:

$\delta
x(t)\approx\frac{\partial\emptyset_{t_{0}}^{t}\left(x_{0}^{p}\right)}{\partial
x_{0}}\delta x_{0},$ (3)

where $\delta x_{0}=x_{0}^{p}-x_{0}^{n}$. The state-transition matrix
$\partial\emptyset_{t_{0}}^{t}\left(x_{0}^{p}\right)/\partial x_{0}$ is also
known as deformation gradient tensor
$\nabla\emptyset_{t_{0}}^{t}\left(x_{0}\right)$ (Shadden, Lekien, and Marsden,
2005). The deformation tensor carries valuable information including the rate
of expansion, compression and rotation. The magnitude of $\delta x(t)$ is

$\displaystyle|\delta x(t)|$ $\displaystyle=\sqrt{\delta x(t)\delta x(t)}$
$\displaystyle=\sqrt{(\nabla\emptyset_{t_{0}}^{t}(x_{0})\delta
x_{0})(\nabla\emptyset_{t_{0}}^{t}(x_{0})\delta x_{0})}$
$\displaystyle=\sqrt{\delta
x_{0}(\nabla\emptyset_{t_{0}}^{t}(x_{0})\nabla\emptyset_{t_{0}}^{t}(x_{0}))\delta
x_{0}},$ (4)

where we define

$\Delta=\nabla\emptyset_{t_{0}}^{t}(x_{0})\nabla\emptyset_{t_{0}}^{t}(x_{0}).$
(5)

$\Delta$ is a symmetric positive definite matrix, also known as the right
Cauchy-Green deformation tensor (Shadden, Lekien, and Marsden, 2005) with
three real and positive eigenvalues in a 3D domain over finite time.

As mentioned before, FTLE measures the rate of stretching between the target
particle and its neighbour. The maximum eigenvalue of the Cauchy-Green tensor
$\lambda_{max}(\Delta)$ shows the expansion or separation in finite time.
Furthermore, the eigenvector corresponding to $\lambda_{max}(\Delta)$
represents the direction of the separation. Eventually, the magnitude of the
maximum displacement can be written as

$|\delta x(t)_{max}|=\sqrt{\delta x_{0}\lambda_{max}\left(\Delta\right)\delta
x_{0}}=\sqrt{\lambda_{max}\left(\Delta\right)}|\delta x_{0}|,$ (6)

and the FTLE value $\Lambda_{t_{0}}^{t}$ is defined as

$\Lambda_{t_{0}}^{t}=\frac{1}{\left|T\right|}\sqrt{\lambda_{max}\left(\Delta\right)}=\frac{1}{\left|T\right|}log\left(\frac{\delta
x(t)}{\delta x(t_{0})}\right).$ (7)

A lower FTLE value means the neighbouring 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 the LCS ridges that divide the
local area into different clusters of coherent particles. With this
formulation, it is possible to index a group of neighbour particles as
coherent or non-coherent with the target particle. As we discussed in section
II, 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. 2b), we start by
fitting a smooth curve over each known neighbour 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, neighbour tracks in the same cluster
with the target particle are classified as coherent neighbours. As illustrated
in Fig. 3, the local region is divided into two blue and red clusters. Only
the neighbouring tracks inside the blue cluster are coherent with the target
particle. All neighbour tracks are coherent neighbours 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 (Lipinski and Mohseni,
2010). 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, Lekien, and Marsden (2005)). After the coherent
neighbour determination, LCTI checks the FTLE value for each possible match
and neighbours. Finally, the most coherent match with the coherent neighbours
will be indexed. This process continues iteratively until no track is found to
be coherent with the tracked poll.

Figure 3:  2D schematic of particle motions inside vortices. Each colour
belongs to a group of coherent clusters. The target dark blue particle with
coherent neighbour particles is located in a clockwise vortex (blue cluster)
while non-coherent particles belong to different clusters. The target particle
is non-coherent with neighbour particles in the red cluster.

##  IV Synthetic evaluation

To quantify and distinguish the LCTI performance from other schemes, we first
applied the tracking process to synthetic data. In this section, we consider a
set of synthetic data for the wake flow downstream of a smooth cylinder. We
first describe the creation of the Lagrangian data by transporting the
synthetic particles in the Eulerian velocity volume. Thereafter, we discuss
how to set characteristic parameters for the synthetic data, including 1-
Particle concentration, 2- Temporal scale, and 3- Noise level. We also address
accuracy analyses of LCTI in comparison with other schemes, followed by the
sensitivity analysis of LCTI to every characteristic parameter. Finally, we
discuss the LCTI performance in the LPT challenge for the wall-bounded wake
flow behind the cylinder (Cfdforpiv.dlr.de, 2020).

###  IV.1 Synthetic data creation

####  IV.1.1 Particle transport

An open-source Direct Numerical Simulation (DNS) code, named Incompact3d, was
employed to simulate the wake downstream of a smooth cylinder at a Reynolds
number equal to $3900$. Incompact3d is part of the Xcompact3d framework
dedicated to the study of turbulent flows using supercomputers (Bartholomew
_et al._ , 2020) and it is dedicated to the incompressible Navier-Stokes
equations. It is based on sixth-order finite-difference schemes on a Cartesian
mesh for spatial discretisation and, for the present simulation, a third-order
Adams–Bashforth scheme for the time advancement. The main originality of
Incompact3d is that the Poisson equation for the incompressibility of the
velocity field is fully solved in spectral space via the use of relevant 3D
Fast Fourier transforms (FFTs). With the help of the concept of modified
wavenumber, the divergence free condition is ensured up to machine accuracy.
The pressure mesh is staggered from the velocity one by half a mesh to avoid
spurious pressure oscillations observed in a fully collocated approach. The
simplicity of the mesh allows an easy implementation of a 2D domain
decomposition based on pencils. More information of the code can be found in
Laizet and Lamballais (2009); Laizet and Li (2011). In the present work, the
smooth cylinder is modelled using a customised immersed boundary method with
an artificial flow inside the cylinder to ensure the smoothness of the
velocity field while imposing a no-slip boundary condition at the cylinder.
Comparisons with experiments for this test case can be found in Chandramouli
_et al._ (2018). A DNS performed with high-order schemes is the natural choice
for the present work, with the highest possible spatial and temporal accuracy
for Eulerian/Lagrangian data. Velocity and pressure Eulerian data are stored
as well as Lagrangian trajectories for a duration of three vortex sheddings.
These data will be made in open access and available to the public.

Lagrangian particle transport is a process to extract trajectories from an
Eulerian velocity volume (Sebille  _et al._ , 2018). In this study, the
Eulerian velocity volume is a fraction of flow around the smooth cylinder of
diameter $D$, starting from $1D$ upstream of the cylinder centre to $3D$
downstream in the streamwise direction to reduce the computation costs and
only focus on the most complex flow regions. Every DNS snapshot (3 velocity
components and pressure field) requires roughly $6\ \text{Gb}$ of storage.
Eulerian data were only recorded every $10$ DNS time steps due to storage
limitation. To quantify the accuracy of particle transport, we compared
trajectories obtained from every $10$ with every $1$ DNS time steps in a
reduced domain. It was found that the mean of position error after $1000$ DNS
time steps are $3.28$ times larger than the Kolmogorov length scale $\eta$.
The standard deviation of position error was $\sigma_{STD}=0.017\ \eta$. It
means that the trajectories we built are not exactly ground truth due to small
position deviation compared with every $1$ DNS time step. However, it has no
impact on the following assessment of the method we propose. More details of
the current synthetic data can be found in Khojasteh  _et al._ (2021). The
domain of interest $L_{x}\times L_{y}\times L_{z}$ has a dimension of
$4D\times\ 2D\times\ 2D$ (from 1D upstream the cylinder). The original
computational domain $20D\times\ 20D\times\ 6D$ is discretised with
$n_{x}\times n_{y}\times n_{z}=1537\times 1025\times 256$ mesh nodes with a
time step of $0.00075D/U_{\infty}$ (where $U_{\infty}$ is the freestream
velocity). The mesh is uniform in the streamwise and spanwise directions while
a non-uniform mesh is used in the vertical direction, with a mesh refinement
towards the centre of the cylinder. The finest mesh size in the vertical
direction is $\Delta y_{min}=0.00563D$. Data are saved for $1500$ snapshots
every $10$ DNS time steps $dt_{\rm DNS}$ corresponding to a duration of nearly
three vortex sheddings. From these data, particles are transported according
to the following steps: First, synthetic random particles are created in the
domain of interest for the first time step; Second, accurate time stepping and
spatial interpolation schemes are used to transport scattered random particles
in time and space (Sebille  _et al._ , 2018). We estimated the Lagrangian
particle velocities by spatial trilinear interpolation from $8$ nearest
Eulerian velocities at each time step. The synthetic particle positions are
updated in time using the 4th order Runge-Kutta scheme. Finally, we collected
the ground truth Lagrangian trajectories by repeating the transport process
for each time step.

Figure 4:  Schematic view of energy spectrum in turbulence. Resolved and
unresolved spatial resolutions of $2,000$, $20,000$, $200,000$ particles in
blue lines with respect to integral($D$), Taylor($l$) and Kolmogorov($\eta$)
length scales in red dash-lines

####  IV.1.2 Particle concentration

In most PIV/PTV studies, particle concentration refers to the number of
particles per pixel (ppp) that is principally an image-related parameter. The
relation between the particle concentration and the turbulence length scales
determines the maximum achievable spatial resolution. Therefore, it is crucial
to address the number of particles corresponding to Kolmogorov ($\eta$),
Taylor ($l$), and Integral ($D$) length scales. To quantify these relations,
we used two volumetric metrics, namely, particles per cubic Kolmogorov scale
($\text{pp}\eta^{3}$) and particles per cubic Integral scale
($\text{pp}D^{3}$), instead of ppp. We chose three low, moderate, and high
particle concentrations, respectively equal to $2,000$, $20,000$, and
$200,000$ particles (see Table 1). Fig. 4 shows the turbulence energy spectrum
marked by the three particle concentrations (blue lines) compared with the
turbulence length scales (red dash lines). Referring to the DNS data in
Section IV.1, we estimated the Kolmogorov length scale as $D/\eta\sim
Re^{3/4}$ that is almost $2.8$ times smaller than the average mesh size in the
$y$-direction. Since the cylinder is inside our domain, it is necessary to
subtract the volume of the cylinder before computing $\text{pp}\eta^{3}$.
Accordingly, the spatial resolution for the case with $2000$ particles leads
to unresolved Taylor and Kolmogorov scales. However, there are enough
particles to resolve the large flow motions. Taylor length scales can be
resolved by adding an order of magnitude more particles, but the smallest
scales are still unresolved with $10^{-4}\ \text{pp}\eta^{3}$. As shown in
Fig. 4, in our case, even $200,000$ particles with $10^{-3}\
\text{pp}\eta^{3}$ are not enough to resolve the smallest turbulence length
scale at Reynolds number equals to $3900$. It is possible to reach up to the
DNS spatial resolution numerically, but very few PTV experimental studies have
reached more than $200,000$ particles in practice.

####  IV.1.3 Temporal scale

Schanz, Gesemann, and Schröder (2016) computed the temporal scale as a
function of the original experiment time sampling rate resulting in a mean 3D
particle displacement of around $6$ pixels for the synthetic data analysis.
However, the temporal scale selection requires satisfying the real experiment
condition and should be characterised by the flow physics. We defined the
temporal resolution as a ratio of turbulence time scales. The non-dimensional
form of the temporal scales can be written as $dt/dt_{\rm DNS}$,
$dt/\tau_{\eta}$, and $dt/T_{D}$, where $dt$ is the temporal scale for either
synthetic or experiment study, and $dt_{\rm DNS}$, $\tau_{\eta}$ and $T_{D}$
are the DNS, Kolmogorov and integral time scales, respectively. By defining
$T_{D}=D/U_{\infty}$, we estimate the Kolmogorov time scale from ratio of the
largest to smallest time scales as $T_{D}/\tau_{\eta}\sim Re^{1/2}$. To mimic
the real experiment condition, we gathered similar wake flow studies (mainly
cylinder wake flows) and plotted the relations of $dt/\tau_{\eta}$ with their
Reynolds numbers in Fig. 5. It is shown that low Reynolds number experiments
can resolve the Kolmogorov time scale (Michaelis  _et al._ , 2006). The
achievable temporal scale increases with the Reynolds number, mainly because
the data acquisition frequency limits the experiments with the order of $1-3\
\text{kHz}$ for the majority of studies mentioned in Fig. 5. In our synthetic
case, the DNS time step is roughly $20$ times smaller than the Kolmogorov time
scale. Such a case with every $10$ DNS time step ($dt/\tau_{\eta}=0.47$) can
resolve the Kolmogorov time scale, but it is unlikely to achieve such a high
acquisition rate in practice, particularly when the fluid is air. Particle
trajectories are very smooth with small displacements in this case. However,
according to Fig. 5, the temporal scale $dt/\tau_{\eta}$ stays relatively
large, between $2.5$ and $5$ for studies close to a Reynolds number of $3900$
that approximately equals to every $50$ and $100$ DNS time steps. As listed in
Table 2, we chose three low, moderate, and high time steps starting from every
$10$, $50$, and $100$ DNS time steps. It is worth mentioning that it takes
$1333$ and $6667$ DNS time steps to reach one integral time scale and one
vortex shedding, respectively.

####  IV.1.4 Noise ratio

In this paper, we created noise (i.e., false particles) in the vicinity of
true positions. A false particle is randomly distributed around the true
position with a maximum displacement radius. Noise ratio (NR) of $0.1$ means
$10\%$ of true particles at every time step have false particles in their
vicinities. A false particle around a single track also impacts the track
detection accuracy for other neighbour tracks, particularly in dense and
intersection situations. In this study, we created three noise ratios, $0$,
$0.1$, and $0.2$.

Figure 5:  Synthetic time scale selection with respect to recent similar wake flow experimental studies and different Reynolds numbers. Each symbol represents a family of flow configuration (Elsinga  _et al._ , 2006; Cfdforpiv.dlr.de, 2020; Neeteson  _et al._ , 2016; Sun and Brücker, 2017; Huhn  _et al._ , 2018; Michaelis  _et al._ , 2006; Schneiders  _et al._ , 2016; Scarano  _et al._ , 2015; He and Liu, 2017; Schröder  _et al._ , 2020; Fahringer, Lynch, and Thurow, 2015; Miozzi  _et al._ , 2015; Hain, Kähler, and Michaelis, 2008; Violato and Scarano, 2011; Booysen, Das, and Ghaemi, 2019) Table 1:  Synthetic particle concentration selection in terms of particles per cubic Kolmogorov and integral scales. Case | particles | $\text{pp}\eta^{3}$ |  $\text{pp}D^{3}$
---|---|---|---
Low | $2,000$ | $10^{-5}$ | 139
Moderate | $20,000$ | $10^{-4}$ | 1386
High | $200,000$ | $10^{-3}$ | 13861
Table 2:  Synthetic time step selection in terms of DNS, Kolmogorov, and integral time scales. Case | ${dt}/{{dt}_{\rm DNS}}$ | ${dt}/{\tau_{\eta}}$ |  ${dt}/{T_{D}}$
---|---|---|---
Low | $10$ | $0.47$ | $0.01$
Moderate | $50$ | $2.34$ | $0.04$
High | 1$00$ | $4.68$ | $0.08$

###  IV.2 Evaluation and sensitivity analyses

In this section, the performance of the LCTI is assessed and compared against
ETI and 4BE-NNI methods. Quantitative accuracy and sensitivity analyses are
performed for different particle concentrations, temporal scale, and noise
ratio compared to the ground truth trajectories (see Section IV.1). After the
initialisation step, the trajectory of each particle is classified as either
untracked, wrong, or true track. A dominant number of true tracks would ease
the 4D-PTV process to converge within short time steps. It is worth mentioning
that a 4D-PTV process is converged if all particles inside the domain are
tracked.

Figure 6:  Comparison performances of three track initialisation techniques in
terms of fraction of true particle detection by changing each characteristic
parameter. Particle concentration is increasing from a) to c).

Fig. 6 compares the ratio of detected true tracks obtained from LCTI, ETI, and
4BE-NNI techniques with different characteristic parameters of
$\text{pp}\eta^{3}$, $dt/\tau_{\eta}$, and NR. We assessed each technique
based on $3\times 3\times 3$ scenarios representing low, moderate and high
levels of each characteristic parameter (see Table 1 and Table 2). All three
techniques performed equally well for low particle concentration and small
temporal scale by reconstructing over $95\%$ of true tracks (see Fig. 6a).
Tracking is not difficult under such scenarios characterised by small particle
displacement with relatively large spatial distance between neighbour tracks.
By increasing $dt/\tau_{\eta}$ in low particle concentration cases, the ratio
of detected true tracks drops down to approximately $90\%$ and $75\%$ in ETI
and 4BE-NNI techniques respectively, whereas LCTI remains stable. The large
relative distance between neighbour trajectories due to low particle
concentration reduces the ambiguity in finding possible tracks. Results of low
particle concentration cases for all three techniques show that the 4D-PTV
process has to recover less than $25\%$ of remaining untracked particles to
converge in the worst case; thereafter, a short convergence time is expected.
The ratio of true tracks drops linearly by increasing the noise ratio (NR) for
all techniques with approximately the same order of magnitude. We found that
the drop in the detected true tracks caused by the noise ratio (NR) is nearly
independent of the other two characteristic parameters. Comparing the three
characteristic parameters reveals that the temporal scale has the most
deterministic impact on the detected true tracks for all techniques. If
$dt/\tau_{\eta}$ stays low, all techniques can cover over $95\%$ of true
tracks regardless of the noise ratio and the particle concentration. Low
temporal scale value means that particle displacements are very small. Even
two frame nearest neighbour techniques can reconstruct the majority of true
tracks. However, in more realistic conditions, when the temporal scale
$dt/\tau_{\eta}$ is high, LCTI performed significantly better than the other
two techniques. Particle concentration also plays an essential role in the
initialisation performance. The ratio of detected true tracks for LCTI stays
consistently above $95\%$ in particle concentrations of
$10^{-4}\text{pp}\eta^{3}$ (see Fig. 6b). In a worse scenario, when the
particle concentration is $10^{-3}\text{pp}\eta^{3}$ associated with high
temporal scale, LCTI still can recover over $75\%$ of true tracks that is
considerably higher than ETI and 4BE-NNI with roughly $50\%$ and $25\%$ true
tracks, respectively (see Fig. 6c).

Figure 7:  Sensitivity analysis of LCTI compared with DNS ground truth by
increasing noise level, particle concentrations, and temporal scale from top
to bottom. Tracks in blue and red colour represent untracked and wrong
trajectories respectively. Gray tracks are true trajectories that the
algorithm built (numbers of grey tracks are scaled down for having a clear
view of red and blue tracks). Figure 8:  Sensitivity of the 4D-PTV convergence
to the number of initialised tracks for the LPT challenge high-density case at
$0.12\ \text{ppp}$ over first $30$ time steps. Number of initialised tracks
after the first four frames varies from $\sim 12,000$ to $\sim 60,000$

The synthetic data evaluation showed that LCTI systematically outperforms
other competing techniques. However, it should be mentioned that the cost of
this achievement is expensive because it computes the coherency for every
possible tracklet. We found that LCTI requires roughly $4$ times more
computation time on a single CPU core than a classic four frame based
initialisation technique without any post-treatment. To this end, an
appropriate initialisation technique should be chosen depending on the
measurement condition. As an example, there is no need to perform a
sophisticated initialisation technique if the temporal scale (i.e., sampling
rate) and the particle concentration are low in comparison with the turbulence
scales. It is important to note that using sophisticated initialisation
techniques such as LCTI is crucial to prevent 4D-PTV failure in challenging
cases with a large temporal scale and high particle concentration.

Figure 9:  LCTI trajectory results for the LPT challenge wake flow at $0.12\
\text{ppp}$ ((Cfdforpiv.dlr.de, 2020; Khojasteh  _et al._ , 2020)); a) slice
view in $y$ direction of particle trajectories in grey and the target track in
red b) Cluster of coherent tracks with the target track in red

Apart from the initialisation performance based on the number of true tracks,
we performed further parametric analyses on LCTI to determine how untracked
and wrong trajectories are sensitive to the characteristic parameters. We
started from a base case with $10^{-4}\ \text{pp}\eta^{3}$ particle
concentration, $dt/\tau_{\eta}=2.34$, and zero noise ratio. Afterwards, we
increased each parameter separately until the fraction of true tracks drops
down to $80\%$. The remained $20\%$ is a mix of untracked and wrong
trajectories. Lagrangian flow maps for increasing of each characteristic
parameter are shown in Fig. 7. If we increase the temporal scale, both
untracked and wrong tracks increase as the number of true tracks drops. Under
such a scenario, the particle displacements are large and comparable with
local distances between the neighbour particles. Therefore, less information
between two-time steps is available, which drastically increases the
possibility of having untracked particles. Depending on the region of the
flow, the majority of untracked trajectories (blue colour in Fig. 7) exist
around the two high shear sides of the wake region. Although the number of
untracked trajectories dominate the whole Lagrangian flow map, more wrong
tracks (red colour in Fig. 7) than untracked trajectories are observed inside
the wake region. By increasing the particle concentration, both untracked and
wrong trajectories raise homogeneously with nearly the same proportion through
the Lagrangian flow map. Interestingly, the number of wrong tracks is still
comparably larger than the number of untracked particles inside the wake
region. It shows that regional flow behaviours can have a direct impact on the
initialisation performance. The impact of increasing the noise ratio is also
shown in Fig. 7. We found a dominated number of wrong tracks in all regions
with increased noise ratio. In this study, we created the noise (i.e., false
particles) in the vicinity of true particles that causes more initialisation
ambiguities. Due to this reason, LCTI yields more wrong trajectories with
increased noise ratio. Besides, wrong indexing of a track takes away at least
one true or false particle that may propagate the wrong detection to another
neighbour track too. In practice, for 4D-PTV, wrong initialised tracks
increase the failure risk and need to be appropriately eliminated in a
subsequent prediction-optimisation step. The untracked particles, on the other
hand, require more iteration and convergence time.

Sensitivity analysis in Fig. 7 implies that LCTI is likely to have more wrong
tracks inside the wake region with increased noise ratio, particle
concentration, and temporal scale. However, different behaviours have been
seen depending on the characteristic parameters in overall. The high
turbulence intensity level at Reynolds $3900$ creates small flow structures
inside the wake region. Consequently, particles inside this region are
coherent with a small number of neighbours and quickly change to different
coherent clusters. This behaviour brings more complexity for the coherency
detection that might be the reason for having a dominating number of wrong
tracks, despite relatively small particle displacements. Schanz, Novara, and
Schröder (2020) observed a similar issue when particles of a specific region
move slowly compared to the rest of the domain. One approach for improving
trajectory in such regions is performing temporal filtering schemes by
adjusting the temporal scale to track the slowest particles (Schanz, Novara,
and Schröder, 2020).

###  IV.3 LPT challenge

LCTI was implemented into KLPT (Yang and Heitz, 2021) to run the whole 4D-PTV
process (see Fig. 1). KLPT featured by LCTI (KLPT-LCTI) was examined on the
time-resolved data from the LPT challenge (Cfdforpiv.dlr.de, 2020; Khojasteh
_et al._ , 2020) at four particle densities (i.e., concentrations) from
$0.005\ \text{ppp}$ up to $0.08\ \text{ppp}$. The challenge cases were
obtained from the wall-bounded wake flow behind a cylinder at a momentum
thickness Reynolds number $Re_{\theta}$ of around $4500$. In terms of
turbulence length scales, particle concentration of the mentioned four cases
varied between $2\times 10^{-7}\ \text{pp}\eta^{3}$ and $3\times 10^{-6}\
\text{pp}\eta^{3}$. The domain of interest was set at $100\ \text{mm}\times\
50\ \text{mm}\times\ 30\ \text{mm}$ downstream of the cylinder. The image
acquisition rate was equal to $600\ \mu s$, resulting in $dt/\tau_{\eta}=2.68$
temporal scale. At the lowest ppp, the proposed method managed to reconstruct
over $99\%$ of true particles accurately. Percentage of true particles
maintained over $99\%$ for higher densities (i.e., $0.025$, $0.05$, and $0.08\
\text{ppp}$). Accordingly, missed and ghost particles were less than $1\%$.
The case studies of the LPT challenge revealed that the positional Root-Mean-
Square error (RMSE) of KLPT-LCTI increased linearly with ppp, but it remained
below $0.0041\ \text{mm}$ for all four particle densities. This illustrates
the reliable performance of the LCTI at particle densities lower than $0.08\
\text{ppp}$, knowing that most of the 4D-PTV real experiments perform at
$0.05\ \text{ppp}$ particle density or lower.

For densities higher than $0.08\ \text{ppp}$, a more accurate initialisation
technique could prevent the 4D-PTV algorithm from failing or improve its
convergence speed. We highlighted that KLPT featuring LCTI succeeded in
reconstructing tracks at the density of $0.12\ \text{ppp}$ while KLPT
featuring NNI failed to converge. Questions have been raised about the 4D-PTV
sensitivity to the number of initialised particles at the beginning. We
illustrated this issue in the LPT challenge case with $0.12\ \text{ppp}$ and
over $120,000$ particles. As shown in Fig. 8, the KLPT-LCTI process reaches no
more than $85,000$ (i.e., $70\%$) final tracks if the process starts with any
number below $30,000$ initialised tracks. However, starting with $60,000$
initialised tracks leads to cover over $99\%$ of final trajectories after $30$
time steps at $0.12\ \text{ppp}$. The evidence from this study indicates that
the number of initialised tracks is one deterministic contributor to the
4D-PTV convergence at high-density scenarios. Without a proper track
initialisation algorithm, a 4D-PTV scheme would not be able to eventually
recover the majority of tracks.

##  V Experiment study

To demonstrate the potential of the LCTI in a practical configuration, we used
the data from a volumetric experiment of liquid-liquid jet impingement on a
circular wall carried out in our laboratory at a Reynolds number equal to
$2500$. Perpendicular impinging the jet into the wall creates significant
deceleration inside the jet core. The deceleration and directional 3D motions
of particles are featured by multi-vortex rings around the jet and secondary
vortex structures after the impingement.

Fig. 10 shows the schematic view of the experiment setup. Four Phantom $M310$
cameras with $1200\times 800$ resolution and the maximum frequency of $3\
\text{kHz}$ empowered. Nikon $105\ \text{mm}$ macro $\text{F}2.8$ (aperture
was set to $\text{F}22$) lenses were employed. As shown in Fig. 10, two
cameras were positioned in $24$ degree with forward scatter light, and the
other two cameras were in backward scattering at $13$ degree. $15\ \text{mJ}$
LDY $300$ laser was set for the illumination with $1\ \text{kHz}$ frequency
($dt/\tau_{\eta}=2.18$) and converted into the rectangular light volume. The
measurement volume was $16\ \text{mm}\times 51\ \text{mm}\times 35\ \text{mm}$
($29\ \text{cm}^{3}$). The seeding particles were hollow glass spheres $9-13\
\mu m$ in diameter, and $1.1\ \text{g}/\text{cm}^{3}$ density, and the
particle concentration was approximately $0.03\ \text{ppp}$ that was
equivalent to $2\times 10^{-6}\ \text{pp}\eta^{3}$. More details in this
experiment can be found in Yang and Heitz (2021).

In this section, we performed the 4D-PTV process without prediction and
optimisation parts to particularly demonstrate the LCTI performance.
Therefore, the particle positions reconstructed by IPR were employed in $20$
consecutive time steps and followed by multi four-frame LCTI processes.
Trajectory results obtained from LCTI are shown in Fig. 11. Away from the jet
impingement region, the trajectories are relatively smooth with small
displacements. As explained earlier in IV.1, there is no critical issue in
such regions, and a simple initialisation technique can track the majority of
particles. Small trajectories away from the jet, therefore, are omitted in
Fig. 11 to concentrate on the most complex regions. As the jet impinges on the
wall, trajectories decelerate highly, turn alongside the wall, but still tend
to keep their coherent local motions. The proposed method effectively detected
these coherent trajectories, although complex behaviours exist. We found that
the trajectories around the jet tend to circulate (see the zoom balloon in
Fig. 11). This reveals strong evidence of the particle coherent motions
impacted by the vortex rings. Near wall trajectory reconstructions show
signatures of the secondary vortices where particles bounce back. The results
of this experiment test case support the idea that each particle can
temporally and spatially behave in coherence with a group of other neighbour
particles. The second significant finding was that the proposed initialisation
technique could cover most tracks in complex and high gradient regions
associated with 3D directional dynamics.

Figure 10:  3D schematic view of the jet impingement experimental, with
snapshot of the experiment in zoom balloon.

(a)

(b)

Figure 11:  a) Side view of the trajectories coloured by their velocity
magnitudes at $0.03\ \text{ppp}$. b) 3D view of the trajectories in the Jet
impingement. Low-velocity tracks away from the jet core are filtered for clear
qualitative view thanks to the colorbar

##  VI Conclusions and outlook

We proposed a novel technique to reconstruct tracklets from four (or multi)
frames by leveraging temporal and local spatial coherency of neighbour tracks.
These tracks should be consistent with the neighbour coherent motions bounded
by LCS ridges. To assess the LCTI performance in various conditions, we have
created an open-access synthetic dataset for the wake flow downstream of a
smooth cylinder obtained from DNS at a Reynolds number equal to $3900$. Future
studies by interested readers should focus on using the current open access
Eulerian velocity and pressure data as well as synthetic Lagrangian
trajectories for algorithm evaluation purposes. We transported particles using
the trilinear spatial interpolation scheme followed by the 4th order Runge
Kutta temporal integration. In the current dataset, temporal and spatial
scales are reported based on turbulence length and time scales.

As a result of the parametric study in section IV.2, we found that a 4D-PTV
process requires more sophisticated initialisation techniques if either one of
the following conditions meets: 1), Particle displacements between two time
steps are relatively large due to either sparse temporal acquisition frequency
or high dynamic gradients of particles (velocity and accelerations); 2), The
particle concentration is high enough to have a length between neighbour
particles with the same order of their trajectory displacement lengths; 3),
The reconstructed particle field is noisy. Under these conditions, LCTI tends
to detect more true tracks than recent ETI and 4BE-NNI techniques. We found
that the temporal scale severely impacts the true track detection yielded by
any initialisation techniques. We also analysed the proportions of untracked
and wrong trajectories in LCTI. The main results indicate that the noise ratio
creates more wrong tracks, the particle concentration homogeneously increases
both untracked and wrong tracks, and finally, the temporal scale causes more
untracked trajectories. Regardless of the characteristic parameters, we found
more wrong trajectories than untracked particles inside the wake region. We
also applied LCTI to the time-resolved dataset in the LPT challenge after
integrating LCTI to a 4D-PTV scheme, KLPT. Our KLPT-LCTI scheme has achieved
state-of-the-art performance for densities up to $0.08\ \text{ppp}$
(Cfdforpiv.dlr.de, 2020; Khojasteh  _et al._ , 2020). LCTI is helpful for high
particle density data where the portion of initialised tracks after the
initialisation stage directly impacts the 4D-PTV’s convergence performance. At
$\text{ppp}=0.12$, KLPT featuring a simple nearest neighbour initialisation
scheme fails to yield any valid results. LCTI, on the contrary, can recover
much more tracks from the starting four frames, therefore provides a more
accurate initialised track field. Besides, LCTI also contributes to following
every frame by bringing more new tracklets (length less than 4) from
triangulated particles on residual images into the tracked poll (see Fig. 1).
This finding is in agreement with the synthetic analysis part in section IV.2
showing the importance of having an advanced initialisation for dense (i.e.,
high particle concentration) conditions. LCTI was then tested on the jet
impingement experiment. Although the flow was characterised by complexities
such as 3D directional motion and trajectory intersections, LCTI successfully
reconstructed the majority of tracks solely on particle fields reconstructed
by IPR, without the necessity of further prediction and optimisation
processes. The trajectory results also showed particles coherent motions in
large scale flow motions such as vortex rings, impinging area, and secondary
vortices. This comprehensive study has demonstrated that coherency based track
initialisation is a robust approach to reconstruct tracks even in complex
situations. The proposed technique can be used either as an embedded module
for the 4D-PTV process, or as a standalone four-frame-based tracker. LCTI
showed that additional physics-based information increases the accuracy and
robustness of the initialisation part. For further investigations, the FTLE
function can be replaced with advanced LCS detection algorithms (see, e.g.,
Filippi  _et al._ (2021)) or other coherent motion detection techniques such
as Coherent Structure Colouring (Martins, Sciacchitano, and Rival, 2021, CSC).
More research is needed to understand and track clusters of coherent particles
in real experiments.

###### Acknowledgements.

The authors would like to greatly thank the LPT challenge organising
committee, particularly Dr Andrea Sciacchitano from TuDelft university, for
providing the time-resolved challenge results. A special thanks from the
authors is given to Anthony Guibert (a former assistant engineer at INRAE) who
conducted the impingement experiment in $2016$. The authors would finally like
to thank EPSRC in the UK for the computational time made available on the UK
supercomputing facility ARCHER2 via the UK Turbulence Consortium
(EP/R029326/1).

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