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.

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.