In the classic 4D- PTV algorithm, predicted particle positions are given to the optimisation process for further corrections. The optimisation can deal with slight deviations between the predicted position and the true position. However, the optimisation fails to find the true position if the deviation is large enough to have multi-candidates for a single particle at tn+1. This shows the importance of having an appropriate prediction in dense and complex motions. The proposed idea is initiated by arguing that predictions in PTV techniques focus on a single particle individually, while this single particle is not acting alone [1]. We propose this approach to locally differentiate coherent and non-coherent motions of neighbour particles around a single particle position to improve prediction accuracy, as shown in Figure 1. On the other hand, the information of coherent particles should be shared with each neighbour particle to predict their behaviour accurately and avoid misprediction. The present study was designed as a complementary function for 4D-PTV algorithms such as STB [2] and KLPT [3].

Briefly, it is assumed that particle positions are known for n time steps (four to five). Afterwards, a mathematical prediction function is implemented to estimate particle positions for time-step n + 1 followed by ”shaking” and position refinement. It should be noted that the ”shaking” process tries to look for a candidate true position very close to the predicted position. If misprediction happens, no matter how many times we perform ”shaking”, the true position is not achievable. This implies the importance of producing accurate predictions. This paper seeks to investigate the possibilities of improvements in motion estimation by adding meaningful physics into the prediction function. A simple prediction approach is a Polynomial function, suggested by Schanz et al. [4], resulting in reasonable predictions and 3D particle position reconstruction in simple flows [4, 5, 6]. However, significant off prediction occurs in case of flow associated with complexities such as high turbulence level, high Reynolds number, and mixing flows [7]. In such conditions, even by increasing the order of the Polynomial predictor functions from 3 to 10 [7], off prediction stays remained. The solution for this challenge is implementing optimal temporal filtering such as the Wiener filter, which has been first examined in 4D-PTV experiments by Schroeder et al. [5]. Since then, this concept became consistent in the STB studies due to its high robustness and accurate motion estimations [8, 7]. As mentioned, the Wiener filter showed robust behaviour in prediction with complex flows but still suffers in high motion gradients. This implies the fact that the prediction function suffers from a lack of information to find true positions. Worth mentioning that these prediction-based techniques rely on one particle individually, excluding it from surroundings. All the information we know from an individual particle is its history. Even if we implement filtering and smoothing schemes such as STB using Wiener filter [8], our information is limited by the history of the target particle, ignoring that every particle is spatially and temporally coherent with a specific group of other particles following the same behaviour. This motivated us to take into account a group of coherent motions for predicting a single particle. We propose to locally determine information of coherent and non-coherent particles during the trajectory procedure by using the Finite-Time Lyapunov Exponent (FTLE). More details of coherent motion detection are discussed in Section 2. After that, we address the prediction function with the minimisation approach in Section 3. In the following Sections 4 and 5, we study and evaluate our proposed technique using synthetic and experimental case studies of the wake over and behind a smooth cylinder at Reynolds number equal to 3900.

where ai is unknown coefficients of the predictor function. yn-m,n is known positions of the last finite frames (in this study m=4). Therefore, the least square cost function is,

The coherent prediction function adds information obtained from temporal / local spatial Lagrangian coherent particles to come up with additional constraints in the polynomial cost function (Equation 3). In the worst-case scenario where there is no coherent neighbour information, the prediction function is just a simple polynomial function without additional constraints. Each particle carries sets of information, including position, first and higher-order derivative values. Assuming positions of at least three time steps m are known. In this function, we impose first and second-order derivatives of each coherent particle into the prediction function as below,

Therefore, each particle will end up with the weighted averaged of local coherent velocity and coherent acceleration values (y ̇LCS , y ̈LCS). In the present study, we take four time step histories of particles to minimise the cost function and then predict the next step. The first and second derivatives of all coherent particles are weighted averaged based on their FTLE level and distance to the target particle. Two weighted averaged values, velocity and acceleration, of each target particle can be obtained for the estimation. Therefore, the modified cost function (i.e. coherent predictor) can be written as,

Velocity constraint controls the direction of the prediction function, while in the case of having high acceleration gradients, a second-order constraint is required to control the acceleration of the prediction. The solution for the cost function in Equation 5 is not only smooth on the history of the target particle but also satisfies local coherent first and second-order derivatives. We compared the performance of the coherent predictor with three other prediction functions, as listed in Table 1. DNS predictor was defined as a reference using the Euler equation to transport particle positions by the ground truth DNS velocity. Predicted particle positions are followed by shaking or other optimisation techniques. Therefore, all particles are either tracked or untracked except for the inlet and outlet trajectories. In every time-step, untracked particles are like additive noises and might gradually cause to collapse of the whole trajectory process. Due to this, untracked particles must be fed by other complementary treatments. To this end, new information can be extracted if any groups of tracked particles are found to be located in the neighbourhood of untracked particles with a time step phase delay (i.e. tn+1). This phase delay means that new tracked particles at time-step tn+1 are locally coherent with one specific untracked particle at time-step tn. Another technique to reduce the number of untracked particles is to use backward prediction, which is well established in classic schemes such as nearest neighbour trajectory. Similarly, we implemented the backward predictor to search for additional information from the coherent particles to estimate in reverse pace followed by backward shaking. On the one hand, surrounding information of an untracked particle can provide the least information to predict in backward pace. In addition, this treatment can also connect spilt tracklets for reconstructing longer particle trajectories. This process is iterative, meaning that every forward step is embedded with a series of backward estimations from the current time-step up to the first step. In the present study, we evaluated and compared the performance of coherent predictor only in forward prediction.