A framework of particle missing compensation for particle tracking velocimetry via global optimization

Abstract

The performance of particle tracking velocimetry (PTV) is constrained by a practical issue, i.e., particle missing in a particle image frame. The randomly appeared loss-of-pair particles will bias the particle pairing relationship that is sought by particle matching algorithms. To handle this issue, this work proposes a general framework for the compensation of particle missing in PTV. Following our previous work (Nie in Exp Fluids 62(4): 68, 2021), we deal with the family of ant colony optimization (ACO) algorithms, which convert the task of particle matching to a global optimization problem for a particular objective function and seek a solution using ACO. To enable particle missing compensation, two core concepts are proposed. The first is to perform two symmetric ACOs from the forth and the back directions on a straddle-frame image pair, and then cross-validate the two sets of solutions to estimate the valid or problematic matching. The second is to make the action of the forth and the back ant colonies work in coordination with each other by sharing their knowledge on the particle matching relationship. This is implemented using an exterior loop, in which the intersection of the solutions obtained by the two colonies of ants, called mutual knowledge, will be learned and passed to the next generation. The first concept relies on an operation of virtual-particle add-on. It leads to the so-called cross-validation ant colony optimization (CVACO) algorithm. The second concept updates CVACO by dynamically adjusting the pheromone factor and the heuristic factor on each candidate particle pair in the next exterior pass, forming the so-called algorithms of pheromone-feedback CVACO (PF-CVACO) and heuristic-feedback CVACO (HF-CVACO), respectively. A synthetic test shows that the proposed algorithms work well in the scenarios of both single-frame particle missing and dual-frame particle missing at low-to-moderate particle missing rates, which cannot be well handled using conventional single-pass ACO.

Graphical Abstract

A Appendix: Influence of the scale factor v

Referring to Eq. (16) and Eq. (18), the scale factor v determines the magnitude of $\text {vof}$, which in turn is a critical parameter affecting the performance of CVACO. The effect of v is empirically tested based on the dataset used in Sect. 4.

Fig. 12 plots the variation of OC and TA as functions of v with $\xi$=0.1 and DPF as the objective function. OC is seen to be independent of v. It can be attributed to the cross-validation strategy, which is capable of identifying valid matching in the output solution. By contrast, TA reaches a plateau in the range of $v=\left[ 1, 3\right]$. Referring to Fig. 3, this observation suggests that the ‘distance’ of virtual particles to real particles, i.e., $\text {vof}$, needs to be comparable to or slightly larger than the mean ‘distance’ of real particle pairs. A smaller $\text {vof}$ will bias ants towards virtual particles, while a larger $\text {vof}$ will reduce the weight of virtual particles to represent missing particles. Both cases lower the algorithm’s productivity, i.e., OR, and thus lead to the deterioration of TA. Based on this observation, the default value of the scale factor is recommended to be $v=2$.

Fig. 12

Dependence of a OC and b TA on the scale factor v in Eq. (18) in CVACO with the objective function of DPF and particle missing rate of $\xi$=0.1. The corresponding value in conventional ACO is shown as the horizontal dashed line for reference