A Kalman filtering approach to particle track filtering and track uncertainty quantification for 3D PTV measurement

Abstract

Three-dimensional Particle Tracking Velocimetry (3D-PTV) is a non-invasive flow measurement technique that computes the velocity field by reconstructing 3D particle positions of individual tracer particles and by subsequently tracking those positions. The particle velocity measurement accuracy depends on the faithful reconstruction of 3D particle positions. The complex measurement chain in 3D-PTV involves several steps, from calibration to 3D position reconstruction and particle position tracking, each having its own source of error. Additionally, higher seeding density increases the uncertainty in particle reconstruction and tracking, which in turn, increases the noise in the estimated tracks. A noisy track decreases the measurement accuracy and amplifies any noise in the PTV-derived quantities of interest, which includes acceleration, pressure and vorticity. Thus, track filtering techniques are critical in a 3D-PTV measurement. Track fitting using polynomial functions, filtering methods adopted from signal processing and object tracking are among the well-established techniques used to achieve smooth position, velocity estimates from reconstructed particle trajectories. The Kalman filter is one such filtering technique that is widely used in various applications. The strength of the Kalman filter lies in its ability to perform noise reduction that is informed by existing physical models and the uncertainty estimates of recorded measurements. However, the measurement uncertainty input to the Kalman filter needs to be known at priori, which in many cases may not be available or could be difficult to estimate. In the literature on Kalman filters and their variants applied to 2D-PIV/PTV, the position uncertainty data fed to the filter is either user-defined or estimated based on global noise levels in the PTV measurements. But instantaneous position and velocity uncertainty quantification for individual particle positions/tracks has been challenging in the 3D PTV community. Recent work by Bhattacharya and Vlachos (2020) provides an estimate of the uncertainty in the reconstructed particle positions for a 3D PTV measurement. This position uncertainty estimate dynamically updates the filter gain for each track and enables the evaluation of the performance of the Kalman filter in 3D PTV track filtering.