A Physics-Informed and Uncertainty-Informed Radial Basis Function Meshless Proper Orthogonal Decomposition for 3D Particle Tracking Velocimetry (PTV)

3D Particle Track Velocimetry offers superior spatial resolution compared to window-based PIV by resolving flow for each tracked particle. However, experimental limitations such as inappropriate particle density and non-uniform illumination can result in incorrect or noisy particle tracks. Additionally, to visualize and evaluate flow quantities, it is essential to interpolate the velocities from unstructured tracks to a regular grid. We propose a Radial Basis Function (RBF) approach utilizing meshless Proper Orthogonal Decomposition (POD) for denoising and interpolating the particle tracks onto a structured grid. Denoising is achieved by rejecting modes with insignificant contributions to the physical flow variance via Student’s t-test. The interpolation is then performed based on the spatial structures of the significant POD modes, guided by the Nyquist frequency. These sparse tracks define the RBFs, whose scaling is determined using velocity and acceleration uncertainty-weighted Least Squares. Mass and momentum conservation are incorporated by enforcing divergence-free and vorticity conservation. Gradient operators up to the third order are computed analytically from infinitely differentiable RBFs. In contrast, traditional polynomial interpolants lose curvature and accuracy beyond the second-order differentiation. We compare our results with standard interpolation methods by evaluating the quality of virtual particle advection and Finite Lyapunov Exponents on Shake-the-Box datasets.