  The three-dimensional Time-Resolved Lagrangian Particle Tracking (3D TR-LPT)
technique has recently advanced flow diagnostics by providing high
spatiotemporal resolution measurements under the Lagrangian framework. To fully
exploit its potential, accurate and robust data processing algorithms are
needed. These algorithms are responsible for reconstructing particle
trajectories, velocities, and differential quantities (e.g., pressure
gradients, strain- and rotation-rate tensors, and coherent structures) from raw
LPT data. In this paper, we propose a three-dimensional (3D) divergence-free
Lagrangian reconstruction method, where three foundation algorithms --
Constrained Least Squares (CLS), stable Radial Basis Function (RBF-QR), and
Partition-of-Unity Method (PUM) -- are integrated into one comprehensive
reconstruction strategy. Our method, named CLS-RBF PUM, is able to (i) directly
reconstruct flow fields at scattered data points, avoiding
Lagrangian-to-Eulerian data conversions; (ii) assimilate the flow diagnostics
in Lagrangian and Eulerian descriptions to achieve high-accuracy flow
reconstruction; (iii) process large-scale LPT data sets with more than hundreds
of thousand particles in two dimensions (2D) or 3D; (iv) enable spatiotemporal
super-resolution while imposing physical constraints (e.g., divergence-free for
incompressible flows) at arbitrary time and location. Validation based on
synthetic and experimental LPT data confirmed that our method can consistently
achieve the above advantages with accuracy and robustness.
