  As most mathematically justifiable Lagrangian coherent structure detection
methods rely on spatial derivatives, their applicability to sparse trajectory
data has been limited. For experimental fluid dynamicists and natural
scientists working with Lagrangian trajectory data via passive tracers in
unsteady flows (e.g. Lagrangian particle tracking or ocean buoys), obtaining
material measures of fluid rotation or stretching is currently only possible
for trajectory concentrations that are often out-of-reach. To facilitate
frame-indifferent investigations in unsteady and sparsely sampled flows, we
present a novel approach to quantify fluid stretching and rotation via relative
Lagrangian velocities. This technique provides a formal objective extension of
quasi-objective metrics to unsteady flows by accounting for mean flow behavior.
For extremely sparse experimental data, fluid structures may be significantly
undersampled, and the mean flow behavior becomes difficult to quantify. We
provide a means to maintain the accuracy of our novel sparse flow diagnostics
in extremely sparse sampling scenarios, such as ocean buoy data and Lagrangian
particle tracking. We use data from multiple numerical and experimental flows
to show that our methods can identify structures beyond existing limits of
sparse, frame-indifferent diagnostics, and exhibit improved interpretability
over common frame-dependent diagnostics.
