  Particle tracking in biological imaging is concerned with reconstructing the
trajectories, locations, or velocities of the targeting particles. The standard
approach of particle tracking consists of two steps: first reconstructing
statically the source locations in each time step, and second applying tracking
techniques to obtain the trajectories and velocities. In contrast, the dynamic
reconstruction seeks to simultaneously recover the source locations and
velocities from all frames, which enjoys certain advantages. In this paper, we
provide a rigorous mathematical analysis for the resolution limit of
reconstructing source number, locations, and velocities by general dynamical
reconstruction in particle tracking problems, by which we demonstrate the
possibility of achieving super-resolution for the dynamic reconstruction. We
show that when the location-velocity pairs of the particles are separated
beyond certain distances (the resolution limits), the number of particles and
the location-velocity pair can be stably recovered. The resolution limits are
related to the cut-off frequency of the imaging system, signal-to-noise ratio,
and the sparsity of the source. By these estimates, we also derive a stability
result for a sparsity-promoting dynamic reconstruction. In addition, we further
show that the reconstruction of velocities has a better resolution limit which
improves constantly as the particles moving. This result is derived by an
observation that the inherent cut-off frequency for the velocity recovery can
be viewed as the total observation time multiplies the cut-off frequency of the
imaging system, which may lead to a better resolution limit as compared to the
one for each diffraction-limited frame. It is anticipated that this observation
can inspire new reconstruction algorithms that improve the resolution of
particle tracking in practice.
