Dynamic Spike Superresolution and Applications to Ultrafast Ultrasound Imaging

We consider the dynamical superresolution problem consisting in the recovery of positions and velocities of moving particles from low-frequency static measurements taken over multiple time steps. The standard approach to this issue is a two-step process: first, at each time step some static reconstruction method is applied to locate the positions of the particles with superresolution, and, second, some tracking technique is applied to obtain the velocities. In this paper we propose a fully dynamical method based on a phase-space lifting of the positions and the velocities of the particles, which are simultaneously reconstructed with superresolution. We provide a rigorous mathematical analysis of the recovery problem, both for the noiseless case and in the presence of noise (in the discrete setting). Several numerical simulations illustrate and validate our method, which shows some advantage over existing techniques. We then discuss the application of this approach to the dynamical superresolution problem in ultrafast ultrasound imaging: blood vessels' locations and blood flow velocities are recovered with superresolution.