Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound Imaging

Abstract: We consider the dynamical super-resolution 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 super-resolution 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 super-resolution. We provide a rigorous mathematical analysis of the recovery problem, both for the noiseless case and in 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 super-resolution problem in ultrafast ultrasound imaging: blood vessels' locations and blood flow velocities are recovered with super-resolution.