Fast algorithms for the reconstruction of images from hyperbolic systems

In this paper, we discuss efficient methods for reconstructing the state variables and inputs of certain hyperbolic systems. We assume that the hyperbolic system is characterized by a system of partial differential equations that it is operating in steady state, and that the observation interval is long enough to reliably compute the Fourier transform (or Fourier series if the data is periodic) with respect to time of the observations. We specifically discuss algorithms applicable to the 2-D wave equation with continuous observations over a rectangle, and a class of hyperbolic systems having one spatial dimension and point sensors placed on the system. It is shown that under realistic stability conditions, these reconstruction algorithms are well-posed. We also present a decentralized implementation of the one spatial dimension, point sensor reconstruction algorithm.