Linear Space-Invariant System Identification and Mismatch Bounds for Estimation of Dynamical Images

For linear space-invariant temporal systems, we provide a lower bound on the penalty incurred by approximating system dynamics in a Kalman filter by a random walk model, a common model when dynamics are unknown. We then present a computationally tractable algorithm for system identification of high-dimensional linear space-invariant dynamical systems, whereby the circulant structure of the state transition operator yields an estimate of the governing dynamics from a small number of temporal steps. By completing all operations in the frequency domain, we efficiently provide an estimate of the system dynamics and the state of the system. The estimation of system dynamics greatly improves the state estimation over the random walk model, suggesting classical estimators may remain applicable in modern imaging tasks.