  Spatiotemporal dynamics is central to a wide range of applications from
climatology, computer vision to neural sciences. From temporal observations
taken on a high-dimensional vector of spatial locations, we seek to derive
knowledge about such dynamics via data assimilation and modeling. It is assumed
that the observed spatiotemporal data represent superimposed lower-rank smooth
oscillations and movements from a generative dynamic system, mixed with
higher-rank random noises. Separating the signals from noises is essential for
us to visualize, model and understand these lower-rank dynamic systems. It is
also often the case that such a lower-rank dynamic system have multiple
independent components, corresponding to different trends or functionalities of
the system under study. In this paper, we present a novel filtering framework
for identifying lower-rank dynamics and its components embedded in a high
dimensional spatiotemporal system. It is based on an approach of structural
decomposition and phase-aligned construction in the frequency domain. In both
our simulated examples and real data applications, we illustrate that the
proposed method is able to separate and identify meaningful lower-rank
movements, while existing methods fail.
