title: The infinitesimal operator for the semigroup of the Frobenius-Perron operator from image sequence data: vector fields and transport barriers from movies.
abstract: In this paper, we present an approach to approximate the Frobenius-Perron transfer operator from a sequence of time-ordered images, that is, a movie dataset. Unlike time-series data, successive images do not provide a direct access to a trajectory of a point in a phase space; more precisely, a pixel in an image plane. Therefore, we reconstruct the velocity field from image sequences based on the infinitesimal generator of the Frobenius-Perron operator. Moreover, we relate this problem to the well-known optical flow problem from the computer vision community and we validate the continuity equation derived from the infinitesimal operator as a constraint equation for the optical flow problem. Once the vector field and then a discrete transfer operator are found, then, in addition, we present a graph modularity method as a tool to discover basin structure in the phase space. Together with a tool to reconstruct a velocity field, this graph-based partition method provides us with a way to study transport behavior and other ergodic properties of measurable dynamical systems captured only through image sequences.
authors.affiliation: Department of Mathematics and Computer Science, Clarkson University, Potsdam, New York 13699-5815, USA.
authors.forename: N
authors.initials: N
authors.lastname: Santitissadeekorn
authors.forename: E M
authors.initials: EM
authors.lastname: Bollt