An Optical Flow Approach to Analyzing Species Density Dynamics and Transport

Abstract

Classical optical fiow techniques were developed for computing virtual motion fields between two images of the same scene, assuming conservation of intensity and a smoothness of the flow field. If these assumptions are dropped, such techniques can be adapted to compute apparent flows between time snapshots of data that do not come from images, even if these fiows are turbulent and divergent, as in the case of flows representing complex spatiotemporal dynamics. While imaging methods have been used to analyze dynamics in experimental applications, they are only beginning to be applied to dynamics computations in settings outside the laboratory, for example in the analysis of species population dynamics from satellite data. In this work we present a variational optical flow approach based on the continuity equation and total variation regularization for computing the flow fields between population densities generated from a two-species predator-prey model for phyto-and Zooplankton interactions. Given the time-varying vector fields produced from the optical flow, computational methods from dynamical systems can be employed to study pseudo-barriers present in the species interaction. This method allows to measure the mixing of the species, as well as the transport of the populations throughout the domain.Mathematics subject classification: 49N45, 49M99, 37M25, 65P99, 68T45.

A framework for estimating potential fluid flow from digital imagery

“…In practice, the derivatives are computed using finite differences in a matrix form, and the adjoint operator reduces to a matrix transpose. Equation (16) can be solved using iterative techniques or direct methods for solving sparse linear systems. In practice, we use a direct LU factorization for solving all linear Euler-Lagrange equations.…”

“…Whereas some image-based techniques for fluid dynamics analysis require rigid experimental conditions-such as PIV-optical flow based techniques can be used in more general settings, and that is one of the primary reasons for developing optical flow-based approaches to fluid dynamics analysis. For example, given satellite data, it could be possible to analyze the dynamics of species transport in the ocean 16 or other kinds of pseudo-transport; 17 analysis that could not be done from methods like PIV. The primary challenge in determining whether optical flow is appropriate to use in a given application is understanding how the 2D image projections of the flow relate to the physical 3D flow, which heavily depends on the application being studied.…”