A Stream Function Approach to Optical Flow with Applications to Fluid Transport Dynamics

Abstract

Optical flow is one of the classical problems in computer vision, but it has recently also been adapted to applications from other fields, such as fluid mechanics and dynamical systems. If the goal is to analyze the dynamics of system whose evolution is governed by a flow field that is the gradient of a potential function -which describes many flows in fluid dynamics -it is natural to approach the optical flow problem by reconstructing the potential function, also called the stream function, rather than reconstructing the components of the flow directly. This alternate approach allows one to impose scientific priors, via regularization, directly on the flow itself rather than on its components independently. We demonstrate the stream function formulation of optical flow and its application to reconstructing an oceanic fluid flow driven by satellite measurements. It is also shown how these flow fields can be used to analyze mixing and mass transport in the fluid system being imaged. Stream Function Optical FlowOptical flow is an image-based method for computing the apparent flow field governing a system evolution. Recently much work has been done using optical flow to analyze fluids, which requires a different physical model from the classical HornSchunk method [12]. Corpetti et al. [6,7] proposed an alternative based on the continuity equation, resulting in an optical flow energy of the formwhere Ω is the spatial domain, R(u, v) is an appropriate regularization scheme, and α > 0 is the regularization parameter. This formulation has become common for analyzing fluid flows, for example in connection with and in comparison to particle image velocimetry (PIV) [5,13], geophysical fluid flows [1,2], and atmospheric motion [4]. While it is possible to incorporate the full Navier-Stokes equations for fluid flows in order to more accurately model the evolution of a fluid system [8], the model (1) is robust and results in more straightforward computational schemes. Under the assumption that the flow being imaged is a "potential flow," i.e. that the flow field is the gradient of a potential, or stream function, it is natural to instead formulate the optical flow problem in terms of reconstructing the stream function directly. Within a Hamiltonian framework, the evolution equation governing the flow is thus I t = −div (I∇ H ψ), where ∇ H ψ = −ψ y , ψ x is the symplectic gradient. The resulting optical flow energy iswhere R(ψ) is a regularization of the stream function. There are two primary goals of regularizing a variational optimization problem. The first goal is to ensure that the optimization problem is well-posed, i.e. that a solution exists, is unique, and is stable with respect to perturbations in the input data. For optical flow problems, this need not happen in general. In particular, for nearly any physically appropriate regularization, there exist data for which the governing flow field is not unique. The second reason for regularization -and one of the main advantages to computing optical flow via (2) -is to im...

Generalized Lagrangian coherent structures

“…Then, in our notation, c(x; t, τ ) gives a function on Ω which expresses the concentration distribution at t, subject to some initial concentration c 0 (x) at a reference/initial time τ . In a given problem of interest, such a scalar c could represent a pollutant concentration or some other kind of transported observable like the sea-surface temperature [56,44,45] or color [84]. A simplest model for the evolution of such scalar fields could be an advection-diffusion equation of the form…”

“…One such quantity is the vorticity, whose evolution is mutually dependent on v, as specified by the vorticity equation that can be derived from the Navier-Stokes equations [54,55]. The temperature might be thought of as either dynamically active (e.g., coupled to the velocity via the Boussinesq approximation in situations such as Rayleigh-Bénard convection [55]), or dynamically passive (e.g., carried by, but not affecting, the flow velocities [56,44,45]) depending on the context. In these examples, relevant "coherent" entities refer to regions of space characterized by the qualitatively similar behavior of the co-evolving variable, for example, regions containing similar tracer concentrations, or the codimension-1 boundaries between these regions, across which there are sharp tracer gradients.…”

“…Unless all these conditions-sufficiently high Pe and compatible initial condition and time-are satisfied, using LCSs as a predictor of GLCSs in this advection-diffusion situation is essentially useless. In some situations in which the FSLE field imputed from oceanic velocities has similarities with the ocean temperature field [56,44,45], presumably the "correct" conditions are approximately satisfied.…”

Bayesian optical flow with uncertainty quantification

“…Given A and b, the problem of inferring x in Eq. (19) is called an inverse problem because rather than direct "forward" computation from the model, it requires a set of indirect, "backward", or "inverse" operations to determine the unknowns [34]. Depending on the rank and conditioning of the matrix A, the problem may be ill-posed or ill-conditioned.…”

“…In particular, consider the model given by Eq. (19) with independent and identically distributed (iid) Gaussian noise of variance λ −1 . It follows that…”