Stochastic models for DIV-CURL optical flow methods

Abstract

In this letter, we consider Suter's DIV-CURL optical flow methods, wherein the Problem of Computing a Velocity field from an image sequence is regularized using smoothness conditions based on the divergence and curl of the Geld. In particular, we develop stochastic formulatiom of DN-CURL splines using the linear smoothing theory of Ad", Willsky, and Levy. Our models are shown to be well posed and thus can be used in both simulating and estimating velocity fields having known stochastic properties. As a special case, our stochastic model reduces to that developed by Rougee, Levy, and Willsky for the classical Horn and Schunck's optical flow.processes. This stochastic model was used by Denney and Psince in [5] and [6] to derive optimal brightness functions for HSOF.In this letter, we first show that Rougee's model is overdetermined, giving a nonphysical model that cannot be used in simulations. We next develop well-posed state-space models for Suter's D W -C W OF methods. Finally, we show that a weu-posed State space model for HSOF is as a special case.

Snakes, shapes, and gradient vector flow

“…It has recently been shown that this term corresponds to an equal penalty on the divergence and curl of the vector field [22]. Therefore, the vector field resulting from this minimization can be expected to be neither entirely irrotational nor entirely solenoidal.…”

Dense Motion Analysis in Fluid Imagery

“…In practice, these extrema are obtained through a simple morphological processing of the potential functions. Both are derived from dense motion field Helmholtz components w ir = ∇φ and w ⊥ so = ∇ψ, using (12). Note that the laminar part of the motion field is assumed to be well approximated by the smooth motion field estimate obtained at the coarsest level of our multigrid setting.…”

“…In addition, an under-estimation of the divergence would be all the more problematic in our case, because the data model includes an explicit use of this quantity. For these reasons, it would seem more appropriate to rely on second-order div-curl regularization [12,22]:…”

A Multi-scale Vector Spline Method for Estimating the Fluids Motion on Satellite Images

“…(1) The conservation equation applies either to the luminance [2] or to the mass [3,4]. (2) The regularity is modelled by the 2nd order div-curl constraint, which provides direct control of the Helmholtz decomposition of the motion field in terms of divergence and vorticity [5]. A recent study [6] formulates the regularity with effective physical constraints: the retrieved flow is supposed steady and satifying the incompressible Stokes equations.…”