Critical point detection in fluid flow images using dynamical system properties

Dense estimation of fluid flows

“…The second-order regularization is replaced by two interleaved first-order regularizations. From the EulerLagrange point-of-view, this amounts to replacing the fourth-order PDE associated with (16), by a coupled PDE of order two. In the next section, we shall see that our minimization, although not posed in terms of PDEs, turns out to be a discrete problem of reasonable complexity as assessed by the locality of the interactions among the unknown variables (or, equivalently, by the bandwidth of the nonlinear sparse system to be solved).…”

“…Based on a dense displacement field previously estimated, such structures can be identified via the extraction and analysis of the critical points (i.e., where the speed vanishes locally) of the flow [11], [16], [17], [18], [33], [35]. This usually requires the fitting of parametric motion models and gives access only to the structure of interest modulo the laminar (i.e., divergence and vorticity free) component of the flow: If a vortex is transported by a global laminar motion, say a translation, its center, being nonstationary, will not be a singular point of the flow, although it is the real location of interest.…”

“…The numerical derivation of these fields provides estimates of the divergence and vorticity [43], whereas numerical integration provides stream functions, velocity potentials [39], [40], and point trajectories [40]. Based on the estimated motion fields, the so-called critical points of the flow can also be recovered and classified [11], [16], [17], [18], [33], [35]. Whatever the domain of application, these points are important to understand, predict, and compactly represent the flows of interest.…”

Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries

“…Other applications are wake flow behind bluff bodies, [12][13][14] flow close to free and viscous surfaces, 15,16 channel flow over a step, 17 axisymmetric flow, 18,19 and the streamline topology of point vortex flow. 20 In visualization experiments a topological approach has been used for pattern recognition 21 and for constructing a wavelet basis for structure identification. 22 When the flow field changes due to unsteadiness of the flow or an external change of parameters or boundary conditions, the streamline topology may also change.…”

Exploration of uncertainty in bidirectional vector fields

“…For example, if the critical points in possible flow fields were automatically detected, such as with Ford's approach, 6 they could be labelled showing the classification confidence. We expect it may be of value to integrate this or the interpreter's confidence directly into the visualization as a decision aid.…”