PDE Methods in Flow Visualization

Description  

Vector field visualization is an important topic in scientific visualization. Its aim is to graphically represent field data in an intuitively understandable and precise way. Two novel methods are described which enable an easy perception of flow data. The texture transport method especially applies to time-dependent velocity fields. Lagrangian coordinates are computed solving the corresponding linear transport equations numerically. Choosing an appropriate texture on the reference frame the coordinate mapping ca n be applied as a suitable texture mapping.

Alternatively, the aligned diffusion methods serves as an appropriate scale space method for the visualization of complicated flow patterns. It is closely related to nonlinear diffusion methods in image analysis where images are smoothed while still retaining and enhancing edges. Here an initial noisy image is smoothed along streamlines, whereas the image is sharpened in the orthogonal direction. 

A generalization of the diffusion based multiscale approach allows for a multiscale visualization of long-time, complex transport problems. Instead of streamline type patterns generated by the original method now streakline patterns are generated and advected. This process obeys a nonlinear transport diffusion equation with typically dominant transport.  Again Starting from some noisy initial image, the diffusion actually generates and enhances patterns which are then transported in the direction of the flow field.  As before the image is simultaneously sharpened in the direction orthogonal to the flow field. A careful adjustment of the model's parameters is derived to balance diffusion and transport effects in a reasonable way. 

The methods have in common that they are based on a continuous model and discretized only in the final implementational step. Therefore, many important properties are naturally established already in the continuous model.

The steady-flow model.

For steady flow fields v on the domain, we consider the following boundary and initial value problem

that delivers a family of images 

with .

Thereby the diffusion tensor A acts smoothing in directions of the vector field and sharpening in orthogonal directions:

.

G(s) is the so called Perona-Malik function that steers the sharpening and clustering process in orthogonal directions:

The evolution is started with a white noise to avoid aliasing artefacts. 

The color coding of the images is arbitrary. In the above examples we chose a 2D color space with randomly distributed colors in that space. We can also colorize the images according to additional information (e.g. velocity, concentration).

The time-dependent-flow model.

The extension of the anisotropic diffusion method adds a transport of the resulting patterns. Thereby we work with the following PDE, where now v = v(t)

.

Since in this model the artificial scale and the time of the vector field data are not separated, we obtain the undesired behavior of moving patterns that grow coarser while the evolution runs. In the diagram below this would correspond to walking along the dashed red line. Our desired behavior would be representations that lie on a vertical line in this diagram. I.e. we would like to fix the scale and then let time run.

Furthermore to enable for a smooth animation of the images, we obtain for different times, we choose an approach that starts different processes of the visualization with coupled scale and time and the blends the results in an appropriate way. This corresponds to the yellow path in the above diagram. 

Figures