Exploring Uncertainty Visualization for Degenerate Tensors in 3D Symmetric Second-Order Tensor Field Ensembles

TL;DR

Novel methods for describing and visualizing degenerate tensor locations in 3D symmetric second-order tensor field ensembles are explored and the results indicate that the interplay of different descriptions for uncertainty can effectively convey information on degenerate tensor locations.

Abstract

Symmetric second-order tensors are fundamental in various scientific and engineering domains, as they can represent properties such as material stresses or diffusion processes in brain tissue. In recent years, several approaches have been introduced and improved to analyze these fields using topological features, such as degenerate tensor locations, i.e., the tensor has repeated eigenvalues, or normal surfaces. Traditionally, the identification of such features has been limited to single tensor fields. However, it has become common to create ensembles to account for uncertainties and variability in simulations and measurements. In this work, we explore novel methods for describing and visualizing degenerate tensor locations in 3D symmetric second-order tensor field ensembles. We base our considerations on the tensor mode and analyze its practicality in characterizing the uncertainty of degenerate tensor locations before proposing a variety of visualization strategies to effectively communicate degenerate tensor information. We demonstrate our techniques for synthetic and simulation data sets. The results indicate that the interplay of different descriptions for uncertainty can effectively convey information on degenerate tensor locations.

Figures (8)

• Figure 1: Tensor mode distribution in a synthetic ensemble. Top row: $\mathop{\mathrm{mode}}\nolimits_{abs}$ values represented as color values using the cool-to-warm color map for the different members. Bottom left: $\mathop{\mathrm{mode}}\nolimits_{abs}$ value of the mean tensor field. Bottom center: mean $\mathop{\mathrm{mode}}\nolimits_{abs}$ of all members. Bottom right: spaghetti plot of extracted degenerate tensor lines.
• Figure 2: Construction of the Ensemble modeTube: Based on the differences between mean mode value at ${{\mathbf p}}_0$ on the meanLine and samples ${{\mathbf p}}_c$ on a circle perpendicular to the line tangent, the point locations are moved towards (mean mode at ${{\mathbf p}}_0$ is higher) or away from (mean mode at ${{\mathbf p}}_0$ is lower) the center, indicating the spatial distribution of mean $\mathop{\mathrm{mode}}\nolimits_{abs}$ values around the meanLine. Mode values are indicated by a divergent cool-to-warm color map.
• Figure 3: Ensemble comprising five members using a synthetic tensor field. Left: Spaghetti plot showing the extracted degenerate tensor lines of all members each represented by a shade of green and the modeTube indicating the distribution of mean mode values around the meanLine. Color and shape both indicate the difference in mean mode values between the sample locations and the meanline. Right: Enhanced meanLine indicating degenerate tensor locations of the mean tensor field and the standard deviation of mode values encoded by color and radius. Further, the three nested iso contours indicate different probabilities ($15\%$, $50\%$, $90\%$) of mode values $\geq0.95$ closing in.
• Figure 4: Ensemble comprising 1000 members each constructed by applying random noise to the components of a synthetic tensor field. Left: Spaghetti plot showing the extracted degenerate tensor lines of 10 randomly selected members. Center: Enhanced meanLine with mode standard deviation encoded as color and probabilityBand encompassing locations, where the probability of finding absolute mode values $\geq0.95$ is $33\%$. Right: Uncertain Tensor visualization gerrits2019towards of tensors located on the meanLine using superquadric tensor glyphs schultz2010superquadric.
• Figure 5: Uncertainty Visualizations for two simulation results describing stresses in an O-ring with varying $p$ parameter. (a) Each member has exactly one degenerate tensor line shown in a different shade of green. (b) The enhanced meanLine shows the location of the degenerate tensor line within the mean tensor field. Radius and color indicate the standard deviation of mode values. (c) probabilityBand indicating locations where mode values have a probability of $50\%$ of a mode value $\geq 0.95$. (d) probabilityBand indicating locations where mode values have a probability of $10\%$ of a mode value $\geq 0.99$. (e) Closeup views show, from top to bottom, an enhanced meanLine, the modeTube, and the probabilityBand visualizations overlaid to the spaghetti plot of degenerate tensor lines of all ensemble members.