Towards Glyphs for Uncertain Symmetric Second‐Order Tensors

Abstract

Measured data often incorporates some amount of uncertainty, which is generally modeled as a distribution of possible samples. In this paper, we consider second‐order symmetric tensors with uncertainty. In the 3D case, this means the tensor data consists of 6 coefficients – uncertainty, however, is encoded by 21 coefficients assuming a multivariate Gaussian distribution as model. The high dimension makes the direct visualization of tensor data with uncertainty a difficult problem, which was until now unsolved. The contribution of this paper consists in the design of glyphs for uncertain second‐order symmetric tensors in 2D and 3D. The construction consists of a standard glyph for the mean tensor that is augmented by a scalar field that represents uncertainty. We show that this scalar field and therefore the displayed glyph encode the uncertainty comprehensively, i.e., there exists a bijective map between the glyph and the parameters of the distribution. Our approach can extend several classes of existing glyphs for symmetric tensors to additionally encode uncertainty and therefore provides a possible foundation for further uncertain tensor glyph design. For demonstration, we choose the well‐known superquadric glyphs, and we show that the uncertainty visualization satisfies all their design constraints.

Uncertainty‐aware Visualization in Medical Imaging ‐ A Survey

“…a) Comparison of two different tensor fields [ZSL∗16]. b) Uncertainty‐aware tensor glyphs with surrounding surfaces [GRT19].…”

“…In addition, the cones are oriented such that the tractography direction can be determined. Gerrits et al [GRT19] presented an uncertainty-aware visualization of tensor glyphs. In their methods, the tensor glyph is surrounded by a transparent hull that indicates the potential variation in the glyph appearance, as shown in Figure 17(b).…”

“…Here, glyphs are a common visualization tool. Ropinski et al [ROP11] provided an overview on glyph‐based spatial multivariate medical data and highlighted the need for uncertainty‐aware tensor glyphs [GRT19].…”

Visualization of Tensor Fields in Mechanics

“…It is based on the estimation and decomposition of the fiber distribution into the main direction and a non-negative residual. The most recent work by Gerrits et al [GRT19] visualized the uncertainty tensor as a set of mean and covariance tensors. They used standard glyph designs for the mean tensor (see Figure 9) and Figure 9: Uncertainty glyph using superquadric (left) and the glyph from Gerrits et al [GRT17] (right) as base for an indefinite mean tensor (image from [GRT19]).…”

Uncertainty in the DTI Visualization Pipeline

“…Unlike Basser et al, Abbasloo et al visualize the confidence interval at each eigenmode separately by glyph overlays and used animation to visualize the differences in each mode. Gerrits et al [35] pointed out the shortcoming in both of these visualization techniques and proposed a generic approach that incorporates all the coefficients of the mean tensor and covariance in a single glyph.…”