Glyphs for Asymmetric Second‐Order 2D Tensors

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.

Section: Related Workmentioning

confidence: 99%

“…This means eigenvectors are generally non-orthogonal, and eigenvalues/eigenvectors may be non-real. Seltzer et al [SK16] introduce glyphs for asymmetric second-order 2D tensors, where texture is used to encode rotational behavior. Gerrits et al…”

Section: Tensor Glyphsmentioning

confidence: 99%

Towards Glyphs for Uncertain Symmetric Second‐Order Tensors

Gerrits

Rössl

Theisel

2019

“…Superquadric tensor glyphs are commonly employed as base geometry for their ability to reduce ambiguity and preserve continuity [Kin04, SK10b, SK16]. A glyph G is constructed following…”

Section: Glyph‐based Overview Visualizationmentioning

confidence: 99%

“…s is an overall scaling function, which takes the tensor trace tr( D ) as input. Unlike the previous work [SK10b, SK16], we can use tr( D ) instead of the Frobenius norm || D || because we normalize eigenvalues with respect to tensor trace as presented in Section 3.1.…”

Section: Glyph‐based Overview Visualizationmentioning

confidence: 99%

Overview + Detail Visualization for Ensembles of Diffusion Tensors

Zhang

Caan

Höllt

et al. 2017

A Diffusion Tensor Imaging (DTI) group study consists of a collection of volumetric diffusion tensor datasets (i.e., an ensemble) acquired from a group of subjects. The multivariate nature of the diffusion tensor imposes challenges on the analysis and the visualization. These challenges are commonly tackled by reducing the diffusion tensors to scalar‐valued quantities that can be analyzed with common statistical tools. However, reducing tensors to scalars poses the risk of losing intrinsic information about the tensor. Visualization of tensor ensemble data without loss of information is still a largely unsolved problem. In this work, we propose an overview + detail visualization to facilitate the tensor ensemble exploration. We define an ensemble representative tensor and variations in terms of the three intrinsic tensor properties (i.e., scale, shape, and orientation) separately. The ensemble summary information is visually encoded into the newly designed aggregate tensor glyph which, in a spatial layout, functions as the overview. The aggregate tensor glyph guides the analyst to interesting areas that would need further detailed inspection. The detail views reveal the original information that is lost during aggregation. It helps the analyst to further understand the sources of variation and formulate hypotheses. To illustrate the applicability of our prototype, we compare with most relevant previous work through a user study and we present a case study on the analysis of a brain diffusion tensor dataset ensemble from healthy volunteers.

“…Glyphs are a common tool to encode multi-dimensional data in space [BKC * 13]. Seltzer and Kindlmann [SK16] and Gerrits et al [GRT17] recently proposed glyphs for general asymmetric second-order tensors in 2D and 3D. They formulated lists of desirable properties that a glyph should fulfill (invariance under isometric domain transformations and scaling, direct encoding of real eigenvalues and eigenvectors, uniqueness and continuity).…”

Section: Visualization Of Inertial Critical Pointsmentioning

confidence: 99%

Flow-Induced Inertial Steady Vector Field Topology

Günther

Groß

2017

Model I: dp = 100 µm Model II: R = 1, St = 0.2 Model I: dp = 100 µm Model II: R = 1, St = 0.2 Figure 1: Critical points in 2D and 3D flows for two different particle models. In contrast to massless flows, inertial particles might oscillate. AbstractTraditionally, vector field visualization is concerned with 2D and 3D flows. Yet, many concepts can be extended to general dynamical systems, including the higher-dimensional problem of modeling the motion of finite-sized objects in fluids. In the steady case, the trajectories of these so-called inertial particles appear as tangent curves of a 4D or 6D vector field. These higher-dimensional flows are difficult to map to lower-dimensional spaces, which makes their visualization a challenging problem. We focus on vector field topology, which allows scientists to study asymptotic particle behavior. As recent work on the 2D case has shown, both extraction and classification of isolated critical points depend on the underlying particle model. In this paper, we aim for a model-independent classification technique, which we apply to two different particle models in not only 2D, but also 3D cases. We show that the classification can be done by performing an eigenanalysis of the spatial derivatives' velocity subspace of the higher-dimensional 4D or 6D flow. We construct glyphs that depict not only the types of critical points, but also encode the directional information given by the eigenvectors. We show that the eigenvalues and eigenvectors of the inertial phase space have sufficient symmetries and structure so that they can be depicted in 2D or 3D, instead of 4D or 6D.