GLYPH- AND TEXTURE-BASED VISUALIZATION

OF SEGMENTED TENSOR FIELDS

Cornelia Auer, Claudia Stripf, Andrea Kratz and Ingrid Hotz

Zuse Institut Berlin, Takustrasse 7, 14195 Berlin, Germany

Keywords:

Tensor Field Visualization, Texture Mapping, Glyph Placement.

Abstract:

In this work we show how the variability of visualization methods like textures and glyphs can be used to

enhance established methods like topology. A topology-based segmentation (Auer et al., 2011) serves as

framework to map textures and place glyph exponents for two dimensional symmetric tensor ﬁelds. The

textures encode physical properties of the underlying ﬁeld and support the understanding of the ﬁeld as a

whole, whereas the glyph exponents can be used to give detailed insight at distinctive locations.

1 INTRODUCTION

Where to Place which Information and how? This

is a crucial question even experts in visualization are

permanently confronted with. An incomplete view

is as undesirable as a cluttered image. Topological

methods have a long tradition and for domain experts

they serve as visualization itself. However, for un-

trained users this might be too abstract – especially

if the visualization is meant to support a ﬁrst under-

standing of the data. In this work, we exploit the

topology-based segmentation of a two-dimensional

tensor ﬁeld (Auer et al., 2011) as basis for texture

generation, as well as for the placement of glyphs.

Textures are a powerful tool to design continu-

ous visualizations. They offer many parameters that

can be used to encode features of interest. This

makes textures especially interesting for the visual-

ization of tensor ﬁelds. Tensor ﬁelds contain direc-

tional and scalar features which can be encoded in

texture parameters. One challenge for many texture-

based methods is the deﬁnition of appropriate tex-

ture coordinates. This is accomplished by the cells of

the topology-based segmentation; they serve as con-

sistent framework for the texture mapping. A dense

visualization of the underlying tensor ﬁeld can be

generated using every pixel of the image to convey

physical properties of the ﬁeld. Furthermore, a di-

versity of visualization options arises: To encode di-

rectional information, for example, stripe patterns can

be used; but also knitting or basketwork patterns are

presented and evaluated towards their applicability to

depict central tensor properties within the segmentat-

ion framework. The goal hereby is not to display all

possible features in one image but to offer a ﬂexible

framework to a user to switch views for the different

points of interest.

The placement of glyphs is an intricate topic for

vector as well as tensor visualization. Here, we use

the tensor ﬁeld segmentation to display glyphs at dis-

tinctive locations in the given cells. The advantages

of the global structural information provided by the

topology and the detailed depiction of representative

tensors via glyphs are combined in a single image.

Tensors are of interest in many application ﬁelds,

for example, engineering, physics but also computer

graphics. Especially for tensors in physics and engi-

neering, even for two dimensional ﬁelds, the develop-

ment of expressive visualizations is still an ongoing

research topic. We will demonstrate our results for

stress tensors which are of high importance in me-

chanical engineering.

2 RELATED WORK

The focus of this review is on visualization meth-

ods designed for tensor ﬁelds arising in applications

such as physics and mechanical engineering. In gen-

eral, the developed visualization methods can be sub-

divided into local and continuous methods.

Local Methods. Local methods use geometric ob-

jects (glyphs) to depict single tensors at discrete

points. Shape, size, color, and texture can be used

to encode tensor properties. The most common glyph

670

Auer C., Stripf C., Kratz A. and Hotz I..

GLYPH- AND TEXTURE-BASED VISUALIZATION OF SEGMENTED TENSOR FIELDS.

DOI: 10.5220/0003857106700677

In Proceedings of the International Conference on Computer Graphics Theory and Applications (IVAPP-2012), pages 670-677

ISBN: 978-989-8565-02-0

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

for tensors is an ellipsoid alligned to the tensor’s

eigenvectors and scaled according to its eigenvalues

(Figure 1). Beside the design of glyphs (see (Hashash

et al., 2003) for an overview of glyphs for stress and

strain tensors), perceptional issues such as visual am-

biguity need to be solved (Schultz and Kindlmann,

2010) when using glyphs for visualization. Another

challenge is the placement of glyphs to maximize the

information that is conveyed in an image. Previous

methods in this context aimed to create sample dis-

tributions that avoid holes and overlaps (Kindlmann

and Westin, 2006), (Hlawitschka et al., 2007), (Feng

et al., 2008), (Kratz et al., 2011). In this work, we ad-

dress the question where to place glyphs by exploiting

a segmentation of the tensor ﬁeld.

Continuous Methods. Closely related to our work

are texture-based methods, which are motivated by

vector ﬁeld visualization. Line integral convolu-

tion (LIC) (Cabral and Leedom, 1993), for example,

has been extended to HyperLIC (Zheng and Pang,

2003). It uses a multi-pass approach to represent

the orientation of one eigenvector ﬁeld and to high-

light anisotropy. Another extension are fabric tex-

tures (Hotz et al., 2004). Two LIC images for every

eigenvector ﬁeld are computed and blended, which

leads to an image that resembles a fabric encoding

central physical properties of stress tensor ﬁelds into

ﬁber thickness (thin = compression, thick = expan-

sion). Recently, a novel method based on anisotropic

sampling was introduced (Kratz et al., 2011). By

computing a generalized Voronoi diagram on the ba-

sis of a tensor-determined local metric, the result-

ing regions can be texturized which results in a large

variety of possibilites to create visualizations. In

general, texture-based methods are restricted to two-

dimensional surfaces.

Three-dimensional visualization methods for

stress tensor ﬁelds are rare (Dick et al., 2009), (Kratz

et al., 2011). They focus on hybrid methods to em-

phasize interesting regions and to avoid clutter. When

features of interest are not known in advance, explo-

rative methods can help (Kratz et al., 2011). To depict

the behavior of a single eigenvector ﬁeld, tensorlines

can be used. These are deﬁned as lines that are tan-

gential to the chosen eigenvector ﬁeld. Hyperstream-

lines (Delmarcelle and Hesselink, 1993) are an exten-

sion, which additionally incorporate the eigenvalues

and other principal directions.

Topological Methods. Tensor ﬁelds can also be

represented by extracting their topology, which is

represented by a graph structure. This topological

graph consists of degenerate points and connecting

tensorlines (Delmarcelle, 1994), (Lavin et al., 1997),

which separate the ﬁeld into regions of similar eigen-

vector behavior. Simpliﬁcation and tracking of the

graph (Tricoche et al., 2001) results in a structure

that is easier to understand and enhances their prac-

tical applicability. This paper builds on a topologi-

cal segmentation (Auer et al., 2011). Therefore, the

topological graph is extracted (Sreevalsan-Nair et al.,

2010) and recursively reﬁned according to eigen-

value characteristics. Thus, a segmentation of the

ﬁeld is achieved that accounts for both, eigenvalue

and eigenvector behavior. Work that is concerned

with topology extraction for three-dimensional ﬁelds

is rare (Hesselink et al., 1997), (Zheng and Pang,

2004), (Tricoche et al., 2008).

3 BASICS AND NOTATION

Figure 1: Illustration of the tensor concept: result (ellipse)

of applying a tensor to an isotropic element (sphere). The

resulting eigenvectors are depicted as arrows and the eigen-

values are reﬂected by the scaling of the arrows.

3.1 Tensors

This work focuses on the visualization of two-

dimensional symmetric tensors. For a ﬁxed coordi-

nate system in R

such a tensor can be described as

2 × 2 matrix of real numbers. For the remainder of

this paper, if not stated differently, the word tensor

refers to two-dimensional symmetric tensors.

Eigenvalues λ

∈ R and eigenvectors

↔

∈ R

, with

i = 1, 2 are important invariants of a tensor. They

are deﬁned by the characteristic equation T · λ

↔

· λ

. The eigenvectors specify the direction of ex-

tremal variation of the quantity encoded by the ten-

sor; the eigenvalues give the magnitude of those ex-

tremal variations (Figure 1). For symmetric tensors

the eigenvalues are real and the eigenvectors mutually

orthogonal. The double arrows allude to the fact that

eigenvectors are bidirectional. A tensor is fully repre-

sented by its eigenvectors and eigenvalues. In the fol-

lowing the eigenvalues are ordered such that λ

≥ λ

They are called major respectively minor eigenvalues

with associated major and minor eigenvector.

3.2 Tensor Fields

A tensor ﬁeld assigns a tensor to each point in a do-

main. In our case the input tensor ﬁelds are given on

GLYPH- AND TEXTURE-BASED VISUALIZATION OF SEGMENTED TENSOR FIELDS

671

a triangulated planar domain, where each vertex of

the triangulation is assigned to a tensor. The tensor

ﬁelds are decomposed into to two eigenvector- and

two eigenvalue ﬁelds (major resp. minor). Integral-

lines that are tangential to an eigenvector ﬁeld every-

where are called tensorlines. To distinguish whether

the tensorlines belong to the major or the minor eigen-

vector ﬁeld, a color-coding is used: blue refers to the

minor ﬁeld and red to the major ﬁeld.

3.3 Topology

The topology is represented by the topological graph,

which consists of degenerate elements and separatri-

ces. Degenerate elements are locations where both

eigenvalues are equal, λ

=λ

, and thus the eigenvec-

tors

↔

are not uniquely deﬁned. Mostly they occur

as points but also as elements of higher dimensional-

ity, such as lines. Separatrices are distinctive tensor-

lines, that radially emerge from degenerate elements

and bound sectors of homogeneous eigenvector be-

havior. For more details we refer to (Delmarcelle and

Hesselink, 1993).

4 TOPOLOGICAL

SEGMENTATION

This work builds on a reﬁned topological graph yield-

ing a complete segmentation of the tensor ﬁeld (Auer

et al., 2011). This segmentation returns cells with

homogeneous eigenvector and eigenvalue behavior.

Starting point for the segmentation is the integral

topological graph combining the topology of both

eigenvector ﬁelds. As the eigenvector ﬁelds are or-

thogonal to each other this graph segments the tensor

ﬁeld into curvilinear cells of homogeneous eigenvec-

tor behavior (Figure 2(a)). Eigenvalues are used to it-

eratively adapt the topological cells until they fulﬁll

predetermined resolution, accuracy, and uniformity

criteria (Figure 2(b,c)). If a cell exhibits dissimilarity

above a predeﬁned accuracy threshold, it gets split by

starting a subdividing tensor line. Further, neighbor-

ing cells that are homogeneous get merged by delet-

ing the connecting edge. To steer the similarity of

eigenvalue characteristics within a cell, one or more

expressive scalar ﬁelds are deﬁned. These can be the

eigenvalues, but also derived tensor invariants. For

stress tensor ﬁelds, e.g., quantities of special interest

arise in context with failure analysis. Exemplary we

use relative shear τ

and maximum shear stress τ as

anisotropy measures.

τ =

− λ

, and τ

− λ

+ λ

+ A

, (1)

where A ∈ R is a positive constant.

The resulting cells are bounded by tensorlines and

possibly degenerated lines. The integration of the ten-

sor lines using a Runge-Kutta-4th-order integration

scheme yields a representation of tensorlines as poly-

lines. Adaptive step sizes are used to maximize the

accuracy of this numerical integration scheme.

5 VISUALIZATION METHOD

In this work we build upon the results of the seg-

mentation (Section 4) and extend it to a texture- and

glyph-based visualization. Mapping textures into the

segmented cells yields a continuous rendering of the

tensor ﬁeld. This facilitates a comprehension of the

ﬁeld’s global nature. The glyph-based approach com-

bines the advantages of the global structural informa-

tion provided by the topology and the local detailed

view of representatives via glyphs.

Cell Structure. Cells that are not adjacent to de-

generate points or the domain boundary are quadran-

gular and bounded by two major and minor tensor-

line segments in alternating order (Figure 2(c),(d)).

Other cells can have more general shapes. The bound-

aries of the extracted cells are stored as polylines (Fig-

ure 3).

Preprocessing. As mentioned in Section 4, the

bounding tensorlines are computed by an integration

scheme with adaptive step size. This guarantees ac-

curate results in the segmentation process but leads to

irregular distances between the vertices of the poly-

lines. To obtain good results for the texturing and

glyph locations, however, a more uniform sampling

of the cell boundaries is favorable. This is achieved

by a pre-processing step deleting respectively adding

vertices if the distance to adjacent vertices does not

fall in a pre-deﬁned distance interval. An angle cri-

terion guarantees that tensorlines are still sufﬁciently

aligned with the eigenvector ﬁelds.

For the texture mapping the segmented cells need

to be triangulated. We use a constrained Delaunay tri-

angulation that maintains non-convex shape.

All pre-processing steps only have to be per-

formed once after the segmentation process.

Eigenvalue Mapping. The goal of this work are in-

tuitive visualizations of stress tensors. These, in gen-

eral, are not positive deﬁnite and thus, have positive as

well as negative eigenvalues. We will use the eigen-

values to control basic texture parameters such as tex-

ture density. Therefore the eigenvalues are mapped

IVAPP 2012 - International Conference on Information Visualization Theory and Applications

672

(a) (b) (c) (d)

Figure 2: (a-c) Schematic illustration of the segmentation process. (a) Step 1: Integral topological graph with degenerate

points as black dots, separatrices as bold lines (major in red, minor in blue). The light red lines depict tensorlines within

the segmented regions and exemplarily illustrate how separatrices aggregate homogeneous eigenvector behavior. (b) Step 2:

Deﬁnition of scalar ﬁeld. (c) Step 3: Reﬁnement of topological graph according to scalar ﬁeld. (d) Texturization of segmented

cell. Mapping of point p

to texture coordinate (s, t) in quadrangular cell.

into a restricted positive interval. We adopt a mapping

that simulates a texture deformation generated by the

underlying tensor ﬁeld (Hotz et al., 2004). Thus,

negative eigenvalues (compression) lead to dense and

positive eigenvalues (expansion) to sparse textures.

Hotz et al. deﬁne:

F(λ) = a + σ · f (λ) . (2)

The function f is chosen to have a large slope in the

neighborhood of zero. In this work, f is the hyper-

bolic tangent, which preserves the differentiation of

negative and positive eigenvalues. The parameter a

relates to an offset and σ is an additional scaling fac-

tor for the slope. Both can be adjusted by the user.

Figure 3: Result of a segmentation, points of the cell bound-

ing polylines are depicted as spheres.

5.1 Segmentation-based Glyph

Placement

The characteristics of the tensor ﬁeld – the eigen-

vectors and eigenvalues – are similar inside each ex-

tracted cell (Section 4). Thus, the essential tensor

properties of each cell can be visualized by one rep-

resentative glyph. The task is to ﬁnd an appropriate

position within each cell to place this representative.

Since most of the segmented cells are non-convex we

follow an algorithm for the computation of barycen-

troids of arbitrarily shaped planar polygons (Rusta-

mov et al., 2009). This algorithm is based on an in-

terior distance measure. The barycentroid is deﬁned

as the point with minimal average interior distance to

the boundary points. Finding this point is a convex

optimization problem and can be solved by standard

gradient descent routines. The barycentroid has the

characteristic that it captures the semantic center of

the polygon and lies inside any arbitrary shaped pla-

nar polygon. See Figure 5 for results.

5.2 Segmentation-based Texturization

Using the segmented cells as basis for texturization

has several beneﬁts. The cells inherently provide the

parametrization for the texture mapping and the un-

derlying topology ensures structural correctness.

Also the segmented cells bounded by tensorlines

already give the eigenvector directions. Simple pro-

cedural stripe textures already depict one eigenvector

ﬁeld. Thus, the use of textures with one or two orthog-

onal dominant directions results in continuous repre-

sentations of the correct eigenvector behavior within

these cells (Figure 4). But also more sophisticated

textures, like knitting patterns, lead to expressive rep-

resentations (Figure 7). The density of the texture pat-

tern will also be used to reﬂect physical properties of

the tensor ﬁeld, such as compression and expansion

(e.g. Figure 4(b)).

Cells containing degenerate elements in their

boundary can have more complex shapes and the

eigenvector behavior cannot be easily represented by

simple stripe patterns. In addition, in the proxim-

ity of degenerate elements the eigenvector behavior

is weakly expressed. For such cells two options are

provided. Either these cells are skipped or textured

with an isotropic noise pattern encoding information

about the isotropic eigenvalues.

GLYPH- AND TEXTURE-BASED VISUALIZATION OF SEGMENTED TENSOR FIELDS

673

For one-directional textures, as stripe patterns, one

image per eigenvector ﬁeld is computed. To depict

both eigenvector ﬁelds in one image the results for

each ﬁeld are blended (Figures 4(c), 6).

The texturization is performed by vertex and fragment

shaders. Texture coordinates (s, t), with s, t ∈ [0, 1] for

quadrangular cells are initially computed by mapping

the points of a cell boundary to a unit square (Fig-

ure 2(d)).

5.2.1 Rendering of Eigenvector Directions

All methods that are presented in the following are

based on textures with linelike elements to depict di-

rections. To ensure that the cell size does not affect

the perception of the pattern, we need a special map-

ping approach that provides an approximately con-

stant pattern frequency (Hummel et al., 2010)(Fig-

ure 4(a)). Hummel et al. adjust the sampling fre-

quency according to the image-space partial deriva-

tives η

, η

at pixel (i, j) of the texture coordinate

(s, t):

(i, j) =



δs

δi





δs

δ j



(i, j) =



δt

δi





δt

δ j



(3)

The initial texture coordinates (s, t) remain un-

changed. The evaluation of the input texture P is mod-

iﬁed according to the variation of η

and η

and steers

the pattern frequency in the ﬁnal image.

gives

the frequency adjusted texture values

(s, t) := P(s ·2

−l

, t ·2

−l

) , (4)

with l

= log

and l

= log

. Hence, short edges

with high partial derivatives yield a low pattern fre-

quency. For large edges this works vice versa. The

resulting pattern frequency also interactively adjusts

to the current zoom level and resolution of the image.

As resolution levels are discretely deﬁned values for

neighboring resolution levels are computed and bilin-

ear ﬁltering applied to achieve a smooth pattern fre-

quency. The evaluation of Equation 3 can be done by

built-in functionality of the rendering system.

5.2.2 Rendering of Eigenvalue Characteristics

Line Frequency. The approach of Hummel et

al. (Hummel et al., 2010) serves as basis to en-

code physical properties like compression and expan-

sion by the pattern frequency (Figure 4(b)). This is

achieved by replacing Equation 3 by the following:

(i, j) =



δs

δi





δs

δ j



(i, j) =



δt

δi





δt

δ j



(5)

where

and

are the mapped eigenvalues. Thus,

the pattern frequency steers the perception of the ﬁeld:

in combination with the mapping (Eq. 2) negative

eigenvalues lead to a higher frequency and allude to

compression. Mapped positive eigenvalues cause a

lower frequency which depicts expansion.

Color Mapping. A simple but effective way of con-

veying additional information is to blend color infor-

mation on the input textures (Figures 4(c), 6(a),(b),

and 7(b),(c)). Coloring can be applied to the eigenvec-

tor ﬁelds (one color for each ﬁeld) but also according

to derived scalar measures such as the magnitude of

one eigenvalue, relative shear, maximum shear stress

(Eq. 1), or the determinant (λ

· λ

Blur by Derived Scalar Quantities. If non relevant

information should be suppressed in the ﬁnal image, a

post-processing step can be applied to the texturized

image. We employ a blur ﬁlter, but also the opac-

ity could be modiﬁed to hide regions of low informa-

tion content. Again quantities like relative and ab-

solute shear stress come into question (Figures 6(b),

and 7(b),(c)) to steer the post-processing. The pixels

of the texturized image are convolved with a Gaus-

sian kernel, where the size of the kernel is propor-

tional to the respective value of the chosen quantity.

For example, if the blur should be determined with

respect to relative shear, isotropic regions are con-

volved with a larger kernel and anisotropic regions

are convolved with a smaller kernel. Thus, regions

with strongly expressed eigenvector directions are en-

hanced and isotropic regions blurred.

6 RESULTS

In this section, the results of the developed visualiza-

tion approaches are presented and evaluated by means

of two simulated datasets: the one-point load and the

two-point load. The one-point load is a solid block

given as a cubic volume with a single load applied to

it. The two-point load is similar, just that two loads of

opposing sign are applied to it. The two datasets are

generated by a Finite Element Method (FEM). The

methods presented above are analyzed on planar cuts

of the volumetric datasets.

IVAPP 2012 - International Conference on Information Visualization Theory and Applications

674

(a) (b) (c) (d)

Figure 4: Data set: Two-point load. Visualization of major eigenvector ﬁeld (a) with even pattern frequency, (b) with encoded

mapped eigenvalues. (c) Superposition of both eigenvector ﬁelds, color by relative shear. (d) Illustration of a hanging node.

6.1 Glyph-based Visualization

(a) (b)

Figure 5: Data set: One-point load. (a) Glyph placement in

the data set. (b) Close-up of pre-computed barycentroids.

Figure 5 shows the results of the segmentation-

based glyph placement (Section 5.1) applied to the

one-point load. The glyphs are placed at the pre-

computed barycentroids, oriented according to the

eigenvectors and scaled by the mapped eigenvalues

(Eq. 2). The color is assigned according to the relative

shear (Eq. 1). Isotropic tensors are encoded in light

blue and spherical geometries. Anisotropic tensors

are encoded in dark blue and result in well-marked

ellipses. This work is not concerned with elaborate

glyph design or similar. With the glyph placement we

rather want to provide a basis for the variety of glyphs

provided in the literature (Section 2). The close-up in

Figure 5(b) nicely shows how the barycentroids cap-

ture the semantic center of non-convex polygons.

6.2 Texture-based Visualization

Figure 4(a) displays the directions of the major

eigenvector ﬁeld. Here the basic approach of ap-

(a) (b)

Figure 6: Data set: One-point load. (a) Superposition of

both eigenvector ﬁelds, color coding according to mapped

eigenvalues; (b) post-processing blur by relative shear.

proximately constant image space line density (Sec-

tion 5.2.1) is applied. Even though the textures are

mapped cell-wise the continuous character of the im-

age is harmonious. Only at transitions of cells with

hanging nodes (Figure 4(d)) slight disruptions are no-

ticeable. Zooming in the image automatically adapts

the texture such that the stripe frequency and the

even appearance is maintained. Figure 4(b) extends

the representation by encoding the physical behav-

ior. The pattern frequency is scaled by the mapped

eigenvalues (Eq. 5). In the lower right corner nega-

tive eigenvalues are predominant and clearly result in

a higher pattern frequency. This resembles to com-

pressive forces and is in contrast to the upper left cor-

ner which is characterized by expansive forces. In

Figure 4(c) the textures for both eigenvector ﬁelds are

blended. The pattern frequency is determined again

by Equation 5 and color coding is applied accord-

ing to the relative shear. Isotropic regions are colored

in blue and characterized by an isotropic pattern fre-

quency. In regions of high anisotropy strongly differ-

ing eigenvalues lead to the unequal pattern frequen-

cies for the two eigenvector ﬁelds, which is addition-

ally emphasized by the red color.

GLYPH- AND TEXTURE-BASED VISUALIZATION OF SEGMENTED TENSOR FIELDS

675

(a) (b) (c) (d)

Figure 7: Data set: Two-point load. Comparison of rendering with different input textures. (a) Bidirectional weave input

pattern, the frequency is adjusted to the mapped eigenvalues. (b) Rendering of the directions of maximal shear. Regions of

high maximal shear stress are emphasized in red. Regions of low maximal shear are blurred. (c) Knitting pattern emphasizing

major eigenvector directions from far, in detail directions of maximal shear are perceivable. Color coding is applied according

to the relative shear stress. (d) Shows from top to bottom the used sample patterns in Figures (a-c).

A similar visualization approach is applied to the

one-point load in Figure 6(a). Here the color cod-

ing corresponds directly to the mapped eigenvalues

(Eq. 2). Anisotropic regions are still discernible as

the superposition generates mixed colors for strongly

differing eigenvalues. Speciﬁc regions can be high-

lighted additionally by applying the post-processing

step: in Figure 6(b) regions with low relative shear

are blurred. The focus of the user is directed to

anisotropic regions, where eigenvalues exhibit a large

difference.

The versatility and power of textures is demon-

strated in Figure 7. For texture samples with inherent

natural variance discontinuities due to hanging nodes

in the original segmentation are less prominent. In

Figure 7(a) a weave input pattern is employed to vi-

sualize the eigenvector directions. Due to the bidi-

rectional nature of the weave pattern both eigenvector

ﬁelds are visualized at once. In Figure 7(b) the user

can switch to investigate directions of maximum shear

of the underlying ﬁeld. Here a texture is used with

line structures illustrating the bisectors of the eigen-

vector directions. Additionally regions of high max-

imal shear stress (Eq. 1) are emphasized by selective

color mapping. The third example, Figure 7(c), gener-

ates a texture that resembles a knitted piece of fabric.

7 CONCLUSIONS

We have combined the accuracy of topological meth-

ods for two dimensional symmetric tensor ﬁelds with

the support of more intuitive visualizations. Our ap-

proach uses the strength of textures for continuous vi-

sualizations and allows to gain insight into detailed

information at discrete locations by placing glyphs.

A speciﬁc topology-based segmentation framework

(Auer et al., 2011) is used to employ these tech-

niques. The cells of this segmentation provide a con-

sistent parametrization for the texture mapping and

the bounding tensorlines correctly predetermine the

the eigenvector directions within. A multitude of pos-

sible textures can be implemented illustrating a har-

monious continuous view on the various tensor prop-

erties. A selection of textures is presented that encode

directional features; simple stripe textures but also

textures with higher inherent variance, that support a

smooth appearance over uneven transitions (hanging

nodes). We believe the latter textures can also be used

for other approaches that aim the mapping of direc-

tional textures region- oder cell-wise without appar-

ent distorted behavior at the boundaries.

Physical properties of the tensor ﬁeld like compres-

sion or expansion are reﬂected in the texture fre-

quency. Selective color mapping and post process-

ing are applied to direct the users attention to scalar

features of interest. Certainly, there remains a large

potential to optimally assist the perceptional habits of

a user. We also presented this work as ﬂexible basis

for further advancement.

ACKNOWLEDGEMENTS

This work was funded by the German Research

Foundation (DFG) through a Junior Research Group

Leader award (Emmy Noether Program).

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676

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