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Joint EUROGRAPHICS - IEEE TCVG Symposium on Visualization (2004)

O. Deussen, C. Hansen, D.A. Keim, D. Saupe (Editors)

Superquadric Tensor Glyphs

Gordon Kindlmann

School of Computing, University of Utah, United States

Abstract

Tensor ﬁeld visualization is a challenging task due in part to the multi-variate nature of individual tensor samples.

Glyphs convey tensor variables by mapping the tensor eigenvectors and eigenvalues to the orientation and shape

of a geometric primitive, such as a cuboid or ellipsoid. Though widespread, cuboids and ellipsoids have problems

of asymmetry and visual ambiguity. Cuboids can display misleading orientation for tensors with underlying rota-

tional symmetry. Ellipsoids differing in shape can be confused, from certain viewpoints, because of similarities in

proﬁle and shading. This paper addresses the problems of asymmetry and ambiguity with a new tunable continuum

of glyphs based on superquadric surfaces. Superquadric tensor glyphs enjoy the necessary symmetry properties of

ellipsoids, while also imitating cuboids and cylinders to better convey shape and orientation, where appropriate.

The new glyphs are demonstrated on ﬁelds of diffusion tensors from the human brain.

Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Curve, surface, solid, and

object representations I.3.8 [Computer Graphics]: Applications

1. Introduction

Scientiﬁc visualization techniques convey structure and in-

formation over a range of scales, from large-scale patterns

spanning an entire dataset, down to the individual samples

comprising the dataset. When working with non-scalar data

from medical imaging, such as diffusion or strain tensors

from MRI [BMB94,AW01], low-level inspection of indi-

vidual tensors is a necessary ﬁrst step in exploring and

understanding the data. Glyphs, or icons, depict multiple

data values by mapping them onto the shape, size, orien-

tation, and surface appearance of a base geometric primi-

tive [PvWPS95]. Ideally, judicious composition of multiple

glyphs from across the tensor ﬁeld can hint at larger-scale

features that may be subsequently explored and extracted

with other visualization techniques, such as hyperstream-

lines [DH95], stream-tubes, or stream-surfaces [ZDL03].

Isotropy (Spherical); Linear anisotropy; Planar anisotropy

Figure 1: Three basic diffusion tensor shapes.

Diffusion tensors can be represented as symmetric three-

by-three matrices, which have three real, positive eigenval-

ues and three real-valued orthogonal eigenvectors [Str76]. A

diffusion tensor Tcan be factored as T=RΛ

Λ

ΛR−1where

Λ

Λ

Λis a diagonal matrix of eigenvalues (by convention sorted

λ1≥λ2≥λ3), and Ris a rotation matrix that transforms the

standard basis onto the eigenvector basis. In this paper, “ten-

sor shape” and “tensor orientation” refer to the eigenvalues

and eigenvectors, respectively, of the tensor. The anisotropy

of a tensor expresses the amount of variation in the eigenval-

ues. If tensors do not have any anisotropy (λ1=λ2=λ3),

they are considered spherical in shape. Anisotropic diffusion

tensors can have linear shapes (λ1>λ2=λ3), planar shapes

(λ1=λ2>λ3), or some combination; see Figure 1.

Glyph-based tensor visualization transforms an initial

glyph geometry Ginto a tensor glyph GTby

GT=RΛ

Λ

ΛG,(1)

and then translating GTto the ﬁeld location of tensor

T[SML03]. By not applying rotation R−1, the axis-aligned

features of G(such as the edges of a unit cube) become rep-

resentations in GTof the tensor eigenvalues and eigenvec-

tors. Different visualization effects are created by choosing

different glyph geometries G, such as cubes [SML03], cylin-

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Kindlmann / Superquadric Tensor Glyphs

ders [WLW00], or spheres [PB96,BP96,LAK∗98]. Eigen-

vectors are known only up to line orientation (they have no

signed direction), which constrains practical glyph geome-

tries to shapes with 180 degree rotational symmetry. A com-

putational advantage of using spheres (to create ellipsoidal

glyphs) is that tensor diagonalization is not required, only

matrix multiplication: GT=R−1Λ

Λ

ΛG=R−1Λ

Λ

ΛRG=TG.

The focus of this paper is the simple task of creat-

ing better tensor visualizations, with a new glyph geome-

try based on superquadric surfaces [Bar81]. Superquadric

tensor glyphs build on previous research by Shaw et

al. which applies superquadrics to glyph-based visual-

ization [SEK∗98,SHB∗99,ERS∗00,ES01]. They describe

how parameterizing shape variations to encode data vari-

ables should enable effective and intuitive “perceptualiza-

tions”, given that distinguishing shape from contours and

shading is a pre-attentive process [PCC∗92]. Offering a con-

tinuous two-parameter space of shapes, superquadrics are a

natural choice for a tunable geometric primitive. The ability

to discern differences between rendered superquadrics was

experimentally quantiﬁed [SHB∗99], and the superquadric

glyphs were successfully used for document corpus visual-

ization [SEK∗98] and scientiﬁc visualization of magnetohy-

drodynamic ﬂow [ERS∗00,ES01].

The contribution of this paper is to use superquadrics as a

tensor glyph rather than simply a multi-variate glyph. This

requires selecting an intuitive subset of the superquadric pa-

rameter space to encode tensor shape, and ensuring that the

display of tensor orientation faithfully conveys the symme-

tries that can arise in the tensor eigensystem.

2. Motivation

Evaluating existing tensor glyph geometries and their prop-

erties is facilitated with an intuitive domain that spans all

possible tensor shapes. Such a domain is afforded by the

geometric anisotropy metrics of Westin et al. [WPG∗97].

Given the non-negative tensor eigenvalues λ1≥λ2≥λ3, the

metrics quantify the certainty (c) with which a tensor may be

said to have a given shape:

cl=λ1−λ2

λ1+λ2+λ3

cp=2(λ2−λ3)

λ1+λ2+λ3(2)

cs=3λ3

λ1+λ2+λ3

The three metrics add up to unity, and deﬁne a barycen-

tric parameterization of a triangular domain, with the ex-

tremes of linear, planar, and spherical shapes at the three

corners. The barycentric shape space has been used as the

domain of transfer functions for direct volume rendering of

diffusion tensors [KWH00], and as an intuitive basis of com-

parison between various anisotropy metrics [AHK∗00]. The

barycentric shape space is drawn in Figure 2using cuboid

glyphs to emphasize variations in aspect ratio over the trian-

gular domain. Complete isotropy is at the top corner (cs=1),

and anisotropy increases toward the lower edge.

PSfrag replacements

cl=1cp=1

cs=1

Figure 2: Tensor shapes, with cuboids.

Figure 2illustrates a problem with cuboid glyphs: mis-

leading depiction of under-constrained orientation. Because

cp=0⇒λ2=λ3for the linear shapes at the left edge of

the triangle, computation of the corresponding eigenvectors

v2and v3may return any two perpendicular vectors within

the plane normal to the principal eigenvector v1. An analo-

gous problem occurs with the planar shapes along the right

edge of the triangle. The cuboid edges depict orientation

with a visual clarity that is disproportionate to the low nu-

merical accuracy with which the eigenvectors can be calcu-

lated [GL96]. For intermediate shapes, however, the sharp

edges of the cuboids are good at depicting legitimate tensor

orientation.

PSfrag replacements

cl=1cp=1

cs=1

Figure 3: Tensor shapes, with cylinders.

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(a) Eight different tensors, shown with ellipsoid glyphs.

(b) Same eight glyphs, but with a different viewpoint.

Figure 5: From some viewpoints, ellipsoids poorly convey tensor shape.

PSfrag replacements

cl=1cp=1

cs=1

Figure 4: Tensor shapes, with ellipsoids.

Cylinder glyphs resolve this problem by aligning their

axis of rotation along the eigenvector for which the numer-

ical accuracy is greatest, as done in Figure 3. There is un-

fortunately a discontinuity problem, with a seam down the

middle of the shape space. Arbitrarily small changes in the

tensor shape can result in discontinuous changes in the glyph

direction, even though the precise location of the seam is

somewhat arbitrary. An alternate deﬁnition of cl,cp,cs(nor-

malized by λ1instead of λ1+λ2+λ3[WMK∗99]) produces

a slightly different distribution of intermediate shapes within

an otherwise similar barycentric shape domain. In addition,

because cylinders have only one axis of symmetry, cylindri-

cal glyphs depict meaningless orientation for spherical ten-

sors, which have no intrinsic orientation.

Ellipsoidal glyphs, shown in Figure 4, avoid all such sym-

metry problems. There is, however, a problem of visual am-

biguity. Glyphs with differing tensor shapes exhibit similar

image-space shapes, with only shading cues for disambigua-

tion. Figure 5demonstrates a pathological example. A wide

range of tensors rendered with ellipsoid glyphs can appear

similar from one viewpoint (Figure 5(a)), though they are

clearly different when seen from another viewpoint (Fig-

ure 5(b)). This example is important because it demonstrates

that even standard, intuitive glyph geometries can sometimes

dramatically fail to properly convey data attributes.

3. Method

The problems of asymmetry and ambiguity can be addressed

with a glyph geometry that changes according to the under-

lying tensor shape. Ideally, the best of Figures 2,3, and 4

could be combined: cylinders for the linear and planar cases,

spheres for the spherical case, and cuboids for intermediate

cases, with smooth blending in between. The general strat-

egy is that edges on the glyph surface signify anisotropy:

anisotropy implies a difference in eigenvalues, which im-

plies conﬁdence in computing eigenvectors [GL96], which

implies lack of rotational symmetry, which can be visually

highlighted by a strong edge on the glyph surface. When two

eigenvalues are equal, the indeterminacy of the eigenvectors

is conveyed with a circular glyph cross-section.

Superquadrics accomplish this goal. They can be parame-

terized explicitly (for polygonal glyph representation):

qz(θ,φ) =

cosαθsinβφ

sinαθsinβφ

cosβφ

,0≤φ≤π

0≤θ≤2π,(3)

where xα=sgn(x)|x|α, or superquadrics may be represented

implicitly (such as for raytracing):

qz(x,y,z) = x2/α+y2/αα/β+z2/β−1=0.(4)

Figure 6shows how αand βcontrol superquadric shape.

Superquadric tensor glyphs draw from a subset of these pos-

sibilities, deﬁned by β≤α≤1. Note that the formulations

of qzand qzare not symmetric with respect to axis permu-

tation. Aside from the spherical case, the superquadrics may

have continuous rotational symmetry around only the zaxis

(when α=1). Thus, as a counter-part, it is useful to deﬁne

superquadrics around the xaxis:

qx(θ,φ) =

cosβφ

−sinαθsinβφ

cosαθsinβφ

,0≤φ≤π

0≤θ≤2π,(5)

qx(x,y,z) = y2/α+z2/αα/β+x2/β−1=0.(6)

With these ingredients, superquadric tensor glyphs are

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β=4

β=2

β=1

β=1

2

β=1

4

α=1

4α=1

2α=1α=2α=4

Figure 6: Superquadrics deﬁned by Equation 3. The gray

triangle indicates the subset of the shape space employed by

superquadric tensor glyphs. Edges indicate the tessellation

resulting from uniform steps in φand θ.

now deﬁned in terms of the geometric anisotropy metrics

cl,cp, and a user-controlled edge sharpness parameter γ:

cl≥cp=⇒

α= (1−cp)γ

β= (1−cl)γ

q(θ,φ) = qx(θ,φ)

q(x,y,z) = qx(x,y,z)

(7)

cl<cp=⇒

α= (1−cl)γ

β= (1−cp)γ

q(θ,φ) = qz(θ,φ)

q(x,y,z) = qx(x,y,z)

These equations deﬁne a base glyph geometry that is

made into a tensor visualization via Equation 1. Figure 7il-

lustrates superquadric glyphs with the same tensors, lighting,

and viewpoint as used in Figures 2,3, and 4. The glyphs have

the necessary symmetry properties of ellipsoids, but convey

orientation and shape more clearly by imitating cylinders

and cuboids where appropriate. The edge sharpness param-

eter γcontrols how rapidly edges form as cland cpincrease,

allowing the user to control the visual prominence of ori-

entation information at low anisotropy levels. Ideally, appli-

cation characteristics would enable an informed choice of

γ: perhaps visualizations of noisy measurements would use

a lower (more conservative) γthan visualizations of high-

precision simulation data. Note that pure ellipsoids can be

recovered as a special case, with γ=0.

The rationale for how αand βare deﬁned in Equation 7

(a) γ=1.5

(b) γ=3.0

(c) γ=6.0

Figure 7: Tensor shapes, with superquadric glyphs, and

three different values of edge sharpness parameter γ.

can be understood with reference to Figure 6. For tensors

that are more linear than planar (cl≥cp), the glyph shape

becomes more distinctly cylindrical as clincreases and β

decreases. True rotational symmetry is only present when

cp=0⇒α=1. As the planar component increases with

cp, the shape gradually tends away from rotational symmetry

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(a) Same tensors, viewpoint, and lighting as Figure 5(a), but with superquadric glyphs.

(b) Same as Figure 5(b), but with superquadric glyphs.

Figure 9: Superquadrics convey shape differences more reliably than ellipsoids (γ=3).

cl=0.33 cl=0.31 cl=0.29 cl=0.27

cp=0.27 cp=0.29 cp=0.31 cp=0.33

Figure 8: Parameterization change across the linear/planar

seam, from cl>cpto cl<cp(γ=3).

due to lower α, increasing the prominence of edges around

the glyph circumference. Analogous reasoning holds for

cl<cp. When cl=cp,α=β, and qx(x,y,z) = qz(x,y,z),

in which case the xaxis (Equations 5,6) and the zaxis (Equa-

tions 3,4) superquadrics are identical. Thus, like cylinders

(Figure 3), superquadric tensor glyphs do have a seam be-

tween the linear and planar sides of the shape space, but

the seam is mathematically continuous. Figure 8illustrates

how the parameterization change may have an effect on a

tessellation-based surface representation.

Figure 9shows how superquadric glyphs are better at con-

veying shape than the ellipsoid glyphs in Figure 5, using the

same tensors, viewpoint, and lighting. For example, the third

and sixth glyphs from the left have precisely linear (cp=0)

and planar (cl=0) shapes, respectively. The existence and

the orientation of the resulting rotational symmetry is easier

to see with superquadrics than with ellipsoids.

4. Results

Diffusion tensor magnetic resonance imaging (DT-MRI) of

nerve tissue indirectly measures the ﬁbrous structure of

white matter by detecting the directionally constrained dif-

fusion of water molecules within it [BMB94], resulting in a

3-D ﬁeld of tensor values. Some DT-MRI voxels within the

largest white matter structures (such as the corpus callosum)

exhibit purely linear anisotropy, because the whole voxel re-

gion is homogeneously uni-directional. However, the com-

plex branching and crossing of the white matter tracts, com-

bined with the limited resolution of the DT-MRI modal-

ity, produces many measurements with signiﬁcant planar

anisotropy. Visualizing the locations and orientation of pla-

nar anisotropy is a step towards understanding the complex

nature of white matter connectivity [WLW00].

For this task, Figure 10 compares the effectiveness of su-

perquadric tensor glyphs and ellipsoids for visualizing a por-

tion of an axial slice through a diffusion tensor dataset, cen-

tered on the right half of the splenium of the corpus callosum

(the black region is the lateral ventricle). The background

squares represent isotropy levels for each sample (“interest-

ing” anisotropic tensors have a darker background and hence

greater contrast with the glyph). Planarly anisotropic sam-

ples are located near the center of the image. With ellipsoids,

it is difﬁcult to discern which of the glyphs represent pla-

nar anisotropy, and it harder to appreciate the differences in

shape that may occur between neighboring samples. Also,

the straight edges of the linearly anisotropic superquadric

glyphs provide a stronger orientation indication than possi-

ble with the rounded contours of ellipsoids.

In three-dimensional glyph-based visualizations of tensor

volumes, it is important to restrict the number of glyphs, to

avoid creating an illegible mass. In diffusion tensors, glyphs

may be culled according to an anisotropy threshold (such

as cl+cp>0.5) so that isotropic tensors (belonging to

gray matter or cerebral spinal ﬂuid) are hidden, resulting

in a coarse depiction of the major white matter pathways.

Figure 11 uses this method to compare ellipsoid and su-

perquadric glyphs for visualizing half of a diffusion tensor

volume, centered again at the right half of the splenium of

the corpus callosum. The superquadrics depict the amount

and orientation of the planar component in the white matter

more clearly than the ellipsoids. Comparing the planar ori-

entation with the direction of adjacent linear anisotropy is an

example of a visualization query which is better answered by

the new glyph method.

5. Discussion

In comparing Figures 5(a) and 9(a), one could argue that

various rendering effects would help clarify the shape dif-

ferences among the ellipsoids: different lighting, specular

highlights, or surface textures, for example. Interactive ma-

nipulation and stereo rendering would also help. On the

other hand, these modiﬁcations would also beneﬁt the de-

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Kindlmann / Superquadric Tensor Glyphs

(a) Ellipsoids

(b) Superquadrics (γ=3)

Figure 10: Slice of DT-MRI dataset of brain visualized with

ellipsoids (top) and superquadrics (bottom).

piction of superquadrics. The same diffuse lighting and

ﬁxed viewpoint, which failed to distinguish the ellipsoid

shapes, were sufﬁcient to differentiate the superquadrics. Us-

ing data-driven variable geometry (Equation 7), in addition

to the eigenvalue-based scaling, helps superquadric glyphs

convey shape more explicitly than previous tensor glyphs.

Fore-shortening of superquadric glyphs can still create vi-

(a) Ellipsoids

(b) Superquadrics (γ=3)

Figure 11: 3-D region of DT-MRI dataset of brain visualized

with ellipsoids (top) and superquadrics (bottom).

sual ambiguity, although the range of affected viewpoints is

smaller than with ellipsoids.

Starting with a more expressive glyph geometry allows

further effects (color, textures, etc.) to be saved for encod-

ing additional degrees of freedom that may be required in

a more complex visualization application. The best way

to enrich three-dimensionsional glyph-based visualizations

with extra information is an important direction of future

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Kindlmann / Superquadric Tensor Glyphs

work, since there are usually a number of related ﬁeld val-

ues which should be visualized along with tensors. Inspi-

ration may be drawn from artistic methods of painting and

illustration [LAd∗98,LKF∗98,KML99,RLH∗01]. Speciﬁ-

cally, the composition of multiple glyphs into a depiction of

larger-scale structure may beneﬁt from context-sensitive and

multi-scale variation of rendering style [HIK∗01].

The incentive to create sharp edges in the superquadric

glyphs was based on the observation that edges generate a

strong visual cue for orientation. However, it is the mathe-

matical property of rotational symmetry that constrains the

glyph to be cylindrical and spherical according to the ten-

sor eigensystem, and the idea of continuity that informed the

design of an invisible seam through the middle of barycen-

tric shape space. The combination of aesthetic judgment and

mathematical constraint may be useful in the design of other

visual abstractions for multi-variate and tensor visualization.

Acknowledgements

Funding was provided by the University of Utah Research

Foundation PID 2107127 and the National Institutes of

Health/NCRR, 5 P20 HL68566-03 and 5 P41 RR12553-05.

Brain dataset courtesy of Andrew Alexander at the W. M.

Keck Laboratory for Functional Brain Imaging and Behav-

ior, University of Wisconsin-Madison

Every ﬁgure in this paper can be regenerated ex-

actly with open-source software and public datasets; see

<http://www.sci.utah.edu/∼gk/vissym04>.

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