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Superquadric Tensor Glyphs

Authors:

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 rotational 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.
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
ΛR1where
Λ
Λ
Λis a diagonal matrix of eigenvalues (by convention sorted
λ1λ2λ3), 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 R1, 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,LAK98]. 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=R1Λ
Λ
ΛG=R1Λ
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 [SEK98,SHB99,ERS00,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 [PCC92]. 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 [SHB99], and the superquadric
glyphs were successfully used for document corpus visual-
ization [SEK98] and scientific visualization of magnetohy-
drodynamic flow [ERS00,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. [WPG97].
Given the non-negative tensor eigenvalues λ1λ2λ3, the
metrics quantify the certainty (c) with which a tensor may be
said to have a given shape:
cl=λ1λ2
λ1+λ2+λ3
cp=2(λ2λ3)
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 [AHK00]. 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=0λ2=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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Kindlmann / Superquadric Tensor Glyphs
(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[WMK99]) 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
,0φπ
0θ2π,(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/β1=0.(4)
Figure 6shows how αand βcontrol superquadric shape.
Superquadric tensor glyphs draw from a subset of these pos-
sibilities, defined by βα1. 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
,0φπ
0θ2π,(5)
qx(x,y,z) = y2/α+z2/αα/β+x2/β1=0.(6)
With these ingredients, superquadric tensor glyphs are
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Kindlmann / Superquadric Tensor Glyphs
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:
clcp=
α= (1cp)γ
β= (1cl)γ
q(θ,φ) = qx(θ,φ)
q(x,y,z) = qx(x,y,z)
(7)
cl<cp=
α= (1cl)γ
β= (1cp)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 (clcp), the glyph shape
becomes more distinctly cylindrical as clincreases and β
decreases. True rotational symmetry is only present when
cp=0α=1. As the planar component increases with
cp, the shape gradually tends away from rotational symmetry
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Kindlmann / Superquadric Tensor Glyphs
(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 [LAd98,LKF98,KML99,RLH01]. 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 [HIK01].
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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... state of turbulence must lie inside this triangle. The shapes of turbulence classified are therein described together with super-quadratic tensor glyphs visualization [44] taken from [45]. ...
... an equilateral triangle barycentric anisotropy map (BAM) can be constructed to characterize turbulence within the three limiting states (1C,2C,3C) representing the three vertices of this map [6]. Note Figure 2 that displays the BAM with all turbulence states description together with super-quadratic tensor glyphs visualization [44] taken from [45]. ...
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... A composite glyph can directly encode linear, planar, and spherical information [71]. Superquadric glyphs [72] with general principles of usage [73] avoid the shortcomings of spherical or cubical glyphs. Uncertainty-aware visualization is available for augmenting spherical, superquadric glyphs for DTI and fourth-order homogeneous polynomial glyphs for high angular resolution diffusion imaging (HARDI) [63]. ...
... Perception of glyphs is difficult in 3D due to loss of information from projection to the 2D image plane and the ambiguity in the 3D shape representation. There are superquadric glyphs that avoid the ambiguity and improve over spherical and cubical glyphs [72,73]. The glyphs are typically rendered with illuminated surfaces and shadow effects in 3D to further enhance depth perception. ...
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Importance . Medical images are essential for modern medicine and an important research subject in visualization. However, medical experts are often not aware of the many advanced three-dimensional (3D) medical image visualization techniques that could increase their capabilities in data analysis and assist the decision-making process for specific medical problems. Our paper provides a review of 3D visualization techniques for medical images, intending to bridge the gap between medical experts and visualization researchers. Highlights . Fundamental visualization techniques are revisited for various medical imaging modalities, from computational tomography to diffusion tensor imaging, featuring techniques that enhance spatial perception, which is critical for medical practices. The state-of-the-art of medical visualization is reviewed based on a procedure-oriented classification of medical problems for studies of individuals and populations. This paper summarizes free software tools for different modalities of medical images designed for various purposes, including visualization, analysis, and segmentation, and it provides respective Internet links. Conclusions . Visualization techniques are a useful tool for medical experts to tackle specific medical problems in their daily work. Our review provides a quick reference to such techniques given the medical problem and modalities of associated medical images. We summarize fundamental techniques and readily available visualization tools to help medical experts to better understand and utilize medical imaging data. This paper could contribute to the joint effort of the medical and visualization communities to advance precision medicine.
... In some works, Mohr diagrams are used to display the stress tensors [29]. A certain possibility is the use of superquadratic tensor glyphs [30,31]. This method is only suitable for symmetric tensors. ...
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A 2D metric space has a limited number of properties through which it can be described. This metric space may comprise objects such as a scalar, a vector, and a rank-2 tensor. The paper provides a comprehensive description of relations between objects in 2D space using the matrix product of vectors, geometric product, and dot product of complex numbers. These relations are also an integral part of the Lagrange’s identity. The entire structure of derived theoretical relationships describing properties of 2D space draws on the Lagrange’s identity. The description of how geometric algebra and tensor calculus are interconnected is given here in a comprehensive and essentially clear manner, which is the main contribution of this paper. A new term in this regard is the total geometric and matrix product, which—in a simple manner—predetermines and defines the existence of differential relations such as the gradient, the divergence, and the curl of a vector field. In addition, geometric interpretation of tensors is pointed out, expressed through angular parameters known from the literature as a tensor glyph. This angular interpretation of the tensor has an unequivocal analytical form, and the paper shows how it is linked to the classical tensor denoted by indices.
... This represents a significant step forward for future anatomical studies based on 3D-PLI, which are prone to misinterpretations without this information. In the future, visualizations based on less ambiguous superquadrics than ellipsoids could be employed [75,76]. This study was limited to mesoscale data acquired at 64 μm × 64 μm with a section thickness of 70 μm, constrained by the resolution of the available polarimetric setup enabling oblique scans. ...
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In recent years, the microscopy technology referred to as Polarized Light Imaging (3D-PLI) has successfully been established to study the brain’s nerve fiber architecture at the micrometer scale. The myelinated axons of the nervous tissue introduce optical birefringence that can be used to contrast nerve fibers and their tracts from each other. Beyond the generation of contrast, 3D-PLI renders the estimation of local fiber orientations possible. To do so, unstained histological brain sections of 70 μm thickness cut at a cryo-microtome were scanned in a polarimetric setup using rotating polarizing filter elements while keeping the sample unmoved. To address the fundamental question of brain connectivity, i. e., revealing the detailed organizational principles of the brain’s intricate neural networks, the tracing of fiber structures across volumes has to be performed at the microscale. This requires a sound basis for describing the in-plane and out-of-plane orientations of each potential fiber (axis) in each voxel, including information about the confidence level (uncertainty) of the orientation estimates. By this means, complex fiber constellations, e. g., at the white matter to gray matter transition zones or brain regions with low myelination (i. e., low birefringence signal), as can be found in the cerebral cortex, become quantifiable in a reliable manner. Unfortunately, this uncertainty information comes with the high computational price of their underlying Monte-Carlo sampling methods and the lack of a proper visualization. In the presented work, we propose a supervised machine learning approach to estimate the uncertainty of the inferred model parameters. It is shown that the parameter uncertainties strongly correlate with simple, physically explainable features derived from the signal strength. After fitting these correlations using a small sub-sample of the data, the uncertainties can be predicted for the remaining data set with high precision. This reduces the required computation time by more than two orders of magnitude. Additionally, a new visualization of the derived three-dimensional nerve fiber information, including the orientation uncertainty based on ellipsoids, is introduced. This technique makes the derived orientation uncertainty information visually interpretable.
... Glyph-based methods, on the other hand, depict the stress field by a set of well-designed geometric primitivesso-called tensor glyphs. Tensor glyphs were originally designed for glyph-based diffusion tensor visualization [19], and later adapted to visualize positive definite tensors [18], general symmetric tensors [34], as well as asymmetric tensors [11,35]. Glyph-based techniques are problematic when used to visualize 3D stress fields, due to their inherent occlusion effects. ...
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In this paper, we present novel algorithms for visualizing the three mutually orthogonal principal stress directions in 3D solids under load and we discuss the efficient integration of these algorithms into the 3D Trajectory-based Stress Visualizer (3D-TSV), a visual analysis tool for the exploration of the principal stress directions of 3D stress field. In the design of 3D-TSV, several perceptual problems have been solved. We present a novel algorithm for generating a space-filling and evenly spaced set of stress lines. The algorithm obtains a more regular appearance by considering the locations of lines, and enables the extraction of a level-of-detail representation with adjustable sparseness of the trajectories along a certain stress direction. A new combined visualization of two principal directions via oriented ribbons enables to convey ambiguities in the orientation of the principal stress directions. Additional depth cues have been added to improve the perception of the spatial relationships between trajectories. 3D-TSV provides a modular and generic implementation of key algorithms required for a trajectory-based visual analysis of principal stress directions, including the automatic seeding of space-filling stress lines, their extraction using numerical schemes, their mapping to an effective renderable representation, and rendering options to convey structures with special mechanical properties. 3D-TSV is accessible to end users via a C++- and OpenGL-based rendering frontend that is seamlessly connected to a MatLab-based extraction backend. The code (BSD license) of 3D-TSV as well as scripts to make ANSYS and ABAQUS simulation results accessible to the 3D-TSV backend are publicly available.
... Kindlmann and Schultz introduced superquadratic tensor glyphs that satisfy all of the requirements, but only where symmetric tensors are concerned [10,11]. ...
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The rendering of tensor glyphs is a progressive process of visualizing the vector space both in fluid dynamics and the latest medical scanning. Nowadays, the rendering accuracy is ensured by numerical methods based on interpolation of tensor functions. The tensor glyph functions to visualize significant properties of the vector space. Not all these properties are visualized at all times. The number of properties and their unambiguity depend on the method chosen. This work presents a direct analytical expression covering rank two tensors in a plane. Unlike the methods used so far, this method is accurate and unambiguous one for tensor visualization. The method was applied to the simplest tensor type, which presented an advantage for the method’s analytical approach. The analytical approach to the planar case is significant also because it provides instruction on how to expand analytical calculations to cover higher spatial dimensions. In this way, numerical methods for tensor rendering can be replaced with an accurate analytical method.
... Glyph-based methods, on the other hand, depict the stress field by a set of well-designed geometric primitives -so-called tensor glyphs. Tensor glyphs were originally designed for glyph-based diffusion tensor visualization [19], and later adapted to visualize positive definite tensors [18], general symmetric tensors [34], as well as asymmetric tensors [35,11]. Glyph-based techniques are problematic when used to visualize 3D stress fields, due to their inherent occlusion effects. ...
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We present the 3D Trajectory-based Stress Visualizer (3D-TSV), a visual analysis tool for the exploration of the principal stress directions in 3D solids under load. 3D-TSV provides a modular and generic implementation of key algorithms required for a trajectory-based visual analysis of principal stress directions, including the automatic seeding of space-filling stress lines, their extraction using numerical schemes, their mapping to an effective renderable representation, and rendering options to convey structures with special mechanical properties. In the design of 3D-TSV, several perceptual challenges have been addressed when simultaneously visualizing three mutually orthogonal stress directions via lines. We present a novel algorithm for generating a space-filling and evenly spaced set of mutually orthogonal lines. The algorithm further considers the locations of lines to obtain a more regular pattern repetition, and enables the extraction of a Focus+Context representation with user-selected levels of detail for individual stress directions. To further reduce visual clutter, the system provides the combined visualization of two selected principal directions via appropriately oriented ribbons. The rendering uses depth cues to better convey the spatial relationships between trajectories and improves ribbon rendering on steep angles. 3D-TSV is accessible to end users via a C++- and OpenGL-based rendering frontend that is seamlessly connected to a MATLAB-based extraction backend. A TCP/IP-based communication interface supports the flexible integration of alternative rendering frontends. The code (BSD license) of 3D-TSV is made publicly available.
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Combining elements of biology, chemistry, physics, and medicine, the science of human physiology is complex and multifaceted. In this report, we offer a broad and multiscale perspective on key developments and challenges in visualization for physiology. Our literature search process combined standard methods with a state‐of‐the‐art visual analysis search tool to identify surveys and representative individual approaches for physiology. Our resulting taxonomy sorts literature on two levels. The first level categorizes literature according to organizational complexity and ranges from molecule to organ. A second level identifies any of three high‐level visualization tasks within a given work: exploration, analysis, and communication. The findings of this report may be used by visualization researchers to understand the overarching trends, challenges, and opportunities in visualization for physiology and to provide a foundation for discussion and future research directions in this area.
Chapter
Prior to the latter part of the twentieth century, the field of scientific visualization was both technical illustration [1, 2] and analytic geometry [3] when used to depict scientific enterprise. It was not directly involved in the emergence of computational theory [4]. However, computers became an essential tool for interactive scientific visualization, via the use of the vacuum tube display.
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Visualization of multi-dimensional data is a challenging task. The goal is not the display of multiple data dimensions, but user comprehension of the multi-dimensional data. This paper explores several techniques for perceptually motivated procedural generation of shapes to increase the comprehension of multi-dimensional data. Our glyph-based system allows the visualization of both regular and irregular grids of volumetric data. A glyph's location, 3D size, color, and opacity encode up to 8 attributes of scalar data per glyph. We have extended the system's capabilities to explore shape variation as a visualization attribute. We use procedural shape generation techniques because they allow flexibility, data abstraction, and freedom from specification of detailed shapes. We have explored three procedural shape generation techniques: fractal detail generation, superquadrics, and implicit surfaces. These techniques allow from 1 to 14 additional data dimensions to be visualized using glyph shape.
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Presents a conceptual framework and a process model for feature extraction and iconic visualization. Feature extraction is viewed as a process of data abstraction, which can proceed in multiple stages, and corresponding data abstraction levels. The features are represented by attribute sets, which play a key role in the visualization process. Icons are symbolic parametric objects, designed as visual representations of features. The attributes are mapped to the parameters (or degrees of freedom) of an icon. We describe some generic techniques to generate attribute sets, such as volume integrals and medial axis transforms. A simple but powerful modeling language was developed to create icons, and to link the attributes to the icon parameters. We present illustrative examples of iconic visualization created with the techniques described, showing the effectiveness of this approach
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Mixed echo train acquisition displacement encoding with stimulated echoes (meta-DENSE) is a phase-based displacement mapping technique suitable for imaging myocardial function. This method has been optimized for use with patients who have a history of myocardial infarction. The total scan time is 12–14 heartbeats for an in-plane resolution of 2.8 × 2.8 mm2. Myocardial strain is mapped at this resolution with an accuracy of 2% strain in vivo. Compared to standard stimulated echo (STE) methods, both data acquisition speed and resolution are improved with inversion-recovery FID suppression and the meta-DENSE readout scheme. Data processing requires minimal user intervention and provides a rapid quantitative feedback on the MRI scanner for evaluating cardiac function. Magn Reson Med 46:523–534, 2001. Published 2001 Wiley-Liss, Inc.