Volume Deformation For Tensor Visualization
Xiaoqiang Zheng and Alex Pang
Computer Science Department
University of California, Santa Cruz, CA 95064
zhengxq@cse.ucsc.edu, pang@cse.ucsc.edu
ABSTRACT
Visualizing second-order 3D tensor fields continue to be a chal-
lenging task. Although there are several algorithms that have been
presented, no single algorithm by itself is sufficient for the analysis
because of the complex nature of tensor fields. In this paper, we
present two new methods, based on volume deformation, to show
the effects of the tensor field upon its underlying media. We focus
on providing a continuous representation of the nature of the tensor
fields. Each of these visualization algorithms is good at displaying
some particular properties of the tensor field.
Key Words and Phrases: stress, strain, shear, symmetric ten-
sors, anti-symmetric tensors, anisotropic tensors.
1 INTRODUCTION
The analysis of tensor data with various orders is important in many
medical and physical applications. For example, tensors can be
found in medical imaging (e.g. diffusion tensor images [9]), fluid
flow (e.g. velocity gradient [4]), stress simulation and tectonics. Al-
though scientists often have to deal with tensor data sets, the tools
available for visualizing them are quite limited. In this paper, we
restrict our discussion to second order 3D tensor fields. Visualizing
such tensor fields is difficult because each tensor in the field con-
tains nine unique quantities. To incorporate so much information
in a single representation is a challenging task. Current methods
either show detailed information about the tensor field but only at a
few local, discrete points using glyphs, or show partial information
about the tensor field but over a global, continuous domain.
In this paper, we present two new techniques based on deforma-
tion of the tensor volume. Deformation is an intuitive way to rep-
resent a tensor. We can observe the tensile, compressive and shear
effects of a tensor value through the transformation of the volume.
The conceptual idea of this algorithm is to consider the tensor field
as a force field that deforms an object. The local deformation of the
object represents the tensor value at that point in the tensor field.
Both of the two techniques are designed to visualize some par-
ticular properties of the tensor field. First, the normal vector de-
formation algorithm is good at showing the directional information
of the tensor field. Second, the anisotropic deformation is capable
of illustrating the compressing and shearing property of the tensor
field.
The rest of this paper is organized as follows: Section 2 reviews
some previous tensor visualization techniques; Section 3 reviews
some tensor decomposition methods to aid with the visualization
task; Section 4 presents the two new volume deformation tech-
nique; Section 5 discusses some implementation issues; Section 6
illustrates the methods and analyzes the results; Section 7 draws
conclusions for the two volume deformation techniques and pro-
poses future works for tensor visualization.
2 PREVIOUS WORK
There are a few methods for visualizing tensors such as pseudo-
coloring, tensor glyphs [2, 7, 8], hyperstreamlines [3], and defor-
mation [1, 10, 12].
Pseudo-coloring is the simplest but also least effective method.
For a tensor field with 9 independent components, a planar slice
is made through the tensor volume and each of the 9 components
on that slice are pseudo-colored. The tensor information for that
slice is then presented as a 3 by 3 collage. The main drawback of
this technique is that users have to mentally integrate the physical
interaction of the 9 tensor components. A more popular method
involves the use of glyphs. The simplest one is the tensor ellip-
soid. Here, the tensor is decomposed into three orthogonal eigen-
vectors, with their corresponding eigenvalues indicating the magni-
tude along each eigenvector. An ellipsoid is then constructed ori-
ented according to the 3 eigenvectors and scaled according to the
3 eigenvalues. This works for symmetric tensor fields. More com-
plex glyphs are constructed to show additional features in the tensor
fields using flow probes [2]. The general drawback of glyphs is their
discrete nature. They do not capture the global or continuous nature
of tensor fields. And they also require judicious placement to avoid
clutter.
Another well known tensor visualization approach is with hyper-
streamlines or tensor field lines [3]. For symmetric tensor fields, the
3 eigenvectors at each point are sorted by their eigenvalues and clas-
sified as the major, medium, and minor eigenvectors. Hyperstream-
lines are then generated by integrating along one of these eigenvec-
tor fields, and letting the two other eigenvector fields to modify the
shape of an ellipse that is swept along the principal hyperstream-
line. For non-symmetric tensor fields, where the 3 eigenvectors are
not necessarily orthogonal to each other, the tensor field is first de-
composed into a symmetric and an axial component. Ribbons along
the hyperstreamline are then added to show the rotational effects of
the axial component. Because one of the eigenvector fields is used
for integrating the hyperstreamline, there are two other possible hy-
perstreamlines that can be generated from each seed location. The
understanding of the tensor field is therefore not complete without
these two. The drawback of this approach then is that users have to
integrate the 3 different views.
An alternative approach is to use deformation. Simple objects
such as polygons [12] or cubes [10] are advected by a flow field.
The deformations on the objects show the local stretch, shear, and
rigid body rotation at a point. This is generalized in [1] to include
idealized objects such as lines, planes, sub-volumes. It allows users
to interactively modify an interrogation vector to see how the ten-
sor field would deform the object. While the user can now see the
continuity over the field, the main drawback of this approach is that
users can only see the information in one direction at a time. Our