Automatic Cross-Sectioning Using 3D Field Topology Analysis
Yuki Mori*

Shigeo Takahashi†

Takeo Igarashi‡

Yuriko Takeshima§

Issei Fujishiro§

The Univ. of Tokyo

The Univ. of Tokyo

The Univ. of Tokyo / JST PRESTO

Tohoku University

Tohoku University

1Introduction

It is difficult to comprehend the inner structures of volumedatasets. Several techniques have been proposed to visualize these
inner structures, such as volume rendering and isosurface
extraction. However, these approaches are still limited in theability to visualize the detailed structures and users frequently
take cross-sections to inspect such detailed structures. The
problem is that existing systems require the users to search for an
ideal cross-section manually, which involves tweaking manyparameters and is tedious and time-consuming. To address thisproblem, we present a method for automatically generating crosssections that reveals the characteristic structures of a dataset.

2The Volume Skeleton Tree

We use a 3D field topology analysis technique to analyze volumedatasets. The Volume Skeleton Tree (VST) [Takahashi et al.
2004] is a level set graph that delineates volumetric structures. It
represents the splitting and merging of isosurfaces with respect tothe filed value. Each node of the graph represents a critical point
where the topology of the evolving isosurface changes, and eachlink represents a region between the isosurfaces associated withthe critical points.
For example, we consider the analytic volume function asshown in Fig. 1(a) [Takahashi et al. 2004]. This volume dataset
includes six critical points: P1, P2 (appearance), P3, P4 (merging),
P5 (splitting), and P6 (disappearance). Figure 1(b) represents theVST of the volume dataset. If a dataset is noisy, the resulting VST
becomes complex and it becomes difficult to see the globalstructure. In such cases, we simplify the VST before computing
cross-sections to obtain a clean VST.

P6P5P3

P1

P1

P2P3P4P5P6

P2(a)
(b)
Figure 1: Topological analysis of the analytic function volume: (a) Achange in the isosurface and critical points; (b) The corresponding VST.

3(a)
(b)
(c)
Figure 2: Cross-section samples of the analytic function volume:(a) A cross-section passing through many important critical points;(b), (c) Cross-sections passing through axes of symmetry.

4Results

We applied our method to various volume datasets taken fromquantum science and medical science. Figure 3 shows volumerendered images of three target datasets using accentuated transfer
functions. The top images show the entire datasets, and the bottom
images show the datasets on the far side of the cutting planes. We
can see that the bottom images better illustrates the detailed innerstructures.

scalar field value

scalar field value
P4

image shows the dataset on the far side of the cutting plane.
Figure 2(a) is rendered with the best-fitting plane, and Figs 2(b)and (c) are rendered with the next two candidates.

Generating Cross-sections using VST

Our basic idea is to cut the dataset at the plane best-fitting thecritical points of VST. A simple method to generate such a planeis least-square estimation. To make such a plane, we construct a
covariance matrix of critical points and calculate threeeigenvalues and eigenvectors. Then, we make three planes, eachof which passes through the center of gravity of the critical points
and includes any two of the three eigenvectors. The system cuts
the dataset using a plane that includes two eigenvectors pairedwith the two major eigenvalues, but the user can select two othercross-sections. This selectivity is interesting from the viewpoint ofuser interface design.
Figure 2 shows that our method can yield appropriate crosssections of the dataset shown in Fig. 1(a). Each volume-rendered*e-mail: yuki@ui.is.s.u-tokyo.ac.jp†e-mail: takahashis@acm.org‡e-mail: takeo@acm.org§e-mail: {takeshima, fuji}@vis.ifs.tohoku.ac.jp

(a)
(b)
(c)
Figure 3: Volume-rendered images of target datasets: (a) The hydrogenion-atom collision [Fujishiro et al.2000]; (b) The nucleon [Meibner]; (c)NML Tooth volume [Woo].

In the future, we plan to extend our algorithm to generatecross-sections including curved surfaces and multiple planes. We
also want to try other methods to analyze inner structures, such as
the medial axis and generalized symmetry, to apply automaticcross-sectioning to other model representations, mainly surface
models. Finally, we would like to design better user interfaces forinspecting volume datasets.

References
TAKAHASHI, S., TAKESHIMA, Y. AND FUJISHIRO, I. 2004. Topological Volume
Skeletonization and its application to transfer function design. Graphical Models,
66(1), 22-49.
FUJISHIRO, I., AZUMA, T., TAKESHIMA, Y., AND TAKAHASHI, S. 2000. Volume data
mining using 3D field topology analysis. IEEE Computer Graphics and Applications,20(5), 46-51.
MEIBNER, M. Web Page [http://www.volvis.org/].
WOO, T. The National Library of Medicine of the National Institutes of Health[http://visual.nlm.nih.gov/data/].