3D VOLUME DATA SEGMENTATION

FROM SUPERQUADRIC TENSOR ANALYSIS

Sang Min Yoon and Arjan Kuijper

GRIS TU Darmstadt, Fraunhofer IGD & TU Darmstadt

Fraunhoferstrasse 5, Darmstadt, 64283, Germany

Keywords:

3D Model segmentation, Ellipsoidal representation, Diffusion tensor ﬁelds based similarity measure.

Abstract:

The segmentation of 3D target objects into coherent subregions is one of the most important issues in computer

graphics as it is applied in many applications, such as medical model visualization and analysis, 3D model

retrieval and recognition, skeleton extraction, and collision detection. The goal of 3D segmentation is to

separate the volume or mesh data into several subregions which have similar characteristics.

In this paper, we present an efﬁcient and accurate 3D model segmentation methodology by merging and

splitting the subregions in a 3D model. Our innovative 3D model segmentation system consists of two steps:

i) the ellipsoidal decomposition of unorganized 3D object using properties of three dimensional second-order

diffusion tensor ﬁelds, and ii) The iteratively merging and splitting of subregions of the 3D model by measuring

the similarity between neighboring regions. Experimental results are conducted to evaluate the performance

of our methodology using 3D models from well-known databases and 3D target objects that are reconstructed

from image sequences.

1 INTRODUCTION

In the worlds of computer graphics, several mature

methods for an automatic 3D visualization and anal-

ysis of real objects have been heavily studied. In par-

ticular, the segmentation of 3D target objects into co-

herent sub-regions has received many concerns in last

few years because it is very useful in many applica-

tions, such as medical volume rendering and analy-

sis, 3D shape retrieval and action recognition, skele-

ton extraction, and collision detection. The goal of

3D model segmentation is to separate the 3D data into

mutually exclusive homogeneous subregions of inter-

est.

Nevertheless, there is no standard formal deﬁni-

tion of exact 3D model segmentation. Subdivided re-

gions using segmentation methodologies are different

according to their features and clustering techniques.

Recently, Chen et al. (Chen et al., 2009) tried to com-

pare the difference of the automatically segmented 3D

models and human-visual perception based segmenta-

tion. For efﬁcient segmentation of 3D volume data,

extraction of robust features and a similarity mea-

sure between separated neighbor regions are needed.

Previous research to segment 3D meshes which are

composed with numerous triangles (Lien et al., 2006;

Figure 1: Flowchart of our proposed 3D object segmenta-

tion using the characteristics of the 3D ellipsoidal model.

Tierny et al., 2008) focused on their effort for novel

clustering techniques in the the space of scalar or vec-

tor ﬁelds.

In this paper, we present a 3D model segmenta-

tion methodology using an ellipsoidal representation

of the 3D model in the space of diffusion tensor ﬁelds.

Using the properties of diffusion tensor ﬁelds, we it-

eratively merge and split the subregions of 3D target

objects. The eigen-features which come from diffu-

sion tensor ﬁelds are used to measure the similarity

between neighbor subregions. Figure 1 shows the to-

tal ﬂowchart which we will explain in the paper using

Min Yoon S. and Kuijper A. (2010).

3D VOLUME DATA SEGMENTATION FROM SUPERQUADRIC TENSOR ANALYSIS.

In Proceedings of the International Conference on Computer Graphics Theory and Applications, pages 72-77

DOI: 10.5220/0002818800720077

 SciTePress

the example 3D volume data.

Our main contributions are (1) the 3D ellipsoidal

representation of the 3D volume/mesh model using

the properties of diffusion tensor ﬁelds, (2) the itera-

tive merging and splitting the neighboring regions by

measuring the similarity of characteristics of diffusion

tensorial features, and (3) the efﬁcient and robust 3D

segmentation for diverse models or reconstructed ob-

jects from multiple images without any constraints,

prior knowledge, or assumptions on the 3D model.

2 PREVIOUS WORK

Our methodology has been motivated by the concept

of superquadric representation of 3D models, 3D dif-

fusion tensor ﬁelds and its applications, and 3D seg-

mentation techniques. In this section, we survey the

previous work which contributed signiﬁcantly to the

innovation within this area.

2.1 Superquadric Representation

The representation of a 3D model using superquadrics

is used for visualizing its characteristics with only

very few parameters cylinders, geons, superquadrics,

etc. Superquadrics are a family of geometric solids,

which can be interpreted as a generalized of basic

quadric surfaces and solids. With only a few parame-

ters, they can represent a large variety of standard ge-

ometric solids and smooth shapes. Superquadrics are

also very efﬁcient for representing three dimensional

surface data. In contrast to a mesh representation of

an object with thousands of triangles, the same ob-

ject can be represented by a small set of superquadrics

which are uniquely deﬁned by 11 parameters per each

voxel (Zhang et al., 2003).

2.2 Diffusion Tensor Fields

Most 3D volume segmentation and visualization

methodologies are based on vector ﬁelds which are

generated from a given image by different physical

properties. Few works have investigated the extrac-

tion of features within resulting tensor ﬁelds. Basser

et al. (Basser et al., 1994) presented their seminar

work on diffusion tensor magnetic resonance imaging

(DT-MRI). Using this new MRI modality, it was pos-

sible to qualify anisotropic properties of an imaged

tissue by characterizing the water diffusion. In partic-

ular, Kindlmann (Kindlmann, 2004) presented a ten-

sor based superquadric visualization method by map-

ping the tensor eigenvalues and eigenvectors to the

orientation and shape of geometric primitives.

2.3 3D Model Segmentation

Segmentation techniques can be separated into

classes in many ways according to different classiﬁ-

cation schemes, Shamir (Sharmir, 2006) surveyed the

previously 3D segmentation of 3D volumes and sep-

arated the 3D segmentation problem into two types:

as surface-type and part-type segmentation. Surface-

type based segmentation is based on the decompo-

sition of geometric primitives such as planes, cylin-

drical patches, and spherical parts (Wu and Levine,

2005; Attene et al., 2000). The part-type segmenta-

tion decomposes a 3D object into sub-meshes by seg-

menting a surface into connected components (Lien et

al., 2006; Tierny et al., 2008). The segmentation al-

gorithms used various approximating solutions such

as K-means (Shlafman et al., 2002), region growing

(Chazelle et al., 1997), hierarchical clustering (Gar-

land et al., 2001; Inoue et al., 2001), graph-cuts (Katz

and Tal, 2003), iterative clustering (Chevalier et al.,

2003), and spectral analysis (Liu and Zhang, 2004).

3 SUPERQUADRIC

DECOMPOSITION USING

TENSORIAL FEATURES OF 3D

MODEL

Previous 3D segmentation techniques are mainly

based on mesh/triangle based 3D models in the space

of vector or scalar transformations, because similarity

measures within these spaces relate to the perception

of the human eye. Nevertheless, tensorial maps con-

tain and provide more information than scalar ones as

to measure the similarity between neighbor regions.

In this section, we decompose the 3D mesh mod-

els into 3D ellipsoidal models using the properties of

three dimensional second-order symmetric diffusion

tensor ﬁelds.

A tensor is the mathematical identiﬁcation of a ge-

ometric or physical quantity whose analytic descrip-

tion consists of an array of numbers. The tensor ﬁeld

commonly deﬁned as a topological representation of

a 3D symmetric, second-order symmetric tensor ﬁeld

is shown as (Delarcelle and Hesselink, 1994):

T =









, (1)

where T

= T

, T

= T

, and T

= T

since the or-

der of differentiation is free, so that the tensor is a

symmetric positive deﬁnite matrix. This matrix can

3D VOLUME DATA SEGMENTATION FROM SUPERQUADRIC TENSOR ANALYSIS

(a) 3D ellipsoidal representation of each voxel using the extracted eigenvalues and eigenvectors. (b) The

superquadric tensor visualization as the change of the eigenvalues’ values, giving rise to differently oriented

ellipsoids.

Figure 2: The superquadric representation uses the eigenvalues and eigenvectors of the diffusion tensor ﬁelds which determine

the scale and orientation of the superquadric model.

be reduced to principal axes by solving the character-

istic equation

(T − λ

· I)e

= 0, (2)

where I is the identity matrix, λ

are the eigenvalues

of the tensor and e

are the orthonormal eigenvectors

(i = 1, . . . , 3). In this case, the tensor in each pixel can

be represented by an ellipsoid, where the main axis

lengths are proportional to the ordered eigenvalues λ

(λ

≥ λ

> 0).

The evaluation of the tensor ellipsoidal geomet-

rics and their properties is facilitated by an intuitive

domain that spans all possible tensor shapes. Such a

domain is afforded by the geometric anisotropy met-

rics of Kindlmann (Kindlmann, 2004). Given the

non-negative tensor eigenvalues λ

, λ

, the metrics

quantifying the certainty with which a tensor is said to

have a given shape are given by:

−λ

+λ

, c

2(λ

−λ

)

+λ

, c

3λ

+λ

The three metrics add up to unity, and deﬁne a

barycentric parameterization of a triangular domain,

with the extremes of linear, planar, and spherical

shapes at the three corners. The barycentric shape

space has been used as the domain of transfer func-

tions for direct volume rendering of diffusion ten-

sors, and as an intuitive basis of comparison be-

tween various anisotropy metrics. Figure 2 is a con-

ceptual representation of volume data showing how

the superquadric models can be represented with ex-

tracted eigenvalues and eigenvectors. The ratio be-

tween eigenvalues determines the shape of the su-

perquadrics, their sum deﬁnes their scale and its prin-

cipal eigenvector direction deﬁnes the rotation of the

superquadrics as shown in Figure 2b. Using the ba-

sic principal direction of the superquadric represen-

tation method, we can visualize the 3D volume data

Figure 3: The superquadric representation of 3D volume

data such as ”tool” and ”horse” using our proposed method.

Figure 4: The process of segmenting the 3D target object

using our proposed merging and splitting methodology. The

unorganized 3D model is divided by its characteristics of

tensorial features.

into superquadrics. Figure 3 shows some examples of

3D models such as ”horse” and ”tool” and zooms in

to see the detailed shape and orientation of each su-

perquadric model.

GRAPP 2010 - International Conference on Computer Graphics Theory and Applications

Figure 5: Some examples of 3D model segmentation using

our approach using 3D models from the Princeton 3D model

dataset.

(a) Imported 3D volume data (b) Segmentation using

our proposed method (c) Detail of the hair tail.

Figure 6: 3D model segmentation using our proposed

methodology. Even though the hair tail is a small region,

its superquadric characteristics are not similar to the neigh-

boring regions and are therefore not merged with the neigh-

bors.

4 ITERATIVE SEGMENTATION

OF 3D MODEL

Various methods have been proposed to solve the

problems of separating and clustering subregions of

the 3D target object. In this section, we present the

separation the unlabeled 3D model into segmented

subregions using properties of the eigen-features. The

degree of anisotropy of the ellipsoidal 3D model

can be quantiﬁed in a single number called diffusion

anisotropy index, or fractional anisotropy (FA), and

is deﬁned as follows:

FA =

3[(λ

− λ

avg

)

+ (λ

− λ

avg

)

+ (λ

− λ

avg

)

]

2(λ

+ λ

)

(3)

where λ

avg

is the average of λ

, λ

, and λ

. The

FA is used as feature to measure the similarity of

neighbor voxels to segment the 3D model.

The 3D model segmentation procedure is as follows:

STEP0 : Initially, the numbers of subregions

of the human body model is equal to the num-

ber of voxels of the 3D human model. Calculate

i j

(FV

− FV

) × (FV

− FV

), where FV

and

are FA of voxel i and j, respectively.

STEP1 : Progressively merge the neighbor

voxels if FA

i j

is less than threshold and recal-

culate the average FA of the merged subregions,

sub

∑

k=1

, where the merged subregion

which have n voxels. The voxels i and j are splitted

if FA

i j

exceeds the threshold.

STEP2 : Repeat STEP1 until there are no subre-

gions whose the FA

sub

is less than threshold.

5 EXPERIMENTS

We implemented our proposed 3D segmentation

methodology using the superquadric features in the

space of diffusion tensor ﬁelds and conducted exper-

iments on a standard PC with a Pentium 4 1.2 GHz

CPU. First, we have tested our methodology using

the Princeton 3D model database. They provided

the 3D models in Object File Format (OFF) with a

polygonal geometry of the model. We have con-

verted the objects to a voxel based model which has

a 128x128x128 voxel size. Figure 5 shows some ex-

amples, such as an animal, a tool, and a guitar. We

displayed the segmented regions with various colors.

Figure 6 shows a Princeton database example in more

detail. As shown in Figure 6(d), a small subregion

is correctly segmented from the neighboring regions

because its tensorial characteristics are very different

from the neighbor regions.

We also tested our algorithms on 3D reconstructed

objects obtained from multiple images. The images

came from the HumanEva database. We ﬁrst recon-

structed the target object from multiple images by

image based visual hulls using provided camera cal-

ibration data and statistics of background modeling

with 128x128x128 voxel size volumes. We adapted

our method to the 3D reconstructed model as shown

in Figure 7. Using the extracted silhouette of fore-

3D VOLUME DATA SEGMENTATION FROM SUPERQUADRIC TENSOR ANALYSIS

(a) Input images of multiple cameras from the

HumanEva dataset.

(b) 3D reconstructed object from the HumanEva

dataset.

(d) 3D segmentation into distinct parts using our

proposed approach.

Figure 7: 3D model segmentation by using 3D reconstruc-

tion from multiple images.

(a) Unlabeled 3D human brain volume data whose

resolution is 512x512x512. (b) Segmented human

brain volume data by using our approach.

Figure 8: Medical volume segmentation and rendering.

ground and camera calibration information, we recon-

structed the target object using Image-Based Visual

Hull (IBVH) (Matusik et al., 2000).

(a) 3D segmentation of Zhang et al., (b) 3D

segmentation of Lien et al., (c) 3D segmentation of

Tierny et al., (d) consistent segmentation, (e) our

proposed 3D segmentation using iterative merging

and splitting.

Figure 9: Some examples of 3D model segmentation us-

ing our approach from the 3D models from Princeton’ 3D

model dataset.

Our 3D model segmentation technique also ap-

plied to medical volume understanding and analysis

area. Figure 8 shows the segmented human brain vol-

ume data using our approach. Figure 8a is the unla-

beled 3D MRI data whose resolution is 512x512x512.

We can segment the 3D medical volume data into sev-

eral subregions which have similar tensorial charac-

teristic. Our approach does not require priori infor-

mation to segment the 3D volume data, we can apply

to various target objects which have high-degree of

freedom.

We compared our methodology to several other ap-

proaches, see for example Figure 9. It contains the

approaches by Zhang et al. (Zhang et al., 2003) in

Figure 9a, Lien et al. (Lien et al., 2006) in Figure 9b,

Tierny et al. (Tierny et al., 2008) in Figure 9c, and the

consistent segmentation in Figure 9d. The 3D “horse”

model is commonly divided into a main body part, 4

legs, a neck, and a head. Especially the 3D segmen-

tation of Tierny et al. (Tierny et al., 2008) has many

subregions. Note that the intention of our approach is

not to speciﬁc segment the target object as shown in

Figure 8, but a generating distinct subregions. Subre-

gions are divided into smaller region if their tensorial

characteristics are very different to the neighborhood

as shown in Figure 6. Previous approaches are based

on the 3D mesh models which are composed of nu-

merous triangles, but our 3D models are only based

128x128x128 voxel sized volumes. Our results can

be used to deﬁne the subregions in the smooth trian-

gulated versions.

6 CONCLUSIONS AND FUTURE

WORK

In this paper, we presented a 3D volume segmenta-

tion methodology using tensorial properties of dif-

GRAPP 2010 - International Conference on Computer Graphics Theory and Applications

fusion tensor ﬁeld. The superquadric model using

the properties of the 3D second-order diffusion tensor

ﬁelds efﬁciently visualize the characteristics of the 3D

model using only few parameters. We cluster the 3D

volume data by iteratively merging and splitting the

neighboring regions using the similarity measure of

tensorial features.

Experiments of 3D segmentations of the imported

3D models and 3D reconstructed objects from multi-

ple image sequences, we show that our proposed sys-

tem is very efﬁcient and robust in separating the 3D

volume data into several subregions which have sim-

ilar tensorial characteristics. Our 3D model segmen-

tation is very useful in 3D deformable object motion

analysis and medical volume visualization.

This methodology provides a basis for the seg-

mentation and tracking of deformable object seg-

ments. Hence, our future work will focus on consis-

tent 3D model segmentation optimized for speed and

performance as soon as qualitative benchmarks can

be given. The exploitation of additional semantics

together with our methodology could lead to a fast

3D segmentation approach. Vice versa, the derivation

of semantical metadata for subregions of a 3D object

will also be addressed in future work.

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