计算机图形学Lecture08-协同分割论文coseg

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Eurographics Symposium on Geometry Processing 2012

Eitan Grinspun and Niloy Mitra

(Guest Editors)

Volume 31 (2012), Number 5

Co-Segmentation of 3D Shapes via Subspace Clustering

Ruizhen Hu Lubin Fan Ligang Liu†

Department of Mathematics, Zhejiang University, China

State Key Laboratory of CAD&CG, Zhejiang University, China

Figure 1: The co-segmentation result of the Candelabra set by our algorithm. Starting from the over-segmented patches of the

shapes (left), our algorithm automatically obtains the consistent segmentations among these objects by grouping the patches

using subspace clustering in multiple feature spaces. Corresponding parts are shown in the same colors (right).

Abstract

We present a novel algorithm for automatically co-segmenting a set of shapes from a common family into con-

sistent parts. Starting from over-segmentations of shapes, our approach generates the segmentations by grouping

the primitive patches of the shapes directly and obtains their correspondences simultaneously. The core of the

algorithm is to compute an afﬁnity matrix where each entry encodes the similarity between two patches, which

is measured based on the geometric features of patches. Instead of concatenating the different features into one

feature descriptor, we formulate co-segmentation into a subspace clustering problem in multiple feature spaces.

Speciﬁcally, to fuse multiple features, we propose a new formulation of optimization with a consistent penalty,

which facilitates both the identiﬁcation of most similar patches and selection of master features for two similar

patches. Therefore the afﬁnity matrices for various features are sparsity-consistent and the similarity between a

pair of patches may be determined by part of (instead of all) features. Experimental results have shown how our

algorithm jointly extracts consistent parts across the collection in a good manner.

Categories and Subject Descriptors (according to ACM CCS): I.4.6 [Computer Graphics]: Segmentation—

†Corresponding: ligang.liu@gmail.com

1. Introduction

Segmentation of a 3D shape into semantic parts is a fundamen-

tal task in high-level shape analysis and processing [AKM∗06,

2012 The Author(s)

Computer Graphics Forum c

2012 The Eurographics Association and Blackwell Publish-

ing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ,

UK and 350 Main Street, Malden, MA 02148, USA.

DOI: 10.1111/j.1467-8659.2012.03175.x

R. Hu, L. Fan, L. Liu / Co-Segmentation of 3D Shapes via Subspace Clustering

Sha08,CGF09]. In recent years, co-segmentation of a set of

3D shapes, i.e., segmentation of the shapes as a whole into

consistent semantic parts with correspondences, has received

increased attention [GF09,XLZ∗10,HKG11,SvKK∗11]. It

has been demonstrated that more knowledge can be in-

ferred from multiple shapes rather than an individual shape

and co-analysis of the set produces better segmentations

than the single-shape algorithms [GF09,vKZHCO10,HKG11,

SvKK∗11]. However, extraction of appropriate knowledge in-

herent to multiple shapes for consistent segmentation remains

challenging.

In this paper, we propose a novel unsupervised framework

to consistently segment a set of 3D shapes from the same class.

We consider the co-segmentation as a clustering problem by

grouping the primitive patches of all shapes into correspond-

ing parts (see Figure 1). The core solution paradigm of our al-

gorithm is a subspace clustering optimization [Vid10] on the

patches.

Multiple feature descriptors, which respect shape geometry

and context, are used to measure similarity of patches. Ac-

cording to our observation, two parts of models perceived as

corresponding may not necessarily be similar in all features

and may even signiﬁcantly differ on some. Figure 2shows an

example of two table models, of which the legs are seman-

tically in correspondence. Their averaged geodesic distance

(AGD) features [HSKK01] are similar, while their shape di-

ameter function (SDF) features [SSS∗10] are quite different,

as shown in the colormaps in Figure 2(right). Hence, con-

catenating the features into a higher dimensional descriptor

may confuse clustering algorithms by augmenting the differ-

ence between two similar parts, as shown in Figure 2(lower

left).

Inspired by recent works in image processing [Vid10,

CLW∗11], we propose a new subspace clustering formula-

tion of optimization with a consistent multi-feature penalty

to guarantee the consistency of co-segmentation results ac-

cording to various features. By solving this optimization we

identify the most similar patch pairs consistent within mul-

tiple feature spaces and the prominent features contributing

to measuring the similarity of each patch pair. Figure 2(up-

per left) shows the co-segmentation results with the consistent

multi-feature penalty .

To the best of our knowledge, we are the ﬁrst to explore the

fusion of multiple features in shape segmentation and geome-

try processing to this extent. It is worthwhile pointing out that

popping-up of the prominent features which are used to mea-

sure the similarity between patches is different with the fea-

ture selection scheme in the learning based approach [KHS10]

where features are selected by JointBoost in order to classify

each training face into its corresponding label. As an unsu-

pervised method, our approach does not require training data.

Instead of selecting the features to satisfy some conditions, we

allow the features which contribute most to similarity between

two patches to pop up by themselves in the results.

Figure 2: The legs of two table models are semantically in

correspondence (upper left). They are quite similar in AGD

features (upper right) while they differ a lot in SDF features

(lower right). Hence, two parts of models perceived as corre-

sponding may not necessarily be similar in all features. Upper

left: result with the consistent multi-feature penalty; Lower

left: result with the concatenated feature descriptor.

We evaluate our proposed approach on various 3D shape

categories and make comparisons with state-of-the-art ap-

proaches. These results demonstrate that our approach

achieves comparable performance to the supervised approach

and produces better results than the others. Moreover, our ap-

proach is more efﬁcient than the previous approaches as the

convex optimization we used can be effectively solved. Be-

sides, as a patch-level approach, it takes much less time for

our method to get the satisfactory results than any previous

face-level approach and is more ﬂexible than segment-level

approaches.

Contributions. Our contributions are twofold.

•We propose a novel framework for shape co-segmentation

based on subspace clustering. It simultaneously generates

the segments and their correspondences from the over-

segmented patches within multiple feature spaces. Various

features can be introduced in our framework, which makes

our algorithm ﬂexible and feasible for co-segmenting dif-

ferent shapes.

•We propose a consistent multi-feature penalty in the sub-

space clustering optimization, which guarantees the consis-

tency of the co-segmentation results according to various

features and makes prominent features used to measure the

similarity between each patch pair stand out actively.

2. Related work

A large variety of approaches have been proposed for

segmenting single shape into meaningful parts [AKM∗06,

Sha08]. It has been shown [CGF09] that no segmentation al-

gorithm always performs well for all models because the ge-

2012 The Author(s)

2012 The Eurographics Association and Blackwell Publishing Ltd.

1704

R. Hu, L. Fan, L. Liu / Co-Segmentation of 3D Shapes via Subspace Clustering

ometry of an individual shape may lack sufﬁcient cues to iden-

tify all parts that would be perceived as meaningful to a human

observer.

Co-segmentation of 3D shapes. There have been various re-

cent works on consistently segmenting a set of 3D shapes from

the same class into semantic parts. Kraevoy et al. [KJS07] per-

form an initial segmentation of each model into parts and then

create a consistent segmentation by matching the parts and

ﬁnding their correspondences. Huang et al. [HKG11] present

a linear programming based approach for jointly segmenting a

heterogenous database of shapes, which signiﬁcantly outper-

forms single-shape segmentation techniques. However, these

two methods provide only mutually consistent segmentation

but cannot guarantee the consistency of the ﬁnal segmenta-

tions across the set.

Golovinskiy and Funkhouser [GF09] consider the consis-

tent segmentation as a graph clustering problem. By assum-

ing that there is a global rigid alignment between matching

shapes, their approach builds connection among the corre-

sponding parts using ICP, and thus can handle limited model

types only. To deal with non-homogeneous part scales, Xu et

al. [XLZ∗10] classify the shapes based on their styles and then

establish part correspondences in each style group. However,

the graph generation process is computationally expensive.

Kalogerakis et al. [KHS10] present a supervised learning

based technique that employs information from shapes in a

training set to segment a given shape, demonstrating signiﬁ-

cant improvement over single-shape segmentation algorithms.

van Kaick et al. [vKTS∗11] further incorporate prior knowl-

edge learned from a set of pre-segmented and labeled shapes

for performing part correspondences. Consistent segmenta-

tion may be established based on individual labeling of the

shapes, however, a large number of manually segmented train-

ing shapes are needed in these learning based approaches,

while our approach is entirely unsupervised and does not re-

quire such training data.

Sidi et al. [SvKK∗11] propose an unsupervised approach

for co-segmenting a set of shapes with large variation. The

correspondences between dissimilar parts could be built via

linking through third-parties. However, this approach needs

initial segmentations for the shapes, which might result in un-

satisfactory results when the per-shape segmentation cannot

reﬂect the semantic parts well. On the other hand, our ap-

proach simultaneously generates the segmentations and their

correspondences from the over-segmented patches, which is

more ﬂexible.

In an independent recent work, Meng et al. [MXLH12] also

present an unsupervised algorithm for co-segmenting a set of

similar 3D shapes by clustering the oversegmentated prim-

itive patches and empolying the multi-label optimization to

improve the results. Unlike our method, they adopt only two

shape descriptors to cluster the patches which might fail in

cases of dissimilar objects.

Subspace clustering. Part of our research is inspired by the

recent works of subspace clustering methods [Vid10].

Subspace clustering

aims to cluster the high-

dimensional datasets into

multiple low-dimensional

linear subspaces simul-

taneously (see the ﬁgure

on the right) and has

been widely used in com-

puter vision, image processing, and data regression, etc.

(see [PHL04,Vid10] and references therein). We treat the

co-segmentation of a set of shapes as a subspace clustering

problem within multiple feature spaces by introducing a

consistent multi-feature penalty in the optimization.

3. Overview

Our co-segmentation algorithm takes a set of meshes from an

object category as input and produces a consistent segmenta-

tion of these meshes as output. First, we independently com-

pute a set of primitive patches for each input mesh. Then, we

calculate a few feature vectors for each patch. Finally, we per-

form subspace clustering on all patches in multiple feature

spaces and obtain co-segmentations of these meshes.

Over-segmentation. Similar to the idea of superpixels in

image segmentation [SM00,RM03] which are used to bal-

ance segmentation quality and computational cost, we per-

form an over-segmentation on each shape by partitioning it

into primitive patches [HKG11]. We employ the normalized

cuts (NCuts) [GF08] to generate the primitive patches for each

shape. The number of patches per shape is set to be p=50

in our implementation. See Figure 1for two examples of the

over-segmentation results.

Feature descriptors. We rely on geometric features to cluster

the patches into parts among the set of shapes. Thus we prefer

to choose a set of feature descriptors that is as informative as

possible to distinguish patches of different parts.

We select H=5 feature descriptors in our algorithm. Four

descriptors, i.e., Gaussian curvature (GC) [GCO06], shape di-

ameter function (SDF) [SSS∗10], average geodesic distance

(AGD) [HSKK01], and shape contexts (SC) [BMP02], are se-

lected based on a study on feature selections in the learning

approach (see Figure 5 in [KHS10]). The ﬁfth descriptor is

the conformal factor (CF) introduced in [BCG08]. All these

feature descriptors are deﬁned and computed on mesh trian-

gles.

For each feature descriptor, we deﬁne a feature vector for

each patch by computing a histogram capturing the distribu-

tion of the feature measurement on the triangles of this patch

(see examples of AGD feature vectors shown in Figure 3). The

number of bins for the histogram is set to be d=100 in our

implementation. Note that the SC feature vector of the patch is

computed differently. We ﬁrst compute and normalize the SC

2012 The Author(s)

2012 The Eurographics Association and Blackwell Publishing Ltd.

1705

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摘要：

EurographicsSymposiumonGeometryProcessing2012EitanGrinspunandNiloyMitra(GuestEditors)Volume31(2012),Number5Co-Segmentationof3DShapesviaSubspaceClusteringRuizhenHuLubinFanLigangLiuyDepartmentofMathematics,ZhejiangUniversity,ChinaStateKeyLaboratoryofCAD&CG,ZhejiangUniversity,ChinaFigure1:Theco-segment...

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