Ryutarou Ohbuchi, Keisuke Banba, Supervised, geometry-aware segmentation of 3D mesh models, accepted, ICME2012
Workshop on Human-Focused Communications in the 3D Continuum, July 9, 2012, Melbourne Australia

SUPERVISED, GEOMETRY-AWARE SEGMENTATION OF 3D MESH MODELS
Keisuke Bamba

Ryutarou Ohbuchi

University of Yamanashi
Kofu, Yamanashi, Japan
g09mk029AT_yamanashi.ac.jp

University of Yamanashi
Kofu, Yamanashi, Japan
ohbuchiAT_yamanashi.ac.jp

Abstract—Segmentation of 3D model models has applications,e.g., in mesh editing and 3D model retrieval. Unsupervised,automatic segmentation of 3D models can be useful. However,someapplications
requireuser-guided,interactive
segmentation that captures user intention. This paper presents
a supervised, local-geometry aware segmentation algorithm for3D mesh models. The algorithm segments manifold meshes
based on interactive guidance from users. The method casts
user-guided mesh segmentation as a semi-supervised learning
problem that propagates segmentation labels given to a subset
of faces to the unlabeled faces of a 3D model. The proposed
algorithm employs Zhou’s Manifold Ranking [18] algorithm,
which takes both local and global consistency in highdimensional feature space for the label propagation.
Evaluation using a 3D model segmentation benchmark dataset
has shown that the method is effective, although achieving
interactivity for a large and complex mesh requires some work.

label is propagated to unlabeled faces by using a semisupervised learning algorithm based on Manifold Ranking(MR) [18], which takes into account both local and globalchanges of local geometrical features.
Supervised
segmentation
Supervision A

Supervision B
Supervised
segmentation

Index Terms— Geometric modeling, manifold ranking,diffusion distance, computer graphics.

Figure 1. Supervised segmentation of 3D models is necessary as user
intention to segment a 3D model may differ by user, by purpose of
segmentation, etc.

I. INTRODUCTION

II. RELATED WORK

Segmentation of a (manifold) 3D mesh model partitions
a polygonal mesh surface into different regions of coherent
properties based on local geometrical feature (e.g.,“smoothness”), semantics associated with local geometry(e.g., “arm”, “leg”), and other properties (e.g., color). Such
3D model segmentation has applications in such areas asmesh transmission [7] and part-based 3D model retrieval [2].
Majority of existing 3D mesh segmentation algorithms areof off-line, fully-automatic kind [10, 8, 4, 15, 16, 3, 9, 12,11]. Recently, interactive mesh segmentation algorithms
have been gaining attention [15, 5, 17]. While fully
automatic segmentation algorithms can be useful, interactive
segmentation is quite important for some applications. For
example, a user may need to segment a part of a 3D model
to be a query for a 3D model retrieval system. In such a case,the segmentation may be quite intentional and specific.
Automatic segmentation probably won’t be enough for suchan application.
This paper proposes an interactive, geometry aware
segmentation algorithm. A set of faces, or a region, on a 3D
mesh model a user considers to be similar are interactively
labeled by a set of brush strokes. (See Figure 1.) Then, theWhile there have been many proposal on meshsegmentation algorithms, majority of existing 3D meshsegmentation algorithms are of off-line, fully-automatic
kind [10, 8, 4, 15, 16, 3, 9, 12, 11]. Automatic segmentation
results, however, may not be what a user wants. While offline learning from a corpus, e.g., [8], helps to incorporate
human perception collectively, such an approach won’t beable to reflect one’s intention. Thus, depending on
application scenarios, interactive segmentation algorithms
such as [15, 5, 17] are quite important.
Figure 2 shows an example of supervised segmentationby using the proposed algorithm. The mesh is marked bythree colors that are the three labels of segmentation regions(a). The system segments the model by propagating thelabels (b). If initial segmentation is not what the user wants,
part of the segmentation may be undone (c) (gray part in theimage) to redo labeling for a new segmentation (d) with an
improved result. (Color to region correspondence changed
from labeling (a) to segmentation results (b~d) due to an
implementation error.)Interactive segmentation methods can be classified byvarious criteria [12]. One criterion is locality of thealgorithm. Local approaches, e.g., [11], are fast, but results

Ryutarou Ohbuchi, Keisuke Banba, Supervised, geometry-aware segmentation of 3D mesh models, accepted, ICME2012
Workshop on Human-Focused Communications in the 3D Continuum, July 9, 2012, Melbourne Australia(a) Labeling regions.

(b) Initial segmentation.

(c) Undoing a part.

(d) 2nd segmentation.

Figure 2. Interactive and iterative segmentation using the proposed
algorithm. Undesirable part may be undoed to be labeled and segmented
again. (Color to region mapping changed from initial labeling (a) tosegmentation results (b~d) due to an implementation error.)

may not be as good as desired, for they only consider local
properties. Local search in feature space often get stuck in
local minima/maxima.
Inspired by semi-supervised image labeling [13], wecast user-guided mesh segmentation as a semi-supervised
learning problem that propagates segmentation labels given
to a subset of faces to the unlabeled faces of a 3D model. As
with [13], our algorithm tries to overcome the issue of local
maxima/minima by employing a powerful diffusion-basedlearning algorithm called Manifold Ranking [18]. While ouralgorithm uses local geometrical feature, determination ofsegmentation regions takes both local and global changes of
the geometrical features into account.
III. ALGORITHM
Proposed algorithm aims at supervised, geometry-aware
segmentation of 3D mesh models. The algorithm assumes
singly-connected manifold mesh 3D model as its input. The
user labels a subset of the 3D mesh model marked by using
an interface similar to 3D painting system. The algorithm
first computes, for each face, n-dimensional local
geometrical feature. The algorithm then propagates labelsfrom marked faces to unlabeled faces over 2-manifold of the
3D mesh (actually, a dual graph of it) embedded in the ndimensional feature space.
The segmentation follows the steps below;

1. Extract local feature: Extract n-dimensional local
feature hi (i=1…m) at every face fi (i=1…m) of the 3Dmodel to be segmented.
2. Construct face connectivity graph G: Given a mesh of
the 3D model, construct a manifold mesh G representing
the connectivity of faces fi of the 3D mesh model. That is,
G is a dual graph of the vertex connectivity of theoriginal 3D mesh model. For example, a triangular face
fi of a 3D model has tree adjacent faces. Thus, fi is avertex having valence 3 in the mesh G. As mentioned
above, fi is associated with local geometrical feature hi.
3. Compute edge weights of G: For each edge connecting
fi and fj in graph G, compute edge weight Gi, j based on
the similarity of features hi and h j . In the label
propagation state (step 5) labels propagate smoothly andquickly through an edge having similar features, that is,a higher “diffusion coefficient”. Conversely, if featureson an edge is different, label propagation is impeded at
the edge.
4. Interactively seed regions: Label a subset of faces
interactively as a set of seed faces of a region having
label l, (l=1…c). There may be c distinct region labels.
Labeling is done by using an interface similar to a meshpainting tool.
5. Propagate labels: Propagates labels over the mesh G byusing Manifold Ranking algorithm by Zhou, et al. [18].
If there are c distinct labels, each vertex has c relevancevalues, each of which indicates likelihood of the vertex
having region label l.
6. Select a label for vertex: After label propagation step
terminates, each vertex i of the graph G is associatedwith c relevance values corresponding to c region labels.
The fi assumes a label having the highest relevancevalues among the c labels. Present the segmentation
result to user, for example, by visualizing labels by
distinct colors of the faces.
7. Repeat segmentation until satisfactory: If resulting
segmentation is not satisfactory, undo undesired part ofthe segmentation. Then go back to step 4. Repeat theinteractive segmentation process while necessary.
Following subsections explain details ofgeometrical feature and manifold ranking algorithm.

local

A. Per face Local Geometrical Feature
A feature vector hi of a face fi (i=1...m) is concatenation
hi=(xi, qi) of its 3D Euclidian coordinate xi and local
geometrical feature qi of barycenter of the face fi. The local
geometrical feature, which we call Local Statistical Featureor LSF, is a 4D joint histogram of angles and distance
between a pair of points [LSF_icme2012].
An LSF for a face is computed from a set of oriented
points within a sphere of influence of radius r at centered atbarycenter of the face (Figure 4). A surface based model isfirst sampled by thousands of oriented points placed at
quasi-random locations on the faces. Number of samples perRyutarou Ohbuchi, Keisuke Banba, Supervised, geometry-aware segmentation of 3D mesh models, accepted, ICME2012
Workshop on Human-Focused Communications in the 3D Continuum, July 9, 2012, Melbourne Australia
face is made so that area density is uniform over the 3Dmodel. Orientation of a point is the surface normal of theface on which the point is generated at. An LSF for a face iscomputed by using points within a sphere of radius r of thebarycenter of the face. Assume that the vertex of interest is
and its normal vector is . Assume also that point
lie inside a sphere of radius r of theand its normal vector
vertex . Usingand , a 4-tuple (, , , ) of distance and angle-related values , ,  is computed as follows(See Figure 5);that lie inside a sphere of influence. The SPRH, on the other
hand, computes a histogram among all the pair of points.
In the experiments below, we compute LSF with 2,048
oriented points per 3D model. We use joint histogram
having 3 bins for each value of the 4-tuple. Thus, if flattened
to a 1D vector, LSF qi is a 3
81 dimensional vector.
Combined with the coodinate of the face xi, the feature
vector per face hi has dimensionality n=84.

(1)

Label propagation over G is performed by using
Manifold Ranking (MR) algorithm by Zhou et al [18] [13]adapted to our purpose. MR is a learning algorithm thatlearns distribution of feature points in high-dimensionalfeature space by simulating diffusion of relevance value
from labeled, i.e., source vertices to unlabeled vertices over
G. In our segmentation algorithm, user-labeled faces in G
are the “sources” of label propagation. Assuming there are cdistinct region labels, c distinct diffusion processes would
take place over the mesh G. Each face is thus associatedwith c distinct relevance values, each corresponding tolikelihood of the face having one of c region labels.
Propagation of a label relevance value occurs from a face fi
to the another face fj across an edge shared by the faces on
the original 3D mesh model. On dual graph G of faceconnectivity, this corresponds to propagation of label
relevance value from a vertex fi to fj through an edge having
weight Gi, j .
Original MR [18] forms the mesh G based on similarityof vertices in an ambient high-dimensional feature space,e.g., by k-nearest neighbor connection. Each edge of G is
associated with an affinity value that controls diffusion of
relevance through the edge. The affinity is computed as theaffinity among a pair of features on the edge. In the
proposed segmentation algorithm, however, the connectivityof mesh G is given a priori as the connectivity of faces of agiven 3D mesh.
Let X   x1 ,..., xl , xl 1 ,..., xm  be the set of vertices in G.
Vertices from x1 to xl are (manually) labeled “source”
vertices, and each one has one of the (initial) segmentation
labels in L  1,..., c . The rest of the vertices in X are the(initially unlabeled) vertices to be labeled by MR.
T
Define F   F1 ,..., Fm  as an m  c matrix whose
element Fi is a ranking score computed for a label at vertex
T
i. Also define Y  Y1 ,..., Ym  as an m  c matrix whose
element is Yij  1 if label of the ith vertex yi  j  j  L  ,and Yij  0 otherwise.
On each edge of G, diffusion of relevance value
indicating the likelihood of a vertex assuming a label iscontrolled by the affinity Gij of the edge given as below;

  arctan  wn1 , un2  .
  v n 2(2)

  u   p 2  p1 p 2  p1  .

(3)  p2  p1 .

(4)
where u  n1 , v   p 2  p1   u  p 2  p1   u , w  u  v.
The radius r of the sphere of influence defines locality of
LSF; the smaller the radius, the local the feature is. If there
are p oriented points within the sphere, (p-1) tuples arecomputed, and (p-1) each of values  , , and arecollected into a 4-dimensional joint histogram to become an
LSF feature.
3D modelconsisting ofm faces.

r

LSF computed atbarycenter of each
face
p1

n1

Compute m LSF
histograms
Figure 4. A local geometrical feature LSF is computed for each face frompoints in a sphere of radius r about the face.
v



n2

w

v
p2


w

u



u= n1
p1



Figure 5. An LSF is a histogram of 4-tuples () from a pairs oforiented points within radius of influence r.

LSF feature is similar to Surflet Pair Relation
Histograms (SPRH) [14] by Wahl, et al., but there are twodifferences. First, original SPRH is global, while LSF is
local. Second, given a set of p oriented points, LSF
computes a histogram by accumulating (p-1) tuples between
a point of interest and points other than the point of interest

B. Label Propagation





2
2
if i  j exp  d (hi , h j ) 2Gij  0otherwise

(5)

Ryutarou Ohbuchi, Keisuke Banba, Supervised, geometry-aware segmentation of 3D mesh models, accepted, ICME2012
Workshop on Human-Focused Communications in the 3D Continuum, July 9, 2012, Melbourne Australia
The matrix G is positive symmetric. Note that G ii  0since there is no edge connecting a point with itself. As the
distance d (hi , h j ) , we chose, based on our preliminary
experiments, L0.5-norm [1] below;nd 0.5 (h i , h j ) 

h ia  h ja

0.5

a 1



1 0.5

(6)

From G, a normalized graph Laplacian S is formed;


1



1

(7)
where D is a diagonal matrix in which Dij equals to the sumof the i-th row of G , that is, Dij   j Gij . The ranking
Tvector F   f1,, f n  that indicates the likelihood ofvertices having a label can then be estimated by iterating thefollowing equation until convergence;F(t 1)   SF(t )  (I  a)Y(8)
The parameter   0 affects convergence of theiteration above. Let F* be the limit of the above iteration.
This iteration has a closed form solution as follows;S  D 2 (D  G )D 2

patch is a component-by-component average of the 4D joint
histogram of the faces contained in the patch.
After forming patches, 3D model consisting of Np
patches is segmented in a manner identical to the processdescribed in Section 3.2. Since N p  N f , the cost of,
manifold ranking in the label-segment interaction
loop ismuch smaller, and the iteration could maintain an interactive
response time, e.g., less than a few seconds.
Figure 6 illustrates the hierarchical method. A complex
mesh (a) is pre-partitioned into “macro faces” (b by using
randomly-placed seed points (red marks). Actually, macro
faces in (c) is too large to obtain a reasonable segmentation;
macro faces of the size shown in (c) is more reasonable.
The hierarchical segmentation by using partitioning andmacro faces do degrade segmentation quality. For example,
boundaries of segmentation results are more uneven, for theunit of segmentation is now macro faces.

1F*  1    I  S  Y(9)Given the converged solution F*, a vertex i would havethe label yi which has the highest relevance rank value.
(10)yi  arg max Fijj cComputational cost is an issue to be considered if MR
is to be used in an interactive segmentation loop. The cost of
MR is dominated by the cost of meshing and computing F* .
The cost increases with the size of matrix S, that is, thenumber of faces m. For example, the matrix S takes O(m2)(if it is dense) to store and roughly O(m3) to perform label
propagation via matrix inversion.
C. Patch-Based Segmentation
Due to the high computational cost of manifold ranking,the proposed segmentation algorithm can’t handle a largemesh, e.g., a mesh having 10k or more faces, interactively.
To alleviate this issue, we extend the segmentation
algorithm described above so that the segmentation ishierarchical. In the following, this extension is called“hierarchical method”, and the non-hierarchical version
described in Section 3.2 is called “direct method”.
Basic idea is to run the label propagation algorithm
described in Section 2.2 on a mesh that consists of patches
whose number is much smaller than the original number offaces Nf. Each patch is a set of faces that is topologicallyadjacent and have similar features.
Patches are formed in a pre-processing step. An input(complex) mesh having Nf faces is partition into a set of Np
patches, in which N p  N f . To partition the original mesh
into set of patches, we employ, again, MR based label
propagation guided by feature. Instead of manual labels,
however, labels are propagated from randomly placed seedpoints on the (original) mesh surface. Feature of a created

(a)

(b)

(c)

Figure 6. To reduce computational cost in the interactive label-segmentloop, segmentation may be performed on a 3D model consisting of “macro
faces”.

IV. EXPERIMENTS AND RESULTS
We performed experiments to evaluate the proposed method
in terms of segmentation quality and computational
efficiency.
A. Quality of Direct Method
Quantitative evaluation of segmentation quality for 3D
mesh segmentation algorithms is by itself is a research topic[4][6]. Furthermore, to author’s knowledge, previously
published quantitative evaluation methods are for fully
automatic segmentation methods. We thus settled for a lessthan ideal option of comparing our interactive segmentation
method with automatic segmentation methods by using theevaluation procedure and benchmark database by Chen et al[6]. Chen’s benchmark contains 380 models, each of whichis associated with multiple human-segmented ground truths.
Chen et al [6] also defined multiple indices of segmentation
quality.
We asked 8 volunteers to interactively segment the 95
models by proposed method with a simple graphical userinterface. Given a model to be segmented, each volunteer
selected one of ground truth segmentations, and tried toRyutarou Ohbuchi, Keisuke Banba, Supervised, geometry-aware segmentation of 3D mesh models, accepted, ICME2012
Workshop on Human-Focused Communications in the 3D Continuum, July 9, 2012, Melbourne Australia
copy the segmentation using our system. Interactive labelsegmentation loop might have iterated a few times before
the user is satisfied. Following the protocol of thebenchmark, quality of the segmentation is quantified byusing the “cut discrepancy” measure proposed by Chen, et al[6].
Figure 7 shows the evaluation result for the direct
method. Compared to seven methods, all of which are fully
automatic, the proposed method has significantly lower cut
discrepancy score. Supervised results using out methodshould be better than the results of fully automatic
segmentations. Our proposed method at least achieved thisas it produced the lowest cut discrepancy value.
Proposed

0.068

RandCuts

0.162

ShapeDiam

0.184

NormCuts

0.178

CoreExtra

0.254

RandWalks

0.309

FitPrim

0.237

KMeans

0.305
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Cut Discrepancy

Figure 8 shows several 3D models, their ground truth
segmentations, and the segmentation results based on the
ground truth segmentation.
B. Quality and Efficiency of Hierarchical Method
We evaluated the hierarchical segmentation method
described in Section 6.2 for its quality (qualitatively) andcomputational efficiency. The evaluation used the same 95
models as in the previous section. We first partitioned themodel into parts having Nm=3,000 patches prior to thesegmentation. A user then segmented the model so the result
resembles one of the ground truths the user selected.
Examples of segmentation results and their execution
timings using the hierarchical method are shown in Figure 9,
and in Table 1. The “Chair” model has 31,456 faces, whilethe “Armadillo” model has 50,542 faces. In Table 1, for the
direct method, “pre-processing” consists mostly of feature
extraction. For hierarchical method, “pre-processing” also
include patch formation. For both methods, the “propagate
labels” step consisting mostly of manifold ranking is done in
the interactive label-segment loop.
As the examples show, segmentations using thehierarchical method have lower quality at jagged thesegmentation boundary. However, we observe that thequality of boundary is reasonable considering the model
consists of only 3,000 patches. Reasonable quality ofsegmentation boundary is achieved since quality of patch
Macro faces

Figure 7. Comparison of segmentation quality with various automatic
segmentation methods by using cut discrepancy.
Ground truth

ResultResultChair

ModelsGround truth

Bear
Almadillo

Airplane

Figure 9. Examples of hierarchical segmentation. The chair modelhas 31,456 faces, while the armadillo model has 50,542 faces.
Number of patches Np=3,000.

Head

Table 1. Execution time for patch-based hierarchical segmentation
method for two example models.
DirectModel

Figure 8. Three models (left), their ground truth (middle), and segmentation
results obtained after less than a few iterations (right).

Pre-process

Hierarchical

Propagatelabel

Pre-process

Propagate
label

Chair

183.3s39.2 s221.0 s1.5 s

Almadillo

384.3s51.9 s464.8 s1.3 s

Ryutarou Ohbuchi, Keisuke Banba, Supervised, geometry-aware segmentation of 3D mesh models, accepted, ICME2012
Workshop on Human-Focused Communications in the 3D Continuum, July 9, 2012, Melbourne Australia
boundaries, which are created the same MR-based label
propagation, were good to start with
As for computational cost, time spent on label
propagation during an interactive segmentation loop has
been reduced from 40~50s down to just over a second. The
hierarchical segmentation, on the other hand, spends moretime in the pre-processing stage forming patches.
V. DISCUSSION
We have proposed an interactive, supervised, geometryaware segmentation algorithm for 3D (manifold) meshmodels. A user labels a small subset of faces of eachsegmentation region by using an interface similar to 3Dmodel painting software. Then, the labels are propagated to
non-labeled faces over the manifold of the mesh guided by
local geometrical features computed for each face. The
propagation is performed by using Manifold Ranking
algorithm by Zhou et al. [18]. The proposed method does
not scale to a complex 3D model, as the label propagation
during interactive loop is computationally costly. To reducethe computational cost, we proposed a hierarchicalsegmentation algorithm.
We evaluated the segmentation quality andcomputational cost by using a subset of mesh segmentation
benchmark [6]. Our algorithm has produced better
segmentation quality than all the fully automatic algorithms.
However, further study is needed for a better benchmark forinteractive mesh segmentation algorithms.
In the future, improvement in patch-based hierarchicalsegmentation algorithm is necessary so that thesegmentation quality would be closer to non-hierarchicalones. An approach similar to the one in Katz et al [9] may
be used to obtain smoother boundaries. We are also
considering methods to accelerate manifold ranking itself.
ACKNOWLEDGEMENTS
This research has been funded in part by the Ministry of
Education, Culture, Sports, Sciences, and Technology ofJapan (No. 18300068).
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