3D-NCuts: Adapting Normalized Cuts to 3D Triangulated Surface

Segmentation

Zahra Toony

, Denis Laurendeau

, Philippe Gigu`ere

and Christian Gagn´e

Computer Vision and System Laboratory, Department of Electrical and Computer Engineering, Universit´e Laval,

Qu´ebec, QC, Canada

Department of Computer Science and Software Engineering, Universit´e Laval, Qu´ebec, QC, Canada

Keywords:

3D Segmentation, Spectral 3D Segmentation, Primitive Extraction from Mesh Models, Clustering.

Abstract:

Being able to automatically segment 3D models into meaningful parts is an important goal in 3D shape pro-

cessing. In this paper, we are proposing a fast and easy-to-implement 3D segmentation approach, which is

based on spectral clustering. For this purpose, we deﬁne an improved formulation of the similarity matrix

which allows our algorithm to segment both free-form and CAD (Computer Aided Design) 3D models. In 3D

space, different shapes, such as planes and cylinders, have different surface normal distributions. We deﬁned

the similarity of vertices based on their normals which can segment a 3D model into its geometric features.

Results show the effectiveness and robustness of our method in segmenting a wide range of 3D models. Even

in the case of complex models, our method results in meaningful segmentations. We tested our segmentation

approach on real data segmentation, in the presence of noise and also in comparison with other methods which

provided good results in all cases.

1 INTRODUCTION

In the early 1980s, availability of precise 2D and 3D

cameras has opened a new ﬁeld of computer vision,

3D image acquisition and processing (Blais, 2004).

Nowadays, this ﬁeld has many different applications

such as assembly, inspection, Computer-aided Design

(CAD), reverse engineering, mechanical engineering,

medicine, and entertainment.

Standard 2D cameras produce 2D images from the

surface of objects, in either black-and-white or color,

but 3D cameras provide information on the geome-

try of the object surface. Today, some 3D cameras

can acquire both the appearance and geometry of ob-

jects concurrently. This results in many 2D process-

ing approaches applied on 3D images: 3D mesh com-

pression (Yu et al., 2012; Jiang et al., 2012; Carnero

et al., 2012), 3D skeleton extraction (Sam et al., 2012;

Zhang et al., 2012; Benhabiles et al., 2012), 3D object

recognition (Karim Baareh et al., 2012; Huber et al.,

2004; Murase and Nayar, 1995; Selinger and Nelson,

1999), and 3D segmentation (Xiao et al., 2011; Liu

and Zhang, 2004; Lavou´e et al., 2012) are instances

of such extensions of 2D approaches to 3D.

One of the challenges in shape processing and un-

derstanding is to segment a 3D object into meaningful

parts (Ning et al., 2010). This implies segmenting a

3D shape, not based on distances, but rather based on

its properties to obtain a result close to a manual seg-

mentation. Many applications, such as object recog-

nition, modelling, or compression depend on mean-

ingful segmentation (Kalogerakis et al., 2010). In

some other applications, segmentation is used to la-

bel parts of the shapes; many of these approaches use

manually segmented shapes. For example, in a human

body one could distinguish “arm”, “leg”, and “head”

from each other (Kalogerakis et al., 2010). Ground-

truth databases for segmentation of 3D-meshes were

created recently, (Benhabiles et al., 2009; Chen et al.,

2009). They offer the opportunity of analyzing and

learning mesh segmentation quantitatively (Benhab-

iles et al., 2011).

Based on what is presented in (Shamir, 2008), al-

gorithms in 3D mesh segmentation can be catego-

rized into ﬁve groups while the paper by (Agathos

et al., 2007) presented a more detailed classi-

ﬁcation of 3D mesh segmentation methods into

twelve groups: “Region growing”, “Watershed-

based”, “Reeb graphs”, “Model-based”, “Skeleton-

based”, “Clustering”, “Spectral analysis”, “Explicit

Boundary Extraction”, “Critical points-based”, “Mul-

tiscale Shape Descriptors”, “Markov Random Fields”

144

Toony Z., Laurendeau D., Giguère P. and Gagné C..

3D-NCuts: Adapting Normalized Cuts to 3D Triangulated Surface Segmentation.

DOI: 10.5220/0004692101440152

In Proceedings of the 9th International Conference on Computer Graphics Theory and Applications (GRAPP-2014), pages 144-152

ISBN: 978-989-758-002-4

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

and “Direct segmentation”. To ﬁnd more details on

methods in each category refer to (Shamir, 2008;

Agathos et al., 2007). Each of these groups can per-

form the segmentation in two main ways: surface-

based and part-based. A surface-based 3D mesh seg-

mentation method attempts to segment a model into

different regions under some criteria that can be esti-

mated by simple shapes such as planes or cylinders.

On the other hand, a part-based approach attempts to

segment it into meaningful volumetric parts (Agathos

et al., 2007).

In this work, we study the segmentation of a 3D

model with a spectral analysis approach in a surface-

based paradigm. Spectral analysis methods are based

on a similarity matrix. Our focus here is to deﬁne a

similarity matrix that avoids thresholding and works

without using training data. We introduce two param-

eters in order to improve the quality of segmentation

results. It can discriminate different parts with differ-

ent geometries very well, thus resulting in meaningful

mesh segmentation.

The rest of the paper is organized as follows. The

related works are represented in Section 2. In Sec-

tion3, we explain our problem and present the details

of our novel 3D Normalized Cuts approach. Section 4

shows the efﬁciency of our approach and gives the ex-

perimental results obtained in different situations and

in comparison with other methods. Finally, we con-

clude the paper in Section 5.

2 RELATED WORK

Conducting a meaningfulsegmentation of a 3D model

is generally application dependent. In our case, we

call a segmentation meaningful when it can separate

different geometries from each other as a human does.

Among different types of mesh segmentation meth-

ods, spectral analysis has attracted much attention.

The theory of spectral graphs was introduced

by (Chung, 1997) and was then applied on image seg-

mentation by (Shi and Malik, 2000) as Normalized

Cuts (NCuts). 3D spectral analysis approaches ﬁrst

deﬁne a similarity matrix on the data, based on prop-

erties of mesh connectivity, geometry, etc. Then, a

Laplacian matrix is computed based on the deﬁned

similarity and its eigenvectors are determined in or-

der to segment the 3D model. Since the Laplacian

matrix and eigenvectors are found based on the sim-

ilarity matrix, the most important step of the spectral

analysis methods is how the similarity is deﬁned such

that good segmentations are then obtained.

In 2004, Liu and Zhang (Liu and Zhang, 2004)

applied spectral analysis on 3D models. They de-

ﬁned the similarity matrix as the likelihood that faces

belong to the same segment. In order to prevent

merging faces with different concavities, they used

the distance matrix presented in (Shlafman et al.,

2002; Katz and Tal, 2003). They constructed the

normalized similarity matrix on the faces and then

identiﬁed the k eigenvectors with the largest asso-

ciated eigenvalues for further processing. Finally,

they applied k-means on the processed eigenvectors.

They segmented a model by emphasizing concavities,

which sometimes resulted in segmenting unmean-

ingful concavities (Agathos et al., 2007). Another

method (Zhang and Liu, 2005) used the same similar-

ity matrix with recursive spectral cut and Nystr¨om ap-

proximation (Fowlkes et al., 2004), which gave them

the opportunity of deﬁning a partial similarity ma-

trix. The extracted eigenvectors are then used in a

line search algorithm to ﬁnd the most prominent cut

in order to segment the 3D model. In (Liu and Zhang,

2007), a spectral analysis algorithm based on spectral

embedding and 2D contour analysis was proposed.

The 3D model was projected into the 2D domain in

order to apply contour analysis. The algorithm re-

quired that some parameters be selected, and tended

to fail in the presence of noise.

Some other types of segmentation algorithms

are based on region growth. In these methods, a

seed point is ﬁrst selected and then grown, in or-

der to ﬁnd ﬁnal regions. This often leads to over-

segmentation (Zhang et al., 2002). Watershed-based

approaches are also subject to over-segmentation of

the 3D models (Agathos et al., 2007).

Some other methods, such as Randomized

Cuts (Golovinskiy and Funkhouser, 2008), take the

advantages of using different segmentation meth-

ods to gain both random and non-random features.

In order to segment a 3D model, Golovinskiy and

Funkhouser (Golovinskiy and Funkhouser, 2008)

generate a pool of segmented parts using different ap-

proaches such as k-means (Shlafman et al., 2002),

hierarchical Normalized Cuts and Min Cuts. After-

wards, based on the reputation of each edge on a seg-

mentation boundary, they deﬁne a “partition function”

on edges. The “consistent cuts” are then found and the

model is segmented through these cuts.

For the k-means approach, they set the number

of clusters and repeat the k-means algorithm several

times in order to generate different segmentation re-

sults. In the hierarchical Normalized Cuts method,

they start with every face in a separate segment and

then try to merge them based on an area-normalized

cut cost function until the desired number of clusters

is obtained. The cost function is the sum of the cut

costs of segments divided by their areas. They are

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145

using this cost function, called modiﬁed version of

Normalized Cuts, instead of using only the cost of all

edges in a segment which is utilized as Normalized

Cuts in (Shi and Malik, 2000). This algorithm is re-

peated to obtain different results. Finally, in the Min

Cuts approach a set of k seeds is selected and a weight

value is deﬁned for each edge in order to maintain the

cuts at a large distance from the seeds. This method is

also repeated to provide more different segmentation

results.

In (Lai et al., 2008) a random walk approach is

utilized in order to achieve a mesh segmentation. The

authors compute the dual graph of the mesh and se-

lect the seeds automatically; the numbers of seeds are

greater than the desired number of clusters. They then

try to associate each face with a seed with the high-

est probability which results in an over segmented

mesh. Hence, a hierarchical method is then applied

to merge the similar segments based on the length of

common boundaries and overall perimeter of adjacent

segments (Chen et al., 2009).

The hierarchical ﬁtting primitive approach that is

proposed in (Attene et al., 2006) considers each face

as a segment and starts to merge them until the de-

sired number of clusters is obtained. At each itera-

tion, a primitive like plane, sphere and etc. is ﬁtted on

each pair of faces and then the best ﬁtted pair will be

merged as a segment (Chen et al., 2009).

Another approach which is called Shape Diameter

Function (SDF) (Shapira et al., 2008) deﬁnes a scalar

function for every face of the mesh and produces a 1-

D histogram. Then, a Gaussian Mixure Model is ap-

plied on the histogram to ﬁt k Gaussian functions on

each face and determine the probability of assigning

faces to each SDF cluster. Finally an alpha-expansion

graph cut method is used to minimize a deﬁned en-

ergy function in order to ﬁnd the appropriate segmen-

tation. This method determines the number of clusters

automatically (Chen et al., 2009).

Recently, a mesh segmentation method for CAD

models was presented in (Xiao et al., 2011). At ﬁrst,

the CAD model is clustered into sparse and dense

triangle regions. Using the Gauss map of triangular

faces and Hough transformation, sparse triangle re-

gions are separated into planar regions, cylindrical re-

gions and conical regions. Dense triangle regions are

also segmented by a mean shift operation which is

applied on the mean curvature ﬁeld of mesh surfaces.

One of the problems of this method is that it deals

poorly with planar regions composed of non-uniform

parts combining sparse and dense triangulations. This

method will split the plane at ﬁrst, without consider-

ing the geometry of the surface before splitting.

There are also some 3D segmentation methods

based on learning algorithms (Kalogerakis et al.,

2010; Benhabiles et al., 2011), which require a

database of labeled meshes, a constraint that cannot

be satisﬁed in many cases. Learning-based meth-

ods might achieve better segmentation results than

other methods, however our approach does not use

any training step or training data but rather consid-

ers the differences between geometries. As we aim at

applying Normalized Cuts (Shi and Malik, 2000) in

3D space, the next sections explain the details of this

method in 2D space.

2.1 2D-NCuts

Normalized Cuts (NCuts) (Shi and Malik, 2000) as-

sumes that a 2D model is described as a graph G =

(V,E), where V is the set of vertices in our model and

E is the set of edges (which transposes in 3D as a set

of faces). The cut, or degree of dissimilarity between

two segments, A and B, can be found by deﬁning

cut(A,B) =

∑

u∈A,v∈B

w(u,v), (1)

where, w(u,v) is the similarity between vertex u and v.

Normalized Cuts aims at minimizing the Ncut value

Ncut(A,B) =

cut(A,B)

assoc(A,V)

cut(A,B)

assoc(B,V)

, (2)

where, assoc(A,V) =

∑

u∈A,t∈V

w(u,t) is the total simi-

larity from all vertices in A to all vertices in the graph.

This minimization is achieved by solving the follow-

ing generalized eigenvalue system:

(D− W)y = λDy, (3)

where, W is the matrix of similarity between each pair

of vertices and D is a diagonal matrix with:

d(i,i) =

∑

w(i, j). (4)

After solving Equation (3), the eigenvector related

to the second smallest eigenvalue of the system is se-

lected. It is claimed that this eigenvector keeps two

vertices i and j in a segment when they have a large

similarity value (large w(i,j)). The problem then con-

sists in determining the splitting point of the eigen-

vector in such a way that the cut value between the

two split parts is minimized. Shi and Malik presented

an optimized loop for achieving this. Another way of

splitting the eigenvector into different parts is apply-

ing a clustering method, such as k-means, to segment

the vector into different regions.

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3 PROPOSED IMPROVEMENTS

TO NCUTS

Our goal is to separate a 3D model into meaningful

parts. We desire to detect different parts based on dif-

ferent geometric primitives such as planes, cylinders,

cubes, and rods. So, the differences between these ge-

ometric primitives must be described through a simi-

larity matrix. In 3D space, it is clear that all vertices

on a plane have normals in the same direction; all ver-

tices on a cylinder have normals rotating uniformly on

a circle, as can be seen in Figure 1b.

(a) (b)

Figure 1: (a) 3D model, (b) normals to the surface of 3D

model.

Therefore we argue that deﬁning the similarity

based on normals is a good strategy to discriminate

and to separate 3D geometric primitives. We thus de-

ﬁne the similarity matrix as a relation between normal

vectors that can reveal sharp edges and boundaries

(see Figure 2). The similarity between two vertices

is then deﬁned as:

w(i, j) = cos ≺ (~n

, ~n

), (5)

where, ~n

and ~n

are normal vectors at vertices i and

j, respectively. As shown in Figure2, we observe

large values of w(i,j) where surface normals are simi-

lar (red), but really small values at the boundaries be-

tween primitives(green) which can help us to separate

different primitives. With this deﬁnition of similar-

ity matrix, we then solve an eigenvalue system in or-

der to ﬁnd the segmentation results. This contrast the

approach proposed in (Golovinskiy and Funkhouser,

2008), where they used the modiﬁed Normalized Cuts

idea to ﬁnd the cost function of a hierarchical clus-

tering approach in order to merge faces. Here, we

produce the segmentation on vertices which, with an

accurate deﬁnition of normals, can else be applicable

to point clouds.

The next sections present the details of our ap-

proach for 3D models.

Algorithm 1: Proposed 3D-NCuts.

Input: similarity matrix W, diagonal matrix D and

number of clusters k.

1. Compute the similarity matrix W,

w(i, j) =



cos ≺ (~n

, ~n

) if i and j are connected by an edge

0 otherwise

2. Evaluate the diagonal matrix D, d(i, i) =

∑

w(i, j).

3. Solve the following generalized eigensystem,

(D− W)y = λDy.

4. Suppose V

eig

as a matrix of the k eigenvectors

associated with the k smallest eigenvalues.

5. Apply a clustering method, such as k-means, on V

eig

achieve k segments.

Output: Clusters C

,...,C

3.1 Our approach: 3D-NCuts

As mentioned in Section 2.1 the Normalized Cuts al-

gorithm begins by deﬁning a similarity matrix on 3D

data. Here we applied this method on 3D shapes and

brought some modiﬁcations to the basic algorithm as

presented in Algorithm 1.

Figure 3b presents the result of applying 3D-

NCuts on a 3D model in order to segment the model

into four clusters. It is observed that the result is

not meaningful, especially on the handle of the mug;

to circumvent this problem, we apply an exponential

function on the weights of similarity to make small

weight values even smaller and large values larger.

This modiﬁcation separates regions very well but, in

order to assign a priority to the neighbours in com-

parison with the points that are far from each other, a

small value η is added to the weights of neighbouring

vertices. This parameter (η) forces the algorithm to

always consider neighboring surfaces to be candidate

for merger. Therefore the similarity matrix is rede-

ﬁned as

w(i, j) =



α cos≺(~n

,~n

)

+ η if i and j are connected by an edge

0 otherwise

(6)

Figure 2: A color-coded representation of the similarity ma-

trix values for two 3D models, where, blue indicates the

lowest value and red indicates the largest value.

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147

As represented in Figure 3c, the modiﬁed sim-

ilarity matrix renders the segmentation meaningful.

When a human wishes to segment this 3D model into

four segments it separates it into the handle, the bot-

tom, the inner part of the mug and the outer part of the

mug. This is precisely what the 3D-NCuts algorithm

does to achieve the mug model. Henceforth, Equation

6 is used in the ﬁrst step of Algorithm 1 and results in

a better segmentation for all other 3D models.

(a)

(b)

(c)

Figure 3: (a) 3D model, (b) result of 3D-NCuts segmenta-

tion with k = 4 and using the similarity matrix presented in

step 1 of Algorithm 1. (c) result of 3D-NCuts segmentation

with the modiﬁed weight Equation 6, with k = 4, α = 15,

η = 0.5 (after replacing the similarity equation in step 1 of

Algorithm 1 by Equation 6).

The experimental results of 3D segmentation are

presented in the next section, which shows the good

performance of our approach.

4 EXPERIMENTAL RESULTS

We tested the performance of our approach on various

3D shapes, for different conditions and in compari-

son with other methods. We evaluated this method

on objects we scanned with a handheld scanner (i.e.,

Creaform Go!SCAN handheld 3D scanner) as well as

on 3D models with artiﬁcial noise.

The free-form 3D models that are used in the

experiments and the results of segmentation by the

approaches presented in Figure 5 are taken from

the Princeton Shape Benchmark for 3D Segmenta-

tion (Chen et al., 2009). The 3D CAD models

are taken from the database of Purdue University

(ESB) (Jayanti et al., 2006).

The values of similarities between vertices (Equa-

tion 5) are between −1 and 1. For parameter α in

αw(i, j)

, choosing a too small value does not create

a large difference between weights on the bound-

aries and the ones on the surface. Choosing a large

value also weakens some useful weights near the high

weights. Since we desire to have a large difference

between the weights on the surface and those on the

boundaries, in all experiments, α is set to an inter-

mediary value of α = 15. The value of η speciﬁes

the impact of neighbour vertices which is set to 0.5 to

keep both dependency and independency of a vertex

to the neighbouring vertices.

(a)

(b)

(c)

3D Model

3D-NCuts

(Xiao et al.,

2011)’s method

k = 5

k = 3

Figure 4: The result of our 3D segmentation approach in

comparison with the method proposed by (Xiao et al., 2011)

on real scanned data. 3D models are presented in column

(a); column (b) and (c) show the result of segmentation by

3D-NCuts and the method presented in (Xiao et al., 2011),

respectively.

Figure 5 shows the segmentation results using our

method in comparison with some recent approaches.

In this ﬁgure, we presented the segmentation re-

sults for four different 3D models. In each row,

column (a) shows the result of our 3D-NCuts seg-

mentation, column (b) presents manual segmentation

which is the ground truth of presented 3D models.

Columns (c), (d), (e), (f), and (g) represent the seg-

mentation results using Randomized Cuts (Golovin-

skiy and Funkhouser, 2008), Normalized Cuts pre-

sented in (Golovinskiy and Funkhouser, 2008), Ran-

dom walk (Lai et al., 2008), ﬁtting primitive (At-

tene et al., 2006) and shape diameter (Shapira et al.,

2008) approaches, respectively. The results show that

in comparison to other methods, our segmentation

approach is able to provide a 3D segmented model

which more clearly resembles the ground truth.

Figure6 shows some results of our method for

CAD models in comparison with a recently pre-

sented CAD model segmentation method (Xiao et al.,

2011). This approach segments the model automati-

cally without considering the number of clusters. The

results of the method proposed by (Xiao et al., 2011)

in Figure 6, are obtained from their prepared GUI.

Since their method is based on Hough transform and

Gauss sphere mapping to detect planes and cylinders,

it can detect these primitives very well but it some-

times fails to associate the parts together (such as the

red line, in Figure6, in the last column of the ﬁrst row,

on cylinder at the bottom part of the object which is

detected as the top cylinder).

As shown in Figure 6, our method can segment

a 3D CAD model very well. The segmentation re-

sults are presented with different number of clusters

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(b) (c) (d)

k = 4 k = 4 k = 4

(a) (e) (f) (g)

k = 4 k = 4 k = 4 k = 8

(b) (c) (d)

k = 8 k = 8 k = 8

(a) (e) (f) (g)

k = 8 k = 8 k = 8 k = 9

(b) (c) (d)

k = 8 k = 8 k = 8

(a) (e) (f) (g)

k = 8 k = 8 k = 8 k = 7

(b) (c) (d)

k = 6 k = 6 k = 6

(a) (e) (f) (g)

k = 6 k = 6 k = 6 k = 16

Figure 5: Comparison to previous segmentation methods. The methods that are compared (a) Our method, (b) Manual

segmentation, (c) Randomized Cuts (Golovinskiy and Funkhouser, 2008), (d) Normalized Cuts presented in (Golovinskiy and

Funkhouser, 2008), (e) Random Walk (Lai et al., 2008), (f) Fittingprimitives (Attene et al., 2006), (g) Shape Diameter (Shapira

et al., 2008). The last method, Shape Diameter (Shapira et al., 2008), is an automatic algorithm which determines the numbers

of clusters.

to show the effectiveness of our method in detecting

more geometric primitives. In the ﬁrst row, where the

object was segmented into three clusters, our method

separated the cylinder from the cube-like shape: the

planes of the cube which are located in the same di-

rection as those of the cylinder have been included

with the cylinder, separating them from the remain-

der of the cube-like shape. In the case of k = 8, it

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149

(a) (b) (c)

3D Model

3D-NCuts (Xiao et al., 2011)’s method

k = 3 k = 8

Figure 6: The result of our 3D segmentation method in comparison with the method proposed by (Xiao et al., 2011). Column

(a) shows the 3D models, column (b) and (c) present the result of segmentation by 3D-NCuts and the method proposed

by (Xiao et al., 2011), respectively. In the ﬁrst row, column (b) shows results for different numbers of clusters.

(a) (b) (c)

3D Noisy/Noiseless Model

Normals of the model 3D-NCuts

k = 8

k = 4

Figure 7: The result of our 3D segmentation approaches in the presence of noise. Column (a) shows the 3D noisy or noiseless

models. The normals of models presented in column (a) are shown in column (b). Column (c) is representing the results of

segmentation by 3D-NCuts for both noisy and noiseless models.

also detects the cylinder, planes and four sides of the

cube-like shape. In the second row, the results of

segmenting the “fandisk” model into three and eight

clusters, show that our method achieves better results

than the method presented in (Xiao et al., 2011). With

an increasing number of clusters our approach detects

more different geometric primitives as well.

In Figure4, we present the segmentationresults for

real 3D data. We scanned two real objects in our lab

to evaluate the efﬁciency of our method on real data.

The models were scanned with an accuracy of 2 mm

with the Go!SCAN scanner. As observed, our method

segments the real CAD models much better than the

approach presented in (Xiao et al., 2011) for CAD

models. This illustrates that the method presented

in (Xiao et al., 2011) is not robust in the presence of

noise, even at low levels.

To further demonstrate the robustness of our algo-

rithm in the presence of noise, we added some Gaus-

sian noise to both free-form and CAD mesh models

and applied our approach to segment these shapes.

These models, with this small amount of vertices, are

sensitive even in the presence of small noise levels.

Figure 7 shows the segmentation results in the case

of noiseless and noisy models. Column (c) of Figure

7 shows the segmentation results obtained using 3D-

NCuts. Considering noisy and pure models, the re-

sults of segmenting a 3D model are almost the same

with or without noise. For the mug model shown in

second column, the 3D-NCuts algorithm segments the

noisy mug into virtually the same four clusters as is

the case for the noiseless mug. These results indicate

that our approach is relatively stable, even in the pres-

ence of noise.

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k = 8 k = 4 k = 3 k = 6 k = 8

k = 15 k = 6 k = 26 k = 10 k = 15

Figure 8: More results on our 3D segmentation approach.

To further illustrate the effectiveness of our ap-

proach more results of segmentation are presented

in Figure 8. As can be observed, our methods are

applicable for both 3D free-form and CAD models.

For example, for the horse model with 10 clusters,

3D-NCuts segments the shape into four different seg-

ments of legs, two segments of ears and one segment

for each of the following: tail, eyes, face and body.

Different meaningful geometric features of the other

models in this ﬁgure are obtained as well.

5 CONCLUSIONS

In this paper, we presented a spectral-based 3D seg-

mentation method. This method, 3D-NCuts, identi-

ﬁes the Laplacian matrix of the 3D data and attempts

to solve the segmentation problem based on the idea

of 2D Normalized Cuts. The most important part of

this method is the deﬁnition of the similarity matrix.

Here, we deﬁned this matrix based on normals of ver-

tices and applied an exponential function with consid-

ering some constants to discriminate different geome-

tries from each other in an object. The results show

the efﬁciency of this approach for both free-form and

CAD 3D models in comparison with other methods.

The results are very similar to manual segmentation

(the ground truth) which reveals our approach is able

to produce meaningful segmentation. From the re-

sults, our method works better or at least similarly to

other presented methods. We also tested our approach

on real scanned data with good results. Even in the

presence of signiﬁcant noise, our approach was able

to produce signiﬁcant segmentation.

ACKNOWLEDGEMENTS

This work was supported by the NSERC-Creaform

Industrial Research Chair on 3D Sensing. Z. Toony

was supported by a FRQ-NT post-graduate scholar-

ship. The authors thank Hongwei Lin for providing

us with the GUI of their method (Xiao et al., 2011).

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