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New Graph Distance based on Stable Marriage

formulation for Deformable 3D Objects Recognition

Kamel Madi

Umanis, Research & Innovation, Levallois-Perret, 92300, France

kmadi@umanis.com

Eric Paquet

National Research Council Canada, Ottawa, Canada

Eric.Paquet@nrc-cnrc.gc.ca

Hamamache Kheddouci

Universit´

e de Lyon, CNRS, LIRIS, UMR5205, France

hamamache.kheddouci@univ-lyon1.fr

Abstract—We propose a novel fast graph matching approach

based on a new formulation of the stable marriage problem, to

measure the distance between graphs. The proposed approach is

optimal in terms of execution time, i.e. quadratic time complexity

O(n2). Our technique is based on the decomposition of graphs

into a set of substructures which are subsequently matched with

the stable marriage algorithm. In this paper, we address the

problem of comparing deformable 3D objects represented by

graphs, we use a triangle-stars decomposition for triangular

tessellations (graphs of 3D shapes). The proposed approach is

based on computing an approximation of Graph Edit Distance

which is fault-tolerant to noise and distortion which makes our

method especially relevant for deformable 3D shapes comparison.

We analyze and determine its time complexity. The proposed

method is evaluated against benchmark databases under dif-

ferent evaluation criteria. Our experimental results consistently

demonstrate the effectiveness and the high performances of our

approach.

Index Terms—Graph matching, Graph edit distance, Graph

decomposition, Stable Marriage, Pattern recognition, 3D object

recognition, Deformable object recognition.

I. INTRODUCTION

3D object recognition and classiﬁcation are one of the

fundamental challenges in computer vision, which have been

studied during many decades [1]. In the last years, there has a

growing interest on the 3D objects analysis. The high advances

in different ﬁelds of technology (especially in the ﬁeld of

3D) and the increased availability of 3D data, engender a

high growing need of automated approaches for 3D shapes

recognition and classiﬁcation. Shapes represented by graphs,

may be compared using graph matching techniques. Graph

matching is the process of measuring a similarity (or a

dissimilarity) between two graphs by ﬁnding a correspondence

between vertices and edges of two graphs that satisﬁes a

set of constraints, ensuring that substructures in one graph

are mapped to similar substructures in the other. In order

to solve the graph matching problem, various methods have

been proposed [2], [3], [4]. Graph Edit Distance (GED) is one

of the most famous measures to determine such a distance

[5], [6], [7] and [8]. GED is deﬁned as the minimum-cost

sequence of edit operations needed to transform one graph into

another. The tolerance to noise and distortion represent one

of the most important advantages of graph edit distance. Un-

fortunately, GED has a high computational complexity which

grows exponentially with the number of vertices [9]. In this

paper, we address the problem of comparing deformable 3D

objects represented by graphs. We propose a novel fast graph

matching approach based on a new formulation of the stable

marriage problem [10]. The proposed approach is optimal

in terms of execution time, i.e., quadratic time complexity

O(n2). This approach is based on graph decomposition into

a set of substructures called triangle-stars [11], [12] which

are subsequently matched with the help of the stable marriage

algorithm.

The remainder of the paper is organized as follows. In

Section II, we brieﬂy review some related works. Our approach

is described in Section III. The time complexity is determined

in Section IV. In Section V, we present and discuss our exper-

imental results while comparing them with some benchmark

shape-matching algorithms. Finally, Section VI concludes the

paper.

II. RE LATE D WO RK S

The state of the art includes a wide range of 3D objects

recognition approaches, we present in this section a brief

review of 3D shapes recognition methods, with a particular

focusing on graph-based approaches. We can generally di-

vide existing methods, for 3D object comparison techniques,

on three main categories [13]: feature-based methods, graph

based methods and others. In the ﬁrst category, feature based

methods, a set of features associated with the geometrical and

topological properties of objects are used for their comparison.

These features may be global, local or spatial maps. The entire

object is represented by a unique descriptor, in the case of

global features and spatial maps, while, in the case of local

features, a descriptor is either associated with the vertices, the

triangles or any substructure of interest. Various feature-based

approaches have been proposed in the state of the art [14],

[15], [16], [17]. In the second category, graph representations

and related approaches are considered. Graphs constitute a

powerful and ﬂexible modeling tool allowing both the descrip-

tion of properties of a shape and the relationships between

a set of shapes. In a graph, the properties are associated

with the vertices while the relationships are represented by

the edges. Vertices, edges and their attributes are speciﬁed

according to the underlying application; for instance, the

vertices may represent points, regions of interest or any other

substructure obtained by applying a data reduction process

such as segmentation. Edges are associated with the vertice’s

connectivity, deﬁning a topological relationship between them,

such as proximity, adjacency, etc. Depending to the kind of

graph considered, several graph-based techniques have been

proposed for 3D objects comparison [13]. For instance, some

approaches reduce shapes to skeletons through a thinning

process [18], [19]. The resulting skeletons are compared using

graph matching techniques. Other approaches rely on Reeb

graphs which are constructed from mapping functions deﬁned

on the shape manifolds [20], [21]. Various segmentation-based

techniques have been proposed in the literature, in which the

shapes are segmented into a ﬁnite set of components from

which a graph is constructed: the segments corresponding to

the vertices while their topological relationships are repre-

sented by the edges. These graphs may be compared with

graph matching techniques as well [22], [23]. In the last

category, others approaches have been proposed [13]: view

based similarity, volumetric error based similarity, weighted

point set based similarity, etc. In the case of view based sim-

ilarity, two shapes with similar projections, from all possible

viewpoints, are considered similar [24], [25]. For volumetric

error based similarity, the distance between two objects is

measured by estimating the volumetric error [26], [27]. In the

case of weighted point set based similarity, a distance between

two objects is measured using a set of descriptors which

consist of weighted 3D points. The shapes are decomposed

into substructures represented by weighted points which are

subsequently matched. The matching process depends on the

weight considered. The volume of a component is considered

as a weight in [28] and a measure for the curvature represent

a weight in [29]. The authors of [30] consider a hierarchy of

weighted point sets, representing spherical shape approxima-

tions. For a more exhaustive review, we refer the reader to [13]

and [31] for 3D object recognition methods and to [9], [32],

[33], [34] and [35] for pattern recognition and graph matching

techniques.

III. ALGORITHM OVERVIEW

We propose a novel fast graph matching approach based

on a new formulation of the stable marriage problem. The

proposed approach is based on graph decomposition into a set

of substructures which are matched with the stable marriage

algorithm [10]. We address the problem of 3D deformable

object recognition using triangle-stars decomposition [11],

[12]. In this section, we ﬁrstly introduce the stable marriage

problem [10], then we present the triangle-stars decomposition

as well the distance between triangle-stars [11], [12]. Finally,

we describe the proposed approach.

A. Stable marriage problem

The stable marriage problem is the problem of ﬁnding a

stable matching between two sets of elements of equal size,

based on an order of preferences; the latter being associated

to each pair of elements. Here, stability means that every

matched couple of elements prefers to stay together rather

than to be ”married” to another one [10]. Originally, the stable

marriage problem was introduced by Gale and Shapley in

1962 in order to ﬁnd a matching between men and women,

based on preference lists in which each person (man and

woman) presents his or her preferences regarding the other

candidates. The principle is summarized in the following

algorithm (Algo. 1). The problem has then adapted in several

research areas, such as in mathematics, economics, game

theory, computer science, etc. We refer the readers to [36]

for further details.

Algorithm 1 Stable marriage algorithm

1: Begin

2: Initialize all men m∈Mand women w∈Wto free

3: while (∃m∈Mfree) do

4: w=W[0] ﬁrst woman on the list of mto whom m

has not yet proposed;

5: if (wis free)

6: wand mare engaged ;

7: else

8: if (wprefer mto her engaged m0)

9: wand mare engaged and m0is free;

10: else

11: wrejects mand mis free ;

12: end if

13: end if

14: end while

15: return the stable marriage which is the ncouples ;

16: End

B. Triangle-stars decomposition

The triangle-stars decomposition [11], [12] is a decomposi-

tion method of a triangular tessellations (of a 3d shape) into

a set of connected components called triangle-stars. This de-

composition aimed to reduce the number of components while

covering larger neighborhoods depending on the considered

neighborhood order Nk. From a triangle-star representation, a

description is deﬁned which is invariant or at least oblivious

under most common deformations. Prior to the decomposition,

a strict total order on the triangles is established, in order

to reduce the number of triangle-stars and guaranteeing the

uniqueness of the decomposition. The authors of [11], [12]

introduce the triangle-stars using the following deﬁnitions:

a) Deﬁnition 1 (Nk-neighborhood of a triangle):two

triangles are neighbors (k= 1), if they share at least a common

vertex. Two triangles t0and tkare Nk-neighbors, if between

t0and tkthere is a chain of at most (k−1) distinct triangles,

which are pairwise consecutive neighbors. Formally, t0and tk

are Nk-neighbors ⇔ ∃ ti=1..k−1where :

∀i∈1...(k−1), tiand ti+1 are neighbors [11], [12].

b) Deﬁnition 2 (Nk-triangle-star):A triangle-star ts

(k= 1) is a labeled subgraph, deﬁned by a triangle and the

set formed by its neighbors. Formally, a triangle-star ts is a

three tuple ts = (tr, T 0, θ), where: tris the root triangle, T0

is the set of adjacent triangles and θ:T→LTis the triangle

labelling function while LTis a set of labels. A Nk-triangle-

star Nk-ts is triangle-star deﬁned by a triangle and the set of

its Nk-neighbors [11], [12].

c) Triangle-star vector representation:Each triangle-

star ts is characterized by a set of descriptors represented

by a vector consisting of the global area AG and the

global perimeter P G of ts, the area Aand the perime-

ter Pof each triangles belonging to ts, the weights

associated with their edges Was well as the degrees

deg associated with their vertices. This vector is given

by: {AG(ts), P G(ts),{A(ti), P (ti), W (ti, j=1...3),

deg(ti, j=1...3)}i=kT(ts)k

i=1 }. The variables are described in Ta-

ble I. The triangles belonging to ts are ranked according to

their areas in descending order. The weights and the degrees

are ranked by descending order as well. All T S vectors have

the same size: size = 2 + (8 Γ). If a ts has a number of

triangles inferior to Γ, the unassigned entries are completed

with zeros [11], [12].

d) Triangle-stars decomposition process:A descend-

ing strict total order, based on the number of neighbors

kneighborskand the vertex’s coordinates, is applied on the

set of triangles prior to their decomposition into triangle-stars;

in order to ensure the uniqueness of the decomposition and

to further reduce the number of triangle-stars. The decom-

position process is performed as follows: according to the

order established for the triangles (strict descending order),

the ﬁrst Nk-triangle-star is constructed from the ﬁrst triangle

and its corresponding Nk-neighbors; that is, the triangles not

belonging to any other Nk-triangle-stars. Then, the set of

triangles and the set of the resulting Nk-triangle-stars are

updated. The process is repeated until each triangle of the

graph is associated with a Nk-triangle-star [11], [12].

C. Distance between triangle-stars

In this section, we present the graph distance between two

triangle-stars [11], [12]. The similarity measure dbetween two

triangle-stars tsiand tsjis deﬁned as:

d(tsi, tsj) = 1 −Pk=6

k=1 simk(tsi, tsj)

Pk=6

k=1 αk

(1)

The similarity measure drequires six auxiliary functions

simk:

dsimk(tsi, tsj) = αk∗

|AG(tsi)−AG(tsj)|

AGMAX if k= 1

|P G(tsi)−P G(tsj)|

P GMAX if k= 2

Pl=Γ

l=1 |A(T(tsi)l)−A(T(tsj)l)|

AMAX Γif k= 3

Pl=Γ

l=1 |P(ti,l)−P(tj,l )|

PMAX Γif k= 4

Pl=Γ

l=1 Pk=3

k=1 |Wi,l,k −Wj,l,k |

3WMAX Γif k= 5

Pl=Γ

l=1 Pk=3

k=1 |Degi,l,k −Degj,l,k |

3DegM AX Γif k= 6

(2)

The associated symbols are deﬁned in Table I.

Symbol Description

ti,l Triangle tlbelonging to the triangle-star tsi:tl∈tsi

Wi,l,k Weight (Euclidean distance) of edge ekbelonging to triangle tl∈tsi

Degi,l,k Degree of vertex vkbelonging to triangle tl∈tsi

ΓMax number of triangles in the triangle-stars

αk=1...6Parameters associated with the descriptors αk∈Nand

Pk=6

k=1 αk>0

A(ti)Area of triangle i.

P(ti)Perimeter of triangle i.

AG(tsi)Area of triangle-star i.AG(tsi) = Pj=kT(tsj)k

j=1 A(tj)

P G(tsi)Perimeter of triangle-star i.P G(tsi) = Pj=kT(tsj)k

j=1 P(tj)

TABLE I

SYM BOL S AS SOC IATE D WIT H TH E SIM IL ARI TY M EAS URE A ND T HEI R

DESCRIPTION.

D. Graph Distance based on the Stable Marriage Algorithm

(GDStM)

The ﬁrst step of our approach is to decompose two graphs

G1and G2into two sets of substructures S1and S2. Since

3D deformable object recognition is addressed in this paper, a

triangle-stars decomposition is employed for the graphs (see

Section III-B). The next step consists of associating to each

substructure s1,i ∈S1, a vector of preferences regarding the

other substructures s2,j ∈S2. The vectors associated to s1,i

∈S1(respectively s2,j ∈S2) contain the set of substructures

S2(respectively S1) ordered following the preferences with

a descending order. The preference P(s1,i, s2,j )between s1,i

∈S1and s2,j ∈S2is measured using the distance deﬁned

in Eq. 1. All the vectors have the same size t, which corre-

sponds to the maximum size, i.e.,t= max(||S1||,||S2||).

If ||S1|| 6=||S2||, the smaller vector is completed by

(max(||S1||,||S2||)−min(||S1||,||S2||)) empty substructures

(ε), with their corresponding preferences P(sk,i, ε). From

these vectors, we construct a square preference matrix Dt,t

in which each entry Di,j contains a substructure (or an

empty substructure ε) as well as the corresponding preferences

Pi,j . In the third step, the algorithm of stable marriage is

employed in order to ﬁnd the best match between the different

substructures S1and S2based on their preferences vectors,

in quadratic time O(n2). Finally, the normalized distance is

estimated based on the preference values, using the following

formula:

GDStM(S1, S2)=1−Pi,j=t

i,j=1 P(s1,i , s2,j )

max(||S1||,||S2||)(3)

Algorithm 2 Graph Distance based on Stable Marriage Algo-

rithm (GDStM)

1: Inputs: Two graphs g1and g2.

2: Outputs: The distance d(g1, g2)between g1and g2.

3: Begin

4: S1=triangle-stars of g1;

5: S2=triangle-stars of g2;

6: For each si∈S1and sj∈S2do

7: si←− a vector of preferences regarding sj∈S2

with descending order using (Eq. 1);

8: sj←− a vector of preferences regarding si∈S1

with descending order using (Eq. 1);

9: end For each

10: Use stable marriage algorithm (Algo. 1);

11: return GDStM (S1, S2)and (si, sj)∈(S1×S2);

12: End

The algorithm of stable marriage is asymmetrical and the

considered version favors men. In order to ensure the symme-

try in our approach (GDStM(S1, S2) = GDStM(S2, S1)),

we consider the set of substructures (S1or S2) with higher

cardinality as the ﬁrst set (men) and the other set as second

(women). In the case of equality (||S2|| =||S1||), we consider

the average of the two distances. The symmetrized distance

between S1and S2is computed as follows:

GDStM (S1, S2) =

GDStM (S1,S2)+GDStM (S2,S1)

2if ||S2|| =||S1||

GDStM (S1, S2)if ||S1|| >||S2||

GDStM (S2, S1)if ||S2|| >||S1||

(4)

Our approach is summarized in Algorithm 2. An illustration

of our approach is proposed in Example 2.

a) Example 2.:Let S1and S2be two set of substruc-

tures (triangle-Stars) associated with two graphs G1and G2.

kS1k=kS2k= 4. Let Dbe the matrix of similarities

between S1and S2calculated using (Eq. 1).

s2,0s2,1s2,2s2,3

s1,00.11 0.90 0.25 0.21

s1,10.10 0.15 0.65 0.89

s1,20.67 0.03 0.51 0.17

s1,30.66 0.88 0.33 0.99

Firstly, we associate to each substructure s1,i=0..3∈S1

and s2,j=0..3∈S2, a vector of preferences with respect to

the other set of substructures s2,j=0..3∈S2and s1,i=0..3∈

S1respectively, with a descending order.

Figure 1(a) shows the four vectors representing the preferences

of s1,i=0..3∈S1with respect to s2,j=0..3∈S2. Figure 1(b)

shows the four vectors representing the preferences of

s2,j=0..3∈S2with respect to s1,i=0..3∈S1. Secondly, we

use the stable marriage algorithm in order to ﬁnd the best

match between the different substructures S1and S2based

on their preferences vectors. We obtain the following couples:

{(s1,0, s2,1); (s1,1, s2,2); (s1,2, s2,0); (s1,3, s2,3)}. Finally, we

calculate the distance corresponding to the couples selected.

The score obtained is 3.21 and the corresponding normalized

distance is GDStM(S1, S2)=1−3.21

4= 0.1975.

S2,1

S2,2

S2,3

S2,0

0,90

0,25

0,21

0,11

S2,3

S2,2

S2,1

S2,0

0,89

0,65

0,15

0,10

S2,0

S2,2

S2,3

S2,1

0,67

0,51

0,17

0,03

S2,3

S2,1

S2,0

S2,2

0,99

0,88

0,66

0,33

S1,0

S1,1

S1,2

S1,3

S1,2

S1,3

S1,0

S1,1

0,67

0,66

0,11

0,10

S1,0

S1,3

S1,1

S1,2

0,90

0,88

0,15

0,03

S1,1

S1,2

S1,3

S1,0

0,65

0,51

0,33

0,25

S1,3

S1,1

S1,0

S1,2

0,99

0,89

0,21

0,17

S2,0

S2,1

S2,2

S2,3

(a) S1preferences (b) S2preferences

Fig. 1. Vectors of preferences of s1,i=0..3∈S1regarding to s2,j=0..3∈

S2and vectors of preferences of s2,j=0..3∈S2regarding to s1,i=0..3∈

S1, in descending order.

IV. COM PL EX IT Y OF T HE P ROPOSED APPROACH

In terms of time complexity, the most demanding part of

the algorithm, is the one solving the assignment problem. The

Stable Marriage algorithm [10] is employed in order to ﬁnd

the best assignment in O(n2)time, where nis the maximum

number of components in the two graphs. For triangle-stars

decomposition (3D), we consider n= max(kV1k,kV2k)and

n0= max(kT S1k,kT S2k), where Viis the set of vertices

and T Siis the set of triangle-stars in gtr i. Any triangle-

star has at least one triangle. Consequently, in the worst case,

we have n0=n

3, which means that the complexity is in the

order of O(0.037 n2). However, the number of triangle-stars

depends on the structure of the underlying graph as well as

the neighborhood order Nkconsidered. Indeed, the number

of triangle-stars decreases when the neighborhood order Nk

increases.

Table II summarizes the relation between the computational

complexity and the neighborhood order Nk=1...6, for the

TOSCA database [37], [38]. The complexity is in the order

of O(α[n

log(n)]2), where α∈[1.80 ∗10−7,0.74] for

Nk=1...6. Table II shows that the complexity decreases when

the neighborhood order Nkincreases. The neighborhood order

Nk=1 has the highest complexity while the neighborhood order

Nk=6 has the lowest.

NkComplexity

N1O(0.74 [ n

log(n)]2)

N2O(0.0001 [ n

log(n)]2)

N3O(5.6115 ∗10−6[n

log(n)]2)

N4O(9.84 ∗10−7[n

log(n)]2)

N5O(3.59 ∗10−7[n

log(n)]2)

N6O(1.80 ∗10−7[n

log(n)]2)

TABLE II

TIME COMPLEXITY ACCORDING TO THE NEIGHBORHOOD ORDER Nk,FOR

TH E TOSCA DATABASE .

We propose a novel fast graph distance, which is an approx-

imation of Graph Edit Distance based on a new formulation

Class Pose 1 Pose 2 Pose 3

Centaur

Gorilla

TABLE III

SMA LL SA MP LE FRO M TH E TOSCA DATABASE .

of the stable marriage problem using triangle-star (TS) de-

composition. The considered dissimilarity measure d(tsi, tsj)

(Eq. 1) is equivalent to a substitution edit operation of two

triangle-stars tsiand tsj. The insertion and deletion edit

operations are equivalent to replace an empty triangle-star ε

(tsid(ε, tsi)or ε d(tsi, ε)). Consequently, the dimensions of

the cost matrix Dare reduced to n×n, where nis given

by n= max(kT S1k,kT S2k)(kT SkkVk). Consequently,

our approach has a better time complexity than the state of

the art. Indeed, in [39] and [40], the complexity achieved is

O((n+m)3), where nand mare the number of vertices in

the two graphs. In [41] and [42], the obtained time complexity

is O((max(n, m))3). However, the Jonker-Volgenant linear

solver [43] presents some convergence problems for some

kind of cost matrices. The proposed algorithm in [44] has

a quadratic time complexity O(n∗m). However, the obtained

correspondence between the vertices of the two graphs is not

bijective. In [45], the complexity achieved is O((min(n, m))2

max(n, m)). In [11], [12], the computational complexity

achieved, in general, is O(0.037 n3)and for the TOSCA

database, is of the order of O(α[n

log(n)]3), where α∈

[1.80 ∗10−7,0.74] for Nk=1...6.

V. EXPERIMENTS

In order to evaluate our approach, we undertook a set of

experiments in which our approach is compared with some

benchmark algorithms for shape-matching for the TOSCA

database [37], [38].

A. Description of the TOSCA database

The TOSCA database [37], [38] consists of 148 three-

dimensional shapes. Each shape is represented by a triangular

tessellation. The database is organized on 12 classes. Each

class is composed of the same shape submitted to isometric

or quasi-isometric deformations. The database is unbalanced

(from 1to 24 objects per class). On average, the number of

vertices is 3154 while the number of triangles is 6220. Table III

shows a small sample of 3D shapes belonging to the TOSCA

database.

B. Benchmark for shape-matching algorithms

For the evaluation of our approach, we performed a set

of experiments and we compare the proposed method with

some state-of-the-art shape-matching algorithms associated

with TOSCA database [37], [38]. Consequently, our method

GDStM has been compared with the following algorithms:

TSM [11], [12]: An approach based on an approximation

of Graph Edit Distance using triangle-stars decomposition,

then followed by a matching of these substructures using the

Hungarian algorithm [46].

CAM [47]: An approach in which surfaces are represented by

3D-curves extracted around feature points.

GeodesicD2 [48]: A global description which consists of the

distribution of the geodesic distances associated with a given

3D shape.

DSR [49]: The Hybrid Feature Vector is a combination of two

view-based descriptors: a depth buffer for the silhouette and a

radialized extent function descriptor.

RSH [50]: The Ray-based approach with Spherical Harmonic

Representation is a method which aligns the models into a

canonical position, extracts the maximal extents and applies a

spherical harmonic decomposition.

TD [51]: The Temperature Distribution descriptor is a shape

descriptor based on the heat kernel. The L2 norm is used in

order to evaluate the distance between the descriptors.

Shape-DNA [52]: The Shape-DNA is a numerical ﬁngerprint

obtained by evaluating the eigenvalues of the Laplace-Beltrami

operator associated with the manifold. The matching in be-

tween two objects is obtained by comparing their respective

eigenvalues.

SRCP-TD [53]: The SRCP-TD is a method based on a sparse

representation of a scale-invariant heat kernel. The authors

use Laplace-Beltrami eigenfunctions in order to detect critical

points on the manifold. The descriptor is constructed from the

values of heat kernel at these points. A sparse representation

is employed in order to reduce the dimensionality of the

descriptor.

C. Experimental results

In this section, we compare our approach with the tech-

niques previously described. We consider the following met-

rics: Precision and Recall, Accuracy, True Positive Rate TPR,

True Negative Rate TNR, F-measure and Runtime [54]. Our

approach GDStM is parameterized through a set of variables

αkwhich determine the weights attributed to the various at-

tributes. The default value for these parameters is: αk= 1,∀k.

In addition, a threshold is determined in order to perform

classiﬁcation. The weights and the threshold may be speciﬁed

by inspection or by using machine learning techniques.

The distance between each pair of triangular tessellations

was evaluated using the similarity measure GDStM , for

various neighborhood orders Nk=1...6. For each neighborhood

order, a n×nconfusion matrix was calculated. Two triangular

tessellations were considered similar if their GDStM distance

was below a certain threshold. The results differ according

to the values of the parameters αk, the level of the threshold

and the neighborhood order Nk. Table IV shows the Accuracy,

TPR and T N R obtained by our approach GDS tM and

TSM for the TOSCA database, for six neighborhood order

Nk=1...6. For our approach GDStM , a direct neighborhood

Nk=1 with a threshold = 0.066, has the highest accuracy

(78%), followed by Nk=4 (68%), Nk=3,Nk=5 (66%), Nk=6

and Nk=2 (62%). Table IV shows that GDStM and TSM have

approximatively the same performances with TSM performing

slightly better mainly for Nk=1 where TSM has an accuracy

of 83% as opposed to 78% for GDStM.

Methods NkThreshold Accuracy T P R T NR

GDStM 1 0.066 78.44 % 78.80 % 78.40 %

GDStM 2 0.135 62.09 % 62.74 % 62.02 %

GDStM 3 0.192 66.50 % 66.06 % 66.55 %

GDStM 4 0.231 68.28 % 69.80 % 68.11 %

GDStM 5 0.265 65.27 % 64.40 % 65.38 %

GDStM 6 0.273 62.44 % 61.43 % 62.57 %

TSM 1 0.06 83.24 % 82.29 % 83.35 %

TSM 2 0.13 60.56 % 61.34 % 60.47 %

TSM 3 0.19 65.78 % 66.93 % 65.65 %

TSM 4 0.22 68.65 % 67.89 % 68.74 %

TSM 5 0.264 65.01 % 64.31 % 65.09 %

TSM 6 0.273 62.11 % 61.52 % 62.18 %

TABLE IV

GDStM A ND TSM AC CUR ACY,T P R A ND T N R FOR T HE TOSCA

DATABAS E FO R VARIO US NE IG HBO RH OOD O RD ERS .

Some results are displayed in Figure 2. The ﬁrst column

represents the query objects, while the other columns are asso-

ciated with the ﬁrst four objects retrieved. All the objects were

properly classiﬁed. Table V shows the resulting correspon-

dences in terms of triangle-stars, using our approach GDStM

with different neighborhood orders, between two poses of four

objects (dog, horse, lioness and michael) belonging to the

TOSCA Database.

dog2 ×dog10 -N8horse9 ×horse0 -N8

michael19 ×michael15 -N6lioness13 ×lioness15 -N8

TABLE V

THE CORRESPONDENCE,US ING O UR AP PRO ACH GDS tM WITH

DIFFERENT NEIGHBORHOOD ORDERS,BETW EE N TWO P OS ES OF F OU R

OBJECTS BELONGING TO THE TOS CA DATABA SE.

Figure 3 shows the precision-recall curves for the six queries

appearing in Figure 2 when a neighborhood order of Nk=1

is employed. A threshold of 0.066 was selected for these

experiments. This implies in turn that only the objects for

which dissimilarity ≤threshold were considered when

evaluating the recall and precision curves. Indeed, when the

dissimilarity is greater than the threshold, the objects are

automatically classiﬁed as dissimilar by our algorithm. As

Query

Result 1

Result 2

Result 3

Result 4

Fig. 2. Some retrieval results for the TOSCA database with a neighborhood

order of Nk=1.

showed in Figure 3, we have obtained excellent precision-

recall curves which are consistent with the results shown in

Figure 2. For instance, centaur0,gorilla0,horse0,lioness0

and seahorse0have a precision and recall of 100%.

Fig. 3. Precision-recall curves for six distinct 3D-objects of the TOSCA

database with a neighborhood order of Nk=1.

We also compared our method GDStM with four

other SHREC benchmark methods namely TSM, CAM,

GeodesicD2, DSR and RSH. Figure 4 shows the precision and

recall curves for our approach with a neighborhood order of

Nk=1, as well as for the methods previously mentioned. The

precision of our approach is essentially always superior than

the one associated with the other methods, except TSM: we

conclude that our algorithm outperforms the others except for

TSM. Nonetheless, our method GDStM is faster as shown in

Figure 5 (logarithmic scale). Figure 5 shows the total runtime

of our approach GDStM and the TSM method according to

the neighborhood order Nkconsidered. Figure 5 further shows

that better performances, in terms of the runtime, are obtained

when using a larger neighborhood. The runtime performances,

as reported in Figure 5, conﬁrm the theoretical quadratic time

complexity O(n2)for our approach GDStM and cubic O(n3)

for TSM; nbeing the number of nodes associated with the

graphs. A comparison in terms of F-measure was performed in

Fig. 4. Comparison of our approach GDStM with the TSM, CAM,

GeodesicD2, DSR and RSH methods, in terms of precision and recall curves,

for the TOSCA database.

Fig. 5. Comparison of our approach GDStM with the TSM method, in

terms of total runtime, for the TOSCA database.

between our method and four SHREC benchmark algorithms

namely TSM, TD, Shape-DNA and SRCP-TD. Our results

are reported in Table VI. Table VI clearly indicates that our

GDStM method has excellent results in terms of F-measure.

Table VI shows that our method GDStM outperforms the

other methods except for TSM. Nonetheless, our method

GDStM is faster as shown in Figure 5.

Methods GDStM TSM TD Shape-DNA SRCP-TD

F-measure 0.74 0.90 0.67 0.45 0.44

TABLE VI

F-MEASURE RESULTS WITH OUR GDStM METHOD FOR Nk=1

NEIGHBORHOOD AS COMPARED WITH FOUR OTHER BENCHMARK

METHODS FOR THE TOSCA DATABAS E.

D. Discussion

Our experimentation against the TOSCA, shows the perfor-

mances and the robustness of our approach GDStM . Indeed,

we obtained excellent results in terms of Accuracy, TPR

and T N R for the TOSCA database. Our predicted quadratic

time complexity has been systematically conﬁrmed by our

experiments. GDStM with quadratic time complexity O(n2)

is faster than TSM by one order of magnitude. Our method

GDStM outperforms CAM, GeodesicD2, DSR and RSH

while retaining the same level of performance as TSM in terms

of precision-recall and F-measure.

VI. CONCLUSIONS

In this paper, we presented a novel graph matching approach

based on a new formulation of the stable marriage problem,

to measure the distance between graphs. The proposed ap-

proach is optimal in terms of execution time, i.e. quadratic

time complexity O(n2). Our approach is based on graph

decomposition into a set of substructures which are matched

with the stable marriage algorithm. A triangle-stars decompo-

sition is employed for triangular tessellations (graphs of 3D

shapes). The present approach is based on an approximation

of the Graph Edit Distance which is fault-tolerant to noise

and distortion: making our technique particularly suitable for

the comparison of deformable 3D objects. Our experimental

results with deformable shapes (TOSCA database) and the

time complexity analysis conﬁrm the performances and the

accuracy of our algorithm.

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