An Efficient 2D Curve Matching Algorithm
under Affine Transformations
Sinda Elghoul and Faouzi Ghorbel
CRISTAL Laboratory, Pole Grift, Campus University, Manouba 2010, Tunisia
Keywords:
2D Curve Matching, Affine Transformation, Partially Occluded, Motion Estimation, Affine Arc Length,
Pseudo-inverse Matrix.
Abstract:
Most of the existing works on partially occluded shape recognition are suited for Euclidean transformations.
As a result, the performance would be degraded in the affine and perspective transformation. This paper
presents a new estimation and matching method of the 2D partially occluded recognition under affine transfor-
mation including translation, rotation, scaling, and shearing. The proposed al gorithm is designed to estimate
the motion between two open 2D shapes based on an affine curve matching algorithms (ACMA). This ACMA
considers the normalized affine arc length coordinated to the 2D contour. Then, it wil l correlate them in order
to minimize the L
2
distance according to any planar affine transformation by means of a method based upon
a pseudo-inverse matrix. Experiments are carried on the Multiview Curve Dataset (MCD). They demonstrate
that our algorithm outperforms other methods proposed in the state-of-the-art.
1 INTRODUCTION
The motion estimation and the matching of planar
shapes that are subjected to certain deform ation and
viewing transformatio ns is one of the most impor-
tant goals in computer vision and pattern reco gnition,
done through different applications such as robotic vi-
sion, Medical Image Registration (Bronstein et al.,
2006), 3D reconstruction, Optical Character Recog-
nition (Belongie et al., 2002), Object Classification
(Adamek and O’Connor, 2004) (Alajlan et al., 2008)
(Baseski et al., 2009), and c ontent-b ased image re-
trieval (Bronstein et al., 2008). However, despite the
progress of the research, remains a challenging task
that makes shape recognition more c omplicated. It
is presented by two critical factors: (a) images ta-
ken from different viewpoints of the same object suf -
fer from perspective d istortions and (b) the partially
occluded shapes sometimes make the recognition pro-
blem more challenging (Turney et al., 1985). So the
matching methods should have the ability to handle
the different cases.
For example, the silhouette tracking application
which records the movement of objects or people,
consist of matching curves extracted from two succes-
sive images at two different instants which would lead
to many problems due to several factors suc h as lo-
cal deforma tions, articulations, missed and extrane-
ous contour portions owing to errors in shape ex-
traction. Under these conditions, it is known that a
perspective transformation between two images o f an
object can be a pprox imated by a two-dimensional af-
fine transformation (Forsyth et al., 1991) when the ob-
ject is far from the camera-since the slight distortion
that may result from th e more general projection-can
be regarded as part of a def ormation.Therefore, local
deformations have been treated in the literature by al-
lowing some leeway in the matching of curve points
via methods like Chamfer and Hausdorff distance.
Also, loc a l geometric corrections of affine transfor-
mation have been a pplied to handle more severe dis-
tortions and articulations. However, the issue is to
specify which portions of the shape should be used
for the geome tric corrections, although some methods
have been tried to solve this problem, presented in the
next section.
Towards the solution of this ch allenging problem,
our contribution aim s to recognizing and curve ma-
tching of partially occluded 2D shape under affine
transformations. The ACMA algorithm is applied to
estimate the motion of two contours and matching
them. First, a curve re-parameterization is defined, in-
spired by the expression of the normalized affine arc
length (Spivak, 1981),( G horbel, 1998). Subsequently,
sampling this part of curves at constant equivalence
lengths which is represented by a sufficiently large
474
Elghoul, S. and Ghorbel, F.
An Efficient 2D Curve Matching Algorithm under Affine Transformations.
DOI: 10.5220/0006719504740480
In Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 4: VISAPP, pages
474-480
ISBN: 978-989-758-290-5
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

set of points that makes the number of equations hig -
her than the unknowns. Finally, an affine part-to-part
curve matching is obtained by the computation of the
pseudo- inverse matrix which makes it possible to mi-
nimize the L
2
distance. This algorithm ACMA has
the ability to handle object recognitio n under affine
distortions, partial occlusions and outperforms other
methods in terms of r egistration and recognition a ccu-
racy.
The remainde r of the paper is organized as fol-
lows: section 2 introduce the re la te d work to our ap-
proach . Then the detailed descriptions of ACMA and
the new curve matching algorithm will be presented
in the next section where we will briefly recall the
affine a rc length reparameter iz ation method and cal-
culate the pseudo-inverse matrix . Section 4 investiga-
tes the effectiveness of the proposed approach through
experiments and analyses. Finally, the last section gi-
ves the conclusion.
2 RELATED WORK
In this section, we focus on work that serves to place
this paper relative to the state o f the art. There-
fore, various affine invariant shape matching met-
hods have been developed (Latecki et al., 2000), (Ma-
weheb et al., 2016), (Chaieb and Ghorbel, 2008) .
They are able to address difficult problems like ma-
tching under noise conditio n, affine transformations
and so on. In this context, the most well-known-
researched shape descrip tion and shape matching
methods include affine invariant Fourier descriptors
method (Arbter et al., 1990), (Osowski et al., 200 2),
(Chaker et al., 2008), affine curvature scale spac e
(ACSS) method (Mokhtarian and Abbasi, 2001) , in -
dependent component analysis (ICA) method (Huang
et al., 2005), curvature tree method (Alajlan et al.,
2008), shape contexts (SCs) methods (Mori et al.,
2005), (Ling and Jacobs, 2007), moments invariants
methods (Huang and Co hen, 1996), (Zhao and Che n,
1997), symbolic representation method (Da liri and
Torr e , 2008) and so on. However, they treat only the
closed-to-c losed shape matching and assuming that
the whole shape is always visible in im a ges. On the
other side, it is possible th at the shape to be recogni-
zed is on ly partially visible in real applications, which
makes th e reco gnition problem, far more difficult than
that of closed shapes.
Only some approaches of shape matching under
partially occluded 2D shapes have been suggested.
However, the most of them work only for sha pes up
to a similarity transform. The work pre sented in (De-
mirci, 2010) proposes a new indexing structure under
partial matching. Shan (Shan et al., 2006) proposes a
method to present model objects using histograms and
then matches the histogram between model a nd ob-
ject to be recognized. Their method can match partial
occluded objects. O rrite (Orrite and H errero, 200 4)
estimates projective transform using alignment appro-
ach and extractes the invariant points bitange nts, this
method is able to deal with partial occluded and p er-
spective transform. However it requires a complete
searching match so that it is time co nsuming. Zhang
(Zhang et al., 2015) presents a method dealing with
recogn ition of pa rtially occluded and affine distortion
objects. Their method was d e signed for objects with
planar polygon shap es, but many objects cannot be
approximated by polygons.
In order to ha ndle local affine changes, Gopa-
lan et al. (Gopalan et al., 2010) proposed a shape-
decomp osition technique that divides a shape into
convex parts using Normalized Cuts. These parts
were then individually affines normalized and combi-
ned into a sin gle shape that was matched using the In-
ner Distance Shape Context (IDSC). As a result, this
method is able to capture more deformations of lo-
cal portions, such as a 3D part articulation that may
be modeled by a 2D affine transformation of its pro-
jection. It nevertheless assumes an a-priori shape de-
composition from a single shape that may be inc on-
sistent in the presence of occlusions or noise in shape
extraction. Furthermo re, the matching is still global
and hence we will be unable to handle partial occlusi-
ons of the shapes. Also, Mai e t a l. (Mai e t a l., 2010)
proposed a method for partial matching and affine dis-
tortions shapes, whe re the shape is described by a se-
quence of ordered affine-invariant segments based on
the prop erties of curvature scale space (CSS) shape
descriptor. Then Smith-Waterman algorithm is a p-
plied to match these sequences. This idea is deve-
loped by Huijing et al. (Fu et al., 2013) where an
affine-invariant curve descriptor (AICD) based on a
new-defined affine-invariant signature and its unsig-
ned sum is proposed to represen t the local shape of
a curve with high distinctiveness. The comparison of
our method to these meth ods will be highlighted in
Section 4.
3 AFFINE MOTION ESTIMATION
AND CURVE MATCHING
In this section, we will present a contour matching ba-
sed on the affine curves matching algorithm (ACMA),
Fig.1, we will illustrate the main steps performed to
obtain the proposed algor ithm. First, an affine arc-
length re- parametrization is performed to meet invari-
An Efficient 2D Curve Matching Algorithm under Affine Transformations
475

ant parametrization. Then, the L
2
distance is minimi-
zed by the c a lc ulation of pseudo-inverse matrix to es-
timate the translation vector b and the linear transfor-
mation A. Finally, Affine curve matching algorithm is
obtained.
Figure 1: Block diagram of ACMA alghorithm.
3.1 Affine Arc-Length
Reparametrization
We focus on planar shapes r e presented by an open 2D
continuous curves Γ
1
and Γ
2
which we can o btain one
of them from other by a planar affine transforma tion.
Lets consider f (t) and h(t
′
) two resp ective parame-
trizations of curves Γ
1
and Γ
2
where their relation is
defined by:
h(t
′
) = A f (t) + b
with b is a translation vector and A is a linea r trans-
formation.
It is obvious that a given curve can be represented
with various parameterizations. So, we can’t com-
pare different views of a p lanar contour and assume
that th e parameterizations ar e the same. To avoid this
problem we must ensure that the parameterization is
indepen dent of transformations.
For this aim, we need to normalize the num ber of
sampled points of the curves. The un derlying idea
is to do an affine re-parameteriza tion of these curves
by applying an affine arc length function L(t) defined
by:
L(t) =
1
l
a
Z
t
0
3
p
| det( f
′
(u), f
′′
(u)) | du (1)
Where the total affine arc length L
a
of the considered
curve presented by:
L
a
=
Z
T
0
3
p
| det( f
′
(u), f
′′
(u)) | du (2)
With f
′
and f
′′
denote, respectively, first and second
derivative of f , while det represents the determinant
operator.
3.2 Calculation of the P seudo-inverse
Matrix
After re-parametrization by the affine arc length, the
estimate of the apparent motion is equivalent to ex-
tracting the p arameters of A and the translation vector
b.







h(l
a1
) = A f (l
a1
) + B
h(l
a2
) = A f (l
a2
) + B
....
h(l
aN
) = f
(
l
aN
) + B
with f (l
a
) and h(l
a
) are the reparametrization, re-
spectively ,of two contours f (t) and h(t
′
). Our goal
is to minimize the error between the two contours by
the estimation of A and b which is defined by:
min
(A,b)
=k A f (l
a
) + b − h(l
a
) k
2
≈ e
Explainin g this system of 2N equations and 6 unkno-
wns we obtain the following set of systems:









h
x
(l
a1
) = f
x
(l
a1
)a
11
+ f
y
(l
a1
)a
12
+ B
x
h
y
(l
a1
) = f
x
(l
a1
)a
21
+ f
y
(l
a1
)a
22
+ B
y
....
h
x
(l
aN
) = f
x
(l
aN
)a
11
+ f
y
(l
aN
)a
12
+ B
x
h
y
(l
aN
) = f
x
(l
aN
)a
21
+ f
y
(l
aN
)a
22
+ B
y
This system can be written in matrix notatio n:
H = DU
VISAPP 2018 - International Conference on Computer Vision Theory and Applications
476

with U = [a
11
a
12
a
21
a
22
B
x
B
y
]
t
, H =
[h
x
l
1
h
y
l
1
h
x
l
2
h
y
l
2
....h
x
l
N
h
y
l
N
] and
D =











f
x
(l
a1
) f
y
(l
a1
) 0 0 1 0
0 0 f
x
(l
a1
) f
y
(l
a1
) 0 1
f
x
(l
a2
) f
y
(l
a2
) 0 0 1 0
0 0 f
x
(l
a2
) f
y
(l
a2
) 0 1
. . . . . .
. . . . . .
f
x
(l
aN
) f
y
(l
aN
) 0 0 1 0
0 0 f
x
(l
aN
) f
y
(l
aN
) 0 1











(3)
The idea of the method of least squares is to solve the
overdetermined system of lin ear equa tions when the
numbers of equatio ns are more than unknowns. So,
the resolution of this rectangular system can b e done
by minimizing the error via inverting the system by
using pseu do-inverse of the matrix D.
U = (D
t
D)
−1
D
t
H
The instability of the reconstructio n of the movement
sometimes ar ises from the po or conditioning of the
normal matrix (D
t
D). There are stabilization me thods
to reduce the effect of poor conditioning (when the
conditioning value of the inverted matrix becomes
high). We suggest in this case:
-To use the classical method which is ob ta ined by
means of multiplication by appropriately chosen
diagona l matric e s.
- To realize the best choice of the set pairs of points in
correspo ndence by reducing to the best con ditioning.
The matrix to be inverted is a normal matrix whose
expression is:
D
t
D = N
6








¯
X
2
¯
XY 0 0
¯
X 0
¯
XY
¯
Y
2
0 0
¯
Y 0
0 0
¯
X
2
¯
XY 0
¯
X
0 0
¯
XY
¯
Y
2
0
¯
Y
¯
X
¯
Y 0 0 1 0
0 0
¯
X
¯
Y 0 1








(4)
and
¯
X=
1
N
∑
N
i=1
( f
x
(l
i
)) ,
¯
Y =
1
N
∑
N
i=1
( f
y
(l
i
))
¯
X
2
=
1
N
∑
N
i=1
( f
x
(l
i
))
2
,
¯
Y
2
=
1
N
∑
N
i=1
( f
y
(l
i
))
2
¯
XY =
1
N
∑
N
i=1
( f
x
(l
i
) f
y
(l
i
))
For mathematical proof the reader can be referred to
(Ghorbel, 20 13).
3.3 ACMA Algorithm
the procedure of matching the apparent affine parti-
ally occluded curves can be described b y the follo -
wing algorithm:
Algorithm
Step1: take two contours of partially affine shape.
Step2: re-parametrize the two contours by the normalized
affine arc length f
∗
and h
∗
.
Step3: sample at constant equivalence lengths in Npoints.
Step4: calculate
¯
X
2
,
¯
Y
2
,
¯
XY ,
¯
X,
¯
Y .
Step5: write the matrices D, D
t
D and H.
Step6: reverse D
t
D .
Step7: calculate U by performing (D
t
D)
−1
D
t
H.
Step8: reconstruct
ˆ
A and
ˆ
B from U .
Step9: apply
ˆ
A and
ˆ
B to f
∗
to obtain (
ˆ
A f
∗
+
ˆ
B ) .
Step10: superimpose
ˆ
A f
∗
+
ˆ
B a h
∗
by the
maximization of the correlation.
Step11: calculate the distance L
2
between
ˆ
A f
∗
+
ˆ
B and h
∗
.
4 EXPERIMENTS
In this section, we test our algorithm for shape ma-
tching and estimation where shapes are represen-
ted only by contours for the task of shapes recogni-
tion and retrieval. Our experimenta tions were con-
ducted on th e Mu ltiview Curve Dataset (MCD) (Zuli-
ani et al., 2004) which is composed of 40 shape c las-
ses taken from MPEG-7 database. Each class con-
tains 14 curve samples that correspond to different
perspective distortions of the original curve. Samples
of shapes from MCD da ta bases are shown in figure 2.
Figure 2: Different shape images from the MCD dataset,
two images from each class.
In the initial MCD d atabase, all shapes are presented
by closed curves. So, to make it open and partially
visible, we remove a few parts of the con tour. In our
experiments, we make three types of test to improve
An Efficient 2D Curve Matching Algorithm under Affine Transformations
477

the performance of our algorithm in matching: (a)
Whole-to-whole matching(Fig.3), (b) whole- to-part
(Fig.4) and (c) Part-to-part matching (Fig.5)
Figure 3: Bird pair: (a) and (b) are the initial curves; (c)
and (d) show the original shape overlaid with the matched
shape.
Figure 4: Butterfly pair:(a) and (b) are the initial curves; (c)
and (d) show the original shape overlaid with the matched
shape.
Figure 5: Key pair: (a) and (b) are t he initial curves; (c)
and (d) show the original shape overlaid with the matched
shape.
So from this result we can co nclude that o ur algorithm
works well for all the three cases and it is robust to
both partial occlusions and affine transformation.
4.1 Alignment Error Calculation
The shape registration is one of the important appli-
cations to evaluate the robustness of our alg orithm
under partial occlusion, where the estimated affine
transformation align the two different curves of the
same shape. So, we calculate th e percentage of non-
overlapping areas to obtain the alignment error bet-
ween the common part of these two 2D open curves.
Then, w e compare our result with different methods
presented in the literature. O ur average alignment er-
ror is 8.9 5 % which is smaller than 12.13 % , 13. 12 %
and 49.41 %, respectively, the average error of the re-
ference approach (Fu et al., 2013), (Mai et al., 2010)
and (Petra kis et al., 200 2).
4.2 Image Database Retrieval
Another significant application to test our algorithm is
the shape retrieval. Several techniques for shape data-
base retrieval exist in the literature, among which FD
is one of the m ost well-kn own descriptors and a state-
of-the- a rt algorithm for affine-invariant shape retrie -
val . Therefore, we select two Affine-Invariant Fou-
rier Descriptors (Chaker ’s FD (Chaker et al., 2007)
and Arber’s FD (Arbter et al., 1990)) as reference
methods. Besides, we select another three reference
methods for shape retrieval: wavelet-based m ethod
(El Rube et al., 2006) , ICA-based me thod (Huang
et al., 2005) and Mai ’s method (Mai e t al., 2010). Ta-
ble 1 compares the retrieval average rates for the first
10 shapes (apple, bell, bone, bird, butterfly, bottle,
bat, brick, camel and insect) of the MCD dataset using
our suggest with the five ref erence methods. The dif-
ferent methods results for this d a ta set are collected
from the respective p apers. In terms of the average
rates performance, our approach perform s reasonably
well as compar e d to many othe r techniques. Figure
6 , shows the retrieval results of 10 random queries
from MCD databases based on our algorithm.
Table 1: Retrieval results on the entire MCD dataset.
Methods Average
Arber (Arbter et al., 1990) 41 %
Huang (Huang et al., 2005) 71 %
Chaker (Chaker et al., 2007) 76 %
Rube (El Rube et al., 2006) 79 %
Mai (Ma i et al., 2010) 89 %
Our algo rithm 94 %
VISAPP 2018 - International Conference on Computer Vision Theory and Applications
478

Figure 6: 10 random retrieval results from MCD database.
5 CONCLUSIONS
This paper presents a general a ffine motion estimation
algorithm based on Affine Curve Matching Algorithm
(ACMA). The re -parameterization of the contours ba-
sed on the affine arc length is indispensable when the
movement is assum ed affines. Under this hypothe-
sis, we recover the affine param e te rs by the compu-
tation of the pseudo-inverse matrix which minimizes
the error. Our experimen ts indicate that our algorithm
works well on the MCD database compared to many
existing techniques, particularly in the case of partial
occlusions th at might arise in many situations. While
the results on this dataset are interesting, but there is
no guarantee that th e same ordering of the methods
would be obtained with other datasets or other met-
hods. So, in the future, we intend to compare our
method with other approaches and other datasets in
terms of both performance s under perspective distor-
tion and complexity.
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