![](bg3.png)
However, this calculus is relatively heavy. An-
other way to obtain this value is to use the Natural
Axis as in (Shen et al., 1999). The technic principle in
a general framework is as follow: One starts from an
ordered set S = {s
i
, 1 ≤ i ≤ N} (in our case S is the
set of values I
l
(i)), one associates to S the necklace of
radius one and N pearls. To each pearl correspond a
weight s
i
. The distance between two successive pearls
is equal to 2π/N. The natural axis of the set S of the
N pearls necklace is represented by a vector with an
origin in the center of the necklace, and the extremity
is a point with coordinates (X
natural
,Y
natural
) defined
by:
X
natural
=
N
∑
i=1
s
i
cos(
2π
N
(i− 1));
Y
natural
=
N
∑
i=1
s
i
sin(
2π
N
(i− 1)) (10)
5 THE CONTOUR DESCRIPTOR
In the above section 3, we showed how to obtain
the affine invariant I
l
(i) from the SWT coefficients
and we also how to obtain the correspondence be-
tween (
˜
I
l
(i))
1≤i≤N
and (I
l
( j))
1≤ j≤N
using the nat-
ural axis orientation. The SWT transform is a non
decimated version of the DWT, so it is redundant as
the I
l
invariant is. To avoid this redundancy and to
keep the invariant values useful for the contour de-
scription, we remove from the invariant I
l
all values
which not correspond to the invariant obtained from
the DWT on the contour (x
i
,y
i
)
1≤i≤N
. In (Misiti et al.,
2003), the author showed that giving a SWT set coef-
ficients, it is possible to get all the DWT ε-decimated
for any sequence ε = [ε
1
ε
2
...ε
n
] with ε
j
= 0 or 1 for
all 1 ≤ j ≤ n. In a same way, it is possible to obtain
the descriptors ε-decimated of the invariant I
l
by the
following relation: D
l
= (D
l
( j))
1≤ j≤2
n−l
=
(I
l
(ind),I
l
(ind + 2
l
),...,I
l
(ind + (2
n−l
− 1)2
l
)) with
ind = 1 +
∑
l
i=1
ε
i
2
i−1
. Note that there is N = 2
n
pos-
sible descriptors. Furthermore, to make our algorithm
more efficient, only a subset of descriptors with scale
from K to L, K ≤ L, is used. The selection of these
scales are automatically performed, using the level
histogram, such the vector magnitude (I
l
) 1 ≤ l ≤ n
of equation (7) is maximal.
6 THE PARTIAL-SHAPE
MATCHING
The method described above is not adapted to open
curves. In real applications and depending of the
image quality, the contours are generally open even
using effective segmentation methods. The opening
may also be due to objects occlusion. Much efforts
has been devoted of finding effective methods for
recognition of partially occluded objects. The dis-
parity matrix to perform similarity matching of oc-
cluded objects modeled by line segments has been
used by many authors as in (Price, 1984) and (Bhanu
and Ming, 1987). Khalil and Bayoumi in (Khalil and
Bayoumi, 2002) proposed to use maxima lines of the
continuous wavelet transform and recognize occluded
objects by identifying singularities on their bound-
aries. In this paper, we use the method of partial shape
matching based on features extracted by the wavelet
transform. Actually, the shape descriptor used is the
affine wavelet descriptor, and its coefficients are used
in the sub-matrix matching algorithm proposed by E.
Saber et al in (Saber et al., 2005). In the following
section, we recall this method.
6.1 Distance Matrix
The distance Matrix represents the proximity of fea-
ture points within each example template or potential
object region in order to determine complete or partial
correspondances between two sets of feature points.
Let (Xi,Yi), i = 1,2,...,n, be feature points for a po-
tential object region or example template contour; the
distance matrix D for the contour is defined as
D =
d
11
d
12
··· d
1n
d
21
d
22
··· d
2n
.
.
.
.
.
.
.
.
.
.
.
.
d
n1
d
n1
··· d
nn
(11)
where d
kl
=
p
(X
k
− X
l
)
2
+ (Y
k
−Y
l
)
2
, k,l =
1,2, ...,n denotes the distance between the feature
points k and l along the contour. The distance ma-
trix is a symmetric matrix,and is invariant to transla-
tion and rotation by definition, since it only depends
on distances between feature points; (2) invariant to
isometric scale variation of a contour, i.e. zoom or
contraction, since that corresponds to scaling all dis-
tances by a constant factor; (3) reflection of a con-
tour is equivalent to reordering of feature points in the
clockwise or counter clockwise direction essentially
inverting the initial ordering, and (4) if two contours
partially match, their distance matrices have match-
ing sub-matrices. The distance matrix depends on
WAVELET TRANSFORM FOR PARTIAL SHAPE RECOGNITION USING SUB-MATRIX MATCHING
515