Digital Signal Processing Reference
In-Depth Information
The Zernike moment
if the
radial polynomial in Eq. (7) is used to compute the polynomial with the condition,
n-|m|= even, eliminated. So
in Eq. (3) becomes Pseudo-Zernike moment
Z
P
nm
nm
can be expressed as:
P
nm
Q
S
³
³
3
I
U
T
5
U
H[S
LP
T
UGUG
T
(8)
QP
S
QP
3.2
Distance Measure of Pseudo-Zernike Moments
In the classical way to measure the similarity of two Pseudo-Zernike moments, just like
Eq. (9), Euclidean distance also only takes the moment magnitude information into
account.
I
nm
and
J
nm
represent Pseudo-Zernike moments of two different images I
P
P
and J.
¦¦
,
QP
-
QP
(9)
G
3
3
Q
P
'
3=0V
Since the Pseudo-Zernike polynomials are also a complete set of functions orthogonal
on the unit disk, Revaud's improvement idea of calculating both magnitude and phase
is also applicable to improve the distance measure of Pseudo-Zernike moments. We can
describe the minimum distance between image I and J using the following expression:
G
3=0V
PLQ
G
M
Q
P
'
(10)
where
is computed as:
2
d
ϕ
)
S
ª
º
¦¦
,
QP
-
QP
,
QP
-
QP
-
QP
,
QP
G
M
3
3
3
3
FRV
P
M
>
3
@
>
3
@
¬
¼
(11)
Q
[
\
d
DQG
Q
P
'
4
Experiments
In this section, we will use a trademark database including 1500 binary trademark
images of BMP format with 111 × 111 resolution, of which 100 images are generated
via 10 kinds of deformation of 10 images in the database. The 100 deformed images
and their original images are shown in Fig. 1[15].
We will do two experiments to compare the trademark retrieval performance be-
tween the classical and the improved distance measure of Zernike moments and
Pseudo-Zernike moments.
One of the experiments is to proof-test the ability of finding original images
through using the 100 deformed images as query images.
The other experiment is to use 10 original images to retrieve their deformed
images in the database.
