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The similarity function in Eq. (
6.3
) has been demonstrated by the related works
[
168
-
170
]. In comparison, the method introduced in the current work extracts ECAs
before computing the similarity. This increases the discriminating power of the
correlator in identifying the firearm. The method for extraction of ECAs is explained
in Sect.
6.3
.
6.2
Image Registration Using Phase-Correlation Method
Image registration is the process of determining the correspondence between two or
more images in a point-by-point manner. These images may be taken of a scene at
different times, by different sensors, or from different viewpoints. The parameters
that make up the registration transformation consist of translation in
x
and
y
, rotation
angle around
z
, and scaling. These can be computed directly, or determined by
finding an optimum of some functions defined on the parameter space. In the
domain of image analysis for ballistic identification, Chu et al. [
167
] calculated
these parameters by optimizing a specified similarity metric using the Newton-
Raphson method. Chu et al.'s work [
167
] is an attempt to register pairs of topography
measurements of standard cartridge cases. The image registration is applied to align
the master standard cartridge case and its replicas before the similarity between
them is calculated.
In the current work, the phase correlation technique demonstrated by Reddy and
Chatterji [
172
] is adopted to obtain parameters for image registration. Unlike the
registration method in Chu et al. [
167
] that uses image pixel values directly, the
current work proposes the adoption of the algorithm that uses the frequency domain.
The Fourier domain approach is used to match images that are translated, rotated,
and scaled with respect to one another. The algorithm searches for the optimal match
according to information in the frequency domain. The mathematical algorithms for
translation, rotation, and scaling are described in the following sections.
6.2.1
Parameter Estimation for Translation
Let
{
f
[
i
,
k
]
}
denote the reference image, 0
≤
i
≤
M
−
1, and 0
≤
k
≤
N
−
1, for an
image with
M
row pixels and
N
column pixels. Let
be the reference image
with shifted position by
a
pixels in row direction and
b
pixels in column direction,
that is:
{
g
[
i
,
k
]
}
g
[
i
,
k
]=
f
[
i
−
a
,
k
−
b
]
(6.4)
According to Gonzalez and Woods [
173
], we can find the Fourier transform of image
{
g
[
i
,
k
]
}
using:
M
1
i
=
0
−
N
1
k
=
0
g
[
i
,
k
]
e
−
j
2ˀ
(
m
−
1
MN
i
M
+
n
N
)
G
[
m
,
n
]=
(6.5)
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