Digital Signal Processing Reference
In-Depth Information
However, given a threshold value “
T
” (
∈
]0,1[
), the second group is composed of
pixels with low non-null membership degrees (≤
T
) and thus with high membership
degrees to the bin
B-1
or to the bin
B+1
. In this case, we define for each pixel
j
of the
second group, the normalized distance
d(j,B+l)
between this pixel and the bin
B-1
, or
the bin
B+1,
and the default bin
B
in order to obtain the final classification by
defining a final membership decision
FMC
i
(j)
(4) according to the nearest neighbors
principle.
Fig. 3.
Definition of the fuzzy membership degree
µ
B
(j)
of a pixel
j
into a bin
B
|
EC
(
j
)
−
EC
(
c
) |
i
i
(4)
Bl
+
djB
(,
)
=
+
l
n
l
where,
l
∈
{-1, 0, 1},
n
l
is the number of pixels in the bin
B+l
and
c
B+l
is the center of
the bin
B+l
. A low value of the normalized distance indicates that the pixel
j
and the
set of pixels in
B+l
are very similar, what indicates a low ambiguity between them.
As a consequence, we modify the belonging of the pixel
j
to the bin
B+l
, depending
on its position inside the bin
B
. Once the final membership matrixes are defined, the
next step consists to compare the produced histograms of the two images, given the
three color channels, while maximizing the similarity between them in order to obtain
approximately
the same foreground for each image (“matched image 1” and “matched
image 2” in Fig. 2).
In fact, two pixels in
I
1
×
I
2
are similar only if they belong both to
the same
RGB
bin
H
k
. This returns to resolve the following optimization problem:
K
K
K
(5)
max
ab
, such that
a
=
CardF
(
) and
b
=
CardF
(
),
kk
k
1
k
2
k
=
1
k
=
1
k
=
1
where,
a
k
(
resp.
b
k
) is the number of pixels in the foreground
F
1
(
resp.
F
2
) of
I
1
(
resp.
I
2
) which were associated to the bin
H
k
.
Once the two matched images are defined, the
co-segmentation model also includes a global constraint to enforce consistency
among the two foreground histograms in addition to the MRF segmentation terms for
each image. Thus, the co-segmentation energy is expressed by (6):
2
K
min
dx
+
wy
−
λ
z
,
ij ij
,
,
jq ijq
,
jq
i
=
1
jV q
∈
()
k
=
1
(,)(
jq
∈∩
×∩
)
(6)
IH
,
I H
,
1
k
2
k
{}
∀∈
i
1,2 ,
x
−
x
≤
y
and
x
−
x
≤
y
;
ij
,
iq
,
ijq
,
iq
,
ij
,
iqj
,
such that,
∀∈
iI z
,
≤
x
and
∀∈
iI z
,
≤
x
.
1
jq
i j
,
2
jq
i q
,
