Biomedical Engineering Reference
In-Depth Information
The structure of the arrows between the other columns repeats the structure
between the first column and the second one.
Let us consider the properties of these permutations. Let the feature
F
s
be
detected in two locations,
P
1
(
i
1
,
j
1
) and
P
2
(
i
2
,
j
2
). Let
dx
¼
j
j
2
j
1
j
;
dy
¼
j
. The feature mask corresponds to the vector
U
. After the
X
permutations,
the vectors
U
1
and
U
2
will be different. Let
i
2
i
1
j
n
be the Hamming distance between
D
U
1
and
U
2
. The mean value of
D
n
can be calculated with the approximate equation:
K
dx
D
n
2
D
c
;
if
ð
dx
<
D
c
Þ;
and
n
2
K
;
if
ð
dx
>¼
D
c
Þ:
(4.11)
D
After
Y
permutations, the vectors
U
1
*
and
U
2
*
will have Hamming distance
n
1
,
D
which could be estimated as:
D
n
1
¼
2
K
D
n
;
(4.12)
where
dx
D
c
dy
D
c
n
1
1
1
;
if
dx
ð
<
D
c
Þ
and
dy
ð
<
D
c
Þ;
D
and
n
1if
ð
dx
>¼
D
c
Þ
or
ð
dy
<¼
D
c
Þ:
D
Thus, vectors
U
1
*
and
U
2
*
are correlated only if
dx
<
D
c
and
dy
<
D
c
. The
correlation increases if
dx
and
dy
are decreasing.
It is seen in Fig.
4.9
that different components of vector
U
could be placed in the
same cell after the accomplished permutations. For example, after the
X
permuta-
tions, the components
U
1
and
U
2
are allocated in the cells
U
5
;
U
3
and
U
4
are
allocated in the cell
U
9
. Such events are undesirable.
Let us consider three cases of component values:
1.
U
1
¼
1;
U
2
¼
0;
2.
U
1
¼
0;
U
2
¼
1;
3.
U
1
¼
1;
U
2
¼
1.
1. We term this
event “absorption of 1s.” Absorption leads to the partial loss of information; therefore,
it is interesting to estimate the probability of absorption in the permutation process.
All these cases will give the same result after permutations:
U
5
¼
