Biomedical Engineering Reference
In-Depth Information
N
−1+
min
{
i
,
j
}−
max
{
i
,
j
}
min
{
N
-
i
,
j
}−1+
N
−
max
{
N
−
i
,
j
}
N
−1+
min
{
i
,
j
}−
max
{
i
,
j
}
(
i
,
j
)
(
i
,
j
)
Figure 4.18
Illustration of the conditions that lead to incorrect classification of candidate defects for
(a) single-diagonal diagnosis and (b) cross-diagonal diagnosis.
defect in the same column as the electrode; (2) there is a defect in the same
row as the electrode; and (3) there are defects on one or both the diagonals
on which the electrode lies. These situations are illustrated in Figures 4.18a
and 4.18b.
If a single-diagonal diagnosis is carried out as in Figure 4.18a, the probability
P
incorrect
of an incorrect classification for an electrode is given by:
P
incorrect
= (1 −
p
)(1 − (1 −
p
)
N−1
)
2
(1 − (1 −
p
)
N−1+min{i, j}+-max{i, j}
)
If a cross-diagonal diagnosis is carried out as in Figure 4.18b, the probability
of a false defect is
(1 −
p
)(1 − (1 −
p
)
N−1
)
2
(1 − (1 −
p
)
N−1 + min{i,j}− max{i,j}
)(1 − (1 −
p
)
N−1+min{N−i,j}− max{N−i,j}
)
Using these equations, we calculate the probability of a false defect occur-
rence under different probabilities of a defect occurrence. The results are
shown in Figure 4.19.
In Figure 4.19, it can be seen that a significant increase in diagnostic reso-
lution, that is, the ratio of the number of actual defects to the number of
classified defects, is achieved by carrying out a single-diagonal diagnosis.
Further improvement can be achieved when a cross-diagonal diagnosis is
applied. However, the difference in the results for cross-diagonal diagnosis
and single-diagonal diagnosis is less apparent for smaller values of the
defect occurrence probability.
4.3.3 Occurrence Probability of untestable Sites
Next, we analyze the probability of the occurrence of untestable sites. Again,
assume that each electrode is failing with the same probability
p
. The elec-
trode is untestable if there is one real defect in each of its four directions, as
