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In practice, this means after optimizing the parameters,
β
becomes insignificant
compared to the values
α
f
i
. A solution to this problem is given in the next
section.
3 A Custom Potential Function
One way to resolve the issue described above, would be to adapt the color seg-
mentation to obtain a more intelligent labeling. However, this means dealing with
an issue we are trying to avoid as much as possible: setting hard thresholds in
color space. As an alternative way to improve the false positive rate, we propose
a novel potential function. Rather than specifying a constant value for
β
, it will
now depend on the relative occurrence of the particular clique in a foreground
(fire) and background model. Note that this means a departure from the MLL
theory. For every possible pair of color labels, a potential value is now calculated
beforehand, based on training data. Let
C
f
(
f
i
,f
j
) denote the number of times
a pairwise clique consisting of the labels
f
i
and
f
j
occurred over all fire areas in
the training data, and likewise
C
b
(
f
i
,f
j
) the number of times it occurred over
all background areas. We will then estimate the occurrence probabilities of the
clique in foreground and background as
1+
C
f
(
f
i
,f
j
)
f
i
,f
j
∈L
P
f
(
f
i
,f
j
)=
(8)
C
f
(
f
i
,f
j
)+
|
L
|
2
1+
C
b
(
f
i
,f
j
)
f
i
,f
j
∈L
P
b
(
f
i
,f
j
)=
.
(9)
C
b
(
f
i
,f
j
)+
|
L
|
2
Note that we added 1 to the occurrence counts of each clique to avoid probabili-
ties of zero, as is common practice (e.g. for training a Bayes classifier). This gives
rise to the term
2
in the denominator. The potential function
V
2
we propose
|
L
|
is then given by
⎧
⎨
P
b
(
f
i
,f
j
)
|
f
i
=
f
j
P
b
(
f
i
,f
j
)+
P
f
(
f
i
,f
j
)
V
2
(
f
i
,f
i
)=
(10)
⎩
f
i
,f
j
)
P
b
(
f
i
,f
j
)+
P
f
(
f
i
,f
j
)
P
f
(
−
|
f
i
=
f
j
The value in the first case is the probability that, if this particular clique occurs,
it is caused by the background model. Likewise the value in the second case is
the probability that it is caused by the fire model. While this potential function
is obviously heuristic, it implements the functionality we require:
- cliques of uniform color are penalized, but more so for unlikely fire colors,
- cliques of different color are encouraged, but more so for typical fire combi-
nations.
Experiments show that with the new potential function
V
2
, the areas near the
edges of flames generate very low energy, while the entire background results in
much higher energy values. The interior part of the flame falls in between, on av-
erage generating more energy than the flame edge but less than the background.
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