Geoscience Reference
In-Depth Information
probability, which can be expressed in terms of the entries of the numerator using the law of total
probability.
Assuming class-conditional independence between the spatial and spectral information, that is,
, one can write:
p
[(),
xu c
| ()
c
u
==
c
]
p
[()| ()
xu
c
u
=◊
c
] [
p
c
| ()
c
u
=
c
]
g
k
k
g
k
p
[()| ()
xu
c
u
= ◊
c
] [
p
c
| ()
c
u
= ◊
c
]
p
k
g
k
k
pc
[( )| ( ),
uxuc
]
=
(11.8)
k
g
p
[ ( ),
xu c
]
g
suffices to
model the spectral information independently from the spatial information, and vice versa. Although
conditional independence is rarely checked in practice, it has been extensively used in the literature
because it renders the computation of the conditional probability tractable. It appears in evidential
reasoning theory (Bonham-Carter, 1994), in multisource fusion (Benediktsson et al., 1990; Bene-
diktsson and Swain, 1992), and in spatial statistics (Cressie, 1993). The consequence of this
assumption is that one can combine spectrally derived and spatially derived probabilities without
accounting for the interaction of spectral and spatial information.
Using Bayes' rule, one arrives at the final form of posterior probability under conditional
independence (Lee et al., 1987; Benediktsson and Swain, 1992):
Class-conditional independence implies that the actual class
at pixel
u
c
()
u
=
c
k
pc
[()| ()]
uxu
◊
pc
[()| ]
uc
k
k
g
p
k
pc
[( )| ( ),
uxuc
]
=
(11.9)
k
g
pc
[()| ()]
uxu
◊
pc
[()| ]
uc
pc
[()| ()]
uxu
◊
pc
[()| ]
uc
k
k
g
k
k
g
+
p
p
k
k
where
-th class and denotes the prior
probability for that event. In the case of three mutually exclusive and exhaustive classes, forest,
shrub, and rangeland, for example, if the
denotes the complement event of the
k
c
()
u
=
c
k
p
k
-th class corresponds to forest then the complement event
is the absence of forest (i.e., presence of either shrub or rangeland), and the probability for that
complement event is the sum of the shrub and rangeland probabilities.
In words, the final posterior probability that accounts for both sources of
information (spectral and spatial) under conditional independence is a simple product of the spectra-
based conditional probability and the space-based conditional probability
divided by the prior class probability
k
pc
k
[( )| ( ),
uxuc
]
g
pc
k
[( )| ( )]
uxu
p
k
pc
k
[( )|
uc
]
. Each resulting probability
g
K
Â
1
pc
k
[( )| ( ),
uxuc
]
is finally standardized by the sum
pc
k
[( )| ( ),
uxuc
]
of all resulting prob-
g
g
k
=
abilities over all
classes to ensure a unit sum.
A more intuitive version of the above fusion equation is easily obtained as:
K
pc
p
[( )|
uc
]
pc
p
[( )| ( )]
uxu
k
g
k
pc
[( )| ( ),
uxuc
]
µ
◊
◊
p
(11.10)
k
g
k
k
k
K
Â
1
where the proportionality constant is still the sum
pc
k
[( )| ( ),
uxuc
]
of all resulting probabil-
g
ities, which ensures that they sum to 1.
This version of the posterior probability equation entails that the ratio
of the final posterior probability
k
=
pc
[( )| ( ),
uxuc
]/
p
k
g
k
pc
k
[( )| ( ),
uxuc
]
to the prior probability
p
k
is simply the product
g
of the ratio
pc
[( )| ( )]/
uxu
p
of the spectrally derived preposterior probability
pc
k
[( )| ( )]
uxu
k
k
