Geoscience Reference
In-Depth Information
{
}
-
{
}
s
k
( )
h
=
EI
(
u
+ ◊
h
)
I
( )
u
EI
(
u
+ ◊
h
)}
EI
{
( )
u
k
k
k
k
(11.3)
{
}
-
{
}
=
Prob I
(
uh
+
)
=
1
,
I
( )
u
=
1
Prob I
(
uh
+
)
= ◊
1
}
Prob I
{
( )
u
=
1
k
k
k
k
The indicator covariance
quantifies the frequency of occurrence of any two pixels of
s
k
()
h
the same category
becomes
larger, that frequency of occurrence would decrease. Note that the indicator covariance is related
to the bivariate probability
k
, found
h
distance units apart. Intuitively, as the modulus of vector
h
{
}
of two pixels of the same
k
-th category
Prob I
(
uh
+=
)
1
,
I
( )
u
=
1
k
k
being
distance units apart, and is thus related to joint count statistics. For an application of joint
count statistics in remote sensing accuracy assessment, the reader is referred to Congalton (1988).
Under second-order stationarity, the sample indicator covariance
h
s
*
()
of the
k
-th category
h
for a separation vector
h
is inferred as:
G
()
h
1
Â
*
2
s
k
()
h
=
i
(
uh u
+ ◊
)
i
(
)
-
p
(11.4)
k
g
k
g
k
G
()
h
g
=
1
where
denotes the number of training samples separated by
h
.
G
()
h
A plot of the modulus
(in the isotropic case) of several vectors
vs. the
h
l
{,
h
l
l
= 1
, ,}
…
L
{
}
corresponding covariance values
*
( ,
constitutes the sample covariance function.
s
kl
h
l
=
1
, ,
…
L
{
}
Parametric and positive definite covariance models
for any arbitrary vector
h
S
k
=
s
()
hh
"
k
are then fitted to the sample covariance functions. The parameters of these functions (e.g., covariance
function type, relative nugget, or range) might be different from one category to another, indicating
different spatial patterns of, say, land-cover types. For a particular separation vector
h
, the corre-
sponding model-derived indicator covariance is denoted as .
The spatial information of the training pixels is encoded partially in the indicator covariance
model
s
k
()
h
-th category and partially in their actual location and class label. In Fourier
analysis jargon, the covariance model provides amplitude information (i.e., textural infor-
mation), whereas the actual locations of the training samples and their class labels provide phase
information (i.e., location information). Taken together, locations and covariance of training pixels
provide contextual information that can be used in the classification procedure.
Ordinary indicator kriging (OIK) is a nonparametric approximation to the conditional PMF
for the
for the
k
s
k
()
h
s
k
()
h
k
-th class to occur at pixel
u
, given the spatial infor-
pc
[ ( ) |
uc
]
=
Prob{ ( )
C
u
=
c
|
c
}
k
g
k
g
mation encapsulated in the
G
training samples
; see Van der Meer (1996),
c
=
[ (
c
u
),
g
=
1
,
…
,
G
]'
g
g
and Goovaerts (1997) for details. The OIK estimate
*
[( )|
for the conditional PMF
pc
k
uc
]
g
that the
k
-th class prevails at pixel
u
is expressed as a weighted linear combination
pc
k
[( )|
uc
G
()
]
g
of the
sample indicators
k
for the same
k
-th class found in a
u
i
=
[
i
(
u
),
g
=
1
,
…
,
G
( )]'
u
k
g
neighborhood
centered at pixel
u
:
N
()
u
G
()
u
Â
1
*
*
k
*
k
pc
[ ( ) |
uc
]
ª
pc
[ ( ) |
ui
]
=
Prob { ( )
C
u
=
c
|
i
}
=
w
(
u
)
◊
i
(
u
)
(11.5)
k
g
k
k
k
g
k
g
g
=
G
()
u
Â
under the constraint
w
k
()
u
=
1
; this latter constraint allows for local, within-neighborhood
g
g
=
1
N
()
u
, departures of the class proportion from the prior (constant) proportion
. In the previous
p
k
equation,
w
k
()
u
denotes the weight assigned to the
g
-th training sample indicator of the
k
-th category
i
k
()
u
for estimation of
pc
k
[( )|
uc
]
for the same
k
-th category at pixel
u
. The size of the neigh-
g
borhood
N
()
u
is typically identified to the range of correlation of the indicator covariance model
S
k
.
