Geoscience Reference
In-Depth Information
Once posterior probabilities of class occurrence are derived at each pixel, they can be converted
to classification accuracy values. In this chapter, we distinguish between classification uncertainty
and classification accuracy: a measure of classification uncertainty, such as the posterior probability
of class occurrence, at a particular pixel does not pertain to the allocated class label at that pixel,
whereas a measure of classification accuracy pertains precisely to the particular class label allocated
at that pixel. We propose a simple procedure for converting posterior probability values to classi-
fication accuracy values, and we illustrate its application in the case study section of this chapter
using a realistically simulated data set.
11.2 METHODS
Let
denote a categorical random variable (RV) at a pixel with 2D coordinate vector
within a study area
C
()
u
A
. The RV
can take
K
mutually exclusive and exhaustive
u
=
(, )
u
12
C
()
u
outcomes (realizations):
alternative land-
cover types. In this chapter, we do not consider fuzzy classes, i.e., we assume that each pixel
, which might correspond to
K
{( )
c
u
=
c
,
k
=
1
,
…
, }
K
k
u
is
composed only of a single class and do not consider the case of mixed pixels.
Let
p
[ ( )]
c
u
=
Prob C
{
( )
u
=
c
}
denote the probability mass function (PMF) modeling uncer-
k
k
tainty about the
k
-th class
c
at location
u
. In the absence of any relevant information, this
k
probability
pc
k
[( )]
u
is deemed constant within the study area
A
, i.e.,
pc
*
[( )]
u
=
p
. For the set
k
k
of
K
classes, these
K
probabilities are typically estimated from the class proportions based on a
G
1
*
set of
G
training samples
c
=
[ (
c
u
),
g
=
1
,
…
,
G
]'
within the study area
A
, as
p
=
i
()
u
,
g
g
k
k
g
G
g
=
1
where
-th class, 0 if not (superscript denotes transposition).
In a Bayesian classification framework of remotely sensed imagery, these
i
k
()
u
=
1
if pixel
u
g
belongs to the
k
'
K
probabilities
{,
pk
=
1
, ,}
…
K
are termed
prior probabilities
, because they are derived before the remote sensing
k
information is accounted for.
11.2.1
Classification Based on Remotely Sensed Data
Traditional classification algorithms, such as the maximum likelihood (ML) algorithm, update
the prior probability
p
k
of each class by accounting for local information at each pixel
u
derived
from reflectance data recorded in various spectral bands. Given a vector
of reflectance values at a pixel
xu
( )
=
[
x
( ),
u
…
,
x
B
( )]'
u
1
u
in the study area, an estimate of the conditional (or posterior)
probability
p
[()| ()]
c
uxu
=
Prob C
{()
u
=
c
| ()}
xu
for a pixel
u
to belong to the
k
-th class can be
k
k
derived via Bayes' rule as:
*
*
p
[()| ()
xu
c
u
xu
=
c
]
p
*
*
k
k
p
[()| ()]
c
uxu
=
Prob
{()
C
u
=
c
| ()}
xu
=
(11.1)
k
k
*
p
[()]
*
*
where denotes the class-
conditional multivariate likelihood function, that is, the PDF for the particular spectral combination
to occur at pixel
p
[()|()
xu
c
u
== =
c
]
Prob
{ ()
X
u
x
(), ,
u
…
X
()
u
= =
x
()|()
u
c
u
c
}
k
1
1
B
B
k
xu
( )
=
[
x
( ),
u
…
,
x
B
( )]'
u
u
, given that the pixel belongs to class
k
. In the
1
*
*
denominator,
p
[ ( )]
xu
=
Prob
{
X
( )
u
=
x
( ),
u
…
x
()
,
X
( )
u
=
x
( )}
u
denotes the unconditional (mar-
1
1
B
B
ginal) PDF for the same spectral combination
to occur at the same pixel. For a particular
