Geoscience Reference
InDepth Information
{
}
{
}
mean vectors
o
. A set of
K
simulated (6
¥
1) vectors
of class
m
X

,
k
=
1
,
…
,
K
m
X

,
k
k
=
1
,
…
,
K
k
conditional means are generated from a sixvariate Gaussian distribution with mean
and
m
X
{
}
covariance . In the case study, simulated classconditional mean vectors
were used instead of their original counterparts in order to introduce class
confusion. Simulated reflectance values are then generated for each pixel in the reference classifi
cation from the appropriate classconditional distribution, which is assumed Gaussian with mean
, and covariance
S
m
X

,
k
k
=
1
,
…
,
K
{
}
o
m
X

,
k
=
1
,
…
,
K
k
o
. For example, if a pixel in the reference classification has LC forest
m
X

k
S
X

k
(
1), six simulated reflectance values are simulated at that pixel from a Gaussian distribution
with mean and covariance . A similar procedure for generating synthetic satellite imagery
(but without the simulation of classconditional mean values ) was adopted by
Swain et al. (1981) and Haralick and Joo (1986). The simulated reflectance values are further
degraded by introducing white noise generated by a sixvariate Gaussian distribution with mean
k =
o
m
X
1
S
X
1
{
}
m
X

,
k
k
=
1
,
…
,
K
0
and (co)variance 0.2
; this entails that the simulated noise is correlated from one spectral band
S
to another.
Independent simulation of reflectance values from one pixel to another implies the nonrealistic
feature of low spatial correlation in the simulated reflectance values. In the case study, in order to
enhance spatial correlation as well as positional error, typical of real images, a motion blur filter
with a horizontal motion of 21 pixels in the 45˚ direction was applied to each band to simulate
the linear motion of a camera. The resulting reflectance values were further degraded by addition
of a realization of an independent multivariate white noise process, which implies correlated noise
from one spectral band to another. This latter realization was generated using a multivariate Gaussian
distribution with mean
and (co)variance 0.05 . To avoid edge effects introduced by the motion
blur filter, the results of Gaussian maximum likelihood classification, as well as those for indicator
kriging, were reported on a smaller (cropped) subscene.
The last step in the simulated TM data generation consists of a bandbyband histogram
transformation: the histogram of reflectance values for each spectral band in the simulated image
is transformed to the histogram of the original TM reflectance values for that band through histogram
equalization. The purpose of this transformation is to force the simulated TM imagery to have the
same histogram as that of the original TM imagery, as well as similar covariance among bands.
The (transformed) simulated reflectance values are finally rounded to preserve the integer digital
nature of the data.
0
S
11.3 RESULTS
To illustrate the proposed methodology for fusing spatial and spectral information for mapping
thematic classification uncertainty, a case study was conducted using simulated imagery based on
a Landsat Thematic Mapper subscene from path 41/row 27 in western Montana, and the procedure
described in Section 11.2.5. The TM imagery, collected on September 27, 1993, was supplied by
the U.S. Geological Survey's (USGS) Earth Resources Observation Systems (EROS) Data Center
and is one of a set from the MultiResolution Land Characteristics (MRLC) program (Vogelmann
et al., 1998). The study site consisted of a subscene covering a portion of the Lolo National Forest
(541
414 pixels). The original 30m TM data served as the basis for generating the simulated
TM imagery used in this case study.
The subscene was classified into
¥
L =
150
clusters using the ISODATA algorithm, and these
L
clusters were assigned to
K =
3 classes: forest (
k =
1), shrub (
k =
2), and rangeland (
k =
3). The
resulting classification was smoothed using MAP selection within a 5
¥
5 window around each
pixel
. The resulting LC map is regarded as the exhaustive reference classification (unavailable
in practice). A small subset (
u
414 pixels (0.14% of the total population) was
selected as training pixels through stratified random sampling. The sample and reference class
proportions of forest, shrub, and rangeland were
G =
314) of the 541
¥
,
, and
, respec
p
1
=
065
.
p
2
=
021
.
p
3
=
014
.