An MRF Model Based Approachfor Sub-pixel Mapping from Hyperspectral Data
259
optimum sub-pixel mapping approach are provided in Sect. 11.3. Experimental
results are discussed in Sect. 11.4 followed by a summary of the chapter in
Sect. 11.5.
11.2
MRF Model for Sub-pixel Mapping
Let Y be the observed coarse spatial resolution image having M x N pixels and
X be the fine resolution sub-pixel map (SPM) having aM x aN pixels where a is
the scale factor of the SPM. This means that a particular pixel in the observed
coarse resolution image contains a 2 pixels of the SPM. We assume that the pixels
in the fine resolution image are pure and that mixed pixels can only occur in the
observed coarse resolution image. Thus, more than one class can occupy a pixel
in a coarse resolution image. In general, a can be any positive real number, but,
for simplicity, here it is assumed to be a positive integer number. Also, let .8
and 7 denote the sets of all sites (i. e. pixels) belonging to the observed image
and the SPM, respectively. Thus, the number of sites belonging to 7 will be a 2
times the number of sites in the set .8. Let 7 j = {~, ... , ~2} represent the set
of all the pixels in 7 that correspond to the same area as the pixel Sj in .8. The
observed coarse resolution multi or hyperspectral image is usually represented
in vector form so that, Y(Sj) E IRK for the pixel Sj where IR denotes the set of
real numbers (e. g. intensity values) and K is the number of spectral bands. As
stated earlier, each pixel in the SPM is assumed pure. That is, its configuration
x(t) (i. e. attribute) denotes one and only one class. Hence, x(t) E {I, ... , L}
can only take an integer value corresponding to the class at a pixel t in the
actual scene, where L is the number of classes. There can be L a
2 MN dissimilar
sub-pixel maps X (7) E {1, ... , L} T each having a different class allocation in
at least one pixel.
It is further assumed that the SPM has the MRF property, i. e., the conditional probability of a configuration (i. e. intensity value) of a pixel given the
configurations of the entire image excluding the pixel of interest is equal to
the conditional probability of the configuration of that pixel given the configurations of its neighboring pixels. This can mathematically be represented
as
Pr (x(t) IX (7 - (t})) = Pr (x(t) IX(Nt )) ,
(1l.1)
where 7 - It} is the set of all the pixels in 7 excluding the pixel t, and Nt is
the set of pixels in the neighborhood of pixel t. For example, in the context of
classification of remotely sensed images, this property implies that the same
class is more likely to occur in connected regions than at isolated pixels. Hence,
the conditional probability density functions (pdfs) in (I l.1) have a higher
value if the configuration of a pixel t is similar to the configurations of its
neighboring pixels than the cases when it is not. From Winkler (1995) and
Bermaud (1999), the marginal PDF of X takes the form of Gibbs distribution,
259
optimum sub-pixel mapping approach are provided in Sect. 11.3. Experimental
results are discussed in Sect. 11.4 followed by a summary of the chapter in
Sect. 11.5.
11.2
MRF Model for Sub-pixel Mapping
Let Y be the observed coarse spatial resolution image having M x N pixels and
X be the fine resolution sub-pixel map (SPM) having aM x aN pixels where a is
the scale factor of the SPM. This means that a particular pixel in the observed
coarse resolution image contains a 2 pixels of the SPM. We assume that the pixels
in the fine resolution image are pure and that mixed pixels can only occur in the
observed coarse resolution image. Thus, more than one class can occupy a pixel
in a coarse resolution image. In general, a can be any positive real number, but,
for simplicity, here it is assumed to be a positive integer number. Also, let .8
and 7 denote the sets of all sites (i. e. pixels) belonging to the observed image
and the SPM, respectively. Thus, the number of sites belonging to 7 will be a 2
times the number of sites in the set .8. Let 7 j = {~, ... , ~2} represent the set
of all the pixels in 7 that correspond to the same area as the pixel Sj in .8. The
observed coarse resolution multi or hyperspectral image is usually represented
in vector form so that, Y(Sj) E IRK for the pixel Sj where IR denotes the set of
real numbers (e. g. intensity values) and K is the number of spectral bands. As
stated earlier, each pixel in the SPM is assumed pure. That is, its configuration
x(t) (i. e. attribute) denotes one and only one class. Hence, x(t) E {I, ... , L}
can only take an integer value corresponding to the class at a pixel t in the
actual scene, where L is the number of classes. There can be L a
2 MN dissimilar
sub-pixel maps X (7) E {1, ... , L} T each having a different class allocation in
at least one pixel.
It is further assumed that the SPM has the MRF property, i. e., the conditional probability of a configuration (i. e. intensity value) of a pixel given the
configurations of the entire image excluding the pixel of interest is equal to
the conditional probability of the configuration of that pixel given the configurations of its neighboring pixels. This can mathematically be represented
as
Pr (x(t) IX (7 - (t})) = Pr (x(t) IX(Nt )) ,
(1l.1)
where 7 - It} is the set of all the pixels in 7 excluding the pixel t, and Nt is
the set of pixels in the neighborhood of pixel t. For example, in the context of
classification of remotely sensed images, this property implies that the same
class is more likely to occur in connected regions than at isolated pixels. Hence,
the conditional probability density functions (pdfs) in (I l.1) have a higher
value if the configuration of a pixel t is similar to the configurations of its
neighboring pixels than the cases when it is not. From Winkler (1995) and
Bermaud (1999), the marginal PDF of X takes the form of Gibbs distribution,