Independent Component Analysis
125
Fig. 4.5. Model of an n-band multispectralJhyperspectral data set as a random vector. Each
band is associated with a component of the random vector. Each realization of the random
vector corresponds to a pixel vector from the image cube
and near infrared). Same conclusions may be drawn by inspecting data in
more than two dimensions. However, the visualization becomes difficult, if not
impossible, once the dimensionality exceeds three.
Thus, a hyperspectral image cube can be associated with a random vector
with each pixel vector corresponding to an observation. The number of observations is, therefore, equal to the size (in pixels) of the image. Individually,
for each component (or feature) of the random vector, the observations are
the intensity values of the pixels in the associated spectral band (see Fig. 4.5).
This model can then be used for extracting features containing most of the
information by applying ICA, as is done in PCA (Richards and Jia 1999).
4.4.2
Linear Mixture Model Based Model
A second model can be drawn from the linear mixture model (LMM). In
LMM, each pixel vector is assumed to be a linear combination of a finite
set of endmembers. For a specific endmember, the contribution (abundance)
corresponding to each pixel vector in the hyperspectral image can be seen as
an image band itself. The relationship between LMM and ICA is achieved by
considering the columns in the ICA mixing matrix to represent endmembers
as described in the linear mixture model, and each independent component as
the abundance of an endmember (Chang et al. 2002).
A precise equivalence between LMM and ICA is not possible. In LMM, the
abundances of the endmembers are not required to be independent. However,
since in LMM we are looking for the most representative pixel vectors, it
seems natural to modify the model by assuming that the abundance of one
endmember in a specific pixel does not provide any information regarding the
abundance of other endmembers for that pixel. Another major difference is
the presence of noise in the LMM, which is not considered in ICA. However, we
may include noise in ICA as Gaussian noise with its abundances assumed to be
independent of one of the endmembers. This satisfies one of the restrictions
125
Fig. 4.5. Model of an n-band multispectralJhyperspectral data set as a random vector. Each
band is associated with a component of the random vector. Each realization of the random
vector corresponds to a pixel vector from the image cube
and near infrared). Same conclusions may be drawn by inspecting data in
more than two dimensions. However, the visualization becomes difficult, if not
impossible, once the dimensionality exceeds three.
Thus, a hyperspectral image cube can be associated with a random vector
with each pixel vector corresponding to an observation. The number of observations is, therefore, equal to the size (in pixels) of the image. Individually,
for each component (or feature) of the random vector, the observations are
the intensity values of the pixels in the associated spectral band (see Fig. 4.5).
This model can then be used for extracting features containing most of the
information by applying ICA, as is done in PCA (Richards and Jia 1999).
4.4.2
Linear Mixture Model Based Model
A second model can be drawn from the linear mixture model (LMM). In
LMM, each pixel vector is assumed to be a linear combination of a finite
set of endmembers. For a specific endmember, the contribution (abundance)
corresponding to each pixel vector in the hyperspectral image can be seen as
an image band itself. The relationship between LMM and ICA is achieved by
considering the columns in the ICA mixing matrix to represent endmembers
as described in the linear mixture model, and each independent component as
the abundance of an endmember (Chang et al. 2002).
A precise equivalence between LMM and ICA is not possible. In LMM, the
abundances of the endmembers are not required to be independent. However,
since in LMM we are looking for the most representative pixel vectors, it
seems natural to modify the model by assuming that the abundance of one
endmember in a specific pixel does not provide any information regarding the
abundance of other endmembers for that pixel. Another major difference is
the presence of noise in the LMM, which is not considered in ICA. However, we
may include noise in ICA as Gaussian noise with its abundances assumed to be
independent of one of the endmembers. This satisfies one of the restrictions