Independent Component Analysis
129
information related to the classes has concentrated in only few of the bands.
Figure 4.9a-d display four of the resulting ICA components.
Next, we analyzed four different pixel locations in the image (indicated by
arrows in Fig. 4.7a), corresponding to four of the sixteen classes road, soybean,
grass and hay. For each location, we have plotted the pixel vectors for the data
produced by ICA. For comparison, the pixel vectors from the PCA-produced
data have also been plotted (Fig. 4.10). There are significant differences between
the plots. In peA, the pixel vectors seem to lack a clear organization, i. e., none
of the features seem to have information related to the type of class. In contrast,
in ICA, we notice information belonging to one class has been concentrated
only in one or very few components.
Further visual inspection reveals that most of the pixel vectors from the
image cube display close to zero values and only one or very few peaks. This
behavior resembles that of the vectors of abundances for the pixel location.
However, since the current algorithm is designed to recover the same number
of endmembers as the number of bands, most of these endmembers will not
contribute to the image and will be associated with noise.
We note that the range of the abundances varies widely and is not restricted
to the interval [0,1], as required by the linear mixture model. Scalar multiplication, as well as shifting may solve this problem. Unfortunately, there is no clear
indication as to how the scaling may be done when dealing with components
that show up reversed. This aspect was also mentioned in general as affecting
ICA (Hyvarinen et al. 2001). The fact that the LMM conditions cannot always
be satisfied turns out to be a limitation ofICA as a linear unmixing tool.
4.5
Summary
In this chapter, we have provided theoretical background ofICA and shown its
utility as an important data processing tool for hyperspectral imaging. From
the point of view of the linear mixture model, the ability of ICA to recover an
approximation of the endmember abundances is the key indicator that ICA is
a powerful tool for hyperspectral image processing. However, the independent
components are identifiable only up to scaling and shifting with scalars. This
indicates that, in fact, the endmembers as well as their abundances can not be
determined exactly. Moreover, the number of endmembers to be determined
is fixed by the number of spectral bands available. Since there are usually
hundreds of spectral bands, it is improbable that there will be that many
distinct endmembers. This is also indicated by the result of the experiments
where we obtained only a few significant components, the rest being mostly
noise. Given this fact, one may try to find only the components that contain
useful information. Unfortunately, there is no clear method for selecting the
bands that contain useful information.
The feature-based model provides a better understanding of the usefulness
of I CA in the context of hyperspectral imagery. If we consider I CA as a feature
extraction method, the resulting features will be statistically independent. This
129
information related to the classes has concentrated in only few of the bands.
Figure 4.9a-d display four of the resulting ICA components.
Next, we analyzed four different pixel locations in the image (indicated by
arrows in Fig. 4.7a), corresponding to four of the sixteen classes road, soybean,
grass and hay. For each location, we have plotted the pixel vectors for the data
produced by ICA. For comparison, the pixel vectors from the PCA-produced
data have also been plotted (Fig. 4.10). There are significant differences between
the plots. In peA, the pixel vectors seem to lack a clear organization, i. e., none
of the features seem to have information related to the type of class. In contrast,
in ICA, we notice information belonging to one class has been concentrated
only in one or very few components.
Further visual inspection reveals that most of the pixel vectors from the
image cube display close to zero values and only one or very few peaks. This
behavior resembles that of the vectors of abundances for the pixel location.
However, since the current algorithm is designed to recover the same number
of endmembers as the number of bands, most of these endmembers will not
contribute to the image and will be associated with noise.
We note that the range of the abundances varies widely and is not restricted
to the interval [0,1], as required by the linear mixture model. Scalar multiplication, as well as shifting may solve this problem. Unfortunately, there is no clear
indication as to how the scaling may be done when dealing with components
that show up reversed. This aspect was also mentioned in general as affecting
ICA (Hyvarinen et al. 2001). The fact that the LMM conditions cannot always
be satisfied turns out to be a limitation ofICA as a linear unmixing tool.
4.5
Summary
In this chapter, we have provided theoretical background ofICA and shown its
utility as an important data processing tool for hyperspectral imaging. From
the point of view of the linear mixture model, the ability of ICA to recover an
approximation of the endmember abundances is the key indicator that ICA is
a powerful tool for hyperspectral image processing. However, the independent
components are identifiable only up to scaling and shifting with scalars. This
indicates that, in fact, the endmembers as well as their abundances can not be
determined exactly. Moreover, the number of endmembers to be determined
is fixed by the number of spectral bands available. Since there are usually
hundreds of spectral bands, it is improbable that there will be that many
distinct endmembers. This is also indicated by the result of the experiments
where we obtained only a few significant components, the rest being mostly
noise. Given this fact, one may try to find only the components that contain
useful information. Unfortunately, there is no clear method for selecting the
bands that contain useful information.
The feature-based model provides a better understanding of the usefulness
of I CA in the context of hyperspectral imagery. If we consider I CA as a feature
extraction method, the resulting features will be statistically independent. This