Independent Component Analysis
123
In the second case, the neg entropy is approximated using skewness and
kurtosis:
1
1
J(x) = -E {x 3 ) + -K(x)2 ,
12
48
(4.47)
where E {x 3 ) estimates the skewness and K ( .) denotes the kurtosis.
If the data is initially whitened, E {x 3 ) = 0, (4.47) becomes:
J(x) = K(x)2 .
(4.48)
In most of the ICA algorithms that are based on non-Gaussianity maximization, the components of u in (4.42) are generated in a sequential order with
each component constrained to be orthogonal to the ones previously generated. Other algorithms that allow simultaneous generation of the components
have also been proposed (Hyvarinen et al. 2001; Chang et al. 2002).
To summarize, ICA is an active area of research and new solutions are being
continuously developed. Therefore, only the most common approaches to
solving the ICA problem have been presented here. For additional approaches,
see Cichocki and Amari (2002); Haykin (2000); Hyvarinen et al. (2001); Lee
1998).
4.4
Application of ICA to Hyperspectrallmagery
A limited amount of research on the use of ICA to process hyperspectral data
has been reported in the literature (e. g. Tu et al. 2001; Chang et al. 2002; Chen
and Zhang 1999; Robila and Varshney 2002; Bayliss et al. 1997; Robila et al.
2000; Parra et al. 2000 Healy and Kuan 2002). There are many reasons for the
scarcity of research in this important area. First, both ICA and hyperspectral
imaging have evolved mostly in the last few years and the research connecting
them is still at the incipient stage. Second, the size of the hyperspectral data
to be processed coupled with the iterative nature of lCA leads to very large
computational times, reducing the appeal of the method. Third, the initial
research has focused on direct application ofICA without considering how the
hyperspectral data is modeled, how the resulting components are interpreted
and whether modifications of the original algorithms are required (Tu et al.
2001; Chen and Zhang 1999; Bayliss et al. 1997). Only recently, ICA has been
linked with the linear mixing model used in hyperspectral image processing
where it has been shown to provide an approximation of the solution to the
linear un mixing problem (Chang et al. 2002).
Most of the research investigating the applicability of ICA to hyperspectral imagery has been focused on the maximization of non-Gaussianity (i. e.
kurtosis) of each component. This approach is similar to projection pursuit
(Hyvarinen et al. 2001) and provides a good approximation of the ICA solution
only when the most non-Gaussian components correspond to the independent
123
In the second case, the neg entropy is approximated using skewness and
kurtosis:
1
1
J(x) = -E {x 3 ) + -K(x)2 ,
12
48
(4.47)
where E {x 3 ) estimates the skewness and K ( .) denotes the kurtosis.
If the data is initially whitened, E {x 3 ) = 0, (4.47) becomes:
J(x) = K(x)2 .
(4.48)
In most of the ICA algorithms that are based on non-Gaussianity maximization, the components of u in (4.42) are generated in a sequential order with
each component constrained to be orthogonal to the ones previously generated. Other algorithms that allow simultaneous generation of the components
have also been proposed (Hyvarinen et al. 2001; Chang et al. 2002).
To summarize, ICA is an active area of research and new solutions are being
continuously developed. Therefore, only the most common approaches to
solving the ICA problem have been presented here. For additional approaches,
see Cichocki and Amari (2002); Haykin (2000); Hyvarinen et al. (2001); Lee
1998).
4.4
Application of ICA to Hyperspectrallmagery
A limited amount of research on the use of ICA to process hyperspectral data
has been reported in the literature (e. g. Tu et al. 2001; Chang et al. 2002; Chen
and Zhang 1999; Robila and Varshney 2002; Bayliss et al. 1997; Robila et al.
2000; Parra et al. 2000 Healy and Kuan 2002). There are many reasons for the
scarcity of research in this important area. First, both ICA and hyperspectral
imaging have evolved mostly in the last few years and the research connecting
them is still at the incipient stage. Second, the size of the hyperspectral data
to be processed coupled with the iterative nature of lCA leads to very large
computational times, reducing the appeal of the method. Third, the initial
research has focused on direct application ofICA without considering how the
hyperspectral data is modeled, how the resulting components are interpreted
and whether modifications of the original algorithms are required (Tu et al.
2001; Chen and Zhang 1999; Bayliss et al. 1997). Only recently, ICA has been
linked with the linear mixing model used in hyperspectral image processing
where it has been shown to provide an approximation of the solution to the
linear un mixing problem (Chang et al. 2002).
Most of the research investigating the applicability of ICA to hyperspectral imagery has been focused on the maximization of non-Gaussianity (i. e.
kurtosis) of each component. This approach is similar to projection pursuit
(Hyvarinen et al. 2001) and provides a good approximation of the ICA solution
only when the most non-Gaussian components correspond to the independent