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9: Chintan A. Shah
the ICA model (Lee et al. 1999) with the addition of an N-dimensional bias
vector bj as,
(9.3)
In (9.3), vector Xt, corresponds to the N-dimensional random vector with its
dimensionality reduced by an appropriate feature extraction technique such
that N = M, where M is the number of unobserved independent sources.
Aj is a N x N, full rank, real-valued mixing matrix. Additionally, the vector
Sj,t = [Sj,l,T, ••. , Sj,N,T J' for class j in (9.3) corresponds to the random vector
of N independent, unobserved, real-valued, source signals 5 = [51> ••• , SN]' as
defined for the ICA model (Lee et al. 1999) (see Chap. 4 for more details).
Since the hyperspectral data X is modeled as a mixture of K classes, and each
of these K classes are described by a linear combination of N independent,
non-Gaussian sources, the generative model in (9.3) depicts one of the K ways
for generating the random pixel vector Xt. Thus, depending on the values of
the class parameters OJ = {Aj, bj}, and the unobserved independent sources
Sj,t corresponding to each class j, there are K ways for viewing Xt (i. e. Xt =
A1S1,t+bl, .•• ,Xt = Ajsj,t+bj, .•• , Xt = AKSK,t+bK). Additionally, it is significant
to note that all the assumptions and restrictions (see Sect. 4.2) necessary for
the identifiability of the ICA solution are implicitly inherited by the ICAMM
algorithm as well.
Expressing (9.3) as
(9.4)
we get further insight into the generative model. Here we subtract the bias for
class j from the random pixel vector Xt> which is then linearly transformed by
the unmixing matrix W; = Ai 1 , to obtain the unobserved independent sources
Sj,t> corresponding to class j. As mentioned before, we assume that the observed
hyperspectral sensor observations X are a mixture of several mutually exclusive
classes, and that the maximum likelihood estimation results in the model that
best fits the data.
The steps involved in the ICAMM algorithm for an unsupervised classification of remote sensing images are provided in the following discussion.
9.2.1
ICAMM Classification Algorithm
A detailed mathematical derivation of the ICAMM algorithm can be found in
(Shah 2003; Lee et al. 2000; Lee and Lewicki 2002). In the present discussion,
we only provide the steps involved in the ICAMM classification algorithm.
Step 1:
l. Initialize the total number of classes K.
9: Chintan A. Shah
the ICA model (Lee et al. 1999) with the addition of an N-dimensional bias
vector bj as,
(9.3)
In (9.3), vector Xt, corresponds to the N-dimensional random vector with its
dimensionality reduced by an appropriate feature extraction technique such
that N = M, where M is the number of unobserved independent sources.
Aj is a N x N, full rank, real-valued mixing matrix. Additionally, the vector
Sj,t = [Sj,l,T, ••. , Sj,N,T J' for class j in (9.3) corresponds to the random vector
of N independent, unobserved, real-valued, source signals 5 = [51> ••• , SN]' as
defined for the ICA model (Lee et al. 1999) (see Chap. 4 for more details).
Since the hyperspectral data X is modeled as a mixture of K classes, and each
of these K classes are described by a linear combination of N independent,
non-Gaussian sources, the generative model in (9.3) depicts one of the K ways
for generating the random pixel vector Xt. Thus, depending on the values of
the class parameters OJ = {Aj, bj}, and the unobserved independent sources
Sj,t corresponding to each class j, there are K ways for viewing Xt (i. e. Xt =
A1S1,t+bl, .•• ,Xt = Ajsj,t+bj, .•• , Xt = AKSK,t+bK). Additionally, it is significant
to note that all the assumptions and restrictions (see Sect. 4.2) necessary for
the identifiability of the ICA solution are implicitly inherited by the ICAMM
algorithm as well.
Expressing (9.3) as
(9.4)
we get further insight into the generative model. Here we subtract the bias for
class j from the random pixel vector Xt> which is then linearly transformed by
the unmixing matrix W; = Ai 1 , to obtain the unobserved independent sources
Sj,t> corresponding to class j. As mentioned before, we assume that the observed
hyperspectral sensor observations X are a mixture of several mutually exclusive
classes, and that the maximum likelihood estimation results in the model that
best fits the data.
The steps involved in the ICAMM algorithm for an unsupervised classification of remote sensing images are provided in the following discussion.
9.2.1
ICAMM Classification Algorithm
A detailed mathematical derivation of the ICAMM algorithm can be found in
(Shah 2003; Lee et al. 2000; Lee and Lewicki 2002). In the present discussion,
we only provide the steps involved in the ICAMM classification algorithm.
Step 1:
l. Initialize the total number of classes K.