Hyperspectral Classification Using ICA Based Mixture Model
219
9.2
Independent Component Analysis Mixture Model (lCAMM) - Theory
Given only sensor observations that are assumed to be linear mixtures of the unobserved, statistically independent source signals, the problem of blind source
separation is to recover these independent source signals. The term "blind"
indicates that both the source signals and the way the signals were mixed are
unknown. ICA provides a solution to the blind source separation problem (see
Sect. 4.1 in Chap. 4). The goal of ICA is to perform a linear transformation
of the observed sensor signals, such that the resulting transformed signals
are as statistically independent from each other as possible. When compared
to correlation-based transformations such as PCA, ICA not only decorrelates
the sensor observations composed of mixed signals (in terms of second order
statistics), but also reduces the higher order statistical dependencies between
them. Chapter 4 provided a detailed description of ICA, where various approaches for the implementation of ICA were discussed. In this chapter, we
will employ an extended version of the information maximization (infomax)
algorithm known as the extended infomax algorithm. A detailed description
of this algorithm can be found in Lee et al. (1999).
Consider a random vector XI> whose elements are the intensity values Xi,t
of the pixel t in the spectral band i of a hyperspectral image such that Xt =
[XI,t, ... , xN,tl', where [.]' denotes the transpose. Assume that the pixel vectors
(for t = 1 to T) are obtained probabilisticallyfrom the set of classes {WI, ... ' WK}.
Class Wj is selected with prior class probability P (Wj) , so that the probability
density function for Xt may be expressed by the mixture density model as
(Duda et al. 2000),
K
P (xtl e ) = I) (xtlWj, OJ) P (Wj) ,
j=1
(9.1)
where, e = (0[, ... ,0 K) are the K class parameter vectors. The class-conditional
densities P (Xt IWj' OJ) are referred to as class-component densities, and the
prior class probabilities P (Wj) are the mixing parameters. Let, X = {XI, •• ·, XT}
be a set of T unlabeled pixel vectors, drawn independently from the mixture
model in (9.1). The likelihood of these observed samples, by definition, is the
joint density, given by,
T
P (Xle) = TIp (xtl e ) .
(9.2)
1=1
The maximum likelihood estimate of e is that value of e which maximizes
P (xle). When the class-component densities in (9.1) are modeled as multivariate Gaussian, the mixture density model is known as a Gaussian mixture
model. However, in the ICAMM algorithm, the class-component densities are
assumed to be non-Gaussian. The data within each class can be described by
219
9.2
Independent Component Analysis Mixture Model (lCAMM) - Theory
Given only sensor observations that are assumed to be linear mixtures of the unobserved, statistically independent source signals, the problem of blind source
separation is to recover these independent source signals. The term "blind"
indicates that both the source signals and the way the signals were mixed are
unknown. ICA provides a solution to the blind source separation problem (see
Sect. 4.1 in Chap. 4). The goal of ICA is to perform a linear transformation
of the observed sensor signals, such that the resulting transformed signals
are as statistically independent from each other as possible. When compared
to correlation-based transformations such as PCA, ICA not only decorrelates
the sensor observations composed of mixed signals (in terms of second order
statistics), but also reduces the higher order statistical dependencies between
them. Chapter 4 provided a detailed description of ICA, where various approaches for the implementation of ICA were discussed. In this chapter, we
will employ an extended version of the information maximization (infomax)
algorithm known as the extended infomax algorithm. A detailed description
of this algorithm can be found in Lee et al. (1999).
Consider a random vector XI> whose elements are the intensity values Xi,t
of the pixel t in the spectral band i of a hyperspectral image such that Xt =
[XI,t, ... , xN,tl', where [.]' denotes the transpose. Assume that the pixel vectors
(for t = 1 to T) are obtained probabilisticallyfrom the set of classes {WI, ... ' WK}.
Class Wj is selected with prior class probability P (Wj) , so that the probability
density function for Xt may be expressed by the mixture density model as
(Duda et al. 2000),
K
P (xtl e ) = I) (xtlWj, OJ) P (Wj) ,
j=1
(9.1)
where, e = (0[, ... ,0 K) are the K class parameter vectors. The class-conditional
densities P (Xt IWj' OJ) are referred to as class-component densities, and the
prior class probabilities P (Wj) are the mixing parameters. Let, X = {XI, •• ·, XT}
be a set of T unlabeled pixel vectors, drawn independently from the mixture
model in (9.1). The likelihood of these observed samples, by definition, is the
joint density, given by,
T
P (Xle) = TIp (xtl e ) .
(9.2)
1=1
The maximum likelihood estimate of e is that value of e which maximizes
P (xle). When the class-component densities in (9.1) are modeled as multivariate Gaussian, the mixture density model is known as a Gaussian mixture
model. However, in the ICAMM algorithm, the class-component densities are
assumed to be non-Gaussian. The data within each class can be described by