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8: Stefan A. Robila, Pramod K. Varshney
feature extraction, it is assumed that the unmixing will lead to identification
of independent features or components.
We have developed two feature extraction algorithms based on independent
component analysis that are presented next.
8.3
Independent Component Analysis Based
Feature Extraction Algorithm (lCA-FE)
When applied to hyperspectral imagery, ICA produces only a few independent
components that contain important information. Most of the other components
are mainly associated with either noise or artifacts introduced by the sensor
or the data acquisition conditions. In the ICA-FE algorithm, first the data
dimensionality is reduced using PCA and then ICA is applied. This is due to
the fact that only a reduced number of components are relevant for further
processing by ICA, which can be recovered from the first few components of
the PCA processed data. The underlying assumption is that these components
contribute significantly to the variance, and are thus pushed in the highest
eigenvalue PCA components (Swain and Davis 1978). In this case, reduction
of the number of bands after PCA should not significantly affect the recovery
of the classes. We can, therefore, proceed to apply the ICA algorithm to the
first few principal components. The result will have the bands (components)
as independent as possible.
The steps involved in the independent component analysis feature extraction
algorithm (lCA-FE) are shown in Fig. 8.1. First, PCA is applied to the n-band
data for dimensionality reduction on the basis of eigenvalues. Following the
computation of the eigenvalues and eigenvectors for the covariance matrix of
x, the number of bands to be retained is determined by taking the highest
eigenvalues that make up a pre-specified percentage of the sum of all the
eigenvalues.
The corresponding principal components can be directly obtained by transforming the data through the eigenvectors associated with the selected eigenvalues. The ICA step described in Chap. 4 (see Sect. 4.3.2) is then applied. The
algorithm results in m-independent features (bands).
The ICA-FE algorithm is very fast, compared to the direct application ofICA
to the full data. In both cases, PCA is used as the preprocessing step. When
PCA is employed in conjunction with band reduction, the complexity of an I CA
iteration is reduced from O(n 2 p) to O(m 2 p) where n is the number of original
bands, m is the number of resulting PCA components, and p is the number of
pixel vectors.
Another difference between the ICA-FE algorithm and ICA applied on the
full data set is the fact that we no longer need to perform band selection on
the results obtained by ICA. This may, however, be a major drawback. Since
we are using only the high variance bands, we assume that all the independent
components contribute to them. In the case of a low variance independent
component, it is possible that, its contribution may be relegated to a lower
8: Stefan A. Robila, Pramod K. Varshney
feature extraction, it is assumed that the unmixing will lead to identification
of independent features or components.
We have developed two feature extraction algorithms based on independent
component analysis that are presented next.
8.3
Independent Component Analysis Based
Feature Extraction Algorithm (lCA-FE)
When applied to hyperspectral imagery, ICA produces only a few independent
components that contain important information. Most of the other components
are mainly associated with either noise or artifacts introduced by the sensor
or the data acquisition conditions. In the ICA-FE algorithm, first the data
dimensionality is reduced using PCA and then ICA is applied. This is due to
the fact that only a reduced number of components are relevant for further
processing by ICA, which can be recovered from the first few components of
the PCA processed data. The underlying assumption is that these components
contribute significantly to the variance, and are thus pushed in the highest
eigenvalue PCA components (Swain and Davis 1978). In this case, reduction
of the number of bands after PCA should not significantly affect the recovery
of the classes. We can, therefore, proceed to apply the ICA algorithm to the
first few principal components. The result will have the bands (components)
as independent as possible.
The steps involved in the independent component analysis feature extraction
algorithm (lCA-FE) are shown in Fig. 8.1. First, PCA is applied to the n-band
data for dimensionality reduction on the basis of eigenvalues. Following the
computation of the eigenvalues and eigenvectors for the covariance matrix of
x, the number of bands to be retained is determined by taking the highest
eigenvalues that make up a pre-specified percentage of the sum of all the
eigenvalues.
The corresponding principal components can be directly obtained by transforming the data through the eigenvectors associated with the selected eigenvalues. The ICA step described in Chap. 4 (see Sect. 4.3.2) is then applied. The
algorithm results in m-independent features (bands).
The ICA-FE algorithm is very fast, compared to the direct application ofICA
to the full data. In both cases, PCA is used as the preprocessing step. When
PCA is employed in conjunction with band reduction, the complexity of an I CA
iteration is reduced from O(n 2 p) to O(m 2 p) where n is the number of original
bands, m is the number of resulting PCA components, and p is the number of
pixel vectors.
Another difference between the ICA-FE algorithm and ICA applied on the
full data set is the fact that we no longer need to perform band selection on
the results obtained by ICA. This may, however, be a major drawback. Since
we are using only the high variance bands, we assume that all the independent
components contribute to them. In the case of a low variance independent
component, it is possible that, its contribution may be relegated to a lower