Hyperspectral Data Processing: Algorithm Design and Analysis

6.4 Dimensionality Reduction by Infinite-Order Statistics-Based Components Analysis Transforms

In Section 6.3, transforms using the kth order of statistics with any as an optimal criterion were presented. When the k becomes infinite, that is, statistics of infinite order, the approaches using (6.44), (6.47), and (6.52) in Section 6.3 are no longer applicable for . To address this issue, two approaches have been investigated. One is Projection Pursuit (PP) discussed in Chapter 16 in Chang (2003a) that uses a projection index as a criterion to find an optimal projection vector. When a projection index is specified by a criterion of the kth order of statistics for , the PP is then reduced to transforms in Sections 6.2 and 6.3, particularly, PCA for k = 2, skewness for k = 3, and kurtosis for k = 6. The other is independent component analysis (ICA) that uses mutual information to de-correlate statistical dependency. Since a probability distribution can be fully described by its moment-generating function with infinite number of moments, theoretically ICA can be viewed as a transform using an infinite-order logical extension of any kth high-order component transforms and will be considered ...