Hyperspectral imaging data are typically treated as vectors in a high-dimensional space. Dimensionality reduction refers to a variety of techniques that enable the representation of vector data using a lower number of components without significant loss of information. The information in a data set is conveyed by the geometrical arrangement of points in the p-dimensional scatter plot rather than by the system of coordinates used for their specification. Therefore, besides the natural coordinate system defined by the original variables (spectral channels), we may wish to analyze the data in other coordinate systems with more desirable properties. Typically, the new coordinates are derived variables without any physical meaning. In this chapter we introduce linear spectral transformations for dimensionality reduction and “feature enhancement” of hyperspectral imaging applications. The most widely used technique is principal component analysis, followed by discriminant analysis, and canonical correlation analysis. We conclude with the related subject of spectral band prediction and its applications.

Introduction

The dimension of hyperspectral imaging data space is equal to the number of the spectral channels used by the sensor, which is typically in the range of a few hundred channels. High dimensionality has two important implications in hyperspectral data processing and exploitation. First, the resulting huge volume of data requires tremendous storage and processing resources. Second, the high-dimensionality of the feature space leads to a large increase in the amount of data required for statistically oriented detection and classification algorithms.

Given a hyperspectral data cube, dimensionality reduction can be achieved in essentially two different approaches. The first, which is called band selection or feature selection, attempts to identify a subset of the original bands that contribute to performance. Given a set of p bands, what is the best subset of size m? To solve this problem, we should evaluate the adopted optimality criterion for all possible combinations of m bands out of p and select the combination that minimizes or maximizes the criterion. The main problem is that the number of possible combinations, which is given by p!/[(p-m)!m!], the selected subset is extremely large, even for small values of m and p. There exist both optimum and suboptimum search methods, but they are, in general, computationally demanding or infeasible (Webb and Copsey, 2011).