Hyperspectral Data Processing: Algorithm Design and Analysis

5.1 Introduction

How to represent multidimensional data in an appropriate form in some sense of optimality is very challenging. There are two key issues involved. One is how to determine the number of basic elements, referred to as p, present in the data. The other is how to find these p basic elements. While these two issues may be addressed separately, they can also be treated as one issue if they are tied together in a certain form. This chapter investigates VD from these two aspects.

The VD proposed in Chang (2003a) is essentially developed from the first aspect without constructing the basic elements. According to the definition provided in Fukunaga (1990) for ID, it is defined as the minimal number of parameters required to account for the observed properties of the data that must be specifically defined. As a matter of fact, the properties observed in the data are generally determined by these basic elements that vary with applications. In other words, when high-dimensional data sets are observed, the data sample vectors are generally represented in a certain form of dimensionality. Most commonly it is the dimensionality of a data sample vector, referred to as data dimensionality, which is defined by the number of coordinates used to represent the data sample vector. As an alternative, in many applications in order for a data set to be represented more effectively, the original data dimensions are usually transformed to components via a transformation where each data component ...