Hyperspectral Data Processing: Algorithm Design and Analysis

6.6 Dimensionality Reduction by Feature Extraction-Based Transforms

Sections 6.2–6.46.2–6.4 present component transforms using criteria of statistics of any order. These transforms find a new set of component images that can represent the image data where each component image is specified by a projection vector produced by an appropriately selected criterion. In this section, we consider another type of transforms, feature transforms, to perform DR. Instead of representing image data by a set of component images, a feature transform projects image data into a feature space specified by a set of feature vectors that are obtained by a feature extraction-based criterion where each image data sample can be expressed in terms of its generated feature vectors. Two feature-based transforms, discriminant analysis and classification, are of interest and will be discussed in the following sections.

6.6.1 Fisher's Linear Discriminant Analysis

PCA uses data variances as an indication to point out the directions where the data cloud will be centered, but it does not necessarily point out the directions where different classes can be best separated. In order to resolve this issue, an approach called canonical analysis is developed (Richards and Jia, 1999) which uses a feature selection-based criterion that is the ratio of among-class variances to within-class variances so as to achieve best possible class separability. For two-class classification for image thresholding, the canonical analysis ...