Hyperspectral Data Processing: Algorithm Design and Analysis

29.2 Wavelet Analysis

Wavelet analysis is a widely used technique in signal processing and communications, where its applications range from one-dimensional (1D) signal processing, such as speech, sonar, and audio processing, to multidimensional signal processing, such as two-dimensional (2D) image processing and three-dimensional (3D) video processing. One of the major features of wavelet analysis is the use of the so-called scaling function to generate a set of wavelets that decompose signals in multiple pair-wise disjoint orthogonal representations, referred to as signal resolutions. When the signals to be considered are one dimensional, the multiple signal resolutions are referred in this chapter to as multiple signal scales. With this interpretation, the resulting multiple pair-wise disjoint orthogonal representations are then called multiscale signal representation. On the other hand, if the signals to be considered are 2D or 3D images, the multiple signal resolutions are referred to as image resolutions and the resulting multiple pair-wise disjoint orthogonal representations are then called multiple image resolutions. Since the main focus of this chapter is 1D signal processing, the term “multiscale” will be used throughout this chapter.

29.2.1 Multiscale Approximation

The idea of multiscale approximation of wavelet analysis is briefly reviewed in this section. First, let Z and R denote the sets of integers and real numbers, respectively, and L²(R) denote the vector space ...