Mathematics of Multilevel Systems

The following sections are included:

• Preface
• About the Authors
• Road-map
• Contents

The following sections are included:

• Organization of the Book: A Bird’s Eye View, and Tips for the Reader
• Motivation

This chapter presents how wavelet theory is implemented as algorithms for wavelet digital color image decomposition and reconstruction in image compression. We take various wavelets as concrete examples, discuss their mathematical properties such as orthogonality, vanishing moments, etc., and show the mathematical algorithm for wavelet image decomposition and reconstruction with image matrix, and finally present the MATLAB wavelet image compression results [Son06a, Son06b, MMOP02] (Figure 2.1).

In Chapter 2, we discussed discrete wavelet transforms explicitly and their applications in wavelet image compression. In this chapter we outline a number of direct links between the two cases of wavelet analysis, continuous and discrete. The theme of the first is perhaps best known, for example the creation of compactly supported wavelets in L²(ℝⁿ) with suitable properties such as localization, vanishing moments, and differentiability. The second (discrete) deals with computation, with sparse matrices, and with algorithms for encoding digitized information such as speech and images. This is centered on constructive approaches to subdivision-filters, to their matrix representation (by sparse matrices), and corresponding fast algorithms. For both approaches, we outline computational transforms; but our emphasis is on effective and direct links between computational analysis of discrete filters, on the one hand, and on continuous wavelets, on the other. By the latter we include both L²(ℝⁿ)-analysis as well as fractal analysis. To facilitate the discussion of the interplay between discrete (used by engineers) and continuous (harmonic analysis), we include a list of terminology commonly used in the two areas; and we include comments on translation between them. The novice might find the following references helpful: [Coh03a, JMR01, Son06b].

Continuing from Chapter 3, in this chapter we outline several points of view on the interplay between discrete and continuous wavelet transforms; stressing both pure and applied aspects. We outline some new links between the two transform technologies based on the theory of representations of generators and relations. By this we mean a finite system of generators which are represented by operators in Hilbert space. We further outline how these representations yield subband filter banks for signal and image processing algorithms. This chapter is inspired by [JS14a]. This builds more mathematical theory to discrete wavelet transforms discussed in Chapter 3 with wavelet image compression examples…

In this chapter, we discuss entropy encoding schemes in wavelet image decomposition step using Karhunen–Loève Transforms, some of which are “borrowed” from [JS07, Son07]. Entropy encoding is mentioned in Chapter 3, and is performed after the wavelet image decomposition step. Also, we discuss Karhunen–Loève Transforms, which are also called principal component analysis (PCA) by engineers in theory and algorithms. We show specific examples to illustrate PCA in Section 5.9.

In this chapter, we discuss matrix factorizations and lifting schemes used in engineering applications that are closely related to computer engineering applications in image processing from our old paper [JS10]. Filter functions are crucial in wavelets and multiresolutions, and we explore ways to perform filter function factorization and use lifting schemes. Further results can be seen in the next chapter.

Motivated by the previous chapter and [JS10], we further discuss filters and matrix factorizations that are closely related to wavelet filters and multiresolutions, which are results from our old paper [JS14b].

The following sections are included:

• Appendix A. Hilbert Space Basics
• Appendix B. Factorization of Matrices, Algorithms, and Wavelets
• Appendix C. Georg Cantor’s Chaos
• Appendix D. Markov Chains and Generalized Wavelet Multiresolutions

The following sections are included:

• References
• Index