MATHEMATICS OF MULTILEVEL SYSTEMS

This book presents the mathematics of wavelet theory and its applications in a broader sense, comprising entropy encoding, lifting scheme, matrix factorization, and fractals. It also encompasses image compression examples using wavelet transform and includes the principal component analysis which is a hot topic on data dimension reduction in machine learning.

Readers will find equal coverage on the following three themes:

The book entails a varied choice of diverse interdisciplinary themes. While the topics can be found in various parts of the pure and applied literature, this book fulfills the need for an accessible presentation which cuts across the fields.

As the target audience is wide-ranging, a detailed and systematic discussion of issues involving infinite dimensions and Hilbert space is presented in later chapters on wavelets, transform theory and, entropy encoding and probability. For the problems addressed there, the case of infinite dimension will be more natural, and well-motivated.

Contents:

• Preface
• About the Authors
• Road Map
• Introduction
• Wavelet Color Image Compression
• Wavelets as Multiresolutions
• Discrete and Continuous Wavelet Transforms
• Entropy Encoding, Hilbert Space, and Karhunen–Loève Transforms
• Matrix Factorization and Lifting
• Filters and Matrix Factorization
• Appendices:
  • Hilbert Space Basics
  • Factorization of Matrices, Algorithms, and Wavelets
  • Georg Cantor's Chaos
  • Markov Chains and Generalized Wavelet Multiresolutions
• References
• Index

Readership: Undergraduate students in wavelet analysis, image compression, and applied mathematics. Master's level wavelet analysis course.

Key Features:

• The book features wavelet algorithms, multiresolutions in both mathematics and engineering, resolution and detailed data, fractals, entropy encoding and principal component analysis which are diverse and powerful tools to make it more accessible to students
• The text offers new viewpoints, and with hands-on projects; ideas and projects the authors have tested with students
• The connection between multiresolutions as they arise in both mathematics, and in their incarnations in applications, for example, in the processing of the algorithm to greyscale numbers in digital images. (In color images, the algorithm further entails a mix of the three basic colors.)
• Simple matrix examples for wavelet decomposition and reconstruction as well as principal component analysis to make the presentation student friendly. It is inspired by a variety of student projects the authors have directed over the past decade
• In order not to interrupt the flow of topics inside the book, background-theory topics are appended at the end of the book