Mathematics Of Multilevel Systems: Data, Scaling, Images, Signals, And Fractals (Contemporary Mathematics And Its Applications: Monographs, Expositions And Lecture Notes)

Description

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: a selection of practical projects and algorithms; the theory underpinning the subjects; the important interplay between theory and applications. 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 Chapter 1. Introduction 1.1 Organization of the Book: A Bird’s Eye View, and Tips for the Reader 1.2 Motivation Chapter 2. Wavelet Color Image Compression 2.1 Introduction 2.1.1 Overview 2.1.2 Motivation 2.2 Wavelet Color Image Compression 2.2.1 Methods 2.2.1.1 Forward wavelet transform 2.2.1.2 Thresholding 2.2.1.3 Entropy encoding 2.2.1.4 Reconstruction of wavelet transformed image 2.2.1.5 Inverse wavelet transformation 2.2.2 Wavelets 2.2.2.1 Multiresolution analysis 2.2.2.2 Symmetry 2.2.2.3 Vanishing moments 2.2.2.4 Size of the filters 2.2.3 Various wavelets 2.2.3.1 Haar Wavelet 2.2.3.2 Daubechies wavelets constructions 2.2.3.3 Coiflets 2.2.3.4 Biorthogonal wavelets 2.2.3.5 Symlets 2.3 Digital Image Representation and Mathematics Behind It 2.3.1 Digital image representation 2.3.1.1 Coordinate convention 2.3.1.2 Images as matrices 2.3.1.3 Color image representation in MATLAB 2.3.1.4 Indexed images 2.3.1.5 The basics of color image processing 2.3.2 Reading images 2.3.3 Wavelet decomposition of an image 2.3.3.1 Color conversion 2.3.4 Mathematical insights 2.4 Haar Wavelet Matrix Example for Wavelet Decomposition and Reconstruction 2.4.1 Examples 2.5 Wavelet Color Image Compression Results and Discussion 2.5.1 Implementation of the program 2.5.2 Discussion of the results 2.5.2.1 Wavelet lossy image compression 2.5.2.2 Barbara image compression using different wavelets and threshold values Chapter 3. Wavelets as Multiresolutions 3.1 Multiresolutions: History, Applications, Examples, Discussion, and Algorithms 3.2 Glossary Chapter 4. Discrete and Continuous Wavelet Transforms 4.1 Introduction 4.2 The Discrete vs. Continuous Wavelet Algorithms 4.2.1 The discrete wavelet transform 4.3 The Continuous Wavelet Transform 4.3.1 Some background on Hilbert space 4.3.1.1 Increasing the dimension 4.4 Connections to Group Theory 4.5 Tools from Mathematics 4.6 A Transfer Operator 4.7 Future Directions 4.7.1 Orthonormal bases generated by Cuntz algebras 4.7.1.1 Piecewise exponential bases on fractals 4.7.1.2 Walsh bases 4.8 List of Names and Discoveries 4.9 History 4.10 Literature Chapter 5. Entropy Encoding, Hilbert Space, and Karhunen–Lo`eve Transforms 5.1 Introduction 5.1.1 Digital image compression 5.2 General Background 5.2.1 From data to Hilbert space 5.2.1.1 The measures Pf 5.2.1.2 The measures PT 5.3 The Karhunen–Lo`eve Transform 5.4 Frame Bounds and Subspaces 5.4.1 Supplement 5.5 Splitting Off Rank-One Operators 5.5.1 Perron–Frobenius 5.6 Weighted Frames and Weighted Frame Operators 5.6.1 B(H) = T(H)∗ 5.7 Localization 5.8 Engineering Applications 5.8.1 Entropy encoding 5.8.1.1 Some terminology 5.8.2 The algorithm 5.8.3 Benefits of entropy encoding 5.9 Principal Component Analysis 5.9.1 The algorithm for a digital image application 5.9.2 Principal component analysis in a digital image 5.9.3 A matrix example 5.9.4 Digital image compression using principal component analysis Chapter 6. Matrix Factorization and Lifting 6.1 Introduction 6.2 Matrix-Valued Functions 6.3 Systems of Filters 6.3.1 Operations on time signals 6.4 Groups of Matrix Functions 6.5 Group Actions 6.5.1 Factorizations 6.5.2 Notational conventions 6.6 Divisibility and Residues for Matrix Functions 6.6.1 The 2 × 2 case 6.6.2 The 3 × 3 case 6.6.3 The N × N case 6.7 Quantization Chapter 7. Filters and Matrix Factorization 7.1 Introduction 7.2 Factorization Algorithm 7.3 Factorization Cases 7.3.1 Factorizations 7.3.2 The 2 × 2 case: polynomials 7.3.3 The 3 × 3 case 7.3.3.1 Comments 7.3.4 The N × N case 7.3.5 L∞(T)-matrix entries 7.3.6 Optimal factorization in the case of SLN(L∞(T)) Appendix A. Hilbert Space Basics Appendix B. Factorization of Matrices, Algorithms, and Wavelets B.1 Resolution and Detail B.2 From Haar to Daubechies B.3 Groups of Wavelets B.4 Examples B.4.1 The world of the spectrum B.4.2 Wavelets on the interval [0, 5] and their associated scaling functions Appendix C. Georg Cantor’s Chaos C.0.1 Haar Meets Cantor Appendix D. Markov Chains and Generalized Wavelet Multiresolutions D.1 Introduction and Setting D.2 Julia Sets D.3 Multiresolutions and Generalized Wavelet Representations References Index