1st Edition

# Wavelet Analysis with Applications to Image Processing

304 Pages
by CRC Press

304 Pages
by CRC Press

Also available as eBook on:

Wavelet analysis is among the newest additions to the arsenals of mathematicians, scientists, and engineers, and offers common solutions to diverse problems. However, students and professionals in some areas of engineering and science, intimidated by the mathematical background necessary to explore this subject, have been unable to use this powerful tool.

The first book on the topic for readers with minimal mathematical backgrounds, Wavelet Analysis with Applications to Image Processing provides a thorough introduction to wavelets with applications in image processing. Unlike most other works on this subject, which are often collections of papers or research advances, this book offers students and researchers without an extensive math background a step-by-step introduction to the power of wavelet transforms and applications to image processing.
The first four chapters introduce the basic topics of analysis that are vital to understanding the mathematics of wavelet transforms. Subsequent chapters build on the information presented earlier to cover the major themes of wavelet analysis and its applications to image processing. This is an ideal introduction to the subject for students, and a valuable reference guide for professionals working in image processing.

PRELIMINARIES
Notation and Abbreviations
Basic Set Operations
Cardinality of Sets - Finite, Countable, and Uncountable Sets
Rings and Algebras of Sets
LINEAR, METRIC, AND HILBERT SPACES
Linear Spaces
Subspaces
Factor Spaces (Quotient Spaces)
Linear Functionals
Null Space (Kernel) of a Functional --Hyperplanes
Geometric Interpretation of Linear Functions
Normed Linear Spaces
Metric Spaces
Continuous Mappings
Convergence
Dense Subsets
Closed Sets
Open Sets
Complex Metric Spaces
Completion of Metric Spaces
Norm-Induced Metric and Banach Spaces
Euclidean Spaces
Scalar Products, Orthogonality, and Bases
Existence of an Orthogonal Basis
Bessel's Inequality, Closed Orthogonal Systems
Complete Euclidean Spaces, Riesz-Fischer Theorem
Hilbert Spaces
Subspaces, Orthogonal Complements, and Direct Sums
Characterization of Euclidean Spaces
INTEGRATION
The Riemann Integral
Upper and Lower Riemann Integrals
Riemann Integration vs. Lebesgue Integration
The Lebesgue Measure on R
Measurable Functions
Simple Functions
Convergence of Measurable Functions
Lebesgue Integration
Some Properties of the Lebesgue Integral
FOURIER ANALYSIS
The Spaces L1(c? and L2(c)
The Space L1(c?
The Space L2(c?
Fourier Series
Fourier Series of Square Integrable Functions
Fourier Series of Absolutely Integrable Functions
The Convolution Product on L1(S1)
Fourier Transforms
Fourier Transforms of Functions in L2 (R)
Fourier Transforms of Functions in L1(R)
Poisson Summation Formula
WAVELET ANALYSIS
Time-Frequency Analysis and the Windowed Fourier Transform
Heisenberg's Uncertainty Principle
The Integral Wavelet Transform
The Discrete Wavelet Transform
Multiresolution Analysis (MRA) of L2(R)
Constructing an MRA from a Scaling Function
Wavelet Decomposition and Reconstruction of Functions
Multiresolution Decomposition and Reconstruction of Functions in L2 (R)
The Fast Wavelet Algorithm
CONSTRUCTION OF WAVELETS
The Battle-Lemarié Family of Wavelets
Cardinal B-Splines
Cardinal B-Spline MRA of L2(R)
Subband Filtering Schemes
Bandlimited Functions
Discrete Filtering
CQFs Arising from MRAs
Compactly Supported Orthonormal Wavelet Bases
The Structure of M0
Necessary and Sufficient Conditions for Orthonormality
Biorthogonal Wavelets
Linear Phase FIR Filters
Compactly Supported o.n. Wavelets are Asymmetric
Dual FIR Filters with Exact Reconstruction
Dual Scaling Functions and Wavelets
Biorthogonal Riesz Bases of Wavelets and Associated MRAs
Conditions for Biorthogonality
Symmetry for m0 and ˜m0
Biorthogonal Spline Wavelets with Compact Support
WAVELETS IN IMAGE PROCESSING
The Smoothing Function H?
Mallat's Wavelet-Based Pyramidal Decomposition Scheme
The 1-D Fast Wavelet Algorithm
An MRA of L2(R2)
The Two-Dimensional Wavelet Algorithm
Multiscale Edge Representation of Images
Signal Reconstruction from its 1-D Dyadic Wavelet Transform
Method of Alternate Projections
The Dyadic Wavelet Transform of Images
Image Reconstruction from its 2-D Dyadic Wavelet Transform
Image Reconstruction from its 2-D Dyadic Wavelet Transform
Method of Alternate Projections in 2-D
The Discrete Finite Dyadic Wavelet Transform
Double-Layered Image Encoding
Multiscale Edge-Based Image Encoding
Image Reconstruction from its 2-D Dyadic Wavelet Transform
Texture-Based Image Encoding
Appendix
Index