  Wavelet Analysis with Applications to Image Processing

1st Edition

CRC Press

304 pages

Purchasing Options:\$ = USD
Hardback: 9780849331695
pub: 1997-06-16
SAVE ~\$35.00
Currently out of stock
\$175.00
\$140.00
x

FREE Standard Shipping!

Description

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

Conjugate Quadrature Filter (CQF)

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 Burt-Adelson Pyramidal Decomposition Scheme

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

The 1-D Dyadic Wavelet Algorithm

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