1st Edition

Wavelet Analysis with Applications to Image Processing

    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.

    Notation and Abbreviations
    Basic Set Operations
    Cardinality of Sets - Finite, Countable, and Uncountable Sets
    Rings and Algebras of Sets
    Linear Spaces
    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
    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
    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
    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
    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
    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
    The Cascade Algorithm
    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
    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


    Prasad\, Lakshman; Iyengar\, S. Sitharama