Wavelet Analysis with Applications to Image Processing

TLDR: In this paper, the authors present an image encoder based on the Burt-Adelson Pyramidal Decomposition Scheme and the Windowed Fourier Transform (WFT).

Abstract: 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-Lemarie 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 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