Wavelets

Wavelets continue to be powerful mathematical tools that can be used to solve problems for which the Fourier (spectral) method does not perform well or cannot handle. This book is for engineers, applied mathematicians, and other scientists who want to learn about using wavelets to analyze, process, and synthesize images and signals. Applications are described in detail and there are step-by-step instructions about how to construct and apply wavelets. The only mathematically rigorous monograph written by a mathematician specifically for nonspecialists, it describes the basic concepts of these mathematical techniques, outlines the procedures for using them, compares the performance of various approaches, and provides information for problem solving, putting the reader at the forefront of current research. Written for an interdisciplinary audience of engineers, applied mathematicians, and other scientists with little or no knowledge of wavelets, readers with an undergraduate background in applied mathematics or engineering will appreciate this book.

Foreword; Preface; Software; Notation; 1. What are wavelets? Waveform modeling and segmentation; Time-frequency analysis; Fast algorithms and filter banks; 2. Time-Frequency Localization. Analog filters; RMS bandwidths; The short-time Fourier transform; The integral wavelet transform; Modeling the cochlea; 3. Multiresolution Analysis. Signal spaces with finite RMS bandwidth; Two simple mathematical representations; Multiresolution analysis; Cardinal splines; 4. Orthonormal Wavelets. Orthogonal wavelet spaces; Wavelets of Haar, Shannon, and Meyer; Spline wavelets of Battle-Lemarié and Strömberg; The Daubechies wavelets; 5. Biorthogonal Wavelets. The need for duals; Compactly supported spline wavelets; The duality principle; Total positivity and optimality of time-frequency windows; 6. Algorithms. Signal representations; Orthogonal decompositions and reconstructions; Graphical display of signal representations; Multidimensional wavelet transforms; The need for boundary wavelets; Spline functions on a bounded interval; Boundary spline wavelets with arbitrary knots; 7. Applications. Detection of singularities and feature extraction; Data compression; Numerical solutions of integral equations; Summary and Notes; References; Subject Index.

This book is for engineers, applied mathematicians, and other scientists who want to learn about using wavelets to analyze, process, and synthesize images and signals. Applications are described in detail with step-by-step instructions about how to construct and apply wavelets. The only mathematically rigorous monograph written by a mathematician specifically for nonspecialists, it puts the reader at the forefront of current research.