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Wavelets

A Mathematical Tool for Signal Analysis
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Wavelets continue to be powerful mathematical tools for solving problems where the Fourier (spectral) method does not perform well or cannot be used. This text 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. A mathematically rigorous monograph, the book was written specifically by a mathematician for non-specialists. 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. Methods and topics include: multidimensional and multifrequency wavelet transforms; spline wavelets with uniform or arbitrary knots on a bounded interval; procedures to construct all the well-known wavelets and to find their corresponding filter sequences; and detailed comparisons of the most popular wavelets with tables of values for evaluating their filtering performance.
Foreword Preface Software Notation Chapter 1: What are wavelets? Waveform modeling and segmentation Time-frequency analysis Fast algorithms and filter banks Chapter 2: Time-Frequency Localization. Analog filters RMS bandwidths The short-time Fourier transform The integral wavelet transform Modeling the cochlea Chapter 3: Multiresolution Analysis. Signal spaces with finite RMS bandwidth Two simple mathematical representations Multiresolution analysis Cardinal splines Chapter 4: Orthonormal Wavelets. Orthogonal wavelet spaces Wavelets of Haar, Shannon, and Meyer Spline wavelets of Battle-Lemarie and Stroemberg The Daubechies wavelets Chapter 5: Biorthogonal Wavelets. The need for duals Compactly supported spline wavelets The duality principle Total positivity and optimality of time-frequency windows Chapter 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 Chapter 7: Applications. Detection of singularities and feature extraction Data compression Numerical solutions of integral equations Summary and Notes References Subject Index.
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