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Full Description
A thorough guide to the classical and contemporary mathematical methods of modern signal and image processing Discrete Fourier Analysis and Wavelets presents a thorough introduction to the mathematical foundations of signal and image processing. Key concepts and applications are addressed in a thought-provoking manner and are implemented using vector, matrix, and linear algebra methods. With a balanced focus on mathematical theory and computational techniques, this self-contained book equips readers with the essential knowledge needed to transition smoothly from mathematical models to practical digital data applications.The book first establishes a complete vector space and matrix framework for analyzing signals and images. Classical methods such as the discrete Fourier transform, the discrete cosine transform, and their application to JPEG compression are outlined followed by coverage of the Fourier series and the general theory of inner product spaces and orthogonal bases. The book then addresses convolution, filtering, and windowing techniques for signals and images. Finally, modern approaches are introduced, including wavelets and the theory of filter banks as a means of understanding the multiscale localized analysis underlying the JPEG 2000 compression standard.Throughout the book, examples using image compression demonstrate how mathematical theory translates into application. Additional applications such as progressive transmission of images, image denoising, spectrographic analysis, and edge detection are discussed. Each chapter provides a series of exercises as well as a MATLAB project that allows readers to apply mathematical concepts to solving real problems. Additional MATLAB routines are available via the book's related Web site.With its insightful treatment of the underlying mathematics in image compression and signal processing, Discrete Fourier Analysis and Wavelets is an ideal book for mathematics, engineering, and computer science courses at the upper-undergraduate and beginning graduate levels. It is also a valuable resource for mathematicians, engineers, and other practitioners who would like to learn more about the relevance of mathematics in digital data processing.
Contents
PREFACE xiACKNOWLEDGMENTS xv1 VECTOR SPACES, SIGNALS, AND IMAGES 11.1 Overview 11.2 Some Common Image Processing Problems 11.3 Signals and Images 31.4 Vector Space Models for Signals and Images 91.5 Basic Waveforms The Analog Case 171.6 Sampling and Aliasing 211.7 Basic Waveforms The Discrete Case 261.8 Inner Product Spaces and Orthogonality 291.9 Signal and Image Digitization 411.10 Infinite-dimensional Inner Product Spaces 461.11 Matlab Project 56Exercises 612 THE DISCRETE FOURIER TRANSFORM 712.1 Overview 712.2 The Time Domain and Frequency Domain 722.3 A Motivational Example 732.4 The One-dimensional DFT 782.5 Properties of the DFT 852.6 The Fast Fourier Transform 902.7 The Two-dimensional DFT 932.8 Matlab Project 97Exercises 1013 THE DISCRETE COSINE TRANSFORM 1053.1 Motivation for the DCT Compression 1053.2 Other Compression Issues 1063.3 Initial Examples Thresholding 1073.4 The Discrete Cosine Transform 1133.5 Properties of the DCT 1173.6 The Two-dimensional DCT 1203.7 Block Transforms 1223.8 JPEG Compression 1243.9 Matlab Project 132Exercises 1344 CONVOLUTION AND FILTERING 1384.1 Overview 1384.2 One-dimensional Convolution 1384.3 Convolution Theorem and Filtering 1454.4 2D Convolution Filtering Images 1504.5 Infinite and Bi-infinite Signal Models 1564.6 Matlab Project 171Exercises 1745 WINDOWING AND LOCALIZATION 1825.1 Overview: Nonlocality of the DFT 1825.2 Localization via Windowing 1845.3 Matlab Project 195Exercises 1976 FILTER BANKS 2016.1 Overview 2016.2 The Haar Filter Bank 2026.3 The General One-stage Two-channel Filter Bank 2106.4 Multistage Filter Banks 2146.5 Filter Banks for Finite Length Signals 2186.6 The 2D Discrete Wavelet Transform and JPEG 2000 2316.7 Filter Design 2396.8 Matlab Project 2516.9 Alternate Matlab Project 255Exercises 2587 WAVELETS 2677.1 Overview 2677.2 The Haar Basis 2697.3 Haar Wavelets versus the Haar Filter Bank 2827.4 Orthogonal Wavelets 2927.5 Biorthogonal Wavelets 3147.6 Matlab Project 318Exercises 321REFERENCES 327SOLUTIONS 329INDEX 335
