Discrete Fourier analysis and wavelets : applications to signal and image processing /: applications to signal and image processing. (2017)

• Record Type: Book
• Further Information: Note: S. Allen Broughton, Kurt M. Bryan.
• Authors: Broughton, S. Allen, 1951-; Bryan, Kurt, 1962-
• Contents: Preface xvii Acknowledgments xxi 1 Vector Spaces, Signals, and Images 1 1.1 Overview 1 1.2 Some Common Image Processing Problems 1 1.2.1 Applications 2 1.2.1.1 Compression 2 1.2.1.2 Restoration 2 1.2.1.3 Edge Detection 3 1.2.1.4 Registration 3 1.2.2 Transform-Based Methods 3 1.3 Signals and Images 3 1.3.1 Signals 4 1.3.2 Sampling, Quantization Error, and Noise 5 1.3.3 Grayscale Images 6 1.3.4 Sampling Images 8 1.3.5 Color 9 1.3.6 Quantization and Noise for Images 9 1.4 Vector Space Models for Signals and Images 10 1.4.1 Examples—Discrete Spaces 11 1.4.2 Examples—Function Spaces 14 1.5 Basic Waveforms—The Analog Case 16 1.5.1 The One-Dimensional Waveforms 16 1.5.2 2D Basic Waveforms 19 1.6 Sampling and Aliasing 20 1.6.1 Introduction 20 1.6.2 Aliasing for Complex Exponential Waveforms 22 1.6.3 Aliasing for Sines and Cosines 23 1.6.4 The Nyquist Sampling Rate 24 1.6.5 Aliasing in Images 24 1.7 Basic Waveforms—The Discrete Case 25 1.7.1 Discrete Basic Waveforms for Finite Signals 25 1.7.2 Discrete Basic Waveforms for Images 27 1.8 Inner Product Spaces and Orthogonality 28 1.8.1 Inner Products and Norms 28 1.8.1.1 Inner Products 28 1.8.1.2 Norms 29 1.8.2 Examples 30 1.8.3 Orthogonality 33 1.8.4 The Cauchy–Schwarz Inequality 34 1.8.5 Bases and Orthogonal Decomposition 35 1.8.5.1 Bases 35 1.8.5.2 Orthogonal and Orthonormal Bases 37 1.8.5.3 Parseval’s Identity 39 1.9 Signal and Image Digitization 39 1.9.1 Quantization and Dequantization 40 1.9.1.1 The General Quantization Scheme 41 1.9.1.2 Dequantization 42 1.9.1.3 Measuring Error 42 1.9.2 Quantifying Signal and Image Distortion More Generally 43 1.10 Infinite-Dimensional Inner Product Spaces 45 1.10.1 Example: An Infinite-Dimensional Space 45 1.10.2 Orthogonal Bases in Inner Product Spaces 46 1.10.3 The Cauchy–Schwarz Inequality and Orthogonal Expansions 48 1.10.4 The Basic Waveforms and Fourier Series 49 1.10.4.1 Complex Exponential Fourier Series 49 1.10.4.2 Sines and Cosines 52 1.10.4.3 Fourier Series on Rectangles 53 1.10.5 Hilbert Spaces and L2(a, b ) 53 1.10.5.1 Expanding the Space of Functions 53 1.10.5.2 Complications 54 1.10.5.3 A Converse to Parseval 55 1.11 Matlab Project 55 Exercises 60 2 The Discrete Fourier Transform 71 2.1 Overview 71 2.2 The Time Domain and Frequency Domain 71 2.3 A Motivational Example 73 2.3.1 A Simple Signal 73 2.3.2 Decomposition into BasicWaveforms 74 2.3.3 Energy at Each Frequency 74 2.3.4 Graphing the Results 75 2.3.5 Removing Noise 77 2.4 The One-Dimensional DFT 78 2.4.1 Definition of the DFT 78 2.4.2 Sample Signal and DFT Pairs 80 2.4.2.1 An Aliasing Example 80 2.4.2.2 Square Pulses 81 2.4.2.3 Noise 82 2.4.3 Suggestions on Plotting DFTs 84 2.4.4 An Audio Example 84 2.5 Properties of the DFT 85 2.5.1 Matrix Formulation and Linearity 85 2.5.1.1 The DFT as a Matrix 85 2.5.1.2 The Inverse DFT as a Matrix 87 2.5.2 Symmetries for Real Signals 88 2.6 The Fast Fourier transform 90 2.6.1 DFT Operation Count 90 2.6.2 The FFT 91 2.6.3 The Operation Count 92 2.7 The Two-Dimensional DFT 93 2.7.1 Interpretation and Examples of the 2-D DFT 96 2.8 Matlab Project 97 2.8.1 Audio Explorations 97 2.8.2 Images 99 Exercises 101 3 The Discrete Cosine Transform 105 3.1 Motivation for the DCT—Compression 105 3.2 Other Compression Issues 106 3.3 Initial Examples—Thresholding 107 3.3.1 Compression Example 1: A Smooth Function 108 3.3.2 Compression Example 2: A Discontinuity 109 3.3.3 Compression Example 3 110 3.3.4 Observations 112 3.4 The Discrete Cosine Transform 112 3.4.1 DFT Compression Drawbacks 112 3.4.2 The Discrete Cosine Transform 113 3.4.2.1 Symmetric Reflection 113 3.4.2.2 DFT of the Extension 113 3.4.2.3 DCT/IDCT Derivation 114 3.4.2.4 Definition of the DCT and IDCT 115 3.4.3 Matrix Formulation of the DCT 116 3.5 Properties of the DCT 116 3.5.1 BasicWaveforms for the DCT 116 3.5.2 The Frequency Domain for the DCT 117 3.5.3 DCT and Compression Examples 117 3.6 The Two-Dimensional DCT 120 3.7 Block Transforms 121 3.8 JPEG Compression 123 3.8.1 Overall Outline 123 3.8.2 DCT and Quantization Details 124 3.8.3 The JPEG Dog 128 3.8.4 Sequential versus Progressive Encoding 128 3.9 Matlab Project 131 Exercises 134 4 Convolution and Filtering 139 4.1 Overview 139 4.2 One-Dimensional Convolution 139 4.2.1 Example: Low-Pass Filtering and Noise Removal 139 4.2.2 Convolution 142 4.2.2.1 Convolution Definition 142 4.2.2.2 Convolution Properties 143 4.3 Convolution Theorem and Filtering 146 4.3.1 The Convolution Theorem 146 4.3.2 Filtering and Frequency Response 147 4.3.2.1 Filtering Effect on BasicWaveforms 147 4.3.3 Filter Design 150 4.4 2D Convolution—Filtering Images 152 4.4.1 Two-Dimensional Filtering and Frequency Response 152 4.4.2 Applications of 2D Convolution and Filtering 153 4.4.2.1 Noise Removal and Blurring 153 4.4.2.2 Edge Detection 154 4.5 Infinite and Bi-Infinite Signal Models 156 4.5.1 L2(ℕ) and L2(ℤ) 158 4.5.1.1 The Inner Product Space L2(ℕ) 158 4.5.1.2 The Inner Product Space L2(ℤ) 159 4.5.2 Fourier Analysis in L2(ℤ) and L2(ℕ) 160 4.5.2.1 The Discrete Time Fourier Transform in L2(ℤ) 160 4.5.2.2 Aliasing and the Nyquist Frequency in L2(ℤ) 161 4.5.2.3 The Fourier Transform on L2(ℕ)) 163 4.5.3 Convolution and Filtering in L2(ℤ) and L2(ℕ) 163 4.5.3.1 The Convolution Theorem 164 4.5.4 The z-Transform 166 4.5.4.1 Two Points of View 166 4.5.4.2 Algebra of z-Transforms; Convolution 167 4.5.5 Convolution in ℂN versus L2(ℤ) 168 4.5.5.1 Some Notation 168 4.5.5.2 Circular Convolution and z-Transforms 169 4.5.5.3 Convolution in ℂN from Convolution in L2(ℤ) 170 4.5.6 Some Filter Terminology 171 4.5.7 The Space L2(ℤ × ℤ) 172 4.6 Matlab Project 172 4.6.1 Basic Convolution and Filtering 172 4.6.2 Audio Signals and Noise Removal 174 4.6.3 Filtering Images 175 Exercises 176 5 Windowing and Localization 185 5.1 Overview: Nonlocality of the DFT 185 5.2 Localization via Windowing 187 5.2.1 Windowing 187 5.2.2 Analysis of Windowing 188 5.2.2.1 Step 1: Relation of X and Y 189 5.2.2.2 Step 2: Effect of Index Shift 190 5.2.2.3 Step 3: N-Point versus M-Point DFT 191 5.2.3 Spectrograms 192 5.2.4 Other Types of Windows 196 5.3 Matlab Project 198 5.3.1 Windows 198 5.3.2 Spectrograms 199 Exercises 200 6 Frames 205 6.1 Introduction 205 6.2 Packet Loss 205 6.3 Frames—Using more Dot Products 208 6.4 Analysis and Synthesis with Frames 211 6.4.1 A
• Edition: Second edition
• Publisher Details: Hoboken, New Jersey : John Wiley & Sons, Inc
• Publication Date: 2017
• Extent: 1 online resource
• Subjects: 515.2433; Fourier analysis; Wavelets (Mathematics); Signal processing -- Mathematics; Image processing -- Mathematics
• Languages: English
• ISBNs: 9781119258247
• Related ISBNs: 9781119258230
• Notes: Note: Includes bibliographical references and index.; Note: Description based on CIP data; resource not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
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