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E-raamat: Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing

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(Rose-Hulman Institute of Technology), (Rose-Hulman Institute of Technology)
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  • Ilmumisaeg: 03-Apr-2018
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119258247
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Discrete Fourier Analysis and Wavelets: Applications to Signal and Image ProcessingSuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 03-Apr-2018
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119258247
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Delivers an appropriate mix of theory and applications to help readers understand the process and problems of image and signal analysis

Maintaining a comprehensive and accessible treatment of the concepts, methods, and applications of signal and image data transformation, this Second Edition of Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing features updated and revised coverage throughout with an emphasis on key and recent developments in the field of signal and image processing. Topical coverage includes: vector spaces, signals, and images; the discrete Fourier transform; the discrete cosine transform; convolution and filtering; windowing and localization; spectrograms; frames; filter banks; lifting schemes; and wavelets.

Discrete Fourier Analysis and Wavelets introduces a new chapter on framesa new technology in which signals, images, and other data are redundantly measured. This redundancy allows for more sophisticated signal analysis. The new coverage also expands upon the discussion on spectrograms using a frames approach. In addition, the book includes a new chapter on lifting schemes for wavelets and provides a variation on the original low-pass/high-pass filter bank approach to the design and implementation of wavelets. These new chapters also include appropriate exercises and MATLAB® projects for further experimentation and practice.





Features updated and revised content throughout, continues to emphasize discrete and digital methods, and utilizes MATLAB® to illustrate these concepts Contains two new chapters on frames and lifting schemes, which take into account crucial new advances in the field of signal and image processing Expands the discussion on spectrograms using a frames approach, which is an ideal method for reconstructing signals after information has been lost or corrupted (packet erasure) Maintains a comprehensive treatment of linear signal processing for audio and image signals with a well-balanced and accessible selection of topics that appeal to a diverse audience within mathematics and engineering Focuses on the underlying mathematics, especially the concepts of finite-dimensional vector spaces and matrix methods, and provides a rigorous model for signals and images based on vector spaces and linear algebra methods Supplemented with a companion website containing solution sets and software exploration support for MATLAB and SciPy (Scientific Python)

Thoroughly class-tested over the past fifteen years, Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing is an appropriately self-contained book ideal for a one-semester course on the subject.
Preface xvi
Acknowledgments xx
1 Vector Spaces, Signals, and images 1(70)
1.1 Overview
1(1)
1.2 Some Common Image Processing Problems
1(2)
1.2.1 Applications
2(1)
1.2.1.1 Compression
2(1)
1.2.1.2 Restoration
2(1)
1.2.1.3 Edge Detection
3(1)
1.2.1.4 Registration
3(1)
1.2.2 Transform-Based Methods
3(1)
1.3 Signals and Images
3(7)
1.3.1 Signals
4(1)
1.3.2 Sampling, Quantization Error, and Noise
5(1)
1.3.3 Grayscale Images
6(2)
1.3.4 Sampling Images
8(1)
1.3.5 Color
9(1)
1.3.6 Quantization and Noise for Images
9(1)
1.4 Vector Space Models for Signals and Images
10(6)
1.4.1 Examples-Discrete Spaces
11(3)
1.4.2 Examples-Function Spaces
14(2)
1.5 Basic Waveforms-The Analog Case
16(4)
1.5.1 The One-Dimensional Waveforms
16(3)
1.5.2 2D Basic Waveforms
19(1)
1.6 Sampling and Aliasing
20(5)
1.6.1 Introduction
20(2)
1.6.2 Aliasing for Complex Exponential Waveforms
22(1)
1.6.3 Aliasing for Sines and Cosines
23(1)
1.6.4 The Nyquist Sampling Rate
24(1)
1.6.5 Aliasing in Images
24(1)
1.7 Basic Waveforms-The Discrete Case
25(3)
1.7.1 Discrete Basic Waveforms for Finite Signals
25(2)
1.7.2 Discrete Basic Waveforms for Images
27(1)
1.8 Inner Product Spaces and Orthogonality
28(11)
1.8.1 Inner Products and Norms
28(2)
1.8.1.1 Inner Products
28(1)
1.8.1.2 Norms
29(1)
1.8.2 Examples
30(3)
1.8.3 Orthogonality
33(1)
1.8.4 The Cauchy-Schwarz Inequality
34(1)
1.8.5 Bases and Orthogonal Decomposition
35(4)
1.8.5.1 Bases
35(2)
1.8.5.2 Orthogonal and Orthonormal Bases
37(2)
1.8.5.3 Parseval's Identity
39(1)
1.9 Signal and Image Digitization
39(6)
1.9.1 Quantization and Dequantization
40(3)
1.9.1.1 The General Quantization Scheme
41(1)
1.9.1.2 Dequantization
42(1)
1.9.1.3 Measuring Error
42(1)
1.9.2 Quantifying Signal and Image Distortion More Generally
43(2)
1.10 Infinite-Dimensional Inner Product Spaces
45(10)
1.10.1 Example: An Infinite-Dimensional Space
45(1)
1.10.2 Orthogonal Bases in Inner Product Spaces
46(2)
1.10.3 The Cauchy-Schwarz Inequality and Orthogonal Expansions
48(1)
1.10.4 The Basic Waveforms and Fourier Series
49(4)
1.10.4.1 Complex Exponential Fourier Series
49(3)
1.10.4.2 Sines and Cosines
52(1)
1.10.4.3 Fourier Series on Rectangles
53(1)
1.10.5 Hilbert Spaces and L2(a, b)
53(20)
1.10.5.1 Expanding the Space of Functions
53(1)
1.10.5.2 Complications
54(1)
1.10.5.3 A Converse to Parseval
55(1)
1.11 Matlab Project
55(5)
Exercises
60(11)
2 The Discrete Fourier Transform 71(34)
2.1 Overview
71(1)
2.2 The Time Domain and Frequency Domain
71(2)
2.3 A Motivational Example
73(5)
2.3.1 A Simple Signal
73(1)
2.3.2 Decomposition into Basic Waveforms
74(1)
2.3.3 Energy at Each Frequency
74(1)
2.3.4 Graphing the Results
75(2)
2.3.5 Removing Noise
77(1)
2.4 The One-Dimensional DFT
78(7)
2.4.1 Definition of the DFT
78(2)
2.4.2 Sample Signal and DFT Pairs
80(4)
2.4.2.1 An Aliasing Example
80(1)
2.4.2.2 Square Pulses
81(1)
2.4.2.3 Noise
82(2)
2.4.3 Suggestions on Plotting DFTs
84(1)
2.4.4 An Audio Example
84(1)
2.5 Properties of the DFT
85(5)
2.5.1 Matrix Formulation and Linearity
85(3)
2.5.1.1 The DFT as a Matrix
85(2)
2.5.1.2 The Inverse DFT as a Matrix
87(1)
2.5.2 Symmetries for Real Signals
88(2)
2.6 The Fast Fourier Transform
90(3)
2.6.1 DFT Operation Count
90(1)
2.6.2 The FFT
91(1)
2.6.3 The Operation Count
92(1)
2.7 The Two-Dimensional DFT
93(4)
2.7.1 Interpretation and Examples of the 2-D DFT
96(1)
2.8 Matlab Project
97(4)
2.8.1 Audio Explorations
97(2)
2.8.2 Images
99(2)
Exercises
101(4)
3 The Discrete Cosine Transform 105(34)
3.1 Motivation for the DCT-Compression
105(1)
3.2 Other Compression Issues
106(1)
3.3 Initial Examples-Thresholding
107(5)
3.3.1 Compression Example 1: A Smooth Function
108(1)
3.3.2 Compression Example 2: A Discontinuity
109(1)
3.3.3 Compression Example 3
110(2)
3.3.4 Observations
112(1)
3.4 The Discrete Cosine Transform
112(4)
3.4.1 DFT Compression Drawbacks
112(1)
3.4.2 The Discrete Cosine Transform
113(3)
3.4.2.1 Symmetric Reflection
113(1)
3.4.2.2 DFT of the Extension
113(1)
3.4.2.3 DCT/IDCT Derivation
114(1)
3.4.2.4 Definition of the DCT and IDCT
115(1)
3.4.3 Matrix Formulation of the DCT
116(1)
3.5 Properties of the DCT
116(4)
3.5.1 Basic Waveforms for the DCT
116(1)
3.5.2 The Frequency Domain for the DCT
117(1)
3.5.3 DCT and Compression Examples
117(3)
3.6 The Two-Dimensional DCT
120(1)
3.7 Block Transforms
121(2)
3.8 JPEG Compression
123(8)
3.8.1 Overall Outline of Compression
123(1)
3.8.2 DCT and Quantization Details
124(4)
3.8.3 The JPEG Dog
128(1)
3.8.4 Sequential versus Progressive Encoding
128(3)
3.9 Matlab Project
131(3)
Exercises
134(5)
4 Convolution and Filtering 139(46)
4.1 Overview
139(1)
4.2 One-Dimensional Convolution
139(7)
4.2.1 Example: Low-Pass Filtering and Noise Removal
139(3)
4.2.2 Convolution
142(4)
4.2.2.1 Convolution Definition
142(1)
4.2.2.2 Convolution Properties
143(3)
4.3 Convolution Theorem and Filtering
146(6)
4.3.1 The Convolution Theorem
146(1)
4.3.2 Filtering and Frequency Response
147(3)
4.3.2.1 Filtering Effect on Basic Waveforms
147(3)
4.3.3 Filter Design
150(2)
4.4 2D Convolution-Filtering Images
152(4)
4.4.1 Two-Dimensional Filtering and Frequency Response
152(1)
4.4.2 Applications of 2D Convolution and Filtering
153(3)
4.4.2.1 Noise Removal and Blurring
153(1)
4.4.2.2 Edge Detection
154(2)
4.5 Infinite and Bi-Infinite Signal Models
156(16)
4.5.1 L2(N) and L2(Z)
158(2)
4.5.1.1 The Inner Product Space L2(N)
158(1)
4.5.1.2 The Inner Product Space L2(Z)
159(1)
4.5.2 Fourier Analysis in L2(Z) and L2(N)
160(3)
4.5.2.1 The Discrete Time Fourier Transform in L2(Z)
160(1)
4.5.2.2 Aliasing and the Nyquist Frequency in L2(Z)
161(2)
4.5.2.3 The Fourier Transform on L2(N))
163(1)
4.5.3 Convolution and Filtering in L2(Z) and L2(N)
163(3)
4.5.3.1 The Convolution Theorem
164(2)
4.5.4 The z-Transform
166(2)
4.5.4.1 Two Points of View
166(1)
4.5.4.2 Algebra of z-Transforms; Convolution
167(1)
4.5.5 Convolution in CN versus L2(Z)
168(3)
4.5.5.1 Some Notation
168(1)
4.5.5.2 Circular Convolution and z-Transforms
169(1)
4.5.5.3 Convolution in CN from Convolution in L2(Z)
170(1)
4.5.6 Some Filter Terminology
171(1)
4.5.7 The Space L2(Z x Z)
172(1)
4.6 Matlab Project
172(4)
4.6.1 Basic Convolution and Filtering
172(2)
4.6.2 Audio Signals and Noise Removal
174(1)
4.6.3 Filtering Images
175(1)
Exercises
176(9)
5 Windowing and Localization 185(20)
5.1 Overview: Nonlocality of the DFT
185(2)
5.2 Localization via Windowing
187(11)
5.2.1 Windowing
187(1)
5.2.2 Analysis of Windowing
188(4)
5.2.2.1 Step 1: Relation of X and Y
189(1)
5.2.2.2 Step 2: Effect of Index Shift
190(1)
5.2.2.3 Step 3: N-Point versus M-Point DFT
191(1)
5.2.3 Spectrograms
192(4)
5.2.4 Other Types of Windows
196(2)
5.3 Matlab Project
198(2)
5.3.1 Windows
198(1)
5.3.2 Spectrograms
199(1)
Exercises
200(5)
6 Frames 205(46)
6.1 Introduction
205(1)
6.2 Packet Loss
205(3)
6.3 Frames-Using more Dot Products
208(3)
6.4 Analysis and Synthesis with Frames
211(7)
6.4.1 Analysis and Synthesis
211(2)
6.4.2 Dual Frame and Perfect Reconstruction
213(1)
6.4.3 Partial Reconstruction
214(1)
6.4.4 Other Dual Frames
215(1)
6.4.5 Numerical Concerns
216(2)
6.4.5.1 Condition Number of a Matrix
217(1)
6.5 Initial Examples of Frames
218(4)
6.5.1 Circular Frames in R2
218(1)
6.5.2 Extended DFT Frames and Harmonic Frames
219(2)
6.5.3 Canonical Tight Frame
221(1)
6.5.4 Frames for Images
222(1)
6.6 More on the Frame Operator
222(3)
6.7 Group-Based Frames
225(12)
6.7.1 Unitary Matrix Groups and Frames
225(3)
6.7.2 Initial Examples of Group Frames
228(4)
6.7.2.1 Platonic Frames
228(2)
6.7.2.2 Symmetric Group Frames
230(2)
6.7.2.3 Harmonic Frames
232(1)
6.7.3 Gabor Frames
232(5)
6.7.3.1 Flipped Gabor Frames
237(1)
6.8 Frame Applications
237(5)
6.8.1 Packet Loss
239(1)
6.8.2 Redundancy and other duals
240(1)
6.8.3 Spectrogram
241(1)
6.9 Matlab Project
242(5)
6.9.1 Frames and Frame Operator
243(2)
6.9.2 Analysis and Synthesis
245(1)
6.9.3 Condition Number
246(1)
6.9.4 Packet Loss
246(1)
6.9.5 Gabor Frames
246(1)
Exercises
247(4)
7 Filter Banks 251(68)
7.1 Overview
251(1)
7.2 The Haar Filter Bank
252(8)
7.2.1 The One-Stage Two-Channel Filter Bank
252(4)
7.2.2 Inverting the One-stage Transform
256(1)
7.2.3 Summary of Filter Bank Operation
257(3)
7.3 The General One-stage Two-channel Filter Bank
260(4)
7.3.1 Formulation for Arbitrary FIR Filters
260(1)
7.3.2 Perfect Reconstruction
261(2)
7.3.3 Orthogonal Filter Banks
263(1)
7.4 Multistage Filter Banks
264(3)
7.5 Filter Banks for Finite Length Signals
267(14)
7.5.1 Extension Strategy
267(2)
7.5.2 Analysis of Periodic Extension
269(5)
7.5.2.1 Adapting the Analysis Transform to Finite Length
270(2)
7.5.2.2 Adapting the Synthesis Transform to Finite Length
272(2)
7.5.2.3 Other Extensions
274(1)
7.5.3 Matrix Formulation of the Periodic Case
274(1)
7.5.4 Multistage Transforms
275(6)
7.5.4.1 Iterating the One-stage Transform
275(2)
7.5.4.2 Matrix Formulation of Multistage Transform
277(1)
7.5.4.3 Reconstruction from Approximation Coefficients
278(3)
7.5.5 Matlab Implementation of Discrete Wavelet Transforms
281(1)
7.6 The 2D Discrete Wavelet Transform and JPEG 2000
281(8)
7.6.1 Two-dimensional Transforms
281(1)
7.6.2 Multistage Transforms for Two-dimensional Images
282(4)
7.6.3 Approximations and Details for Images
286(2)
7.6.4 JPEG 2000
288(1)
7.7 Filter Design
289(14)
7.7.1 Filter Banks in the z-domain
290(1)
7.7.1.1 Downsampling and Upsampling in the z-domain
290(1)
7.7.1.2 Filtering in the Frequency Domain
290(1)
7.7.2 Perfect Reconstruction in the z-frequency Domain
290(2)
7.7.3 Filter Design I: Synthesis from Analysis
292(3)
7.7.4 Filter Design II: Product Filters
295(2)
7.7.5 Filter Design III: More Product Filters
297(2)
7.7.6 Orthogonal Filter Banks
299(4)
7.7.6.1 Design Equations for an Orthogonal Bank
299(1)
7.7.6.2 The Product Filter in the Orthogonal Case
300(1)
7.7.6.3 Restrictions on P(z); Spectral Factorization
301(1)
7.7.6.4 Daubechies Filters
301(2)
7.8 Matlab Project
303(3)
7.8.1 Basics
303(1)
7.8.2 Audio Signals
304(1)
7.8.3 Images
305(1)
7.9 Alternate Matlab Project
306(3)
7.9.1 Basics
306(1)
7.9.2 Audio Signals
307(1)
7.9.3 Images
307(2)
Exercises
309(10)
8 Lifting for Filter Banks and Wavelets 319(42)
8.1 Overview
319(1)
8.2 Lifting for the Haar Filter Bank
319(5)
8.2.1 The Polyphase Analysis
320(1)
8.2.2 Inverting the Polyphase Haar Transform
321(1)
8.2.3 Lifting Decomposition for the Haar Transform
322(2)
8.2.4 Inverting the Lifted Haar Transform
324(1)
8.3 The Lifting Theorem
324(6)
8.3.1 A Few Facts About Laurent Polynomials
325(1)
8.3.1.1 The Width of a Laurent Polynomial
325(1)
8.3.1.2 The Division Algorithm
325(1)
8.3.2 The Lifting Theorem
326(4)
8.4 Polyphase Analysis for Filter Banks
330(9)
8.4.1 The Polyphase Decomposition and Convolution
331(2)
8.4.2 The Polyphase Analysis Matrix
333(1)
8.4.3 Inverting the Transform
334(4)
8.4.4 Orthogonal Filters
338(1)
8.5 Lifting
339(12)
8.5.1 Relation Between the Polyphase Matrices
339(2)
8.5.2 Factoring the Le Gall 5/3 Polyphase Matrix
341(2)
8.5.3 Factoring the Haar Polyphase Matrix
343(2)
8.5.4 Efficiency
345(1)
8.5.5 Lifting to Design Transforms
346(5)
8.6 Matlab Project
351(5)
8.6.1 Laurent Polynomials
351(3)
8.6.2 Lifting for CDF(2,2)
354(2)
8.6.3 Lifting the D4 Filter Bank
356(1)
Exercises
356(5)
9 Wavelets 361(60)
9.1 Overview
361(2)
9.1.1
Chapter Outline
361(1)
9.1.2 Continuous from Discrete
361(2)
9.2 The Haar Basis
363(13)
9.2.1 Haar Functions as a Basis for L2(0,1)
364(8)
9.2.1.1 Haar Function Definition and Graphs
364(3)
9.2.1.2 Orthogonality
367(1)
9.2.1.3 Completeness in L2(0,1)
368(4)
9.2.2 Haar Functions as an Orthonormal Basis for L2(R)
372(2)
9.2.3 Projections and Approximations
374(2)
9.3 Haar Wavelets Versus the Haar Filter Bank
376(10)
9.3.1 Single-stage Case
377(3)
9.3.1.1 Functions from Sequences
377(1)
9.3.1.2 Filter Bank Analysis/Synthesis
377(1)
9.3.1.3 Haar Expansion and Filter Bank Parallels
378(2)
9.3.2 Multistage Haar Filter Bank and Multiresolution
380(6)
9.3.2.1 Some Subspaces and Bases
381(1)
9.3.2.2 Multiresolution and Orthogonal Decomposition
381(1)
9.3.2.3 Direct Sums
382(2)
9.3.2.4 Connection to Multistage Haar Filter Banks
384(2)
9.4 Orthogonal Wavelets
386(21)
9.4.1 Essential Ingredients
386(1)
9.4.2 Constructing a Multiresolution Analysis: The Dilation Equation
387(2)
9.4.3 Connection to Orthogonal Filters
389(1)
9.4.4 Computing the Scaling Function
390(4)
9.4.5 Scaling Function Existence and Properties
394(5)
9.4.5.1 Fixed Point Iteration and the Cascade Algorithm
394(1)
9.4.5.2 Existence of the Scaling Function
395(2)
9.4.5.3 The Support of the Scaling Function
397(2)
9.4.5.4 Back to Multiresolution
399(1)
9.4.6 Wavelets
399(5)
9.4.7 Wavelets and the Multiresolution Analysis
404(3)
9.4.7.1 Final Remarks on Orthogonal Wavelets
406(1)
9.5 Biorthogonal Wavelets
407(4)
9.5.1 Biorthogonal Scaling Functions
408(1)
9.5.2 Biorthogonal Wavelets
409(1)
9.5.3 Decomposition of L2(R)
409(2)
9.6 Matlab Project
411(3)
9.6.1 Orthogonal Wavelets
411(3)
9.6.2 Biorthogonal Wavelets
414(1)
Exercises
414(7)
Bibliography 421(2)
Appendix: Solutions to Selected Exercises 423(16)
Index 439
S. Allen Broughton, PhD, is Professor Emeritus of Mathematics at Rose-Hulman Institute of Technology. Dr. Broughton is a member of the American Mathematical Society (AMS) and the Society for the Industrial Applications of Mathematics (SIAM), and his research interests include the mathematics of image and signal processing, and wavelets.

Kurt Bryan, PhD, is Professor of Mathematics at Rose-Hulman Institute of Technology. Dr. Bryan is a member of MAA and SIAM and has authored over twenty peer-reviewed journal articles.
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