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E-raamat: Applied Fourier Analysis: From Signal Processing to Medical Imaging

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  • Formaat: EPUB+DRM
  • Ilmumisaeg: 20-Nov-2017
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781493973934
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Applied Fourier Analysis: From Signal Processing to Medical ImagingSuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 20-Nov-2017
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781493973934
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The first of its kind, this focused textbook serves as a self-contained resource for teaching from scratch the fundamental mathematics of Fourier analysis and illustrating some of its most current, interesting applications, including medical imaging and radar processing. Developed by the author from extensive classroom teaching experience, it provides a breadth of theory that allows students to appreciate the utility of the subject, but at as accessible a depth as possible. With myriad applications included, this book can be adapted to a one or two semester course in Fourier Analysis or serve as the basis for independent study.

Applied Fourier Analysis assumes no prior knowledge of analysis from its readers, and begins by making the transition from linear algebra to functional analysis. It goes on to cover basic Fourier series and Fourier transforms before delving into applications in sampling and interpolation theory, digital communications, radar processing, medi





cal imaging, and heat and wave equations. For all applications, ample practice exercises are given throughout, with collections of more in-depth problems built up into exploratory chapter projects.   Illuminating videos are available on Springer.com and Link.Springer.com that present animated visualizations of several concepts.

The content of the book itself is limited to what students will need to deal with in these fields, and avoids spending undue time studying proofs or building toward more abstract concepts. The book is perhaps best suited for courses aimed at upper division undergraduates and early graduates in mathematics, electrical engineering, mechanical engineering, computer science, physics, and other natural sciences, but in general it is a highly valuable resource for introducing a broad range of students to Fourier analysis.
1 Introduction: From Linear Algebra to Linear Analysis
1(18)
1.1 Three variations of Fourier Analysis
2(3)
1.1.1 Fourier Series
2(1)
1.1.2 The Discrete Fourier Transform
3(1)
1.1.3 The Continuous Fourier Transform
4(1)
1.2 Motivations
5(3)
1.2.1 Exploration and Understanding
5(2)
1.2.2 Manipulation
7(1)
1.2.3 Compression
7(1)
1.3 Linear Algebra, Linear Analysis, and Fourier Analysis
8(11)
1.3.1 The Dot Product, Inner Product, and Orthogonality
9(3)
1.3.2 Eigenvectors and Eigenvalues in Linear Algebra
12(2)
1.3.3 Orthogonal Diagonalization
14(1)
1.3.4 Diagonalization in Linear Analysis: Eigenfunctions
14(2)
1.3.5 Fourier Analysis
16(1)
1.3.6 Notational Differences
16(3)
2 Basic Fourier Series
19(56)
2.1 Fourier Series on L2 [ a, b]
19(11)
2.1.1 Calculating a Fourier Series
22(5)
2.1.2 Periodicity and Equality
27(2)
2.1.3 Problems and Exercises
29(1)
2.2 Orthogonality and Hilbert Spaces
30(5)
2.2.1 Orthogonal Expansions
33(2)
2.2.2 Problems and Exercises
35(1)
2.3 The Pythagorean Theorem
35(11)
2.3.1 The Isometry between L2[ a, b] and l2
37(2)
2.3.2 Complex Notation
39(3)
2.3.3 Estimating Truncation Errors
42(3)
2.3.4 Problems and Exercises
45(1)
2.4 Differentiation and Convergence Rates
46(7)
2.4.1 A Quandary between Calculus and Fourier Analysis
47(1)
2.4.2 Derivatives and Rates of Decay
48(3)
2.4.3 Fourier Derivatives and Induced Discontinuities
51(1)
2.4.4 Problems and Exercises
52(1)
2.5 Sine and Cosine Series
53(6)
2.5.1 Problems and Exercises
58(1)
2.6 Perhaps Cosine Series Only
59(4)
2.6.1 Induced Discontinuities vs. True Discontinuities
61(1)
2.6.2 Problems and Exercises
62(1)
2.7 Gibbs Ringing Phenomenon
63(2)
2.7.1 Problems and Exercises
64(1)
2.8 Convolution and Correlation
65(5)
2.8.1 A couple of classic examples
67(1)
2.8.2 Problems and Exercises
68(2)
2.9
Chapter Project
70(1)
2.10 Summary of Expansions
71(4)
3 The Discrete Fourier Transform
75(46)
3.1 The Fourier Matrix
76(3)
3.1.1 Orthogonality of the Fourier Matrix
77(2)
3.2 The Complex N'th roots of unity and their structure
79(9)
3.2.1 Problems and Exercises
79(2)
3.2.2 The roots of unity and frequency
81(1)
3.2.3 Making the Fourier Matrix Understandable
82(1)
3.2.4 Problems and Exercises
83(1)
3.2.5 Group structure of the roots of unity
84(2)
3.2.6 Subgroups and Coset Representations
86(1)
3.2.7 Problems and Exercises
87(1)
3.3 The Fast Fourier Transform
88(13)
3.3.1 Speed Enabling Algorithm
89(1)
3.3.2 The simplest examples: F2, F4, and F8
89(4)
3.3.3 The FFT in the Classic Case N = 2q
93(5)
3.3.4 The FFT when N = N1N2
98(2)
3.3.5 Problems and Exercises
100(1)
3.4 Discrete Convolution and Correlation
101(4)
3.4.1 Circulant Toeplitz Matrices
103(2)
3.5 Applications of Convolution and Correlation
105(10)
3.5.1 Derivatives via Convolution
105(4)
3.5.2 Averaging
109(1)
3.5.3 Narrowband and Nonlinear filtering
110(2)
3.5.4 Matched Filtering
112(2)
3.5.5 Problems and Exercises
114(1)
3.6 Computing Sine and Cosine Expansions Using the FFT: Compression
115(4)
3.6.1 Cosine expansions
115(2)
3.6.2 Sine expansions
117(1)
3.6.3 Piecewise Cosine and Sine expansions
117(1)
3.6.4 Problems and Exercises
118(1)
3.7
Chapter Project
119(2)
4 The Fourier Transform
121(28)
4.1 From Fourier Series to the Fourier Transform
121(4)
4.2 The Fourier Isometry from L2(R) to L2(R)
125(1)
4.2.1 Problems and Exercises
126(1)
4.3 The Basic Theorems of Fourier Analysis
126(13)
4.3.1 Differentiation
127(1)
4.3.2 Translation
128(1)
4.3.3 Convolution and Correlation
129(4)
4.3.4 Dilation and Uncertainty
133(4)
4.3.5 The Fourier Transform of a Gaussian
137(1)
4.3.6 Problems and Exercises
138(1)
4.4 The Inverse Transform
139(2)
4.4.1 Proving the Fourier Inversion Formula
139(2)
4.4.2 Problems and Exercises
141(1)
4.5 Partition of Unity
141(2)
4.6 Revisiting Gibbs' ringing
143(3)
4.6.1 Problems and Exercises
145(1)
4.7 Suggested Test Review
146(3)
5 Sampling and Interpolation
149(28)
5.1 The Shannon Sampling Theorem
150(6)
5.1.1 Recalling Polynomial Interpolation
154(1)
5.1.2 Shannon's Theorem: Advantages and Disadvantages
155(1)
5.2 Advanced Interpolation Theory
156(11)
5.2.1 Basic Ideas
156(3)
5.2.2 Oversampling and Adaptive Windows
159(2)
5.2.3 Interpolating Window Functions
161(3)
5.2.4 Approximate Window functions
164(3)
5.2.5 Problems and Exercises
167(1)
5.3 Aliasing
167(4)
5.3.1 Anti-Aliasing Filters
170(1)
5.3.2 Problems and Exercises
171(1)
5.4 Numerical Computation of the Shannon Series
171(5)
5.4.1 Example 1: Exact Interpolation
172(1)
5.4.2 Example 2: Approximate Interpolation
173(2)
5.4.3 Problems and Exercises
175(1)
5.5
Chapter Project
176(1)
6 Digital Communications
177(28)
6.1 Introduction and Basic Terms
177(4)
6.1.1 Definitions from Communications
180(1)
6.2 Modulation
181(3)
6.2.1 Analogue vs. Digital
181(1)
6.2.2 AM vs FM
182(1)
6.2.3 RZ vs. NRZ
183(1)
6.3 The Eye Diagram
184(5)
6.3.1 Interference
186(3)
6.4 Fourier Constraints
189(3)
6.4.1 Uncertainty, Dilation, and ISI
190(1)
6.4.2 Uncertainty, Dilation, and CCI
191(1)
6.5 Statistics and Matched Filtering
192(13)
6.5.1 Gaussian White Noise
195(1)
6.5.2 Averaging Noise
196(2)
6.5.3 The Matched Filter
198(3)
6.5.4 Transmitter and Receiver Details
201(1)
6.5.5
Chapter Project
202(3)
7 Radar Processing
205(22)
7.1 Range and Velocity Measurements
205(3)
7.1.1 Range Measurements
205(1)
7.1.2 Simple Range Resolution
206(1)
7.1.3 Maximum Unambiguous Range
207(1)
7.1.4 Velocity Estimation and Resolution
208(1)
7.2 Radar Range Equation
208(3)
7.2.1 Problems and Exercises
210(1)
7.3 Signal to Noise Management in Radar
211(7)
7.3.1 Matched Filter
212(1)
7.3.2 Signal Averaging
213(1)
7.3.3 Heterodyne Receivers
214(2)
7.3.4 IF Band-Pass Filters
216(1)
7.3.5 Problems and Exercises
217(1)
7.4 Revisiting Range Resolution: Uncertainty and Bandwidth
218(3)
7.5 Doppler Measurement and Resolution
221(6)
7.5.1 Doppler from a moving Target
221(2)
7.5.2 Doppler from a moving Source
223(1)
7.5.3 Doppler Resolution
224(2)
7.5.4 Problems and Exercises
226(1)
7.5.5
Chapter Project
226(1)
8 Image Processing
227(28)
8.1 Fourier Analysis in Rn
227(2)
8.2 Edge Detection
229(5)
8.2.1 Horizontal and Vertical Edges: Standard Approximations
230(4)
8.2.2 Exact Partial Derivatives and Directional Derivatives
234(1)
8.3 Interpolation: High-Definition TV and beyond
234(5)
8.3.1 Revisiting Shannon and Advanced Interpolation
235(3)
8.3.2 Regaining a jpg image in MATLAB: Normalization and Format
238(1)
8.4 Compression
239(13)
8.4.1 Shannon Sampling, Compression, and Interpolation
239(1)
8.4.2 Fourier Truncation
240(2)
8.4.3 Periodization and Fourier Truncation
242(2)
8.4.4 Blocking, Compressing, and Smoothing Images
244(5)
8.4.5 Periodization and Blocking
249(1)
8.4.6 Compressing Modern Images
250(2)
8.5
Chapter Project
252(3)
9 Medical Imaging
255(24)
9.1 Computerized Tomography
255(14)
9.1.1 Projections, or Line Integrals of a 2-D function
256(3)
9.1.2 The Radon Tranform, or Central Slice Theorem
259(3)
9.1.3 Filtered Backprojection
262(2)
9.1.4 Non-locality of the Radon transform
264(1)
9.1.5 Localized Imaging
265(4)
9.2 Magnetic Resonance Imaging
269(8)
9.2.1 NMR and the Larmor Frequency: Making the Moments Sing
269(2)
9.2.2 Relaxation times: T1 and T2 times
271(2)
9.2.3 MRI by way of Tomography
273(1)
9.2.4 MRI via Phase Encoding
274(3)
9.3
Chapter Project
277(2)
10 Partial Differential Equations
279(20)
10.1 Ordinary Differential Equations: The Spring Equation
279(1)
10.2 The Wave Equation
280(11)
10.2.1 The Bound String Equation
280(1)
10.2.2 Separation of Variables
281(2)
10.2.3 Revisiting Completeness and Sine Transforms
283(1)
10.2.4 Eigenfunctions: Musical Overtones
284(1)
10.2.5 Problems and Exercises
285(2)
10.2.6 Two-Dimensional Wave Equation: The Square Drum
287(2)
10.2.7 Problems, Exercises, and Projects
289(2)
10.3 The Heat Equation
291(8)
10.3.1 A One-Dimensional Conductor
291(3)
10.3.2 Eigenfunctions and Examples
294(1)
10.3.3 A Two-Dimensional Example: The Heat Equation on a Square Plate
294(3)
10.3.4 Problems, Exercises, and Projects
297(2)
Bibliography 299(2)
Index 301
Tim Olson is an Associate Professor of Mathematics at the University of Florida. His research focuses on applications of Fourier Analysis to medical imaging and radar processing, electromagnetics, and other related fields. He is also an experienced fly-fisherman and a fishing guide in the rivers of Montana.
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