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E-raamat: Fourier Methods in Imaging

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Fourier Methods in Imaging introduces the mathematical tools for modeling linear imaging systems to predict the action of the system or for solving for the input. The chapters are grouped into five sections, the first introduces the imaging tasks (direct, inverse, and system analysis), the basic concepts of linear algebra for vectors and functions, including complex-valued vectors, and inner products of vectors and functions. The second section defines special functions, mathematical operations, and transformations that are useful for describing imaging systems. Among these are the Fourier transforms of 1-D and 2-D function, and the Hankel and Radon transforms. This section also considers approximations of the Fourier transform. The third and fourth sections examine the discrete Fourier transform and the description of imaging systems as linear filters, including the inverse, matched, Wiener and Wiener-Helstrom filters.This book helps students develop an understanding of mathematical tools for describing general one- and two- dimensional linear imaging systems, and will also serve as a reference for engineers and scientists.The Society for Imaging Science and Technology (imaging.org) is an international professional society whose mission is to keep members and other aware of the latest scientific and technological developments in the greater field of imaging. A major objective of the Wiley IS&T series is to advance this goal at the professional level the broad scope of the series focuses on imaging in all its aspects with particular emphasis on digital printing, electronic imaging, image assessment and reproduction, imagine archiving and preservation, color science, pre-press technologies, and hybrid imaging systems. Fourier Methods in Imaging introduces the mathematical tools for modeling linear imaging systems to predict the action of the system or for solving for the input. The chapters are grouped into five sections, the first introduces the imaging “tasks” (direct, inverse, and system analysis), the basic concepts of linear algebra for vectors and functions, including complex-valued vectors, and inner products of vectors and functions. The second section defines special functions, mathematical operations, and transformations that are useful for describing imaging systems. Among these are the Fourier transforms of 1-D and 2-D function, and the Hankel and Radon transforms. This section also considers approximations of the Fourier transform. The third and fourth sections examine the discrete Fourier transform and the description of imaging systems as linear filters, including the inverse, matched, Wiener and Wiener-Helstrom filters. The final section examines applications of linear system models to optical imaging systems, including holography. Provides a unified mathematical description of imaging systems. Develops a consistent mathematical formalism for characterizing imaging systems. Helps the reader develop an intuitive grasp of the most common mathematical methods, useful for describing the action of general linear systems on signals of one or more spatial dimensions. Offers parallel descriptions of continuous and discrete cases. Includes many graphical and pictorial examples to illustrate the concepts. This book helps students develop an understanding of mathematical tools for describing general one- and two-dimensional linear imaging systems, and will also serve as a reference for engineers and scientists

Arvustused

"Overall, this is an excellent text, appropriate for the graduate student approaching this material for the first time, and for the seasoned professional looking for an up-to-date reference." (Journal of Electronic Imaging, 1 April 2011) "This comprehensive textbook represents a practical review of Fourier techniques in imaging methods. It will be very useful for graduate students (in engineering, science, computer science, and applied mathematics) as well as engineers interested in linear imaging systems." (Zentralblatt Math, 2010)

Series Editor's Preface xix
Preface xxiii
1 Introduction
1(14)
1.1 Signals, Operators, and Imaging Systems
1(2)
1.1.1 The Imaging Chain
1(2)
1.2 The Three Imaging Tasks
3(1)
1.3 Examples of Optical Imaging
4(4)
1.3.1 Ray Optics
4(1)
1.3.2 Wave Optics
5(2)
1.3.3 System Evaluation of Hubble Space Telescope
7(1)
1.3.4 Imaging by Ground-Based Telescopes
8(1)
1.4 Imaging Tasks in Medical Imaging
8(7)
1.4.1 Gamma-Ray Imaging
9(2)
1.4.2 Radiography
11(2)
1.4.3 Computed Tomographic Radiography
13(2)
2 Operators and Functions
15(14)
2.1 Classes of Imaging Operators
15(1)
2.1.1 Linearity
15(1)
2.1.2 Shift Invariance
16(1)
2.2 Continuous and Discrete Functions
16(13)
2.2.1 Functions
16(1)
2.2.2 Functions with Continuous and Discrete Domains
17(2)
2.2.3 Continuous and Discrete Ranges
19(1)
2.2.4 Discrete Domain and Range- "Digitized" Functions
20(1)
2.2.5 Periodic, Aperiodic, and Harmonic Functions
21(5)
2.2.6 Symmetry Properties of Functions
26(1)
Problems
27(2)
3 Vectors with Real-Valued Components
29(22)
3.1 Scalar Products
29(5)
3.1.1 Scalar Product of Distinct Vectors
32(2)
3.1.2 Projection of One Vector onto Another
34(1)
3.2 Matrices
34(7)
3.2.1 Simultaneous Evaluation of Multiple Scalar Products
34(2)
3.2.2 Matrix-Matrix Multiplication
36(1)
3.2.3 Square and Diagonal Matrices, Identity Matrix
37(1)
3.2.4 Matrix Transposes
38(1)
3.2.5 Matrix Inverses
39(2)
3.3 Vector Spaces
41(10)
3.3.1 Basis Vectors
43(1)
3.3.2 Vector Subspaces Associated with a System
44(4)
Problems
48(3)
4 Complex Numbers and Functions
51(14)
4.1 Arithmetic of Complex Numbers
52(1)
4.1.1 Equality of Two Complex Numbers
52(1)
4.1.2 Sum and Difference of Two Complex Numbers
52(1)
4.1.3 Product of Two Complex Numbers
53(1)
4.1.4 Reciprocal of a Complex Numbers
53(1)
4.1.5 Ratio of Two Complex Numbers
53(1)
4.2 Graphical Representation of Complex Numbers
53(3)
4.3 Complex Functions
56(6)
4.4 Generalized Spatial Frequency-Negative Frequencies
62(1)
4.5 Argand Diagrams of Complex-Valued Functions
62(3)
Problems
63(2)
5 Complex-Valued Matrices and Systems
65(32)
5.1 Vectors with Complex-Valued Components
65(2)
5.1.1 Inner Product
66(1)
5.1.2 Products of Complex-Valued Matrices and Vectors
67(1)
5.2 Matrix Analogues of Shift-Invariant Systems
67(17)
5.2.1 Eigenvectors and Eigenvalues
70(2)
5.2.2 Projections onto Eigenvectors
72(3)
5.2.3 Diagonalization of a Circulant Matrix
75(5)
5.2.4 Matrix Operators for Shift-Invariant Systems
80(4)
5.2.5 Alternative Ordering of Eigenvectors
84(1)
5.3 Matrix Formulation of Imaging Tasks
84(4)
5.3.1 Inverse Imaging Problem
84(2)
5.3.2 Solution of Inverse Problems via Diagonalization
86(1)
5.3.3 Matrix-Vector Formulation of System Analysis
87(1)
5.4 Continuous Analogues of Vector Operations
88(9)
5.4.1 Inner Product of Continuous Functions
88(3)
5.4.2 Complete Sets of Basis Functions
91(1)
5.4.3 Orthonormal Basis Functions
92(1)
5.4.4 Continuous Analogue of DFT
93(1)
5.4.5 Eigenfunctions of Continuous Operators
93(1)
Problems
94(3)
6 1-D Special Functions
97(74)
6.1 Definitions of 1-D Special Functions
98(28)
6.1.1 Constant Function
99(1)
6.1.2 Rectangle Function
99(2)
6.1.3 Triangle Function
101(1)
6.1.4 Signum Function
101(1)
6.1.5 Step Function
102(2)
6.1.6 Exponential Function
104(1)
6.1.7 Sinusoid
105(4)
6.1.8 SINC Function
109(2)
6.1.9 SINC2 Function
111(1)
6.1.10 Gamma Function
112(3)
6.1.11 Quadratic-Phase Sinusoid- "Chirp" Function
115(2)
6.1.12 Gaussian Function
117(2)
6.1.13 "SuperGaussian" Function
119(2)
6.1.14 Bessel Functions
121(3)
6.1.15 Lorentzian Function
124(1)
6.1.16 Thresholded Functions
125(1)
6.2 1-D Dirac Delta Function
126(16)
6.2.1 1-D Dirac Delta Function Raised to a Power
131(1)
6.2.2 Sifting Property of 1-D Dirac Delta Function
132(1)
6.2.3 Symmetric (Even) Pair of 1-D Dirac Delta Functions
133(1)
6.2.4 Antisymmetric (Odd) Pair of 1-D Dirac Delta Functions
134(1)
6.2.5 COMB Function
135(2)
6.2.6 Derivatives of 1-D Dirac Delta Function
137(2)
6.2.7 Dirac Delta Function with Functional Argument
139(3)
6.3 1-D Complex-Valued Special Functions
142(7)
6.3.1 Complex Linear-Phase Sinusoid
143(1)
6.3.2 Complex Quadratic-Phase Exponential Function
143(2)
6.3.3 "Superchirp" Function
145(2)
6.3.4 Complex-Valued Lorentzian Function
147(2)
6.3.5 Logarithm of the Complex Amplitude
149(1)
6.4 1-D Stochastic Functions-Noise
149(13)
6.4.1 Moments of Probability Distributions
151(1)
6.4.2 Discrete Probability Laws
152(4)
6.4.3 Continuous Probability Distributions
156(4)
6.4.4 Signal-to-Noise Ratio
160(1)
6.4.5 Example: Variance of a Sinusoid
161(1)
6.4.6 Example: Variance of a Square Wave
161(1)
6.4.7 Approximations to SNR
161(1)
6.5 Appendix A: Area of SINC[ x] and SINC2[ x]
162(4)
6.6 Appendix B: Series Solutions for Bessel Functions J0[ x] and J1[ x]
166(5)
Problems
169(2)
7 2-D Special Functions
171(36)
7.1 2-D Separable Functions
171(3)
7.1.1 Rotations of 2-D Separable Functions
172(1)
7.1.2 Rotated Coordinates as Scalar Products
172(2)
7.2 Definitions of 2-D Special Functions
174(8)
7.2.1 2-D Constant Function
174(1)
7.2.2 Rectangle Function
175(1)
7.2.3 Triangle Function
176(1)
7.2.4 2-D Signum and STEP Functions
176(2)
7.2.5 2-D SINC Function
178(1)
7.2.6 SINC2 Function
178(2)
7.2.7 2-D Gaussian Function
180(1)
7.2.8 2-D Sinusoid
180(2)
7.3 2-D Dirac Delta Function and its Relatives
182(13)
7.3.1 2-D Dirac Delta Function in Cartesian Coordinates
183(1)
7.3.2 2-D Dirac Delta Function in Polar Coordinates
184(2)
7.3.3 2-D Separable COMB Function
186(1)
7.3.4 2-D Line Delta Function
187(7)
7.3.5 2-D "Cross" Function
194(1)
7.3.6 "Corral" Function
195(1)
7.4 2-D Functions with Circular Symmetry
195(9)
7.4.1 Cylinder (Circle) Function
196(1)
7.4.2 Circularly Symmetric Gaussian Function
197(1)
7.4.3 Circularly Symmetric Bessel Function of Zero Order
197(3)
7.4.4 Besinc or Sombrero Function
200(1)
7.4.5 Circular Triangle Function
201(1)
7.4.6 Ring Delta Function
202(2)
7.5 Complex-Valued 2-D Functions
204(1)
7.5.1 Complex 2-D Sinusoid
204(1)
7.5.2 Complex Quadratic-Phase Sinusoid
205(1)
7.6 Special Functions of Three (or More) Variables
205(2)
Problems
206(1)
8 Linear Operators
207(32)
8.1 Linear Operators
208(5)
8.2 Shift-Invariant Operators
213(3)
8.3 Linear Shift-Invariant (LSI) Operators
216(6)
8.3.1 Linear Shift-Variant Operators
221(1)
8.4 Calculating Convolutions
222(1)
8.4.1 Examples of Convolutions
223(1)
8.5 Properties of Convolutions
223(3)
8.5.1 Region of Support of Convolutions
225(1)
8.5.2 Area of a Convolution
225(1)
8.5.3 Convolution of Scaled Functions
226(1)
8.6 Autocorrelation
226(3)
8.6.1 Autocorrelation of Stochastic Functions
228(1)
8.6.2 Autocovariance of Stochastic Functions
229(1)
8.7 Crosscorrelation
229(3)
8.8 2-D LSI Operations
232(2)
8.8.1 Line-Spread and Edge-Spread Functions
233(1)
8.9 Crosscorrelations of 2-D Functions
234(1)
8.10 Autocorrelations of 2-D Functions
235(4)
8.10.1 Autocorrelation of the Cylinder Function
236(1)
Problems
236(3)
9 Fourier Transforms of 1-D Functions
239(86)
9.1 Transforms of Continuous-Domain Functions
239(11)
9.1.1 Example 1: Input and Reference Functions are Even Sinusoids
242(3)
9.1.2 Example 2: Even Sinusoid Input, Odd Sinusoid Reference
245(1)
9.1.3 Example 3: Odd Sinusoid Input, Even Sinusoid Reference
246(1)
9.1.4 Example 4: Odd Sinusoid Input and Reference
247(3)
9.2 Linear Combinations of Reference Functions
250(4)
9.2.1 Hartley Transform
251(1)
9.2.2 Examples of the Hartley Transform
251(1)
9.2.3 Inverse of the Hartley Transform
252(2)
9.3 Complex-Valued Reference Functions
254(2)
9.4 Transforms of Complex-Valued Functions
256(3)
9.5 Fourier Analysis of Dirac Delta Functions
259(2)
9.6 Inverse Fourier Transform
261(2)
9.7 Fourier Transforms of 1-D Special Functions
263(17)
9.7.1 Fourier Transform of δ[ x]
264(1)
9.7.2 Fourier Transform of Rectangle
264(2)
9.7.3 Fourier Transforms of Sinusoids
266(2)
9.7.4 Fourier Transform of Signum and Step
268(2)
9.7.5 Fourier Transform of Exponential
270(5)
9.7.6 Fourier Transform of Gaussian
275(1)
9.7.7 Fourier Transforms of Chirp Functions
276(3)
9.7.8 Fourier Transform of COMB Function
279(1)
9.8 Theorems of the Fourier Transform
280(40)
9.8.1 Multiplication by Constant
281(1)
9.8.2 Addition Theorem (Linearity)
281(1)
9.8.3 Fourier Transform of a Fourier Transform
281(3)
9.8.4 Central-Ordinate Theorem
284(1)
9.8.5 Scaling Theorem
284(3)
9.8.6 Shift Theorem
287(2)
9.8.7 Filter Theorem
289(6)
9.8.8 Modulation Theorem
295(2)
9.8.9 Derivative Theorem
297(1)
9.8.10 Fourier Transform of Complex Conjugate
298(1)
9.8.11 Fourier Transform of Crosscorrelation
299(3)
9.8.12 Fourier Transform of Autocorrelation
302(1)
9.8.13 Rayleigh's Theorem
302(2)
9.8.14 Parseval's Theorem
304(2)
9.8.15 Fourier Transform of Periodic Function
306(1)
9.8.16 Spectrum of Sampled Function
307(1)
9.8.17 Spectrum of Discrete Periodic Function
308(1)
9.8.18 Spectra of Stochastic Signals
308(2)
9.8.19 Effect of Nonlinear Operations of Spectra
310(10)
9.9 Appendix: Spectrum of Gaussian via Path Integral
320(5)
Problems
321(4)
10 Multidimensional Fourier Transforms
325(22)
10.1 2-D Fourier Transforms
325(2)
10.1.1 2-D Fourier Synthesis
326(1)
10.2 Spectra of Separable 2-D Functions
327(8)
10.2.1 Fourier Transforms of Separable Functions
328(1)
10.2.2 Fourier Transform of δ[ x, y]
328(2)
10.2.3 Fourier Transform of δ[ x - x0, y - y0]
330(2)
10.2.4 Fourier Transform of RECT[ x, y]
332(1)
10.2.5 Fourier Transform of TRI[ x, y]
332(1)
10.2.6 Fourier Transform of GAUS[ x, y]
332(2)
10.2.7 Fourier Transform of STEP[ x] . STEP[ y]
334(1)
10.2.8 Theorems of Spectra of Separable Functions
334(1)
10.2.9 Superpositions of 2-D Separable Functions
335(1)
10.3 Theorems of 2-D Fourier Transforms
335(12)
10.3.1 2-D "Transform-of-a-Transform" Theorem
336(1)
10.3.2 2-D Scaling Theorem
336(1)
10.3.3 2-D Shift Theorem
336(1)
10.3.4 2-D Filter Theorem
337(1)
10.3.5 2-D Derivative Theorem
338(2)
10.3.6 Spectra of Rotated 2-D Functions
340(1)
10.3.7 Transforms of 2-D Line Delta and Cross Functions
341(4)
Problems
345(2)
11 Spectra of Circular Functions
347(24)
11.1 The Hankel Transform
347(6)
11.1.1 Hankel Transform of Dirac Delta Function
351(2)
11.2 Inverse Hankel Transform
353(1)
11.3 Theorems of Hankel Transforms
354(2)
11.3.1 Scaling Theorem
354(1)
11.3.2 Shift Theorem
354(1)
11.3.3 Central-Ordinate Theorem
354(1)
11.3.4 Filter and Crosscorrelation Theorems
355(1)
11.3.5 "Transform-of-a-Transform" Theorem
355(1)
11.3.6 Derivative Theorem
355(1)
11.3.7 Laplacian of Circularly Symmetric Function
356(1)
11.4 Hankel Transforms of Special Functions
356(9)
11.4.1 Hankel Transform of J0(2πrρ0)
356(2)
11.4.2 Hankel Transform of CYL(r)
358(2)
11.4.3 Hankel Transform of r-1
360(1)
11.4.4 Hankel Transform from 2-D Fourier Transforms
361(2)
11.4.5 Hankel Transform of r2 GAUS(r)
363(1)
11.4.6 Hankel Transform of CTRI(r)
364(1)
11.5 Appendix: Derivations of Equations (11.12) and (11.14)
365(6)
Problems
369(2)
12 The Radon Transform
371(50)
12.1 Line-Integral Projections onto Radial Axes
371(9)
12.1.1 Radon Transform of Dirac Delta Function
377(2)
12.1.2 Radon Transform of Arbitrary Function
379(1)
12.2 Radon Transforms of Special Functions
380(7)
12.2.1 Cylinder Function CYL(r)
380(2)
12.2.2 Ring Delta Function δ(r - r0)
382(2)
12.2.3 Rectangle Function RECT[ x, y]
384(1)
12.2.4 Corral Function COR[ x, y]
385(2)
12.3 Theorems of the Radon Transform
387(4)
12.3.1 Radon Transform of a Superposition
387(1)
12.3.2 Radon Transform of Scaled Function
388(1)
12.3.3 Radon Transform of Translated Function
389(1)
12.3.4 Central-Slice Theorem
389(1)
12.3.5 Filter Theorem of the Radon Transform
390(1)
12.4 Inverse Radon Transform
391(11)
12.4.1 Recovery of Dirac Delta Function from Projections
392(6)
12.4.2 Summation of Projections over Azimuths
398(4)
12.5 Central-Slice Transform
402(8)
12.5.1 Radial "Slices" of ƒ[ x, y]
402(1)
12.5.2 Central-Slice Transforms of Special Functions
403(6)
12.5.3 Inverse Central-Slice Transform
409(1)
12.6 Three Transforms of Four Functions
410(9)
12.7 Fourier and Radon Transforms of Images
419(2)
Problems
420(1)
13 Approximations to Fourier Transforms
421(38)
13.1 Moment Theorem
421(15)
13.1.1 First Moment-Centroid
424(1)
13.1.2 Second Moment-Moment of Inertia
424(1)
13.1.3 Central Moments-Variance
425(2)
13.1.4 Evaluation of 1-D Spectra from Moments
427(4)
13.1.5 Spectra of 1-D Superchirps via Moments
431(2)
13.1.6 2-D Moment Theorem
433(2)
13.1.7 Moments of Circularly Symmetric Functions
435(1)
13.2 1-D Spectra via Method of Stationary Phase
436(16)
13.2.1 Examples of Spectra via Stationary Phase
440(12)
13.3 Central-Limit Theorem
452(2)
13.4 Width Metrics and Uncertainty Relations
454(5)
13.4.1 Equivalent Width
454(1)
13.4.2 Uncertainty Relation for Equivalent Width
455(1)
13.4.3 Variance as a Measure of Width
455(2)
Problems
457(2)
14 Discrete Systems, Sampling, and Quantization
459(52)
14.1 Ideal Sampling
460(7)
14.1.1 Ideal Sampling of 2-D Functions
461(1)
14.1.2 Is Sampling a Linear Operation?
462(1)
14.1.3 Is the Sampling Operation Shift Invariant?
462(3)
14.1.4 Aliasing Artifacts
465(2)
14.1.5 Operations Similar to Ideal Sampling
467(1)
14.2 Ideal Sampling of Special Functions
467(5)
14.2.1 Ideal Sampling of δ[ x] and COMB[ x]
470(2)
14.3 Interpolation of Sampled Functions
472(7)
14.3.1 Examples of Interpolation
478(1)
14.4 Whittaker-Shannon Sampling Theorem
479(1)
14.5 Aliasing and Interpolation
480(3)
14.5.1 Frequency Recovered from Aliased Samples
480(2)
14.5.2 "Unwrapping" the Phase of Sampled Functions
482(1)
14.6 "Prefiltering" to Prevent Aliasing
483(3)
14.6.1 Prefiltered Images Recovered from Samples
484(1)
14.6.2 Sampling and Reconstruction of Audio Signals
485(1)
14.7 Realistic Sampling
486(5)
14.8 Realistic Interpolation
491(9)
14.8.1 Ideal Interpolator for Compact Functions
491(1)
14.8.2 Finite-Support Interpolators in Space Domain
491(4)
14.8.3 Realistic Frequency-Domain Interpolators
495(5)
14.9 Quantization
500(7)
14.9.1 Quantization "Noise"
503(2)
14.9.2 SNR of Quantization
505(2)
14.9.3 Quantizers with Memory - "Error Diffusion"
507(1)
14.10 Discrete Convolution
507(4)
Problems
509(2)
15 Discrete Fourier Transforms
511(62)
15.1 Inverse of the Infinite-Support DFT
513(1)
15.2 DFT over Finite Interval
514(13)
15.2.1 Finite DFT of ƒ[ x] = 1[ x]
522(2)
15.2.2 Scale Factor in DFT
524(2)
15.2.3 Finite DFT of Discrete Dirac Delta Function
526(1)
15.2.4 Summary of Finite DFT
526(1)
15.3 Fourier Series Derived from Fourier Transform
527(2)
15.4 Efficient Evaluation of the Finite DFT
529(5)
15.4.1 DFT of Two Samples-The "Butterfly"
530(1)
15.4.2 DFT of Three Samples
531(1)
15.4.3 DFT of Four Samples
532(1)
15.4.4 DFT of Six Samples
532(1)
15.4.5 DFT of Eight Samples
533(1)
15.4.6 Complex Matrix for Computing 1-D DFT
534(1)
15.5 Practical Considerations for DFT and FFT
534(29)
15.5.1 Computational Intensity
534(2)
15.5.2 "Centered" versus "Uncentered" Arrays
536(2)
15.5.3 Units of Measure in the Two Domains
538(1)
15.5.4 Ensuring Periodicity of Arrays - Data "Windows"
539(6)
15.5.5 A Garden of 1-D FFT Windows
545(6)
15.5.6 Undersampling and Aliasing
551(3)
15.5.7 Phase
554(1)
15.5.8 Zero Padding
554(1)
15.5.9 Discrete Convolution and the Filter Theorem
555(4)
15.5.10 Discrete Transforms of Quantized Functions
559(1)
15.5.11 Parseval's Theorem for DFT
560(2)
15.5.12 Scaling Theorem for Sampled Functions
562(1)
15.6 FFTs of 2-D Arrays
563(4)
15.6.1 Interpretation of 2-D FFTs
564(3)
15.6.2 2-D Hann Window
567(1)
15.7 Discrete Cosine Transform
567(6)
Problems
571(2)
16 Magnitude Filtering
573(30)
16.1 Classes of Filters
574(2)
16.1.1 Magnitude Filters
574(1)
16.1.2 Phase ("Allpass") Filters
575(1)
16.2 Eigenfunctions of Convolution
576(1)
16.3 Power Transmission of Filters
577(2)
16.4 Lowpass Filters
579(6)
16.4.1 1-D Test Object
581(1)
16.4.2 Ideal 1-D Lowpass Filter
581(1)
16.4.3 1-D Uniform Averager
581(2)
16.4.4 2-D Lowpass Filters
583(2)
16.5 Highpass Filters
585(4)
16.5.1 Ideal 1-D Highpass Filter
585(1)
16.5.2 1-D Differentiators
586(1)
16.5.3 2-D Differentiators
587(1)
16.5.4 High-Frequency Boost Filters-Image Sharpeners
588(1)
16.6 Bandpass Filters
589(5)
16.7 Fourier Transform as a Bandpass Filter
594(2)
16.8 Bandboost and Bandstop Filters
596(3)
16.9 Wavelet Transform
599(4)
16.9.1 Tiling of Frequency Domain with Orthogonal Wavelets
600(2)
16.9.2 Example of Wavelet Decomposition
602(1)
Problems
602(1)
17 Allpass (Phase) Filters
603(44)
17.1 Power-Series Expansion for Allpass Filters
604(1)
17.2 Constant-Phase Allpass Filter
605(1)
17.3 Linear-Phase Allpass Filter
606(2)
17.4 Quadratic-Phase Filter
608(7)
17.4.1 Impulse Response and Transfer Function
608(4)
17.4.2 Scaling of Quadratic-Phase Transfer Function
612(3)
17.4.3 Limiting Behavior of the Quadratic-Phase Allpass Filter
615(1)
17.4.4 Impulse Response of Allpass Filters of Order 0, 1, 2
615(1)
17.5 Allpass Filters with Higher-Order Phase
615(4)
17.5.1 Odd-Order Allpass Filters with n≥3
618(1)
17.5.2 Even-Order Allpass Filters with n≥4
619(1)
17.6 Allpass Random-Phase Filter
619(7)
17.6.1 Information Recovery after Random-Phase Filtering
626(1)
17.7 Relative Importance of Magnitude and Phase
626(2)
17.8 Imaging of Phase Objects
628(4)
17.9 Chrip Fourier Transform
632(15)
17.9.1 1-D "M-C-M" Chirp Fourier Transform
632(2)
17.9.2 1-D "C-M-C" Chirp Fourier Transform
634(3)
17.9.3 M-C-M and C-M-C with Opposite-Sign Chirps
637(1)
17.9.4 2-D Chirp Fourier Transform
638(1)
17.9.5 Optical Correlator
638(3)
17.9.6 Optical Chirp Fourier Transformer
641(4)
Problems
645(2)
18 Magnitude-Phase Filters
647(20)
18.1 Transfer Functions of Three Operations
648(5)
18.1.1 Identity Operator
648(1)
18.1.2 Differentiation
648(2)
18.1.3 Integration
650(3)
18.2 Fourier Transform of Ramp Function
653(1)
18.3 Causal Filters
654(4)
18.4 Damped Harmonic Oscillator
658(3)
18.5 Mixed Filters with Linear or Random Phase
661(1)
18.6 Mixed Filter with Quadratic Phase
661(6)
Problems
666(1)
19 Applications of Linear Filters
667(56)
19.1 Linear Filters for the Imaging Tasks
667(2)
19.2 Deconvolution - "Inverse Filtering"
669(10)
19.2.1 Conditions for Exact Recovery via Inverse Filtering
671(1)
19.2.2 Inverse Filter for Uniform Averager
672(3)
19.2.3 Inverse Filter for Ideal Lowpass Filter
675(3)
19.2.4 Inverse Filter for Decaying Exponential
678(1)
19.3 Optimum Estimators for Signals in Noise
679(17)
19.3.1 Wiener Filter
680(8)
19.3.2 Wiener Filter Example
688(1)
19.3.3 Wiener-Helstrom Filter
689(4)
19.3.4 Wiener-Helstrom Filter Example
693(2)
19.3.5 Constrained Least-Squares Filter
695(1)
19.4 Detection of Known Signals - Matched Filter
696(7)
19.4.1 Inputs for Matched Filters
701(2)
19.5 Analogies of Inverse and Matched Filters
703(5)
19.5.1 Wiener and Wiener-Helstrom "Matched" Filter
706(2)
19.6 Approximations to Reciprocal Filters
708(11)
19.6.1 Small-Order Approximations of Reciprocal Filters
711(2)
19.6.2 Examples of Approximate Reciprocal Filters
713(6)
19.7 Inverse Filtering of Shift-Variant Blur
719(4)
Problems
720(3)
20 Filtering in Discrete Systems
723(30)
20.1 Translation, Leakage, and Interpolation
724(4)
20.1.1 1-D Translation
724(2)
20.1.2 2-D Translation
726(2)
20.2 Averaging Operators-Lowpass Filters
728(3)
20.2.1 1-D Averagers
728(2)
20.2.2 2-D Averagers
730(1)
20.3 Differencing Operators - Highpass Filters
731(9)
20.3.1 1-D Derivative
731(1)
20.3.2 2-D Derivative Operators
732(2)
20.3.3 1-D Antisymmetric Differentiation Kernel
734(2)
20.3.4 Second Derivative
736(1)
20.3.5 2-D Second Derivative
736(1)
20.3.6 Laplacian
737(3)
20.4 Discrete Sharpening Operators
740(3)
20.4.1 1-D Sharpeners
740(2)
20.4.2 2-D Sharpening Operators
742(1)
20.5 2-D Gradient
743(1)
20.6 Pattern Matching
744(5)
20.6.1 Normalization of Contrast of Detected Features
747(1)
20.6.2 Amplified Discrete Matched Filters
748(1)
20.7 Approximate Discrete Reciprocal Filters
749(4)
20.7.1 Derivative
749(2)
Problems
751(2)
21 Optical Imaging in Monochromatic Light
753(70)
21.1 Imaging Systems Based on Ray Optics Model
754(8)
21.1.1 Seemingly "Plausible" Models of Light in Imaging
754(4)
21.1.2 Imaging Systems Based on Ray "Selection" by Absorption
758(2)
21.1.3 Imaging System that Selects and Reflects Rays
760(1)
21.1.4 Imaging Systems Based on Refracting Rays
761(1)
21.1.5 Model of Imaging Systems
761(1)
21.2 Mathematical Model of Light Propagation
762(21)
21.2.1 Wave Description of Light
762(3)
21.2.2 Irradiance
765(1)
21.2.3 Propagation of Light
765(7)
21.2.4 Examples of Fresnel Diffraction
772(11)
21.3 Fraunhofer Diffraction
783(7)
21.3.1 Examples of Fraunhofer Diffraction
785(5)
21.4 Imaging System based on Fraunhofer Diffraction
790(2)
21.5 Transmissive Optical Elements
792(4)
21.5.1 Optical Elements with Constant or Linear Phase
793(1)
21.5.2 Lenses with Spherical Surfaces
794(2)
21.6 Monochromatic Optical Systems
796(15)
21.6.1 Single Positive Lens with z1>>0
796(3)
21.6.2 Single-Lens System, Fresnel Description of Both Propagations
799(4)
21.6.3 Amplitude Distribution at Image Point
803(3)
21.6.4 Shift-Invariant Description of Optical Imaging
806(1)
21.6.5 Examples of Single-Lens Imaging Systems
807(4)
21.7 Shift-Variant Imaging Systems
811(12)
21.7.1 Response of System at "Nonimage" Point
811(5)
21.7.2 Chirp Fourier Transform and Fraunhofer Diffraction
816(3)
Problems
819(4)
22 Incoherent Optical Imaging Systems
823(32)
22.1 Coherence
823(15)
22.1.1 Optical Interference
823(5)
22.1.2 Spatial Coherence
828(10)
22.2 Polychromatic Source - Temporal Coherence
838(4)
22.2.1 Coherence Volume
842(1)
22.3 Imaging in Incoherent Light
842(3)
22.4 System Function in Incoherent Light
845(10)
22.4.1 Incoherent MTF
846(1)
22.4.2 Comparison of Coherent and Incoherent Imaging
847(6)
Problems
853(2)
23 Holography
855(62)
23.1 Fraunhofer Holography
856(11)
23.1.1 Two Points: Object and Reference
856(6)
23.1.2 Multiple Object Points
862(2)
23.1.3 Fraunhofer Hologram of Extended Object
864(2)
23.1.4 Nonlinear Fraunhofer Hologram of Extended Object
866(1)
23.2 Holography in Fresnel Diffraction Region
867(18)
23.2.1 Object and Reference Sources in Same Plane
868(4)
23.2.2 Reconstruction of Virtual Image from Hologram with Compact Support
872(1)
23.2.3 Reconstruction of Real Image: z2 > 0
872(1)
23.2.4 Object and Reference Sources in Different Planes
873(5)
23.2.5 Reconstruction of Point Object
878(4)
23.2.6 Extended Object and Planar Reference Wave
882(1)
23.2.7 Interpretation of Fresnel Hologram as Lens
883(2)
23.2.8 Reconstruction of Real Image of 3-D Extended Object
885(1)
23.3 Computer-Generated Holography
885(13)
23.3.1 CGH in the Fraunhofer Diffraction Region
886(4)
23.3.2 Examples of Cell CGHs
890(4)
23.3.3 2-D Lohmann Holograms
894(1)
23.3.4 Error-Diffused Quantization
895(3)
23.4 Matched Filtering with Cell-Type CGH
898(2)
23.5 Synthetic-Aperture Radar (SAR)
900(17)
23.5.1 Range Resolution
904(2)
23.5.2 Azimuthal Resolution
906(1)
23.5.3 SAR System Architecture
907(7)
Problems
914(3)
References 917(4)
Index 921
Professor Roger L. Easton, Jr Chester F. Carlson Center for Imaging Science, Rochester Institute of Technology Professor Easton teaches undergraduate and graduate courses in linear systems, optical imaging, and digital image processing at Rochester Institute of Technology. He received a B.S. degree in Astronomy from Haverford College, an M.S. in physics from the University of Maryland, and an M.S. and Ph.D. degree in Optical Sciences from the University of Arizona. His research interests include the application of digital image processing to text documents and manuscripts. He has contributed to work on the Dead Sea Scrolls and is now part of an imaging team helping scolars to read the original Archimiedes Palimpsest. Professor Easton also conducts research into optical signal processing and computer-generated holography, publishing articles on both.
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