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E-raamat: Limits of Resolution

(Birmingham University, United Kingdom), (King's College, London, United Kingdom)
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Limits of ResolutionSuurem pilt
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"This beautiful book can be read as a novel presenting carefully our quest to get more and more information from our observations and measurements. Its authors are particularly good at relating it." --Pierre C. Sabatier

"This is a unique text - a labor of love pulling together for the first time the remarkably large array of mathematical and statistical techniques used for analysis of resolution in many systems of importance today optical, acoustical, radar, etc. I believe it will find widespread use and value." --Dr. Robert G.W. Brown, Chief Executive Officer, American Institute of Physics

"The mix of physics and mathematics is a unique feature of this book which can be basic not only for PhD students but also for researchers in the area of computational imaging." --Mario Bertero, Professor, University of Geneva

"a tour-de-force covering aspects of history, mathematical theory and practical applications. The authors provide a penetrating insight into the often confused topic of resolution and in doing offer a unifying approach to the subject that is applicable not only to traditional optical systems but also modern day, computer-based systems such as radar and RF communications." --Prof. Ian Proudler, Loughborough University

"a must have for anyone interested in imaging and the spatial resolution of images. This book provides detailed and very readable account of resolution in imaging and organizes the recent history of the subject in excellent fashion. I strongly recommend it." --Michael A. Fiddy, Professor, University of North Carolina at Charlotte

This book brings together the concept of resolution, which limits what we can determine about our physical world, with the theory of linear inverse problems, emphasizing practical applications. The book focuses on methods for solving illposed problems that do not have unique stable solutions. After introducing basic concepts, the contents address problems with "continuous" data in detail before turning to cases of discrete data sets. As one of the unifying principles of the text, the authors explain how non-uniqueness is a feature of measurement problems in science where precision and resolution is essentially always limited by some kind of noise.

Arvustused

"This is a unique text - a labor of love, I suspect - pulling together for the first time the remarkably large array of mathematical and statistical techniques used for analysis of resolution in many systems of importance today optical, acoustical, radar, etc. The final two chapters provide the perspective of practical resolution challenges in a multitude of situations, from light scattering and particle sizing to microscopy and tomography. These chapters will be of great value to those concerned with the fundamental limits in these fields. I believe it will find widespread use and value." Dr. Robert G.W. Brown, Chief Executive Officer, American Institute of Physics

"a tour-de-force covering aspects of history, mathematical theory and practical applications. The authors provide a penetrating insight into the often confused topic of resolution and in doing offer a unifying approach to the subject that is applicable not only to traditional optical systems but also modern day, computer-based systems such as radar and RF communications."

Prof. Ian Proudler, Loughborough University

"This is a book on the ultimate limits of resolution in imaging when digital image processing is considered. Therefore it is also a book on linear inverse problems, a basic tool in computational imaging. It provides a thorough account of this topic showing that solving an inverse problem is a mathematical game which requires a deep understanding of its physical foundations. The mix of physics and mathematics is a unique feature of this book which can be basic not only for PhD students but also for researchers in the area of computational imaging." Mario Bertero, Professor, University of Geneva

"This beautiful book can be read as a novel presenting carefully our quest to get more and more information from our observations and measurements. Its authors are particularly good at relating it." Pierre C. Sabatier

"a must

Preface xv
Authors xxi
1 Early Concepts of Resolution 1(66)
1.1 Introduction
1(30)
1.1.1 Early History
1(7)
1.1.2 A Human Perspective on Resolution
8(1)
1.1.3 Pinhole Camera
9(2)
1.1.4 Coherent and Incoherent Imaging
11(3)
1.1.5 Abbe Theory of the Coherently Illuminated Microscope
14(3)
1.1.6 Digression on the Sine Condition
17(3)
1.1.7 Further Discussion on Abbe's Work
20(3)
1.1.8 Work of Helmholtz
23(2)
1.1.9 Filters, Signals and Fourier Analysis
25(2)
1.1.10 Optical Transfer Functions and Modulation Transfer Functions
27(3)
1.1.11 Some Observations on the Term Spectrum
30(1)
1.2 Resolution and Prior Information
31(1)
1.2.1 One- and Two-Point Resolution
31(1)
1.2.2 Different Two-Point Resolution Criteria
31(1)
1.3 Communication Channels and Information
32(6)
1.3.1 Early Steps towards the 2WT Theorem
33(1)
1.3.2 Nyquist Rate
33(1)
1.3.3 Hartley's Information Capacity
34(1)
1.3.4 Entropy and the Statistical Approach to Information
35(3)
1.4 Shannon Sampling Theorem
38(2)
1.4.1 Sampling Theorem
38(1)
1.4.2 Origins of the Sampling Theorem
39(1)
1.5 The 2WT Theorem
40(3)
1.5.1 'Proof' of the 2WT Theorem
40(1)
1.5.2 Flaw in the 2WT Theorem
41(1)
1.5.3 Shannon Number
41(1)
1.5.4 Gabor's Elementary Signals
42(1)
1.6 Channel Capacity
43(4)
1.6.1 Channel Capacity for a Band-Limited Noisy Channel
43(2)
1.6.2 Information Content of Noisy Images: Gabor's Approach
45(1)
1.6.3 Information Content of Noisy Images: The Fellgett and Linfoot Approach
45(1)
1.6.4 Channel Capacity and Resolution
46(1)
1.7 Super-Directivity and Super-Resolution through Apodisation
47(11)
1.7.1 Introduction
47(1)
1.7.2 Super-Directive Endfire Arrays
48(2)
1.7.3 Radiation from an Aperture
50(4)
1.7.4 Woodward's Method for Beam-Pattern Design
54(1)
1.7.5 Dolph-Chebyshev Beam Pattern
54(1)
1.7.6 Taylor Line-Source Distribution
55(1)
1.7.7 Taylor Disc-Source Distribution
56(2)
1.7.8 Super-Resolving Pupils
58(1)
1.8 Summary
58(1)
Acknowledgements
59(1)
References
59(8)
2 Beyond the 2WT Theorem 67(78)
2.1 Introduction
67(1)
2.2 Simultaneous Concentration of Functions in Time and Frequency
68(6)
2.2.1 Prolate Spheroidal Wave Functions
68(4)
2.2.2 2WT Theorem as a Limit
72(2)
2.3 Higher Dimensions
74(3)
2.3.1 Generalised Prolate Spheroidal Wave Functions
74(1)
2.3.2 Circular Prolate Functions
75(2)
2.4 2WT Theorem and Information for Coherent Imaging
77(1)
2.5 2WT Theorem and Optical Super-Resolution
77(4)
2.5.1 Moire Imaging
79(1)
2.5.2 Digital Super-Resolution
80(1)
2.5.2.1 Microscan
80(1)
2.5.2.2 Super-Resolution Using a Rotating/Reconfigurable Mask
80(1)
2.5.2.3 TOMBO
80(1)
2.5.2.4 Super-Resolution in Panoramic Imaging
81(1)
2.5.2.5 Super-Resolution through Motion
81(1)
2.6 Super-Directivity
81(15)
2.6.1 Introduction
81(1)
2.6.2 Super-Directivity Ratio
82(1)
2.6.3 Digression on Singular Functions
82(1)
2.6.4 Line Sources and the Prolate Spheroidal Wave Functions
83(2)
2.6.5 Realisable Rectangular Aperture Distributions
85(4)
2.6.6 Physical Interpretation of the Super-Directivity Ratio
89(1)
2.6.7 Circular Apertures: The Scalar Theory
90(1)
2.6.8 Realisable Circular Aperture Distributions
91(2)
2.6.9 Discretisation of Continuous Aperture Distributions
93(3)
2.7 Broadband Line Sources
96(4)
2.7.1 Basic Spaces
96(1)
2.7.2 Operators
97(1)
2.7.3 Properties of the Singular Functions
97(3)
2.7.4 Uses of the Singular Functions
100(1)
2.8 Super-Resolution through Apodisation
100(5)
2.8.1 Apodisation Problem of Slepian
100(3)
2.8.2 Super-Resolving Pupils
103(2)
2.8.3 Generalised Gaussian Quadrature and Apodisation
105(1)
2.8.4 Further Developments in Apodisation
105(1)
2.9 Super-Oscillation
105(1)
2.10 Linear Inverse Problems
106(7)
2.10.1 Band-Limited Extrapolation
107(1)
2.10.2 General Inverse Problems
107(1)
2.10.3 Ill-Posedness
108(1)
2.10.4 Linear Inverse Problems
108(4)
2.10.5 Some Ways of Dealing with Ill-Posedness
112(1)
2.11 One-Dimensional Coherent Imaging
113(8)
2.11.1 Eigenfunction Solution
113(2)
2.11.2 Singular-Function Solution
115(5)
2.11.3 Super-Resolution through Restriction of Object Support
120(1)
2.12 One-Dimensional Incoherent Imaging
121(4)
2.12.1 Eigenfunction Approach
121(1)
2.12.2 2WT Theorem and Information for Incoherent Imaging
122(1)
2.12.3 Singular-Function Approach
122(3)
2.13 Two-Dimensional Coherent Imaging
125(3)
2.13.1 Generalised Prolate Spheroidal Wave Functions
125(1)
2.13.2 Case of Square Object and Square Pupil
126(1)
2.13.3 Case of Circular Object and Circular Pupil
126(1)
2.13.4 Super-Resolution
127(1)
2.14 Two-Dimensional Incoherent Imaging
128(2)
2.14.1 Square Object and Square Pupil
128(1)
2.14.2 Circular Object and Circular Pupil
129(1)
2.15 Quantum Limits of Optical Resolution
130(9)
2.15.1 Basic Physics
130(2)
2.15.2 One-Dimensional Super-Resolving Fourier Microscopy
132(3)
2.15.3 Effects of Quantum Fluctuations
135(1)
2.15.4 Squeezed States
136(1)
2.15.5 Extension to Two Dimensions
137(2)
References
139(6)
3 Elementary Functional Analysis 145(40)
3.1 Introduction
145(1)
3.2 Metric Spaces
146(2)
3.2.1 Continuity
147(1)
3.2.2 Basic Topology for Metric Spaces
147(1)
3.3 Measures and Lebesgue Integration
148(4)
3.3.1 Introduction
148(1)
3.3.2 Basic Measure Theory and Borel Sets
148(1)
3.3.3 Lebesgue Measure
149(1)
3.3.4 Measureable Functions
150(1)
3.3.5 Lebesgue Integration
150(2)
3.4 Vector Spaces
152(2)
3.4.1 Operators on Vector Spaces
153(1)
3.5 Finite-Dimensional Vector Spaces and Matrices
154(7)
3.5.1 Finite-Dimensional Normed Spaces
155(2)
3.5.2 Finite-Dimensional Inner-Product Spaces
157(3)
3.5.3 Singular-Value Decomposition
160(1)
3.6 Normed Linear Spaces
161(2)
3.6.1 Operators on Normed Linear Spaces
162(1)
3.6.2 Dual Spaces and Convergence
163(1)
3.7 Banach Spaces
163(3)
3.7.1 Compact Operators
165(1)
3.8 Hilbert Spaces
166(9)
3.8.1 Projection Theorem
167(3)
3.8.2 Riesz Representation Theorem
170(1)
3.8.3 Transpose and Adjoint Operators on Hilbert Spaces
170(2)
3.8.4 Bases for Hilbert Spaces
172(2)
3.8.5 Examples of Hilbert Spaces
174(1)
3.9 Spectral Theory
175(2)
3.9.1 Spectral Theory for Compact Self-Adjoint Operators
176(1)
3.10 Trace-Class and Hilbert-Schmidt Operators
177(4)
3.10.1 Singular Functions
179(2)
3.11 Spectral Theory for Non-Compact Bounded Self-Adjoint Operators
181(1)
3.11.1 Resolutions of the Identity
181(1)
3.11.2 Spectral Representation
181(1)
References
182(3)
4 Resolution and Ill-Posedness 185(54)
4.1 Introduction
185(2)
4.2 Ill-Posedness and Ill-Conditioning
187(1)
4.3 Finite-Dimensional Problems and Linear Systems of Equations
188(7)
4.3.1 Overdetermined and Underdetermined Problems
189(1)
4.3.2 Ill-Conditioned Problems
190(3)
4.3.3 Illustrative Example
193(2)
4.4 Linear Least-Squares Solutions
195(5)
4.4.1 Effect of Errors
199(1)
4.5 Truncated Singular-Value Decomposition
200(1)
4.6 Infinite-Dimensional Problems
200(2)
4.6.1 Generalised Inverses for Compact Operators of Non-Finite Rank
201(1)
4.7 Truncated Singular-Function Expansion
202(10)
4.7.1 Oscillation Properties of the Singular Functions
205(4)
4.7.2 Finite Weierstrass Transform
209(1)
4.7.3 Resolution and the Truncation Point of the Singular-Function Expansion
209(1)
4.7.4 Profiled Singular-Function Expansion
210(2)
4.8 Finite Laplace Transform
212(4)
4.8.1 Commuting Differential Operators
213(2)
4.8.2 Oscillation Properties
215(1)
4.9 Fujita's Equation
216(1)
4.10 Inverse Problem in Magnetostatics
217(1)
4.11 C-Generalised Inverses
218(2)
4.12 Convolution Operators
220(4)
4.12.1 Solution of the Eigenvalue Problem for -id/dx
221(1)
4.12.2 Eigenfunctions of Convolution Operators
222(1)
4.12.3 Resolution and the Band Limit for Convolution Equations
223(1)
4.13 Mellin-Convolution Operators
224(7)
4.13.1 Eigenfunctions of Mellin-Convolution Operators
225(5)
4.13.2 Laplace and Fourier Transforms
230(1)
4.13.3 Resolution and the Band Limit for Mellin-Convolution Equations
231(1)
4.14 Linear Inverse Problems with Discrete Data
231(5)
4.14.1 Normal Solution
232(1)
4.14.2 Generalised Solution
233(1)
4.14.3 C-Generalised Inverse
233(1)
4.14.4 Singular System
234(1)
4.14.5 Oscillation Properties for Discrete-Data Problems
235(1)
4.14.6 Resolution for Discrete-Data Problems
236(1)
References
236(3)
5 Optimisation 239(38)
5.1 Introduction
239(1)
5.1.1 Optimisation and Prior Knowledge
239(1)
5.2 Finite-Dimensional Problems
239(2)
5.3 Unconstrained Optimisation
241(4)
5.3.1 Steepest Descent
241(1)
5.3.2 Newton's Method
241(1)
5.3.3 Levenberg-Marquardt Method
242(1)
5.3.4 Conjugate-Direction Methods
242(1)
5.3.5 Quasi-Newton Methods
243(1)
5.3.6 Conjugate-Gradient Methods
244(1)
5.4 Gradient-Descent Methods for the Linear Problem
245(2)
5.4.1 Steepest-Descent Method
245(1)
5.4.2 Conjugate-Directions Method
245(1)
5.4.3 Conjugate-Gradient Method
246(1)
5.5 Constrained Optimisation with Equality Constraints
247(3)
5.6 Constrained Optimisation with Inequality Constraints
250(5)
5.6.1 Karush-Kuhn-Tucker Conditions
251(1)
5.6.2 Constraint Qualifications
252(1)
5.6.3 Lagrangian Duality
252(2)
5.6.4 Primal-Dual Structure with Positivity Constraints
254(1)
5.7 Ridge Regression
255(2)
5.7.1 Ridge Regression and Lagrange Duality
255(1)
5.7.2 Ridge Regression with Non-Negativity Constraints
256(1)
5.8 Infinite-Dimensional Problems
257(1)
5.9 Calculus on Banach Spaces
258(3)
5.9.1 Frechet Derivatives
258(2)
5.9.2 Gateaux Derivatives
260(1)
5.10 Gradient-Descent Methods for the Infinite-Dimensional Linear Problem
261(3)
5.10.1 Steepest-Descent Method
262(1)
5.10.2 Conjugate-Descent Method
263(1)
5.10.3 Conjugate-Gradient Method
264(1)
5.11 Convex Optimisation and Conjugate Duality
264(4)
5.11.1 Convex Functions
265(1)
5.11.2 Conjugate Functions
266(1)
5.11.3 Perturbed Problems and Duality
266(1)
5.11.4 Lagrangians and Convex Optimisation
267(1)
5.11.5 Fenchel Duality
268(1)
5.12 Partially Finite Convex Programming
268(6)
5.12.1 Fenchel Duality for Partially Finite Problems
269(2)
5.12.2 Elements of Lattice Theory
271(1)
5.12.3 Cone Constraints
272(2)
References
274(3)
6 Deterministic Methods for Linear Inverse Problems 277(54)
6.1 Introduction
277(2)
6.2 Continuity and Stability of the Inverse
279(5)
6.2.1 Restoration of Continuity by Choice of Spaces
280(1)
6.2.2 Continuity of the Inverse Using Compact Sets
280(1)
6.2.3 Least-Squares Using a Prescribed Bound for the Solution
281(2)
6.2.4 Types of Continuity
283(1)
6.3 Regularisation
284(13)
6.3.1 Regularisation Using Spectral Windows
286(1)
6.3.2 Tikhonov Regularisation
287(2)
6.3.3 Regularisation of C-Generalised Inverses
289(1)
6.3.4 Miller's Method
289(2)
6.3.5 Generalised Tikhonov Regularisation
291(1)
6.3.6 Regularisation with Linear Equality Constraints
291(1)
6.3.7 Regularisation through Discretisation
292(1)
6.3.8 Other Regularisation Methods
293(1)
6.3.9 Convergence Rates for Regularisation Algorithms
293(1)
6.3.10 Methods for Choosing the Regularisation Parameter
294(2)
6.3.10.1 The Discrepancy Principle
294(1)
6.3.10.2 Miller's Method and the L-Curve
295(1)
6.3.10.3 The Interactive Method
296(1)
6.3.10.4 Comparisons between the Different Methods
296(1)
6.3.11 Regularisation and Resolution
296(1)
6.4 Iterative Methods
297(5)
6.4.1 Landweber Iteration and the Method of Steepest Descent
297(2)
6.4.2 Krylov Subspace Methods and the Conjugate-Gradient Method
299(1)
6.4.3 Projection onto Convex Sets
300(1)
6.4.4 Iterative Methods, Regularisation and Resolution
301(1)
6.5 Smoothness
302(6)
6.5.1 The Method of Mollifiers
302(1)
6.5.2 Hilbert-Scale Methods
303(3)
6.5.3 Sobolev-Scale Approach
306(2)
6.6 Positivity
308(7)
6.6.1 A Linear Regularisation Method
308(1)
6.6.2 Non-Negative Constrained Tikhonov Regularisation
309(1)
6.6.3 A Dual Method
309(6)
6.6.3.1 The Constraint Qualification
312(1)
6.6.3.2 Resolution and the Dual Method
313(2)
6.7 Sparsity and Other Sets of Basis Functions
315(2)
6.7.1 Compressive Sensing and Sparsity
315(1)
6.7.2 Sparsity in Linear Inverse Problems
316(1)
6.8 Linear Inverse Problems with Discrete Data
317(1)
6.8.1 Singular-Value Decomposition
318(1)
6.8.2 Scanning Singular-Value Decomposition
318(1)
6.9 Regularisation for Linear Inverse Problems with Discrete Data
318(2)
6.9.1 Regularisation of the C-Generalised Inverse
320(1)
6.9.2 Resolution and Finite-Dimensional Regularisation
320(1)
6.10 Iterative Methods for Linear Inverse Problems with Discrete Data
320(1)
6.11 Averaging Kernels
321(1)
6.12 The Backus-Gilbert Method
322(4)
6.12.1 Connections between the Backus-Gilbert Method and Regularisation for Discrete-Data Problems
325(1)
6.12.2 Comparison between the Method of Mollifiers and the Backus-Gilbert Method
326(1)
6.13 Positivity for Linear Inverse Problems with Discrete Data
326(1)
References
327(4)
7 Convolution Equations and Deterministic Spectral Analysis 331(32)
7.1 Introduction
331(2)
7.2 Basic Fourier Theory
333(3)
7.2.1 Periodic Functions and Fourier Series
333(1)
7.2.2 Aperiodic Functions and Fourier Transforms
334(1)
7.2.3 Fourier Analysis of Sequences
335(1)
7.2.4 Discrete Fourier Transform
336(1)
7.3 Convergence and Summability of Fourier Series and Integrals
336(6)
7.3.1 Convergence and Summability of Fourier Series
336(4)
7.3.2 Convergence and Summability of Fourier Integrals
340(2)
7.4 Determination of the Amplitude Spectrum for Continuous-Time Functions
342(1)
7.4.1 Application of Windowing to Time-Limited Continuous-Time Functions
342(1)
7.5 Regularisation and Windowing for Convolution Equations
343(4)
7.5.1 Regularisation for Convolution Equations
343(2)
7.5.2 Windowing and Convolution Equations
345(1)
7.5.3 Averaging Kernel and Resolution
345(1)
7.5.4 Positive Solutions
346(1)
7.5.5 An Extension to the Backus-Gilbert Theory of Averaging Kernels
346(1)
7.6 Regularisation and Windowing for Mellin-Convolution Equations
347(2)
7.7 Determination of the Amplitude Spectrum for Discrete-Time Functions
349(4)
7.7.1 Assorted Windows
349(2)
7.7.2 Spectral Leakage and Windows
351(1)
7.7.3 Figures of Merit for Windows
351(2)
7.8 Discrete Prolate Spheroidal Wave Functions and Sequences
353(3)
7.8.1 Discrete-Time Concentration Problem
354(2)
7.8.1.1 Kaiser-Bessel Window
355(1)
7.8.1.2 Multi-Tapering
356(1)
7.9 Regularisation and Windowing for Convolution Operators on the Circle
356(1)
7.9.1 Positive Solutions to Circular Convolutions
357(1)
7.10 Further Band Limiting
357(4)
7.10.1 Determination of the Amplitude Spectrum for Band-Limited Continuous-Time Functions
358(1)
7.10.2 Determination of the Amplitude Spectrum for Band-Limited Discrete-Time Functions
359(2)
References
361(2)
8 Statistical Methods and Resolution 363(38)
8.1 Introduction
363(1)
8.2 Parameter Estimation
363(3)
8.2.1 Likelihood
364(1)
8.2.2 Cramer-Rao Lower Bound
365(1)
8.2.3 Bayesian Parameter Estimation
366(1)
8.3 Information and Entropy
366(3)
8.4 Single-Source Resolution and Differential-Source Resolution
369(1)
8.5 Two-Point Optical Resolution from a Statistical Viewpoint
369(4)
8.5.1 Decision-Theoretic Approach of Harris
369(3)
8.5.2 Approach of Shahram and Milanfar
372(1)
8.6 Finite-Dimensional Problems
373(5)
8.6.1 The Best Linear Unbiased Estimate
374(1)
8.6.2 Minimum-Variance Linear Estimate
374(1)
8.6.3 Bayesian Estimation
375(1)
8.6.4 Statistical Resolution in a Bayesian Framework
376(1)
8.6.5 Solution in an Ensemble of Smooth Functions
377(1)
8.7 Richardson-Lucy Method
378(1)
8.8 Choice of the Regularisation Parameter for Inverse Problems with a Compact Forward Operator
379(4)
8.8.1 Unbiased Predictive Risk Estimate
380(1)
8.8.2 Cross-Validation and Generalised Cross-Validation
381(1)
8.8.3 Turchin's Method
382(1)
8.8.4 Klein's Method
382(1)
8.8.5 Comparisons between the Different Methods for Finding the Regularisation Parameter
382(1)
8.9 Introduction to Infinite-Dimensional Problems
383(1)
8.10 Probability Theory for Infinite-Dimensional Spaces
383(6)
8.10.1 Cylinder Sets and Borel Sets of a Hilbert Space
383(1)
8.10.2 Hilbert-Space-Valued Random Variables
384(1)
8.10.3 Cylinder-Set Measures
385(1)
8.10.4 Weak Random Variables
386(1)
8.10.5 Cross-Covariance Operators and Joint Measures
387(2)
8.11 Weak Random Variable Approach
389(6)
8.11.1 Comparison with the Miller Method
391(1)
8.11.2 Probabilistic Regularisation
391(4)
8.12 Wiener Deconvolution Filter
395(1)
8.13 Discrete-Data Problems
396(2)
8.13.1 The Best Linear Estimate
396(1)
8.13.2 Bayes and the Discrete-Data Problem
397(1)
8.13.3 Statistical Version of the Backus-Gilbert Approach
397(1)
References
398(3)
9 Some Applications in Scattering and Absorption 401(44)
9.1 Introduction
401(1)
9.2 Particle Sizing by Light Scattering and Extinction
401(13)
9.2.1 Mie Scattering Problem
402(5)
9.2.2 Fraunhofer Diffraction Problem
407(2)
9.2.3 Extinction Problem (Spectral Turbidity)
409(5)
9.3 Photon-Correlation Spectroscopy
414(5)
9.3.1 Particle Sizing by Photon-Correlation Spectroscopy
416(2)
9.3.2 Laser Doppler Velocimetry
418(1)
9.4 Projection Tomography
419(12)
9.4.1 Basic Ideas
419(2)
9.4.2 Inversion Formula
421(1)
9.4.3 Filtered Back-Projection
422(1)
9.4.4 Smoothness of the Radon Transform
423(1)
9.4.5 Singular-Value Analysis
424(7)
9.4.5.1 Some Preliminary Results
424(3)
9.4.5.2 SVD of the Full-Angle Problem
427(2)
9.4.5.3 SVD of the Limited-Angle Problem
429(2)
9.4.6 Discrete-Data Problem
431(1)
9.5 Linearised Inverse Scattering Theory
431(4)
9.5.1 The Born Approximation
433(1)
9.5.2 Prior Discrete Fourier Transform Algorithm
433(1)
9.5.3 Resolution and the Born Approximation
434(1)
9.5.4 Beyond the Born Approximation
435(1)
9.6 Diffraction Tomography
435(5)
9.6.1 Rytov Approximation
436(2)
9.6.2 Generalised Projection Slice Theorem
438(1)
9.6.3 Backpropagation
439(1)
9.6.4 Filtered Backpropagation
439(1)
9.6.5 Super-Resolution in Diffraction Tomography
440(1)
References
440(5)
10 Resolution in Microscopy 445(52)
10.1 Introduction
445(1)
10.2 Scanning Microscopy
445(33)
10.2.1 Object Reconstruction in Two Dimensions
447(1)
10.2.2 One-Dimensional Coherent Case
448(6)
10.2.2.1 Singular System
450(2)
10.2.2.2 Sampling Theory and the Generalised Solution
452(1)
10.2.2.3 Super-Resolving Optical Masks
452(2)
10.2.3 One-Dimensional Incoherent Case
454(10)
10.2.3.1 Null-Space of the Forward Operator
456(1)
10.2.3.2 Projection onto the Null-Space
457(1)
10.2.3.3 Sampling Theory and the Generalised Solution
458(2)
10.2.3.4 Noiseless Impulse Response
460(2)
10.2.3.5 Optical Masks
462(2)
10.2.4 Two-Dimensional Case with Circular Pupil
464(6)
10.2.4.1 Masks
468(2)
10.2.5 Scanning Microscopy in Three Dimensions
470(8)
10.3 Compact Optical Disc
478(1)
10.4 Near-Field Imaging
478(7)
10.4.1 Near-Field Scanning Optical Microscopy
479(2)
10.4.2 Perfect Lens
481(2)
10.4.3 Near-Field Superlenses
483(2)
10.4.4 Hyperlenses
485(1)
10.5 Super-Resolution in Fluorescence Microscopy
485(8)
10.5.1 Introduction
485(1)
10.5.2 Total Internal Reflection Fluorescence Microscopy
486(1)
10.5.3 Multi-Photon Microscope
486(1)
10.5.4 4Pi Microscopy
486(1)
10.5.5 Structured Illumination
486(1)
10.5.6 Methods Based on a Non-Linear Photoresponse
486(4)
10.5.6.1 Stimulated Emission Depletion
487(2)
10.5.6.2 Ground-State Depletion Microscopy
489(1)
10.5.6.3 RESOLFT
489(1)
10.5.6.4 Fluorescence Saturation and Structured Illumination
490(1)
10.5.7 Super-Resolution Optical Fluctuation Imaging
490(1)
10.5.8 Localisation Microscopy
490(1)
10.5.9 Localisation Ultrasound Microscopy
491(2)
Acknowledgements
493(1)
References
493(4)
Appendix A: The Origin of Spectacles 497(10)
Appendix B: Set Theory and Mappings 507(4)
Appendix C: Topological Spaces 511(12)
Appendix D: Basic Probability Theory 523(6)
Appendix E: Wavelets 529(2)
Appendix F: MATLAB® Programme for TM Surface Polaritons 531(2)
Index 533
Geoffrey D de Villiers (M Inst P. C.Phys., FIMA, C.Math) is currently an honorary senior research fellow in the School of Electronic, Electrical and Systems Engineering at the University of Birmingham. He is an applied mathematician with over 30 years of experience in signal processing. His specialty is linear inverse problems with particular emphasis on singular-function methods and resolution enhancement. He has worked on a wide variety of practical inverse problems in photon correlation spectroscopy, radar, sonar, communications, seismology, antenna array design, broadband array processing, computational imaging and, currently, gravitational imaging.

E. Roy Pike FRS has been Clerk-Maxwell Professor for Theoretical Physics at King's College London, and head of its School of Physical Sciences and Engineering, and is currently Emeritus Professor of Physics.
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