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E-grāmata: Multi-wave Medical Imaging: Mathematical Modelling And Imaging Reconstruction

Edited by (Ecole Polytechnique, France), (Ecole Normale Superieure, France), (Inha Univ, Korea), Edited by (Eth Zurich, Switzerland), (Ecole Normale Superieure, France)
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Multi-wave Medical Imaging: Mathematical Modelling And Imaging ReconstructionPalielināt
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Super-Resolution imaging refers to modern techniques of achieving resolution below conventional limits. This book gives a comprehensive overview of mathematical and computational techniques used to achieve this, providing a solid foundation on which to develop the knowledge and skills needed for practical application of techniques. Split into five parts, the first looks at the mathematical and probabilistic tools needed, before moving on to description of different types of imaging; single-wave, anomaly, multi-wave and spectroscopic and nanoparticle.As an important contribution to the understanding of super-resolution techniques in biomedical imaging, this book is a useful resource for scientists and engineers in the fields of biomedical imaging and super-resolution, and is self-contained reference for any newcomers to these fields.
Acknowledgments xix
1 Introduction
1(16)
1.1 Properties of Biological Tissues
2(11)
1.1.1 Dielectric Properties
2(5)
1.1.2 Optical Properties
7(3)
1.1.3 Elastic Properties
10(3)
1.2 Superresolution Biomedical Imaging
13(4)
Part I Mathematical and Probabilistic Tools 17(116)
2 Basic Mathematical Concepts
19(36)
2.1 Special Functions
19(7)
2.1.1 Bessel Functions
19(4)
2.1.2 Hankel Functions
23(3)
2.2 Function Spaces
26(2)
2.3 Fourier Analysis
28(5)
2.3.1 Fourier Transform
28(3)
2.3.2 Shannon's Sampling Theorem
31(2)
2.4 Kramers-Kronig Relations and Causality
33(2)
2.5 Singular Value Decomposition
35(2)
2.6 Compact Operators
37(1)
2.7 Spherical Mean Radon Transform
38(3)
2.8 Regularization of Ill-Posed Problems
41(8)
2.8.1 Stability
41(2)
2.8.2 The Truncated SVD
43(1)
2.8.3 Tikhonov-Phillips Regularization
44(2)
2.8.4 Regularization by Truncated Iterative Methods
46(2)
2.8.5 Regularizations by Nonquadratic Constraints
48(1)
2.9 Optimal Control
49(1)
2.10 Convergence of Nonlinear Landweber Iterations
50(2)
2.11 Level Set Method
52(3)
3 Layer Potential Techniques
55(60)
3.1 The Laplace Equation
56(17)
3.1.1 Fundamental Solution
56(1)
3.1.2 Layer Potentials
57(7)
3.1.3 Invertibility of LambdaI-KD
64(1)
3.1.4 Symmetrization of KD
65(4)
3.1.5 Neumann Function
69(2)
3.1.6 Transmission Problems
71(2)
3.2 Helmholtz Equation
73(23)
3.2.1 Fundamental Solution
74(2)
3.2.2 Layer Potentials
76(2)
3.2.3 Transmission Problem
78(3)
3.2.4 Reciprocity
81(2)
3.2.5 Lippmann-Schwinger Representation Formula
83(1)
3.2.6 The Helmholtz-Kirchhoff Theorem
84(1)
3.2.7 Scattering Amplitude and the Optical Theorem
85(11)
3.3 Elasticity Equations
96(19)
3.3.1 Radiation Condition
101(1)
3.3.2 Integral Representation of Solutions to the Lame System
102(8)
3.3.3 Reciprocity Property and Helmholtz-Kirchhoff Identities
110(2)
3.3.4 Incompressible Limit
112(3)
4 Probabilistic Tools
115(14)
4.1 Random Variables
116(1)
4.2 Random Vectors
117(3)
4.3 Gaussian Random Vectors
120(1)
4.4 Random Processes
121(8)
4.4.1 Gaussian Random Processes
122(1)
4.4.2 Stationary Gaussian Random Processes
123(2)
4.4.3 Local Maxima of a Gaussian Random Field
125(1)
4.4.4 Global Maximum of a Gaussian Random Field
125(1)
4.4.5 The Local Shape of a Local Maximum
126(1)
4.4.6 Realization of a Cluttered Medium
127(2)
5 General Image Characteristics
129(4)
5.1 Spatial Resolution
129(2)
5.1.1 Point Spread Function
129(2)
5.1.2 Rayleigh Resolution Limit
131(1)
5.2 Signal-to-Noise Ratio
131(2)
Part II Single-Wave Imaging 133(56)
6 Electrical Impedance Tomography
135(8)
6.1 Mathematical Model
135(2)
6.2 Ill-Conditioning
137(6)
6.2.1 Static Imaging
138(1)
6.2.2 Dynamic Imaging
138(3)
6.2.3 Electrode Model
141(2)
7 Ultrasound and Microwave Tomographies
143(10)
7.1 Born Approximation
143(1)
7.2 Diffraction Tomography Algorithm
144(2)
7.3 Time-Reversal Techniques
146(7)
7.3.1 Ideal Time-Reversal Imaging Technique
147(4)
7.3.2 A Modified Time-Reversal Imaging Technique
151(2)
8 Time-Harmonic Reverse-Time Imaging with Additive Noise
153(10)
8.1 The Data Set
153(1)
8.2 The Forward Problem
154(2)
8.3 Imaging Functionals
156(1)
8.4 The RT-Imaging Function
157(6)
8.4.1 The Imaging Function without Measurement Noise
157(1)
8.4.2 The Imaging Function with Measurement Noise
158(3)
8.4.3 Localization Error
161(2)
9 Reverse-Time Imaging with Clutter Noise
163(14)
9.1 The Data Set
163(1)
9.2 A Model for the Scattering Medium
164(2)
9.3 The Forward Problem
166(2)
9.4 The Imaging Function
168(9)
9.4.1 The Imaging Function without Clutter Noise
168(3)
9.4.2 The Imaging Function with Clutter Noise
171(6)
10 Optical Coherence Tomography with Clutter Noise
177(12)
10.1 The Principle of Optical Coherence Tomography
177(2)
10.2 The Reference and Sample Beams
179(4)
10.3 The Imaging Function
183(1)
10.4 The Point Spread Function
184(2)
10.5 The Clutter Noise in Optical Coherence Tomography
186(3)
Part III Anomaly Imaging 189(58)
11 Small Volume Expansions
191(28)
11.1 Conductivity Problem
192(4)
11.2 Helmholtz Equation
196(3)
11.3 Asymptotic Formulas for Monopole Sources in Free Space
199(1)
11.3.1 Conductivity Problem
199(1)
11.3.2 Helmholtz Equation
199(1)
11.4 Elasticity Equations
200(12)
11.4.1 Static Regime
202(2)
11.4.2 Time-Harmonic Regime
204(3)
11.4.3 Properties of the EMT
207(5)
11.5 Asymptotic Expansions for Time-Dependent Equations
212(7)
11.5.1 Asymptotic Formulas for the Wave Equation
212(2)
11.5.2 Asymptotic Analysis of Temperature Perturbations
214(5)
12 Anomaly Imaging Algorithms
219(28)
12.1 Direct Imaging for the Conductivity Problem
220(3)
12.1.1 Detection of a Single Inclusion: A Projection-Type Algorithm
220(1)
12.1.2 Detection of Multiple Inclusions: A MUSIC-Type Algorithm
221(2)
12.2 Direct Imaging Algorithms for the Helmholtz Equation
223(12)
12.2.1 Direct Imaging at a Fixed Frequency
223(9)
12.2.2 Direct Imaging at Multiple Frequencies
232(3)
12.3 Direct Elasticity Imaging
235(6)
12.3.1 A MUSIC-Type Method in the Static Regime
235(2)
12.3.2 A MUSIC-Type Method in the Time-Harmonic Regime
237(3)
12.3.3 Reverse-Time Migration and Kirchhoff Imaging in the Time-Harmonic Regime
240(1)
12.4 Time-Domain Anomaly Imaging
241(8)
12.4.1 Wave Imaging of Small Anomalies
241(2)
12.4.2 Thermal Imaging of Small Anomalies
243(4)
Part IV Multi-Wave Imaging 247(226)
13 Photoacoustic Imaging
249(48)
13.1 Introduction
249(2)
13.2 Mathematical Formulation
251(2)
13.3 Photoacoustic Imaging in Free Space
253(16)
13.3.1 Full-View Setting
254(1)
13.3.2 Limited-View Setting
255(2)
13.3.3 Compensation of the Effect of Acoustic Attenuation
257(12)
13.4 Photoacoustic Imaging of Small Absorbers with Imposed Boundary Conditions on the Pressure
269(13)
13.4.1 Reconstruction Methods
269(6)
13.4.2 Backpropagation of the Acoustic Signals
275(2)
13.4.3 Selective Detection
277(5)
13.5 Imaging with Limited-View Data
282(2)
13.5.1 Geometrical Control of the Wave Equation
282(1)
13.5.2 Reconstruction Procedure
283(1)
13.5.3 Implementation of the HUM
284(1)
13.6 Quantitative Photoacoustic Imaging
284(7)
13.6.1 Asymptotic Approach
286(3)
13.6.2 Multi-Wavelength Approach
289(2)
13.7 Coherent Interferometry Algorithms
291(4)
13.8 Concluding Remarks
295(2)
14 Quantitative Thermoacoustic Imaging
297(12)
14.1 Introduction
297(1)
14.2 Measurements
298(1)
14.3 Exact Formula
299(5)
14.4 Optimal Control Approach
304(5)
14.4.1 The Differentiability of the Data Map and Its Inverse
304(3)
14.4.2 Landweber's Iteration
307(2)
15 Ultrasonically Induced Lorentz Force Electrical Impedance Tomography
309(28)
15.1 Introduction
309(2)
15.2 Electric Measurements from Acousto-Magnetic Coupling
311(5)
15.2.1 Electrical Conductivity in Electrolytes
312(1)
15.2.2 Ion Deviation by Lorentz Force
312(1)
15.2.3 Internal Electrical Potential
313(2)
15.2.4 Virtual Potential
315(1)
15.3 Construction of the Virtual Current
316(3)
15.4 Recovering the Conductivity by Optimal Control
319(3)
15.5 The Orthogonal Field Method
322(7)
15.5.1 Uniqueness Result for the Transport Equation
323(4)
15.5.2 The Viscosity-Type Regularization
327(2)
15.6 Numerical Illustrations
329(6)
15.6.1 Deconvolution
329(1)
15.6.2 Conductivity Reconstructions
330(5)
15.7 Concluding Remarks
335(2)
16 Magnetoacoustic Tomography with Magnetic Induction
337(28)
16.1 Introduction
337(2)
16.2 Forward Problem Description
339(4)
16.2.1 Time Scales Involved
339(1)
16.2.2 Electromagnetic Model
339(2)
16.2.3 Acoustic Problem
341(2)
16.3 Reconstruction of the Acoustic Source
343(3)
16.4 Reconstruction of the Conductivity
346(13)
16.4.1 Reconstruction of the Electric Current Density
346(2)
16.4.2 Recovery of the Conductivity from Internal Electric Current Density
348(11)
16.5 Numerical Illustrations
359(5)
16.5.1 Optimal Control
359(1)
16.5.2 Fixed Point Method
360(1)
16.5.3 Orthogonal Field Method
361(3)
16.6 Concluding Remarks
364(1)
17 Impediography
365(10)
17.1 Introduction
365(2)
17.2 Mathematical Model
367(3)
17.3 Substitution Algorithm
370(2)
17.4 Optimal Control Algorithm
372(2)
17.5 Concluding Remarks
374(1)
18 Microwave Imaging by Elastic Deformation
375(16)
18.1 Introduction
375(3)
18.2 Exact Reconstruction Formulas
378(6)
18.3 The Forward Problem and the Differentiability of the Data at a Fixed Frequency
384(4)
18.4 Optimal Control Algorithm
388(3)
19 Ultrasound-Modulated Optical Tomography
391(26)
19.1 Introduction
391(2)
19.2 Preliminaries
393(6)
19.2.1 Acoustic Wave
393(3)
19.2.2 Regularity Results
396(3)
19.3 Reconstruction Algorithms
399(12)
19.3.1 Fixed Point Algorithm
402(6)
19.3.2 Optimal Control Algorithm
408(3)
19.4 Numerical Illustrations
411(6)
19.4.1 Concluding Remarks
415(2)
20 Viscoelastic Modulus Reconstruction
417(18)
20.1 Introduction
417(2)
20.2 Reconstruction Methods
419(11)
20.2.1 Viscoelasticity Model
419(2)
20.2.2 Optimal Control Algorithm
421(5)
20.2.3 Initial Guess
426(3)
20.2.4 Local Reconstruction
429(1)
20.3 Numerical Illustrations
430(4)
20.4 Concluding Remarks
434(1)
21 Mechanical Vibration-Assisted Conductivity Imaging
435(16)
21.1 Introduction
435(1)
21.2 Mathematical Modeling
436(3)
21.3 Vibration-Assisted Anomaly Identification
439(8)
21.3.1 Location Search Method and Asymptotic Expansion
441(4)
21.3.2 Size Estimation and Reconstruction of the Material Parameters
445(2)
21.4 Numerical Illustrations
447(2)
21.4.1 Simulations of the Voltage Difference Map
447(1)
21.4.2 Anomaly Location
448(1)
21.5 Concluding Remarks
449(2)
22 Full-Field Optical Coherence Elastography
451(22)
22.1 Introduction
451(3)
22.2 Preliminaries
454(1)
22.3 Displacement Field Measurements
455(14)
22.3.1 First-Order Approximation
456(3)
22.3.2 Local Recovery via Linearization
459(4)
22.3.3 Minimization of the Discrepancy Functional
463(6)
22.4 Reconstruction of the Shear Modulus
469(1)
22.5 Numerical Illustrations
469(3)
22.6 Concluding Remarks
472(1)
Part V Spectroscopic and Nanoparticle Imaging 473(166)
23 Effective Electrical Tissue Properties
475(52)
23.1 Introduction
475(2)
23.2 Problem Settings and Main Results
477(9)
23.2.1 Periodic Domain
477(1)
23.2.2 Electrical Model of the Cell
478(4)
23.2.3 Governing Equation
482(1)
23.2.4 Main Results
483(3)
23.3 Analysis of the Problem
486(4)
23.3.1 Existence and Uniqueness of a Solution
487(1)
23.3.2 Energy Estimate
488(2)
23.4 Homogenization
490(16)
23.4.1 Two-Scale Asymptotic Expansions
491(5)
23.4.2 Convergence
496(10)
23.5 Effective Admittivity for a Dilute Suspension
506(8)
23.5.1 Computation of the Effective Admittivity
506(4)
23.5.2 Case of Concentric Circular-Shaped Cells: The Maxwell-Wagner-Fricke Formula
510(1)
23.5.3 Debye Relaxation Times
511(1)
23.5.4 Properties of the Membrane Polarization Tensor and the Debye Relaxation Times
512(1)
23.5.5 Anisotropy Measure
513(1)
23.6 Numerical Simulations
514(3)
23.7 Technical Results
517(9)
23.7.1 Extension Lemmas
517(4)
23.7.2 Poincare-Wirtinger Inequality
521(2)
23.7.3 Equivalence of the Two Norms on Wepsilon
523(2)
23.7.4 Existence Result
525(1)
23.8 Concluding Remarks
526(1)
24 Plasmonic Nanoparticle Imaging
527(56)
24.1 Introduction
527(2)
24.2 Layer Potential Formulation for Plasmonic Resonances
529(12)
24.2.1 Problem Formulation and Some Basic Results
529(4)
24.2.2 First-Order Correction to Plasmonic Resonances and Field Behavior at the Plasmonic Resonances
533(8)
24.3 Multiple Plasmonic Nanoparticles
541(11)
24.3.1 Layer Potential Formulation in the Multi-Particle Case
541(1)
24.3.2 First-Order Correction to Plasmonic Resonances and Field Behavior at Plasmonic Resonances in the Multi-Particle Case
542(10)
24.4 Scattering and Absorption Enhancements
552(11)
24.4.1 The Quasi-Static Limit
552(3)
24.4.2 An Upper Bound for the Averaged Extinction Cross-Section
555(8)
24.5 Link with the Scattering Coefficients
563(6)
24.5.1 Scattering Coefficients of Plasmonic Nanoparticles
563(3)
24.5.2 The Leading-Order Term in the Expansion of the Scattering Amplitude
566(3)
24.6 Asymptotic Expansion of the Integral Operators: Single Particle
569(2)
24.7 Asymptotic Expansion of the Integral Operators: Multiple Particles
571(6)
24.8 Sum Rules for the Polarization Tensor
577(3)
24.9 Concluding Remarks
580(3)
25 Nonlinear Harmonic Holography
583(56)
25.1 Introduction
583(2)
25.2 Problem Formulation
585(2)
25.3 Small-Volume Expansions
587(10)
25.3.1 Fundamental Frequency Problem
587(6)
25.3.2 Second-Harmonic Problem
593(4)
25.4 Imaging Functional
597(2)
25.4.1 The Fundamental Frequency Case
597(1)
25.4.2 Second-Harmonic Backpropagation
598(1)
25.5 Statistical Analysis
599(23)
25.5.1 Assumptions on the Random Process µ
600(2)
25.5.2 Standard Backpropagation
602(8)
25.5.3 Second-Harmonic Backpropagation
610(7)
25.5.4 Stability with Respect to Measurement Noise
617(5)
25.6 Numerical Results
622(9)
25.6.1 The Direct Problem
622(1)
25.6.2 The Imaging Functionals and the Effects of the Number of Plane Wave Illuminations
623(3)
25.6.3 Statistical Analysis
626(5)
25.7 Proof of Estimate (25.8)
631(4)
25.8 Proof of Proposition 25.1
635(1)
25.9 Proof of Proposition 25.3
636(1)
25.10 Concluding Remarks
637(2)
References 639(24)
Index 663
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