BOOK
Multi-wave Medical Imaging: Mathematical Modelling And Imaging Reconstruction
Ammari Habib | Garnier Josselin | Nguyen Loc Hoang
(2017)
Additional Information
Book Details
Table of Contents
| Section Title | Page | Action | Price |
|---|---|---|---|
| Contents | v | ||
| Acknowledgments | xix | ||
| 1. Introduction | 1 | ||
| 1.1 Properties of Biological Tissues | 2 | ||
| 1.1.1 Dielectric Properties | 2 | ||
| 1.1.2 Optical Properties | 7 | ||
| 1.1.3 Elastic Properties | 10 | ||
| 1.2 Superresolution Biomedical Imaging | 13 | ||
| Part I Mathematical and Probabilistic Tools | 17 | ||
| 2. Basic Mathematical Concepts | 19 | ||
| 2.1 Special Functions | 19 | ||
| 2.1.1 Bessel Functions | 19 | ||
| 2.1.2 Hankel Functions | 23 | ||
| 2.2 Function Spaces | 26 | ||
| 2.3 Fourier Analysis | 28 | ||
| 2.3.1 Fourier Transform | 28 | ||
| 2.3.2 Shannon’s Sampling Theorem | 31 | ||
| 2.4 Kramers–Kronig Relations and Causality | 33 | ||
| 2.5 Singular Value Decomposition | 35 | ||
| 2.6 Compact Operators | 37 | ||
| 2.7 Spherical Mean Radon Transform | 38 | ||
| 2.8 Regularization of Ill-Posed Problems | 41 | ||
| 2.8.1 Stability | 41 | ||
| 2.8.2 The Truncated SVD | 43 | ||
| 2.8.3 Tikhonov–Phillips Regularization | 44 | ||
| 2.8.4 Regularization by Truncated Iterative Methods | 46 | ||
| 2.8.5 Regularizations by Nonquadratic Constraints | 48 | ||
| 2.9 Optimal Control | 49 | ||
| 2.10 Convergence of Nonlinear Landweber Iterations | 50 | ||
| 2.11 Level Set Method | 52 | ||
| 3. Layer Potential Techniques | 55 | ||
| 3.1 The Laplace Equation | 56 | ||
| 3.1.1 Fundamental Solution | 56 | ||
| 3.1.2 Layer Potentials | 57 | ||
| 3.1.3 Invertibility of λI − K*D | 64 | ||
| 3.1.4 Symmetrization of K*D | 65 | ||
| 3.1.5 Neumann Function | 69 | ||
| 3.1.6 Transmission Problems | 71 | ||
| 3.2 Helmholtz Equation | 73 | ||
| 3.2.1 Fundamental Solution | 74 | ||
| 3.2.2 Layer Potentials | 76 | ||
| 3.2.3 Transmission Problem | 78 | ||
| 3.2.4 Reciprocity | 81 | ||
| 3.2.5 Lippmann–Schwinger Representation Formula | 83 | ||
| 3.2.6 The Helmholtz–Kirchhoff Theorem | 84 | ||
| 3.2.7 Scattering Amplitude and the Optical Theorem | 85 | ||
| 3.3 Elasticity Equations | 96 | ||
| 3.3.1 Radiation Condition | 101 | ||
| 3.3.2 Integral Representation of Solutions to the Lamé System | 102 | ||
| 3.3.3 Reciprocity Property and Helmholtz–Kirchhoff Identities | 110 | ||
| 3.3.4 Incompressible Limit | 112 | ||
| 4. Probabilistic Tools | 115 | ||
| 4.1 Random Variables | 116 | ||
| 4.2 Random Vectors | 117 | ||
| 4.3 Gaussian Random Vectors | 120 | ||
| 4.4 Random Processes | 121 | ||
| 4.4.1 Gaussian Random Processes | 122 | ||
| 4.4.2 Stationary Gaussian Random Processes | 123 | ||
| 4.4.3 Local Maxima of a Gaussian Random Field | 125 | ||
| 4.4.4 Global Maximum of a Gaussian Random Field | 125 | ||
| 4.4.5 The Local Shape of a Local Maximum | 126 | ||
| 4.4.6 Realization of a Cluttered Medium | 127 | ||
| 5. General Image Characteristics | 129 | ||
| 5.1 Spatial Resolution | 129 | ||
| 5.1.1 Point Spread Function | 129 | ||
| 5.1.2 Rayleigh Resolution Limit | 131 | ||
| 5.2 Signal-to-Noise Ratio | 131 | ||
| Part II Single-Wave Imaging | 133 | ||
| 6. Electrical Impedance Tomography | 135 | ||
| 6.1 Mathematical Model | 135 | ||
| 6.2 Ill-Conditioning | 137 | ||
| 6.2.1 Static Imaging | 138 | ||
| 6.2.2 Dynamic Imaging | 138 | ||
| 6.2.3 Electrode Model | 141 | ||
| 7. Ultrasound and Microwave Tomographies | 143 | ||
| 7.1 Born Approximation | 143 | ||
| 7.2 Diffraction Tomography Algorithm | 144 | ||
| 7.3 Time-Reversal Techniques | 146 | ||
| 7.3.1 Ideal Time-Reversal Imaging Technique | 147 | ||
| 7.3.2 A Modified Time-Reversal Imaging Technique | 151 | ||
| 8. Time-Harmonic Reverse-Time Imaging with Additive Noise | 153 | ||
| 8.1 The Data Set | 153 | ||
| 8.2 The Forward Problem | 154 | ||
| 8.3 Imaging Functionals | 156 | ||
| 8.4 The RT-Imaging Function | 157 | ||
| 8.4.1 The Imaging Function without Measurement Noise | 157 | ||
| 8.4.2 The Imaging Function with Measurement Noise | 158 | ||
| 8.4.3 Localization Error | 161 | ||
| 9. Reverse-Time Imaging with Clutter Noise | 163 | ||
| 9.1 The Data Set | 163 | ||
| 9.2 A Model for the Scattering Medium | 164 | ||
| 9.3 The Forward Problem | 166 | ||
| 9.4 The Imaging Function | 168 | ||
| 9.4.1 The Imaging Function without Clutter Noise | 168 | ||
| 9.4.2 The Imaging Function with Clutter Noise | 171 | ||
| 10. Optical Coherence Tomography with Clutter Noise | 177 | ||
| 10.1 The Principle of Optical Coherence Tomography | 177 | ||
| 10.2 The Reference and Sample Beams | 179 | ||
| 10.3 The Imaging Function | 183 | ||
| 10.4 The Point Spread Function | 184 | ||
| 10.5 The Clutter Noise in Optical CoherenceTomography | 186 | ||
| Part III Anomaly Imaging | 189 | ||
| 11. Small Volume Expansions | 191 | ||
| 11.1 Conductivity Problem | 192 | ||
| 11.2 Helmholtz Equation | 196 | ||
| 11.3 Asymptotic Formulas for Monopole Sources in Free Space | 199 | ||
| 11.3.1 Conductivity Problem | 199 | ||
| 11.3.2 Helmholtz Equation | 199 | ||
| 11.4 Elasticity Equations | 200 | ||
| 11.4.1 Static Regime | 202 | ||
| 11.4.2 Time-Harmonic Regime | 204 | ||
| 11.4.3 Properties of the EMT | 207 | ||
| 11.5 Asymptotic Expansions for Time-Dependent Equations | 212 | ||
| 11.5.1 Asymptotic Formulas for the Wave Equation | 212 | ||
| 11.5.2 Asymptotic Analysis of TemperaturePerturbations | 214 | ||
| 12. Anomaly Imaging Algorithms | 219 | ||
| 12.1 Direct Imaging for the Conductivity Problem | 220 | ||
| 12.1.1 Detection of a Single Inclusion: A Projection-Type Algorithm | 220 | ||
| 12.1.2 Detection of Multiple Inclusions: A MUSIC-Type Algorithm | 221 | ||
| 12.2 Direct Imaging Algorithms for the Helmholtz Equation | 223 | ||
| 12.2.1 Direct Imaging at a Fixed Frequency | 223 | ||
| 12.2.2 Direct Imaging at Multiple Frequencies | 232 | ||
| 12.3 Direct Elasticity Imaging | 235 | ||
| 12.3.1 A MUSIC-Type Method in the Static Regime | 235 | ||
| 12.3.2 A MUSIC-Type Method in the Time-Harmonic Regime | 237 | ||
| 12.3.3 Reverse-Time Migration and Kirchhoff Imaging in the Time-Harmonic Regime | 240 | ||
| 12.4 Time-Domain Anomaly Imaging | 241 | ||
| 12.4.1 Wave Imaging of Small Anomalies | 241 | ||
| 12.4.2 Thermal Imaging of Small Anomalies | 243 | ||
| Part IV Multi-Wave Imaging | 247 | ||
| 13. Photoacoustic Imaging | 249 | ||
| 13.1 Introduction | 249 | ||
| 13.2 Mathematical Formulation | 251 | ||
| 13.3 Photoacoustic Imaging in Free Space | 253 | ||
| 13.3.1 Full-View Setting | 254 | ||
| 13.3.2 Limited-View Setting | 255 | ||
| 13.3.3 Compensation of the Effect of Acoustic Attenuation | 257 | ||
| 13.4 Photoacoustic Imaging of Small Absorbers with Imposed Boundary Conditions on the Pressure | 269 | ||
| 13.4.1 Reconstruction Methods | 269 | ||
| 13.4.2 Backpropagation of the Acoustic Signals | 275 | ||
| 13.4.3 Selective Detection | 277 | ||
| 13.5 Imaging with Limited-View Data | 282 | ||
| 13.5.1 Geometrical Control of the Wave Equation | 282 | ||
| 13.5.2 Reconstruction Procedure | 283 | ||
| 13.5.3 Implementation of the HUM | 284 | ||
| 13.6 Quantitative Photoacoustic Imaging | 284 | ||
| 13.6.1 Asymptotic Approach | 286 | ||
| 13.6.2 Multi-Wavelength Approach | 289 | ||
| 13.7 Coherent Interferometry Algorithms | 291 | ||
| 13.8 Concluding Remarks | 295 | ||
| 14. Quantitative Thermoacoustic Imaging | 297 | ||
| 14.1 Introduction | 297 | ||
| 14.2 Measurements | 298 | ||
| 14.3 Exact Formula | 299 | ||
| 14.4 Optimal Control Approach | 304 | ||
| 14.4.1 The Differentiability of the Data Map and Its Inverse | 304 | ||
| 14.4.2 Landweber’s Iteration | 307 | ||
| 15. Ultrasonically Induced Lorentz Force Electrical Impedance Tomography | 309 | ||
| 15.1 Introduction | 309 | ||
| 15.2 Electric Measurements from Acousto-Magnetic Coupling | 311 | ||
| 15.2.1 Electrical Conductivity in Electrolytes | 312 | ||
| 15.2.2 Ion Deviation by Lorentz Force | 312 | ||
| 15.2.3 Internal Electrical Potential | 313 | ||
| 15.2.4 Virtual Potential | 315 | ||
| 15.3 Construction of the Virtual Current | 316 | ||
| 15.4 Recovering the Conductivity by Optimal Control | 319 | ||
| 15.5 The Orthogonal Field Method | 322 | ||
| 15.5.1 Uniqueness Result for the Transport Equation | 323 | ||
| 15.5.2 The Viscosity-Type Regularization | 327 | ||
| 15.6 Numerical Illustrations | 329 | ||
| 15.6.1 Deconvolution | 329 | ||
| 15.6.2 Conductivity Reconstructions | 330 | ||
| 15.7 Concluding Remarks | 335 | ||
| 16. Magnetoacoustic Tomography with Magnetic Induction | 337 | ||
| 16.1 Introduction | 337 | ||
| 16.2 Forward Problem Description | 339 | ||
| 16.2.1 Time Scales Involved | 339 | ||
| 16.2.2 Electromagnetic Model | 339 | ||
| 16.2.3 Acoustic Problem | 341 | ||
| 16.3 Reconstruction of the Acoustic Source | 343 | ||
| 16.4 Reconstruction of the Conductivity | 346 | ||
| 16.4.1 Reconstruction of the Electric Current Density | 346 | ||
| 16.4.2 Recovery of the Conductivity from Internal Electric Current Density | 348 | ||
| 16.5 Numerical Illustrations | 359 | ||
| 16.5.1 Optimal Control | 359 | ||
| 16.5.2 Fixed Point Method | 360 | ||
| 16.5.3 Orthogonal Field Method | 361 | ||
| 16.6 Concluding Remarks | 364 | ||
| 17. Impediography | 365 | ||
| 17.1 Introduction | 365 | ||
| 17.2 Mathematical Model | 367 | ||
| 17.3 Substitution Algorithm | 370 | ||
| 17.4 Optimal Control Algorithm | 372 | ||
| 17.5 Concluding Remarks | 374 | ||
| 18. Microwave Imaging by Elastic Deformation | 375 | ||
| 18.1 Introduction | 375 | ||
| 18.2 Exact Reconstruction Formulas | 378 | ||
| 18.3 The Forward Problem and the Differentiability of the Data at a Fixed Frequency | 384 | ||
| 18.4 Optimal Control Algorithm | 388 | ||
| 19. Ultrasound-Modulated Optical Tomography | 391 | ||
| 19.1 Introduction | 391 | ||
| 19.2 Preliminaries | 393 | ||
| 19.2.1 Acoustic Wave | 393 | ||
| 19.2.2 Regularity Results | 396 | ||
| 19.3 Reconstruction Algorithms | 399 | ||
| 19.3.1 Fixed Point Algorithm | 402 | ||
| 19.3.2 Optimal Control Algorithm | 408 | ||
| 19.4 Numerical Illustrations | 411 | ||
| 19.4.1 Concluding Remarks | 415 | ||
| 20. Viscoelastic Modulus Reconstruction | 417 | ||
| 20.1 Introduction | 417 | ||
| 20.2 Reconstruction Methods | 419 | ||
| 20.2.1 Viscoelasticity Model | 419 | ||
| 20.2.2 Optimal Control Algorithm | 421 | ||
| 20.2.3 Initial Guess | 426 | ||
| 20.2.4 Local Reconstruction | 429 | ||
| 20.3 Numerical Illustrations | 430 | ||
| 20.4 Concluding Remarks | 434 | ||
| 21. Mechanical Vibration-Assisted Conductivity Imaging | 435 | ||
| 21.1 Introduction | 435 | ||
| 21.2 Mathematical Modeling | 436 | ||
| 21.3 Vibration-Assisted Anomaly Identification | 439 | ||
| 21.3.1 Location Search Method and Asymptotic Expansion | 441 | ||
| 21.3.2 Size Estimation and Reconstruction of the Material Parameters | 445 | ||
| 21.4 Numerical Illustrations | 447 | ||
| 21.4.1 Simulations of the Voltage Difference Map | 447 | ||
| 21.4.2 Anomaly Location | 448 | ||
| 21.5 Concluding Remarks | 449 | ||
| 22. Full-Field Optical Coherence Elastography | 451 | ||
| 22.1 Introduction | 451 | ||
| 22.2 Preliminaries | 454 | ||
| 22.3 Displacement Field Measurements | 455 | ||
| 22.3.1 First-Order Approximation | 456 | ||
| 22.3.2 Local Recovery via Linearization | 459 | ||
| 22.3.3 Minimization of the Discrepancy Functional | 463 | ||
| 22.4 Reconstruction of the Shear Modulus | 469 | ||
| 22.5 Numerical Illustrations | 469 | ||
| 22.6 Concluding Remarks | 472 | ||
| Part V Spectroscopic and Nanoparticle Imaging | 473 | ||
| 23. Effective Electrical Tissue Properties | 475 | ||
| 23.1 Introduction | 475 | ||
| 23.2 Problem Settings and Main Results | 477 | ||
| 23.2.1 Periodic Domain | 477 | ||
| 23.2.2 Electrical Model of the Cell | 478 | ||
| 23.2.3 Governing Equation | 482 | ||
| 23.2.4 Main Results | 483 | ||
| 23.3 Analysis of the Problem | 486 | ||
| 23.3.1 Existence and Uniqueness of a Solution | 487 | ||
| 23.3.2 Energy Estimate | 488 | ||
| 23.4 Homogenization | 490 | ||
| 23.4.1 Two-Scale Asymptotic Expansions | 491 | ||
| 23.4.2 Convergence | 496 | ||
| 23.5 Effective Admittivity for a Dilute Suspension | 506 | ||
| 23.5.1 Computation of the Effective Admittivity | 506 | ||
| 23.5.2 Case of Concentric Circular-Shaped Cells: The Maxwell–Wagner–Fricke Formula | 510 | ||
| 23.5.3 Debye Relaxation Times | 511 | ||
| 23.5.4 Properties of the Membrane Polarization Tensor and the Debye Relaxation Times | 512 | ||
| 23.5.5 Anisotropy Measure | 513 | ||
| 23.6 Numerical Simulations | 514 | ||
| 23.7 Technical Results | 517 | ||
| 23.7.1 Extension Lemmas | 517 | ||
| 23.7.2 Poincaré–Wirtinger Inequality | 521 | ||
| 23.7.3 Equivalence of the Two Norms on Wε | 523 | ||
| 23.7.4 Existence Result | 525 | ||
| 23.8 Concluding Remarks | 526 | ||
| 24. Plasmonic Nanoparticle Imaging | 527 | ||
| 24.1 Introduction | 527 | ||
| 24.2 Layer Potential Formulation for Plasmonic Resonances | 529 | ||
| 24.2.1 Problem Formulation and Some Basic Results | 529 | ||
| 24.2.2 First-Order Correction to Plasmonic Resonances and Field Behavior at the Plasmonic Resonances | 533 | ||
| 24.3 Multiple Plasmonic Nanoparticles | 541 | ||
| 24.3.1 Layer Potential Formulation in the Multi-Particle Case | 541 | ||
| 24.3.2 First-Order Correction to Plasmonic Resonances and Field Behavior at Plasmonic Resonances in the Multi-Particle Case | 542 | ||
| 24.4 Scattering and Absorption Enhancements | 552 | ||
| 24.4.1 The Quasi-Static Limit | 552 | ||
| 24.4.2 An Upper Bound for the Averaged Extinction Cross-Section | 555 | ||
| 24.5 Link with the Scattering Coefficients | 563 | ||
| 24.5.1 Scattering Coefficients of Plasmonic Nanoparticles | 563 | ||
| 24.5.2 The Leading-Order Term in the Expansion of the Scattering Amplitude | 566 | ||
| 24.6 Asymptotic Expansion of the Integral Operators: Single Particle | 569 | ||
| 24.7 Asymptotic Expansion of the Integral Operators: Multiple Particles | 571 | ||
| 24.8 Sum Rules for the Polarization Tensor | 577 | ||
| 24.9 Concluding Remarks | 580 | ||
| 25. Nonlinear Harmonic Holography | 583 | ||
| 25.1 Introduction | 583 | ||
| 25.2 Problem Formulation | 585 | ||
| 25.3 Small-Volume Expansions | 587 | ||
| 25.3.1 Fundamental Frequency Problem | 587 | ||
| 25.3.2 Second-Harmonic Problem | 593 | ||
| 25.4 Imaging Functional | 597 | ||
| 25.4.1 The Fundamental Frequency Case | 597 | ||
| 25.4.2 Second-Harmonic Backpropagation | 598 | ||
| 25.5 Statistical Analysis | 599 | ||
| 25.5.1 Assumptions on the Random Process μ | 600 | ||
| 25.5.2 Standard Backpropagation | 602 | ||
| 25.5.3 Second-Harmonic Backpropagation | 610 | ||
| 25.5.4 Stability with Respect to Measurement Noise | 617 | ||
| 25.6 Numerical Results | 622 | ||
| 25.6.1 The Direct Problem | 622 | ||
| 25.6.2 The Imaging Functionals and the Effects of the Number of Plane Wave Illuminations | 623 | ||
| 25.6.3 Statistical Analysis | 626 | ||
| 25.7 Proof of Estimate (25.8) | 631 | ||
| 25.8 Proof of Proposition 25.1 | 635 | ||
| 25.9 Proof of Proposition 25.3 | 636 | ||
| 25.10 Concluding Remarks | 637 | ||
| References | 639 | ||
| Index | 663 |