Tomography

Tomography......Page 5
Table of Contents......Page 7
Preface......Page 19
Notation......Page 23
1.1. Introduction......Page 33
1.2. Observing contrasts......Page 34
1.3. Localization in space and time......Page 39
1.4. Image reconstruction......Page 41
1.5. Application domains......Page 44
1.6. Bibliography......Page 49
Part 1. Image Reconstruction......Page 53
2.1. Introduction......Page 55
2.2.1. Definition and concept of sinogram......Page 57
2.2.2. Fourier slice theorem and data sufficiency condition......Page 58
2.2.3. Inversion by filtered backprojection......Page 59
2.2.4. Choice of filter......Page 60
2.2.5. Frequency–distance principle......Page 63
2.3.1. Definition......Page 64
2.3.3. Reconstruction by filtered backprojection......Page 65
2.3.4. Fast acquisitions......Page 66
2.3.5. 3D helical tomography in fan-beam geometry with a single line detector......Page 67
2.4.1. Definition......Page 69
2.4.2. Fourier slice theorem and data sufficiency conditions......Page 70
2.4.3. Inversion by filtered backprojection......Page 71
2.5.1. Definition......Page 72
2.5.3. Inversion by filtered backprojection......Page 73
2.6.2. Approximate reconstruction by rebinning to transverse slices......Page 74
2.6.3. Direct reconstruction by filtered backprojection......Page 77
2.7.2. Connection to the derivative of the 3D Radon transform and data sufficiency condition......Page 78
2.7.3. Approximate inversion by rebinning to transverse slices......Page 81
2.7.4. Approximate inversion by filtered backprojection......Page 82
2.7.5. Inversion by rebinning in Radon space......Page 84
2.7.6. Katsevich algorithm for helical cone-beam reconstruction......Page 85
2.8.2. 2D dynamic Radon transform......Page 86
2.8.4. Inversion......Page 87
2.9. Bibliography......Page 90
3.1.1. Periodic functions, integrable functions, Fourier transforms......Page 95
3.1.2. Poisson summation formula and sampling of bandlimited functions......Page 97
3.1.3. Sampling of essentially bandlimited functions......Page 98
3.1.4. Efficient sampling......Page 100
3.1.5. Generalization to periodic functions in their first variables......Page 102
3.2.1. Essential support of the 2D Radon transform......Page 103
3.2.2. Sampling conditions and efficient sampling......Page 105
3.2.3.1. Vector tomography......Page 106
3.2.3.2. Generalized, rotation invariant Radon transform......Page 108
3.2.3.3. Exponential and attenuated Radon transform......Page 109
3.3.1. Introduction......Page 111
3.3.2. Sampling of the X-ray transform......Page 112
3.3.3. Numerical results on the sampling of the cone-beam transform......Page 116
3.4. Bibliography......Page 117
4.1. Introduction......Page 121
4.2. Discrete models......Page 122
4.3.1.1. Algebraic methods based on quadratic minimization......Page 124
4.3.1.2. Algorithms......Page 125
4.3.2.1. Algebraic methods based on constraint optimization......Page 128
4.3.2.2. Algorithms......Page 129
4.4.1.1. Bayesian statistical methods......Page 131
4.4.1.3. Markov fields and Gibbs distributions......Page 132
4.4.1.4. Potential function......Page 134
4.4.1.5. Choice of hyperparameters......Page 135
4.4.2.2. MAP-EM algorithm......Page 137
4.4.2.4. Regularization algorithm ARTUR......Page 138
4.4.3.1. MEM method......Page 139
4.4.3.2. A priori distributions......Page 141
4.6. Discussion and conclusion......Page 142
4.7. Bibliography......Page 144
Part 2. Microtomography......Page 149
5.1. Introduction......Page 151
5.2. Projection tomography in electron microscopy......Page 152
5.3. Tomography by optical sectioning......Page 153
5.3.1.1. Principle of confocal microscopy......Page 154
5.3.1.2. Image formation......Page 155
5.3.1.3. Optical sectioning......Page 158
5.3.1.4. Fluorochromes employed in confocal microscopy......Page 160
5.4.1. Fluorograms......Page 161
5.4.2.1. Denoising......Page 162
5.4.2.3. Numerical deconvolution......Page 164
5.4.3. Microscopy by multiphoton absorption......Page 167
5.5. Bibliography......Page 170
6.1. Introduction......Page 173
6.2. Interaction of light with matter......Page 174
6.2.1. Absorption......Page 175
6.2.2. Fluorescence......Page 177
6.2.3.2. Elastic scattering......Page 178
6.3. Propagation of photons in diffuse media......Page 182
6.3.1. Coherent propagation......Page 183
6.3.2. Mixed coherent/incoherent propagation......Page 188
6.3.3. Incoherent propagation......Page 189
6.3.4. Radiative transfer theory......Page 190
6.4. Optical tomography methods......Page 196
6.4.1.1. Diffraction tomography......Page 198
6.4.1.2. Born approximation......Page 201
6.4.1.3. Principle of the reconstruction of the object in diffraction tomography......Page 202
6.4.1.4. Data acquisition......Page 206
6.4.1.5. Optical coherence tomography......Page 208
6.4.1.6. Role of the coherence length......Page 209
6.4.1.8. Lateral scanning......Page 210
6.4.1.9. Coherence tomography in the temporal domain......Page 212
6.5. Optical tomography in highly diffuse media......Page 213
6.5.1. Direct model, inverse model......Page 214
6.5.2.2. Radiance......Page 216
6.5.3. Inverse model......Page 217
6.5.5. Perturbation method......Page 218
6.5.7. Temporal domain......Page 219
6.5.8. Frequency domain......Page 221
6.6. Bibliography......Page 222
7.2.1. Physical principles......Page 229
7.2.2. Advantages of synchrotron radiation for tomography......Page 233
7.3.1. State of the art......Page 234
7.3.2. ESRF system and applications......Page 236
7.4.1. State of the art......Page 238
7.4.2. ESRF system and applications......Page 239
7.5.1. Phase contrast and holographic tomography......Page 242
7.6. Conclusion......Page 243
7.7. Bibliography......Page 244
Part 3. Industrial Tomography......Page 247
8.1. Introduction......Page 249
8.2. Physics of the measurement......Page 250
8.3. Sources of radiation......Page 251
8.4. Detection......Page 252
8.5.3. Reconstruction artifacts......Page 255
8.6.1. Tomodensitometry......Page 256
8.6.3. High-energy tomography......Page 257
8.6.4. CAD models in tomography: reverse engineering and simulation......Page 258
8.6.4.2. Simulation......Page 259
8.6.5. Microtomography......Page 260
8.6.7. Dual-energy tomography......Page 262
8.6.8. Tomosynthesis......Page 264
8.6.9. Scattering tomography......Page 265
8.8. Bibliography......Page 267
9.1.1. Context and objectives......Page 271
9.1.3. Nuclear techniques......Page 272
9.2.1. Two-dimensional flow......Page 274
9.2.2. Flow in a pipe – analysis of a junction......Page 276
9.3.1. Photon transport......Page 279
9.3.2. Principle of the Monte Carlo simulation......Page 280
9.3.3.1. Estimation of projection profiles......Page 281
9.3.3.2. Estimation of the projection matrix......Page 283
9.4.1. Characteristic system parameters......Page 284
9.4.2.1. Underdetermined systems......Page 286
9.6. Bibliography......Page 287
Part 4. Morphological Medical Tomography......Page 289
10.1.1. Definition......Page 291
10.1.2. Evolution of CT......Page 292
10.1.3. Scanners with continuous rotation......Page 293
10.1.4. Multislice scanners......Page 295
10.1.5. Medical applications of CT......Page 296
10.2.1. Projection acquisition system......Page 297
10.2.1.3. X-ray detectors with high dynamics, efficiency, and speed......Page 298
10.2.2.1. Hypotheses......Page 299
10.2.2.2. Interpolation algorithms......Page 300
10.2.2.3. Slice spacing......Page 301
10.2.3.3. Image noise......Page 302
10.2.4. Dose......Page 303
10.3.3. Abdominal applications......Page 304
10.3.4. Thoracic applications......Page 305
10.3.5. Vascular applications......Page 308
10.3.6. Cardiac applications......Page 309
10.4. Conclusion......Page 311
10.5. Bibliography......Page 312
11.1.1. Definition......Page 319
11.1.2. Acquisition systems......Page 320
11.1.3. Positioning with respect to computed tomography......Page 321
11.2. Example of 3D angiography......Page 322
11.2.2.1. Calibration......Page 323
11.2.2.2. Reconstruction algorithm......Page 327
11.2.3. Other reconstruction methods in 3D radiology......Page 328
11.3. Clinical examples......Page 329
11.3.1. High contrast applications......Page 330
11.3.2. Soft tissue contrast applications......Page 331
11.3.3. Cardiac applications......Page 332
11.4. Conclusion......Page 334
11.5. Bibliography......Page 335
12.1. Introduction......Page 339
12.2.2. Nuclear magnetic relaxation and Larmor precession......Page 340
12.2.3. NMR signal formation......Page 341
12.2.4. Instrumentation......Page 343
12.3.1. Information content of the signal’s phase......Page 344
12.3.2.1. Cartesian k-space sampling......Page 346
12.3.2.2. Non-Cartesian k-space sampling......Page 348
12.4.2. Relaxation times and discrimination between soft tissues......Page 350
12.4.4. Relevance of magnetization transfer techniques......Page 352
12.4.5. Flow of matter......Page 353
12.4.6. Diffusion and perfusion effects......Page 354
12.6. Bibliography......Page 355
Part 5. Functional Medical Tomography......Page 359
13.1.2. Functional versus anatomical imaging......Page 361
13.2.1. Vectors......Page 362
13.3.1. General principle of the gamma camera......Page 363
13.3.2. Special features of single photon detection: collimator and spectrometry......Page 364
13.3.3. Principle characteristics of the gamma camera......Page 365
13.3.4. Principle of projection acquisition......Page 366
13.3.5. Transmission measurement system......Page 367
13.4.1. Hypotheses......Page 368
13.4.2.1. Tomographic reconstruction problem in SPECT......Page 369
13.4.2.3. Algebraic reconstruction in SPECT......Page 370
13.4.3. Specific problems of single photon detection......Page 371
13.4.3.2. Attenuation......Page 372
13.4.3.3. Scatter......Page 373
13.4.3.4. Variation of the spatial resolution with depth......Page 374
13.5.1. Indications......Page 375
13.5.3. Reconstruction and interpretation criteria......Page 376
13.5.4. Importance of the accuracy of the projection model......Page 377
13.6. Conclusion......Page 378
13.7. Bibliography......Page 380
14.1.2. PET versus other functional imaging techniques......Page 383
14.2.1.1. Vectors......Page 385
14.2.2.1. Positron annihilation......Page 386
14.2.2.3. Type of detected coincidences......Page 387
14.2.3.1. Detectors......Page 388
14.2.3.2.Detector arrangement......Page 389
14.2.5.1. Two-dimensional (2D) acquisition......Page 390
14.2.5.2. Three-dimensional (3D) acquisition......Page 392
14.2.5.4. LOR and list mode acquisition......Page 393
14.2.5.6 Transmission scan acquisition......Page 394
14.3. Data processing......Page 395
14.3.1. Data correction......Page 396
14.3.2. Reconstruction of corrected data......Page 397
14.3.3. Dynamic studies......Page 398
14.3.3.1. Measurement of glucose metabolism......Page 399
14.3.3.2. Model......Page 400
14.3.3.3. Acquisition protocol......Page 401
14.4.1.2. Acquisition and reconstruction protocol......Page 402
14.4.1.3. Image interpretation: sensitivity and specificity of this medical imaging modality......Page 403
14.4.2.3. Tracers......Page 404
14.5. Conclusion......Page 405
14.6. Bibliography......Page 406
15.1. Introduction......Page 409
15.2. Functional MRI of cerebrovascular responses......Page 410
15.3.1. Biophysical model......Page 412
15.3.2. “Static” conditions......Page 413
15.3.4. “Motional narrowing” conditions......Page 414
15.4. Different protocols......Page 415
15.4.2. Event-related paradigms......Page 416
15.4.3. Fourier paradigms......Page 419
15.5. Bibliography......Page 421
16.1. Introduction......Page 425
16.2.1. Sources of MEG and EEG......Page 426
16.2.3. MEG instrumentation......Page 427
16.2.4. Applications......Page 429
16.3.1. Difficulties of reconstruction......Page 430
16.3.2. Direct problem and different field calculation methods......Page 431
16.3.3. Inverse problem......Page 434
16.3.3.1. Parametric or “dipolar” methods......Page 435
16.3.3.2. Tomographic or “distributed” methods......Page 437
16.4. Conclusion......Page 439
16.5. Bibliography......Page 440
List of Authors......Page 443
Index......Page 449