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Bayesian Approach to Inverse Problems(解逆问题的贝叶斯方法)
Bayesian Approach to Inverse Problems~Jérôme Idier.Wiley.2008.pdf
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Table of Contents
Introduction 15
Jérôme IDIER
PART I FUNDAMENTAL PROBLEMS AND TOOLS 23
Chapter 1 Inverse Problems, Ill-posed Problems 25
Guy DEMOMENT, Jérôme IDIER
1.1Introduction 25
1.2.Basic example26
1.3Ill-posedproblem30
1.3.1.Case of discrete data 31
1.3.2Continuous case 32
1.4.Generalizedinversion 34
1.4.1Pseudo-solutions 35
1.4.2.Generalizedsolutions 35
1.4.3.Example 35
1.5.Discretization and conditioning 36
1.6.Conclusion 38
1.7.Bibliography 39
Chapter 2 Main Approaches to the Regularization of Ill-posed Problems 41
Guy DEMOMENT, Jérôme IDIER
2.1.Regularization 41
2.1.1Dimensionality control 42
2.1.1.1Truncated singular value decomposition 42
2.1.1.2.Change ofdiscretization 43
2.1.1.3Iterative methods 43
2.1.2.Minimizationof a composite criterion 44
2.1.2.1.Euclidiandistances 452.1.2.2Roughness measures 46
2.1.2.3Non-quadratic penalization 47
2.1.2.4Kullback pseudo-distance 47
2.2Criterion descent methods 48
2.2.1.Criterionminimizationfor inversion 48
2.2.2.The quadraticcase 49
2.2.2.1.Non-iterativetechniques 49
2.2.2.2Iterative techniques 50
2.2.3.The convexcase 51
2.2.4.General case 52
2.3.Choice of regularizationcoefficient 53
2.3.1.Residual error energycontrol 53
2.3.2“L-curve”method 53
2.3.3.Cross-validation 54
2.4.Bibliography 56
Chapter 3 Inversion within the Probabilistic Framework 59
Guy DEMOMENT, Yves GOUSSARD
3.1Inversionand inference 59
3.2.Statistical inference60
3.2.1.Noise lawanddirect distributionfor data 61
3.2.2Maximum likelihood estimation 63
3.3.Bayesian approachto inversion 64
3.4Links with deterministic methods 66
3.5Choice of hyperparameters 67
3.6A priorimodel 68
3.7.Choice of criteria 70
3.8.The linear,Gaussian case 71
3.8.1.Statistical propertiesof the solution 71
3.8.2Calculation of marginal likelihood 73
3.8.3.Wienerfiltering 74
3.9.Bibliography 76PART II DECONVOLUTION 79
Chapter 4 Inverse Filtering and Other Linear Methods 81
Guy LE BESNERAIS, Jean-François GIOVANNELLI, Guy DEMOMENT
4.1Introduction 81
4.2Continuous-time deconvolution 82
4.2.1Inversefiltering 82
4.2.2.Wienerfiltering 84
4.3.Discretization of the problem 85
4.3.1Choice of a quadrature method 854.3.2Structure of observation matrix H 87
4.3.3Usual boundary conditions 89
4.3.4.Problemconditioning 89
4.3.4.1.Case of the circulantmatrix 90
4.3.4.2Case of the Toeplitz matrix 90
4.3.4.3Opposition between resolution and conditioning 91
4.3.5.Generalizedinversion 91
4.4.Batch deconvolution 92
4.4.1.Preliminarychoices 92
4.4.2.Matrix formof the estimate 93
4.4.3Hunt’s method (periodic boundary hypothesis) 94
4.4.4Exact inversion methods in the stationary case 96
4.4.5Case of non-stationary signals 98
4.4.6.Results and discussionon examples 98
4.4.6.1Compromise between bias and variance in 1D deconvolution 98
4.4.6.2.Results for 2Dprocessing 100
4.5.Recursive deconvolution 102
4.5.1.Kalmanfiltering 102
4.5.2Degenerate state model and recursive least squares 104
4.5.3.Autoregressivestatemodel 105
4.5.3.1Initialization 106
4.5.3.2Criterion minimized by Kalman smoother 107
4.5.3.3.Exampleof result 108
4.5.4.FastKalmanfiltering 108
4.5.5Asymptotic techniques in the stationary case 110
4.5.5.1Asymptotic Kalman filtering 110
4.5.5.2.Small kernelWienerfilter 111
4.5.6ARMA model and non-standard Kalman filtering 111
4.5.7Case of non-stationary signals 111
4.5.8.On-lineprocessing: 2Dcase 112
4.6.Conclusion 112
4.7.Bibliography 113
Chapter 5 Deconvolution of Spike Trains 117
Frédéric CHAMPAGNAT, Yves GOUSSARD, Stéphane GAUTIER, Jérôme IDIER
5.1Introduction 117
5.2Penalization of reflectivities, L2LP/L2Hy deconvolutions 119
5.2.1.Quadratic regularization 121
5.2.2.Non-quadraticregularization 122
5.2.3L2LPorL2Hy deconvolution 123
5.3Bernoulli-Gaussian deconvolution 124
5.3.1Compound BG model 124
5.3.2.Various strategies for estimation 1245.3.3General expression for marginal likelihood 125
5.3.4An iterative method for BG deconvolution 126
5.3.5Other methods 128
5.4Examples of processing and discussion 130
5.4.1.Natureof the solutions 130
5.4.2Setting the parameters 132
5.4.3.Numerical complexity 133
5.5.Extensions 133
5.5.1Generalization of structures of R and H 134
5.5.2Estimation of the impulse response 134
5.6.Conclusion 136
5.7.Bibliography 137
Chapter 6 Deconvolution of Images 141
Jérôme IDIER, Laure BLANC-FÉRAUD
6.1Introduction 141
6.2Regularization in the Tikhonov sense 142
6.2.1.Principle 142
6.2.1.1Case of a monovariate signal 142
6.2.1.2.Multivariate extensions 143
6.2.1.3.Discrete framework 144
6.2.2Connection with image processing by linear PDE 144
6.2.3Limits of Tikhonov’s approach 145
6.3.Detection-estimation 148
6.3.1.Principle 148
6.3.2.Disadvantages 149
6.4.Non-quadraticapproach 150
6.4.1Detection-estimation and non-convex penalization 154
6.4.2.AnisotropicdiffusionbyPDE 155
6.5.Half-quadraticaugmentedcriteria 156
6.5.1Duality between non-quadratic criteria and HQ criteria 157
6.5.2.MinimizationofHQcriteria 158
6.5.2.1.Principle of relaxation 158
6.5.2.2Case of a convex function φ 159
6.5.2.3Case of a non-convex function φ 159
6.6.Applicationin imagedeconvolution 159
6.6.1.Calculationof the solution 159
6.6.2.Example 161
6.7.Conclusion 164
6.8.Bibliography 165PART III ADVANCED PROBLEMS AND TOOLS 169
Chapter 7 Gibbs-Markov Image Models 171
Jérôme IDIER
7.1Introduction 171
7.2.Bayesian statistical framework 172
7.3.Gibbs-Markovfields 173
7.3.1.Gibbsfields 174
7.3.1.1Definition 174
7.3.1.2.Trivial examples 175
7.3.1.3Pairwise interactions, improper laws 176
7.3.1.4.Markovchains 176
7.3.1.5Minimum cliques, non-uniqueness of potential 177
7.3.2Gibbs-Markovequivalence 177
7.3.2.1Neighborhood relationship 177
7.3.2.2Definition of a Markov field 178
7.3.2.3.AGibbsfield is aMarkovfield 179
7.3.2.4.Hammersley-Cliffordtheorem 179
7.3.3.Posterior lawof aGMRF 180
7.3.4.Gibbs-Markovmodels for images 181
7.3.4.1Pixels with discrete values and label fields for classification 181
7.3.4.2.GaussianGMRF 182
7.3.4.3Edge variables, composite GMRF 183
7.3.4.4Interactive edgevariables 184
7.3.4.5.Non-GaussianGMRFs 185
7.4Statistical tools, stochastic sampling 185
7.4.1.Statistical tools 185
7.4.2.Stochastic sampling 188
7.4.2.1Iterative sampling methods 189
7.4.2.2Monte Carlo method of the MCMC kind 192
7.4.2.3.Simulated annealing 193
7.5.Conclusion 194
7.6.Bibliography 195
Chapter 8 Unsupervised Problems 197
Xavier DESCOMBES, Yves GOUSSARD
8.1Introduction and statement of problem 197
8.2.Directly observedfield 199
8.2.1Likelihood properties 199
8.2.2.Optimization 200
8.2.2.1.Gradientdescent 200
8.2.2.2Importancesampling 200
8.2.3.Approximations 2028.2.3.1Encoding methods 202
8.2.3.2Pseudo-likelihood 203
8.2.3.3.Meanfield 204
8.3Indirectlyobservedfield 205
8.3.1.Statement of problem 205
8.3.2.EMalgorithm 206
8.3.3Application to estimation of the parameters of a GMRF 207
8.3.4.EMalgorithmandgradient 208
8.3.5Linear GMRF relative to hyperparameters 210
8.3.6Extensions and approximations 212
8.3.6.1Generalized maximum likelihood 212
8.3.6.2.FullBayesian approach 213
8.4.Conclusion 215
8.5.Bibliography 216PART IV SOME APPLICATIONS 219
Chapter 9 DeconvolutionApplied toUltrasonicNon-destructive Evaluation 221
Stéphane GAUTIER, Frédéric CHAMPAGNAT, Jérôme IDIER
9.1Introduction 221
9.2Example of evaluation and difficulties of interpretation 222
9.2.1.Descriptionof the part to be inspected 222
9.2.2.Evaluationprinciple 222
9.2.3.Evaluationresults andinterpretation 223
9.2.4Help with interpretation by restoration of discontinuities 224
9.3Definition of direct convolution model 225
9.4.Blind deconvolution 226
9.4.1Overview of approaches for blind deconvolution 226
9.4.1.1.Predictivedeconvolution 226
9.4.1.2Minimum entropy deconvolution 228
9.4.1.3Deconvolution by “multipulse” technique 228
9.4.1.4Sequential estimation: estimation of the kernel, then the input 228
9.4.1.5Joint estimation of kernel and input 229
9.4.2.DL2Hy/DBGdeconvolution 230
9.4.2.1Improveddirectmodel 230
9.4.2.2Prior information on double reflectivity 230
9.4.2.3Double Bernoulli-Gaussian (DBG) deconvolution 230
9.4.2.4Double hyperbolic (DL2Hy) deconvolution 231
9.4.2.5Behavior of DL2Hy/DBG deconvolution methods 231
9.4.3.BlindDL2Hy/DBGdeconvolution 232
9.5.Processing real data 232
9.5.1Processing by blind deconvolution 233
9.5.2.Deconvolutionwith ameasuredwave 2349.5.3Comparison between DL2Hy and DBG 237
9.5.4.Summary 240
9.6.Conclusion 240
9.7.Bibliography 241
Chapter 10 Inversion in Optical Imaging through Atmospheric Turbulence 243
Laurent MUGNIER, Guy LE BESNERAIS, SergeMEIMON
10.1Optical imaging through turbulence 243
10.1.1Introduction 243
10.1.2Image formation 244
10.1.2.1.Diffraction 244
10.1.2.2Principle of optical interferometry 245
10.1.3Effect of turbulence on image formation 246
10.1.3.1.Turbulenceand phase 246
10.1.3.2Long-exposure imaging 247
10.1.3.3Short-exposure imaging 247
10.1.3.4Case of a long-baseline interferometer 248
10.1.4Imagingtechniques 249
10.1.4.1.Speckle techniques 249
10.1.4.2Deconvolution from wavefront sensing (DWFS) 250
10.1.4.3.Adaptiveoptics 251
10.1.4.4.Optical interferometry 251
10.2Inversion approach and regularization criteria used 253
10.3.Measurementof aberrations 254
10.3.1Introduction 254
10.3.2.Hartmann-Shacksensor 255
10.3.3.Phase retrieval and phasediversity 257
10.4Myopic restoration in imaging 258
10.4.1.Motivationand noise statistic 258
10.4.2Data processing in deconvolution from wavefront sensing 259
10.4.2.1Conventional processing of short-exposure images 259
10.4.2.2Myopic deconvolution of short-exposure images 260
10.4.2.3.Simulations 261
10.4.2.4.Experimental results 262
10.4.3Restoration of images corrected by adaptive optics 263
10.4.3.1Myopic deconvolution of images corrected by adaptive optics 263
10.4.3.2.Experimental results 265
10.4.4.Conclusion267
10.5Image reconstruction in optical interferometry (OI) 268
10.5.1.Observationmodel 268
10.5.2.TraditionalBayesian approach 271
10.5.3Myopic modeling 272
10.5.4.Results 27410.5.4.1Processing of synthetic data 274
10.5.4.2Processing of experimental data 276
10.6.Bibliography 277Chapter 11 Spectral Characterization in Ultrasonic Doppler Velocimetry 285
Jean-François GIOVANNELLI, Alain HERMENT
11.1Velocity measurement in medical imaging 285
11.1.1Principle of velocity measurement in ultrasound imaging 286
11.1.2Information carried by Doppler signals 286
11.1.3.Some characteristics andlimitations 288
11.1.4.Data andproblems treated 288
11.2.Adaptive spectral analysis 290
11.2.1Least squares and traditional extensions 290
11.2.2Long AR models – spectral smoothness – spatial continuity 291
11.2.2.1.Spatial regularity 291
11.2.2.2.Spectral smoothness 292
11.2.2.3Regularized least squares 292
11.2.2.4.Optimization 293
11.2.3.Kalman smoothing 293
11.2.3.1State and observation equations 293
11.2.3.2Equivalence between parameterizations 294
11.2.4Estimation of hyperparameters 294
11.2.5Processing results and comparisons 296
11.2.5.1.Hyperparameter tuning 296
11.2.5.2Qualitative comparison 296
11.3.Trackingspectralmoments 297
11.3.1Proposed method 298
11.3.1.1Likelihood 298
11.3.1.2Amplitudes: prior distribution and marginalization 298
11.3.1.3Frequencies: prior law and posterior law 300
11.3.1.4.Viterbi algorithm 302
11.3.2Likelihood of the hyperparameters 302
11.3.2.1Forward-Backward algorithm 302
11.3.2.2Likelihood gradient 303
11.3.3Processing results and comparisons 304
11.3.3.1Tuning the hyperparameters 304
11.3.3.2Qualitative comparison 305
11.4.Conclusion 306
11.5.Bibliography 307
Chapter 12 Tomographic Reconstruction from Few Projections 311
Ali MOHAMMAD-DJAFARI, Jean-Marc DINTEN
12.1Introduction 31112.2.Projectiongenerationmodel 312
12.32D analytical methods 313
12.43D analytical methods 317
12.5Limitations of analytical methods 317
12.6.Discrete approachto reconstruction 319
12.7Choice of criterion and reconstruction methods 321
12.8.Reconstructionalgorithms 323
12.8.1Optimization algorithms for convex criteria 323
12.8.1.1.Gradient algorithms 324
12.8.1.2SIRT (Simultaneous Iterative Relaxation Techniques) 325
12.8.1.3ART (Algebraic Reconstruction Technique) 325
12.8.1.4.ARTby blocks 326
12.8.1.5ICD (Iterative Coordinate Descent) algorithms 326
12.8.1.6Richardson-Lucy algorithm 326
12.8.2Optimization or integration algorithms 327
12.9.Specificmodels forbinaryobjects 328
12.10Illustrations 328
12.10.1.2Dreconstruction 328
12.10.2.3Dreconstruction 329
12.11.Conclusions 331
12.12.Bibliography 332
Chapter 13 Diffraction Tomography 335
Hervé CARFANTAN, AliMOHAMMAD-DJAFARI
13.1Introduction 335
13.2.Modelingthe problem 336
13.2.1Examples of diffraction tomography applications 336
13.2.1.1.Microwave imaging 337
13.2.1.2Non-destructive evaluation of conducting materials using
eddycurrents 337
13.2.1.3Geophysical exploration 338
13.2.2Modeling the direct problem 338
13.2.2.1Equations of propagation in an inhomogeneous medium 338
13.2.2.2Integral modeling of the direct problem 339
13.3.Discretizationof the directproblem340
13.3.1.Choice of algebraic framework 340
13.3.2.Methodofmoments 341
13.3.3Discretization by the method of moments 342
13.4Construction of criteria for solving the inverse problem 343
13.4.1First formulation: estimation of x 344
13.4.2Second formulation: simultaneous estimation of x and φ 345
13.4.3.Properties of the criteria 347
13.5.Solvingthe inverseproblem 34713.5.1.Successive linearizations 348
13.5.1.1.Approximations 348
13.5.1.2.Regularization 349
13.5.1.3Interpretation 349
13.5.2Jointminimization 350
13.5.3.MinimizingMAPcriterion 351
13.6.Conclusion 353
13.7.Bibliography 354
Chapter 14 Imaging from Low-intensity Data 357
Ken SAUER, Jean-Baptiste THIBAULT
14.1Introduction 357
14.2Statistical properties of common low-intensity image data 359
14.2.1Likelihood functions and limiting behavior 359
14.2.2.PurelyPoissonmeasurements 360
14.2.3Inclusion of background counting noise 362
14.2.4Compound noise models with Poisson information 362
14.3Quantum-limited measurements in inverse problems 363
14.3.1Maximum likelihood properties 363
14.3.2.Bayesian estimation 366
14.4Implementation and calculation of Bayesian estimates 368
14.4.1Implementation for pure Poisson model 368
14.4.2Bayesian implementation for a compound data model 370
14.5.Conclusion 372
14.6.Bibliography 372
List of Authors 375
Index 377
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