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Introduction to Inverse Problems

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Introduction to Inverse Problems Introduction to Inverse Problems Guillaume Bal 1 January 29, 2012 1Columbia University, New York NY, 10027; [email protected] ii Contents 1 What constitutes an Inverse Problem1 1.1 Elements of an Inverse Problem (IP)....................1 1.1.1 Injectivity and stability of the Measurement Operator.......1 1.1.2 \Noise", Modeling, and Prior Information.............2 1.1.3 Numerical simulations........................4 1.2 Examples of Measurement Operator.....................6 1.3 IP and Modeling. Application to MRI...................8 1.4 Inverse Problems and Smoothing: Hilbert scale.............. 12 1.4.1 Fourier transforms and well-posedness............... 12 1.4.2 Hilbert scale and degrees of ill-posedness.............. 13 2 Integral Geometry. Radon transforms 19 2.1 Transmission Tomography.......................... 19 2.2 Two dimensional Radon transform..................... 21 2.3 Three dimensional Radon transform..................... 28 2.4 Attenuated Radon Transform........................ 29 2.4.1 Single Photon Emission Computed Tomography.......... 30 2.4.2 Riemann Hilbert problem...................... 31 2.4.3 Inversion of the Attenuated Radon Transform........... 32 2.4.4 Step (i): The @ problem, an elliptic equation............ 33 2.4.5 Step (ii): jump conditions...................... 35 2.4.6 Step (iii): reconstruction formulas.................. 37 3 Integral Geometry. Generalized Ray Transform 39 3.1 Generalized Ray Transform: Setting in two dimensions........... 40 3.1.1 Family of curves............................ 40 3.1.2 Generalized Ray Transform...................... 40 3.1.3 Adjoint operator and rescaled Normal operator.......... 41 3.2 Oscillatory integrals and Fourier Integral Operators............ 43 3.2.1 Symbols, phases, and oscillatory integrals.............. 43 3.2.2 Parameterized oscillatory integrals................. 45 3.2.3 Definition of Fourier Integral Operators............... 46 3.3 Pseudo-differential operators and GRT................... 47 3.3.1 Absence of singularities away from the diagonal x = y ....... 48 3.3.2 Change of variables and phase (x − y) · ξ .............. 49 3.3.3 Choice of a parametrix......................... 50 iii iv CONTENTS 3.3.4 Proof of smoothing by one derivative................ 51 3.3.5 Boundedness of ΨDOs of order 0 in the L2 sense.......... 51 3.3.6 Injectivity and implicit inversion formula............... 53 3.4 Kinematic Inverse Source Problem..................... 55 3.4.1 Transport equation.......................... 56 3.4.2 Variational form and energy estimates............... 56 3.4.3 Injectivity result........................... 58 3.4.4 Summary on GRT........................... 58 3.5 Propagation of singularities for the GRT................... 59 3.5.1 Wave Front Set and Distributions................... 59 3.5.2 Propagation of singularities in FIOs................. 61 4 Inverse wave problems 65 4.1 One dimensional inverse scattering problem................ 66 4.2 Linearized Inverse Scattering problem.................... 68 4.2.1 Setting and linearization....................... 68 4.2.2 Far field data and reconstruction.................. 70 4.2.3 Comparison to X-ray tomography.................. 73 4.3 Inverse source problem in PAT........................ 74 4.3.1 An explicit reconstruction formula for Σ the unit sphere..... 74 4.3.2 An explicit reconstruction for detectors on a plane......... 75 4.4 One dimensional inverse coefficient problem................ 77 5 Inverse Kinematic and Inverse Transport Problems 81 5.1 Inverse Kinematic Problem.......................... 82 5.1.1 Spherical symmetry.......................... 83 5.1.2 Abel integral and Abel transform.................. 85 5.1.3 Kinematic velocity Inverse Problem................. 86 5.2 Forward transport problem.......................... 88 5.3 Inverse transport problem.......................... 92 5.3.1 Decomposition of the albedo operator and uniqueness result... 92 5.3.2 Stability in inverse transport..................... 96 6 Inverse diffusion 99 6.1 Cauchy Problem and Electrocardiac potential............... 99 6.2 Half Space Problem.............................. 100 6.2.1 The well posed problem....................... 101 6.2.2 The electrocardiac application.................... 102 6.2.3 Prior bounds and stability estimates................ 102 6.2.4 Analytic continuation......................... 104 6.3 General two dimensional case........................ 105 6.3.1 Laplace equation on an annulus................... 105 6.3.2 Riemann mapping theorem...................... 106 6.4 Backward Heat Equation........................... 106 CONTENTS v 7 Calder´onproblem 109 7.1 Introduction.................................. 109 7.2 Uniqueness and Stability........................... 110 7.2.1 Reduction to a Sch¨odinger equation................. 110 7.2.2 Proof of injectivity result....................... 112 7.2.3 Proof of the stability result..................... 113 7.3 Complex Geometric Optics Solutions.................... 115 7.4 The Optical Tomography setting...................... 118 8 Coupled-physics IP I: PAT and TE 121 8.1 Introduction to PAT and TE......................... 122 8.1.1 Modeling of photoacoustic tomography............... 122 8.1.2 First step: Inverse wave source problem............... 124 8.1.3 Second step: Inverse problems with internal functionals...... 125 8.1.4 Reconstruction of one coefficient................... 126 8.1.5 Introduction to Transient Elastography............... 127 8.2 Theory of quantitative PAT and TE..................... 127 8.2.1 Uniqueness and stability results in QPAT.............. 127 8.2.2 Application to Quantitative Transient Elastography........ 134 8.3 Well-chosen illuminations in PAT and TE................. 134 8.3.1 The two dimensional case...................... 134 8.3.2 The n dimensional case........................ 135 9 Coupled-physics IP II: UMT 137 9.1 Ultrasound Modulation Tomography.................... 137 9.2 Inverse problems in ultrasound modulation................. 139 9.3 Eliminations and redundant systems of ODEs............... 142 9.3.1 Elimination of F ........................... 142 9.3.2 System of ODEs for Sj ........................ 144 9.3.3 ODE solution and stability estimates................ 146 9.4 Well-chosen illuminations........................... 146 9.4.1 The case n = 2............................ 146 9.4.2 The case n ≥ 3............................ 146 9.5 Remarks on hybrid inverse problems.................... 148 10 Priors and Regularization 151 10.1 Smoothness Regularization.......................... 152 10.1.1 Ill-posed problems and compact operators............. 152 10.1.2 Regularity assumptions and error bound.............. 153 10.1.3 Regularization methods....................... 156 10.2 Sparsity and other Regularization Priors.................. 163 10.2.1 Smoothness Prior and Minimizations................ 163 10.2.2 Sparsity Prior and Minimizations.................. 164 10.3 Bayesian framework and regularization................... 165 10.3.1 Penalization methods and Bayesian framework........... 166 10.3.2 Computational and psychological costs of the Bayesian framework 167 vi CONTENTS 11 Geometric Priors and Parameterizations 171 11.1 Reconstructing the domain of inclusions.................. 171 11.1.1 Forward Problem........................... 172 11.1.2 Factorization method......................... 173 11.1.3 Reconstruction of Σ......................... 177 11.2 Reconstructing small inclusions....................... 178 11.2.1 First-order effects........................... 179 11.2.2 Stability of the reconstruction.................... 181 12 Inverse Problems and Modeling 183 12.1 Imaging in Highly Heterogeneous Media.................. 184 12.1.1 Wave model.............................. 184 12.1.2 Kinetic Model............................. 185 12.1.3 Statistical Stability.......................... 186 12.1.4 Inverse transport problem...................... 188 12.1.5 Random media and Correlations................... 189 12.1.6 Imaging with waves in random media................ 190 12.2 Random fluctuations and Noise models................... 190 Bibliography 195 Chapter 1 What constitutes an Inverse Problem 1.1 Elements of an Inverse Problem (IP) The definition of an inverse problem starts with that of a mapping between objects of interest, which we call parameters, and acquired information about these objects, which we call data or measurements. The mapping, or forward problem, is called the measurement operator (MO). We denote it by M. The MO maps parameters in a functional space X, typically a Banach or Hilbert space, to the space of data Y, typically another Banach or Hilbert space. We write y = M(x) for x 2 X and y 2 Y; (1.1) the correspondence between the parameter x and the data y. Solving the inverse problem amounts to finding point(s) x 2 X from knowledge of the data y 2 Y such that (1.1) or an approximation of (1.1) holds. The MO describes our best effort to construct a model for the available data y, which we assume here depend only on the sought parameters x. The choice of X describes our best effort to characterize the space where we believe the parameters belong. 1.1.1 Injectivity and stability of the Measurement Operator The first question to ask about the MO is whether we have acquired enough data to uniquely reconstruct the parameters. In other words, whether the MO is injective. Injectivity means that M(x1) = M(x2) =) x1 = x2 for all x1; x2 2 X: (1.2) Then the data y, if given in the range of M, uniquely characterize
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