Idier J. Bayesian Approach to Inverse Problems

N.-Y.: Wiley-ISTE, 2008. - 381p.Many scientific, medical or engineering problems raise the issue of recovering some physical quantities from indirect measurements; for instance, detecting or quantifying flaws or cracks within a material from acoustic or electromagnetic measurements at its surface is an essential problem of non-destructive evaluation. The concept of inverse problems precisely originates from the idea of inverting the laws of physics to recover a quantity of interest from measurable data.
Unfortunately, most inverse problems are ill-posed, which means that precise and stable solutions are not easy to devise. Regularization is the key concept to solve inverse problems.
The goal of this book is to deal with inverse problems and regularized solutions using the Bayesian statistical tools, with a particular view to signal and image estimation.
The first three chapters bring the theoretical notions that make it possible to cast inverse problems within a mathematical framework. The next three chapters address the fundamental inverse problem of deconvolution in a comprehensive manner. Chapters 7 and 8 deal with advanced statistical questions linked to image estimation. In the last five chapters, the main tools introduced in the previous chapters are put into a practical context in important applicative areas, such as astronomy or medical imaging.Fundamental Problems and Tools.Basic example.
Ill-posed problem.
Case of discrete data.
Continuous case.
Generalized inversion.
Example.
Discretization and conditioning.Regularization.
Dimensionality control.
Minimization of a composite criterion.
Criterion minimization for inversion.
The quadratic case.
The convex case.
General case.
L-curve method.
Cross-validation.Inversion and inference.
Statistical inference.
Noise law and direct distribution for data.
Maximum likelihood estimation.
Bayesian approach to inversion.
Links with deterministic methods.
Choice of hyperparameters.
A priori model.
Choice of criteria.
Statistical properties of the solution.
Calculation of marginal likelihood.
Wiener filtering.Deconvolution.Inverse filtering.
Wiener filtering.
Choice of a quadrature method.
Structure of observation matrix H.
Problem conditioning.
Generalized inversion.
Preliminary choices.
Matrix form of the estimate.
Hunt’s method (periodic boundary hypothesis).
Exact inversion methods in the stationary case.
Results and discussion on examples.
Kalman filtering.
Degenerate state model and recursive least squares.
Autoregressive state model.
Fast Kalman filtering.
Asymptotic techniques in the stationary case.
Case of non-stationary signals.Penalization of reflectivities, L2LP/L2Hy deconvolutions.
Quadratic regularization.
L2LP or L2Hy deconvolution.
Non-quadratic regularization.
Various strategies for estimation.
General expression for marginal likelihood.
An iterative method for BG deconvolution.
Other methods.
Nature of the solutions.
Setting the parameters.
Extensions.
Estimation of the impulse response.Principle.
Connection with image processing by linear PDE.
Limits of Tikhonov’s approach.
Principle.
Disadvantages.
Non-quadratic approach.
Detection-estimation and non-convex penalization.
Anisotropic diffusion by PDE.
Half-quadratic augmented criteria.
Duality between non-quadratic criteria and HQ criteria.
Minimization of HQ criteria.
Calculation of the solution.
Example.Advanced Problems and Tools.Bayesian statistical framework.
Gibbs-Markov fields.
Gibbs fields.
Gibbs-Markov equivalence.
Posterior law of a GMRF.
Gibbs-Markov models for images.
Statistical tools.
Stochastic sampling.Introduction and statement of problem.
Likelihood properties.
Optimization.
Approximations.
Statement of problem.
EM algorithm.
Application to estimation of the parameters of a GMRF.
EM algorithm and gradient.
Linear GMRF relative to hyperparameters.
Extensions and approximations.Some Applications.Evaluation principle.
Evaluation results and interpretation.
Help with interpretation by restoration of discontinuities.
Definition of direct convolution model.
Overview of approaches for blind deconvolution.
DL2Hy/DBG deconvolution.
Processing real data.
Processing by blind deconvolution.
Deconvolution with a measured wave.
Comparison between DL2Hy and DBG.Image formation.
Effect of turbulence on image formation.
Imaging techniques.
Inversion approach and regularization criteria used.Hartmann-Shack sensor.
Phase retrieval and phase diversity.
Motivation and noise statistic.
Data processing in deconvolution from wavefront se.
Restoration of images corrected by adaptive optics.Observation model.
Traditional Bayesian approach.
Myopic modeling.Velocity measurement in medical imaging.
Information carried by Doppler signals.
Data and problems treated.
Least squares and traditional extensions.
Long AR models – spectral smoothness – spatial continuity.
Kalman smoothing.
Estimation of hyperparameters.
Processing results and comparisons.
Tracking spectral moments.
Proposed method.
Likelihood of the hyperparameters.
Processing results and comparisons.Projection generation model.
2D analytical methods.
Limitations of analytical methods.
Discrete approach to reconstruction.
Choice of criterion and reconstruction methods.
Optimization algorithms for convex criteria.
Optimization or integration algorithms.
2D reconstruction.
3D reconstruction.
Conclusions.Examples of diffraction tomography applications.
Modeling the direct problem.
Choice of algebraic framework.
Method of moments.
Discretization by the method of moments.
Construction of criteria for solving the inverse problem.
First formulation: estimation of x.
Second formulation: simultaneous estimation of x and φ.
Solving the inverse problem.
Successive linearizations.
Joint minimization.
Minimizing MAP criterion.Likelihood functions and limiting behavior.
Purely Poisson measurements.
Compound noise models with Poisson information.
Maximum likelihood properties.
Bayesian estimation.
Implementation for pure Poisson model.
Bayesian implementation for a compound data model.List of Authors.