Bayesian Inference Tools for Inverse Problems

2012

In this paper, first the basics of the Bayesian inference for linear inverse problems are presented. The inverse problems we consider are, for example, signal deconvolution, image restoration or image reconstruction in Computed Tomography (CT). The main point to discuss then is the prior modeling of signals and images. We consider two classes of priors: simple or hierarchical with hidden variables. For practical applications, we need also to consider the estimation of the hyper parameters. Finally, we see that we have to infer simultaneously the unknowns, the hidden variables and the hyper parameters. Very often, the expression of the joint posterior law of all the unknowns is too complex to be handled directly. Indeed, rarely we can obtain analytical solutions to any point estimators such the Maximum A posteriori (MAP) or Posterior Mean (PM). Three main tools can then be used: Laplace approximation (LAP), Markov Chain Monte Carlo (MCMC) and Bayesian Variational Approximations (BVA). To illustrate all these aspects, we will consider a deconvolution problem where we know that the input signal is sparse and propose to use a Student-t distribution for that. Then, to handle the Bayesian computations with this model, we use the property of Student-t which is modelling it via an infinite mixture of Gaussians, introducing thus hidden variables which are the variances. Then, the expression of the joint posterior of the input signal samples, the hidden variables (which are here the inverse variances of those samples) and the hyper-parameters of the problem (for example the variance of the noise) is given. From this point, we will present the joint maximization by alternate optimization and the three possible approximation methods. Finally, the proposed methodology is applied in different applications such as mass spectrometry, spectrum estimation of quasi periodic biological signals and X ray computed tomography.

Proceedings of 3rd IEEE International Conference on Image Processing, 1996

ABSTRACT In this paper we propose a joint estimation of the parame-ters and hyperparameters (the parameters of the prior law) when a Bayesian approach with Maximum Entropy (ME) priors is used to solve the inverse problems which arise in signal and image reconstruction and ...

2002

Traditionally, the MaxEnt workshops start by a tutorial day. This paper summarizes my talk during 2001'th workshop at John Hopkins University. The main idea in this talk is to show how the Bayesian inference can naturally give us all the necessary tools we need to solve real inverse problems: starting by simple inversion where we assume to know exactly the forward model and all the input model parameters up to more realistic advanced problems of myopic or blind inversion where we may be uncertain about the forward model and we may have noisy data.

Bayesian approach has become a commonly used method for inverse prob- lems arising in signal and image processing. One of the main advantages of the Bayesian approach is the possibility to propose unsupervised methods where the likelihood and prior model parameters can be estimated jointly with the main unknowns. In this paper, we propose to consider linear in- verse problems in which the noise may be non stationary and where we are looking for a sparse solution. To consider both of these requirements, we propose to use Student-t prior model both for the noise of the forward model and the unknown signal or image. The main interest of the Student-t prior model is its Infinite Gaussian Scale Mixture (IGSM) property. Using the resulted hierarchical prior models we obtain a joint posterior probability distribution of the unknowns of interest (input signal or image) and their associated hidden variables. To be able to propose practical methods, we use either a Joint Maximum A Posteriori (JMAP) estimator or an appropriate Variational Bayesian Approximation (VBA) technique to compute the Pos- terior Mean (PM) values. The proposed method is applied in many inverse problems such as deconvolution, image restoration and computed tomogra- phy. In this paper, we show only some results in signal deconvolution and in periodic components determination of some biological signals related to dynamic circadian clock period determination for cancer studies.

2013 8th International Workshop on Systems, Signal Processing and their Applications (WoSSPA), 2013

Bayesian approach is nowadays commonly used for inverse problems. Simple prior laws (Gaussian, Generalized Gaussian, Gauss-Markov and more general Markovian priors) are common in modeling and in their use in Bayesian inference methods. But, we need still more appropriate prior models which can account for non stationnarities in signals and for the presence of the contours and homogeneous regions in images. Recently, we proposed a family of hierarchical prior models, called Gauss-Markov-Potts, which seems to be more appropriate for many applications in Imaging systems such as X ray Computed Tomography (CT) or Microwave imaging in Non Destructive Testing (NDT). In this tutorial paper, first some bacgrounds on the Bayesian inference and the tools for assignment of priors and doing efficiently the Bayesian computation is presented. Then, more specifically hiearachical models and particularly the Gauss-Markov-Potts family of prior models are presented. Finally, their real applications in image restoration, in different practical Computed Tomography (CT) or other imaging systems are presented.