Bayesian inference with hierarchical prior models for inverse problems in imaging systems

Abstract

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-MarkovPotts, 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.

Bayesian Inference with Error Variable Splitting and Sparsity Enforcing Priors for Linear Inverse Problems

“…which we can use to infer on f , v and v f . For the expressions of the updating these tilded variables and more details see [30], [4], [31]. Figure 1 shows the generative graphical representations of the supervised and unsupervised models.…”

“…Finally, the third advantage is that we can do better than JMAP by using the Variational Bayesian Approximation methods and for example approximate the joint posterior law p(f , g 0 , v , v ξ , v f |g) by the following separable one (30) using the Kullback-Leibler divergence KL (q : p) = q ln q p (31) which gives rise to the VBA algorithms [32], [33], [4], [34], [35], [36], [37]. The following figure shows the graphical representations of the new forward model.…”

“…For this simple model, nowadays, almost everything has been told, starting by deterministic methods: Least Squares (LS), then Quadratic Regularization (QR), L 1 regularization, Total Variation and all associated optimization algorithms such as Augmented Lagrangian (AL), ADMM, ISTA, FISTA, etc. [1], [2], But, also the probabilistic methods and in particular the Bayesian approach with simple Gaussian models for the noise and Gaussian prior model, Double Exponential prior, Student-t prior [3] to much more sophisticated Hierarchical models [4], [5], [6]. However, in many real applications, it is needed to propose forward models which can account for other sources of uncertainties.…”

Computed tomography reconstruction based on a hierarchical model and variational Bayesian method

“…The details of this minimization are discussed in [17]. Thanks to the conjugacy property of the prior model, we obtain the conjugate distributions for z and the parameters.…”

Bayesian 3D X-ray computed tomography image reconstruction with a scaled Gaussian mixture prior model

“…where the z j is a hidden variable which represents the inverse variance of f j [4,1]. This property of Infinite Gaussian Mixture gives the possibility to propose the following hierarchical model:…”