Tapio Helin

Hierarchical models in statistical inverse problems and the Mumford?Shah functional

The Bayesian methods for linear inverse problems are studied using hierarchical Gaussian models. The problems are considered with different discretizations, and we analyse the phenomena which appear when the discretization becomes finer. A hierarchical solution method for signal restoration problems is introduced and studied with arbitrarily fine discretization. We show that the maximum a posteriori estimate converges to a minimizer of the Mumford–Shah functional, up to a subsequence. A new result regarding the existence of a minimizer of the Mumford–Shah functional is proved. Moreover, we study the inverse problem under different assumptions on the asymptotic behaviour of the noise as discretization becomes finer. We show that the maximum a posteriori and conditional mean estimates converge under different conditions. This paper concentrates on the results regarding the maximum a posteriori estimate. The convergence results of the conditional mean estimate are proven in Helin (2009 Inverse Problems Imaging 3 4).

Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems

We consider the inverse problem of recovering an unknown functional parameter

Atmospheric turbulence profiling with unknown power spectral density

u

A variational reconstruction method for undersampled dynamic x-ray tomography based on physical motion models

in a separable Banach space, from a noisy observation

Limits of turbulence and outer scale profiling with non-Kolmogorov statistics

y

Infinite Photography: New Mathematical Model for High-Resolution Images

of its image through a known possibly non-linear ill-posed map

On the Stability of MAP Estimation with Hierarchical Prior Distributions

{\mathcal G}

An inverse problem for the wave equation with one measurement and the pseudorandom source

. The data

On infinite-dimensional hierarchical probability models in statistical inverse problems

y

Inverse acoustic scattering problem in half-space with anisotropic random impedance

is finite-dimensional and the noise is Gaussian. We adopt a Bayesian approach to the problem and consider Besov space priors (see Lassas et al. 2009), which are well-known for their edge-preserving and sparsity-promoting properties and have recently attracted wide attention especially in the medical imaging community. Our key result is to show that in this non-parametric setup the maximum a posteriori (MAP) estimates are characterized by the minimizers of a generalized Onsager--Machlup functional of the posterior. This is done independently for the so-called weak and strong MAP estimates, which as we show coincide in our context. In addition, we prove a form of weak consistency for the MAP estimators in the infinitely informative data limit. Our results are remarkable for two reasons: first, the prior distribution is non-Gaussian and does not meet the smoothness conditions required in previous research on non-parametric MAP estimates. Second, the result analytically justifies existing uses of the MAP estimate in finite but high dimensional discretizations of Bayesian inverse problems with the considered Besov priors.