62 research outputs found
Discretization and Bayesian modeling in inverse problems and imaging
In this thesis the Bayesian modeling and discretization are studied in inverse problems related to imaging. The treatise consists of four articles which focus on the phenomena that appear when more detailed data or a priori information become available. Novel Bayesian methods for solving ill-posed signal processing problems in edge-preserving manner are introduced and analysed. Furthermore, modeling photographs in image processing problems is studied and a novel model is presented
Inverse scattering for a random potential
In this paper we consider an inverse problem for the -dimensional random
Schr\"{o}dinger equation . We study the scattering of
plane waves in the presence of a potential which is assumed to be a
Gaussian random function such that its covariance is described by a
pseudodifferential operator. Our main result is as follows: given the
backscattered far field, obtained from a single realization of the random
potential , we uniquely determine the principal symbol of the covariance
operator of . Especially, for this result is obtained for the full
non-linear inverse backscattering problem. Finally, we present a physical
scaling regime where the method is of practical importance.Comment: Previous version 48 pages; Current version 51 pages, 3 figures,
several references have been adde
Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency
The Bayesian formulation of inverse problems is attractive for three primary
reasons: it provides a clear modelling framework; means for uncertainty
quantification; and it allows for principled learning of hyperparameters. The
posterior distribution may be explored by sampling methods, but for many
problems it is computationally infeasible to do so. In this situation maximum a
posteriori (MAP) estimators are often sought. Whilst these are relatively cheap
to compute, and have an attractive variational formulation, a key drawback is
their lack of invariance under change of parameterization. This is a
particularly significant issue when hierarchical priors are employed to learn
hyperparameters. In this paper we study the effect of the choice of
parameterization on MAP estimators when a conditionally Gaussian hierarchical
prior distribution is employed. Specifically we consider the centred
parameterization, the natural parameterization in which the unknown state is
solved for directly, and the noncentred parameterization, which works with a
whitened Gaussian as the unknown state variable, and arises when considering
dimension-robust MCMC algorithms; MAP estimation is well-defined in the
nonparametric setting only for the noncentred parameterization. However, we
show that MAP estimates based on the noncentred parameterization are not
consistent as estimators of hyperparameters; conversely, we show that limits of
finite-dimensional centred MAP estimators are consistent as the dimension tends
to infinity. We also consider empirical Bayesian hyperparameter estimation,
show consistency of these estimates, and demonstrate that they are more robust
with respect to noise than centred MAP estimates. An underpinning concept
throughout is that hyperparameters may only be recovered up to measure
equivalence, a well-known phenomenon in the context of the Ornstein-Uhlenbeck
process.Comment: 36 pages, 8 figure
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