Bayesian Optimal Experimental Design in Imaging

An inverse problem is defined as a problem that violates one of the classical criteria of a well posed problem: a solution exists, is unique, and is continuous with respect to the data in some reasonable topology. A problem that is not well posed is called illposed, and the development of tools to tackle illposed problems is the goal of the field of inverse problems research. In imaging, illposedness is often an inevitable consequence of the high dimension of the unknown, compared with the measurement data. In an imaging problem, one aims to reconstruct the spatial two- or three-dimensional structure of an object of interest, leading to unknown parameters in the hundreds of thousands or beyond, while the dimension of the measurement data is determined by the number of sensors, and thus limited by physical constraints to values often at least an order of magnitude lower. Another consequence of the high dimensionality of the problem is the computational cost involved in the computations. In imaging problems, there is also usually a cost involved in acquiring data, and thus one would naturally want to minimize the amount of data collection required. One tool for this is optimal experimental design, where one aims to perform the experiment in a way as to maximize the value of the data obtained. The challenge of this however, is that the search for this optimal design usually leads to a computationally challenging problem, whose size is dependent on the dimension of both the data and the unknown. Overcoming this difficulty is the main objective of this thesis. The problem can be tackled by using Gaussian approximations in the formulation of the imaging problem, which leads to practical solution formulas for the quantities of interest. In this thesis, tools are developed to enable the efficient computation of expected utilities for certain measurement designs, particularily in sequential imaging problems and for non-Gaussian prior models. Additionally, these tools are applied to medical imaging and astronomy.