EC Bayes Seminar: Francesca Crucinio, University of Warwick – A particle method for Fredholm Integral Equations of the First Kind
We present a novel method for the solution of Fredholm integral equations of the first kind, a set of ill-posed inverse problems which model, among others, reconstruction of images from distorted noisy observations and indirect density estimation. This novel method is based upon a non-standard sequential Monte Carlo (SMC) algorithm which provides a stochastic discretisation of a smoothed expectation maximisation scheme (EMS) usually implemented under the assumption of piecewise constant solutions. The stochastic discretisation provided by SMC does not assume piecewise constant signals and results in smooth approximate solutions. We analyse the theoretical properties of the EMS iteration, showing existence of a fixed point, and of the corresponding SMC algorithms. We compare the novel method with alternatives using a simulation study and present results for realistic systems, including motion deblurring and reconstruction of cross-section images of the brain from positron emission tomography.