Fredholm integral equations of the first kind describe a wide set of problems in science (e.g. image processing for motion deblurring and positron emission tomography, inverse boundary problems) in which a signal f has to be recovered from an observed distorted signal when the type of distortion is known.

A popular method to approximate f is an infinite dimensional Expectation-Maximization (EM) algorithm that, given an initial guess, iteratively refines the approximation by including the information given by the distorted signal and the distortion process. We use Sequential Monte Carlo (SMC) to develop a stochastic discretisation of the Expectation-Maximization-Smoothing (EMS) algorithm a regularised variant of EM. We show that the approximations given by the resulting SMC algorithms converge to the solution of the integral equation in the weak topology and we compare the novel method with alternatives using a simulation study and present results for realistic systems.