Diffusion Process Interpolation and Applications in Image Restoration

This project aims to develop new stochastic interpolation techniques for the design of image reconstruction algorithms. In particular, the research will use stochastic bridges having arbitrary marginal distributions constructed via forward/backward stochastic calculus to exploit both the past and future information present in a signal.The notion of stochastic bridges is by now a classic subject in one dimension; however, the researchers will need to extend it to consider the presence of obstacles hiding a signal, that is, they will need to construct bridges conditioned to stay inside a given region of space. To be able to deal with two-dimensional images, they will work on the construction of 2-parameter stochastic bridges given the data of marginal probability laws on the boundary of a domain in the plane.The researchers also plan to develop new stochastic analysis techniques for the validation of the statistical estimators obtained in this way, in particular in the framework of maximum likelihood estimation.