From image restoration to image understanding

Inverse problems are ubiquitous in image processing and applied science in general. Such problems describe the challenge of computing the parameters that characterize a system from the outcomes. While this might seem easy at first for simple systems, many inverse problems share a property that makes them much more intricate: they are ill-posed. This means that either the problem does not have a unique solution or this solution does not depend continuously on the outcomes of the system. Bayesian statistics provides a framework that allows to treat such problems in a systematic way. The missing piece of information is encoded as a prior distribution on the space of possible solutions. In this talk, we will study probabilistic image models as priors for statistical inversion. In particular, we will give a probabilistic interpretation of the classical TV-prior and discuss how this interpretation can be used as a starting point for more complex models. We will see that many important auxiliary quantities such as edges and regions can be incorporated into the model in the form of latent variables. This leads to the conjecture that many image processing tasks, such as denoising and segmentation, should not be considered separately, but instead be treated together.