Inverse Problems in Imaging and Computer Vision - From Regularization Theory to Bayesian Inference

The concept of inverse problems is now a familiar one to most scientists and engineers, particularly in the field of imaging systems and computer vision. Inverse problems arise whenever we want to infer an unknown quantity f(r) which is not directly observable through a measurement system H which gives access to an observable quantity g(s). The mathematical equations linking these two quantities g(s)=H[f(r)](s) are called “forward model”. The problem of predicting g(s) when f(r) is assumed to be known is called “forward problem” and the one of inferring f(r) from the observation of g(s) is called “inverse problem”. Very often forward problems are “well-posed” and almost always the inverse problems are “ill-posed” [HAD 1901]. The classical mathematical tools for such inverse problems are either the deterministic regularization theory [TIK 1963, TIK 1976] or the probabilistic Bayesian inference and estimation theory [HAN 1983, TAR 1982].