Inverse problems, Ill-posedness and regularization - an illustrative example

Whenever one is confronted with the necessity to measure some quantities, which are not accessible directly, however, are linked via a mathematical model to some measurement data, one has to solve an inverse problem. In this context we speak of a direct problem, when expected measurement data are calculated from a mathematical model, when the not directly accessible quantities are given and, on the other hand, of an inverse problem, when these quantities are calculated from measured data via the mathematical model. In this paper the principles of inverse problems are explained on the basis of a one-dimensional image restoration problem, which is a linear problem and hence is easy to understand. Furthermore, the term ill-posedness will be explained and some possibilities to attain reasonable solutions to ill-posed problems are discussed.