The Bayesian Approach to Solving Inverse Ill-Posed Problems

RUBEN D. SPIES

Rosario

Congreso; VII Congreso Italo-Latinoamericano de Matemática Aplicada e Industrial; 2012

Universidad Austral

Inverse ill-posed problems appear in a wide variety of areas of Science and Technology and with them comes the necessity of ?regularization? with the objective of extracting the largest possible amount of information about the exact solution maintaining the stability of the inversion process. Although there are many options for choosing a regularization method, for a number of reasons the Bayesian tools are showing an ever increasing appeal. One of the reasons for this growing popularity is the flexibility that these methods offer when there is available information of qualitative type about the exact solution. In image processing, this is similar to the process naturally followed by the human eye-brain system. It is precisely the possibility of improving the contribution provided by low quality data with information that can be based on our own preconception of what we are observing or in information comingfrom sources external to the data, what makes the statistical regularization methods particularly attractive in certain applications, mainly in signal and image restoration problems. In this talk we will provide a brief introduction to the Bayesian approach to solving inverse ill-posed problems. We will show how information of qualitative or structural nature can be appropriately taken into account in different hierarchical levels, giving rise to what are known as ?hierarchical models? or ?hypermodels?. Relations between classical and statistical methods as well as recent advances in the area will be shown. Several examples in signal and image processing will be presented.