Nonlinear inverse problems in Image Processing and image processing methods for Inverse Problems.

In this proposal we establish a broad coherency between methods developed independently in the mathematical theory of Inverse Problems and in Signal-and Image Processing (especially Computer Vision) which enables us to apply the theory and methodology developed in the field of Image Processing to Inverse Problems and vice versa. Therefore we can stimulate both mathematical fields with new mathematical techniques, which will lead to further developments. In the past twenty 25 years the theory of linear Inverse Problems has significantly contributed to the mathematical theory of Signal-and Image Processing. Since for a long time there has not been a general setting for the solution of nonlinear Inverse Problems and for the solution of nonlinear problems in the mathematical theory of Signal-and Image Processing these two mathematical areas have developed rather independently. In this project we study the synthesis of the two mathematical areas. The synthesis is important for at least two reasons. A general theory provides a deeper understanding. Second there exist several methods which have not been applied in the respective other area. Influence of the proposed work on the development of the field The coherencies between Signal-and Image Processing and Inverse Problems established in this proposal enables us to apply and further to develop new mathematical technologies in the areas of lnverse Problems and Signal-and Image Processing.

In this proposal we establish a broad coherency between methods developed independently in the mathematical theory of Inverse Problems and in Signal-and Image Processing (especially Computer Vision) which enables us to apply the theory and methodology developed in the field of Image Processing to Inverse Problems and vice versa. Therefore we can stimulate both mathematical fields with new mathematical techniques, which will lead to further developments. In the past twenty 25 years the theory of linear Inverse Problems has significantly contributed to the mathematical theory of Signal-and Image Processing. Since for a long time there has not been a general setting for the solution of nonlinear Inverse Problems and for the solution of nonlinear problems in the mathematical theory of Signal-and Image Processing these two mathematical areas have developed rather independently. In this project we study the synthesis of the two mathematical areas. The synthesis is important for at least two reasons. A general theory provides a deeper understanding. Second there exist several methods which have not been applied in the respective other area. Influence of the proposed work on the development of the field. The coherencies between Signal-and Image Processing and Inverse Problems established in this proposal enables us to apply and further to develop new mathematical technologies in the areas of lnverse Problems and Signal-and Image Processing.