Mumford-Shah Models for Tomography

Within the last three decades, mathematical and algorithmic machinery has been applied most successfully to the problem of reconstructing the density distributions within a body from radiation data. The now classical algorithms provide an extremely fast and reliable way to solve the problem of inversion of tomography data. On the other hand, the development of new scanning devices and the increasing demand for quantitative and functional information rather that mere schematic information calls for the development of new mathematical inversion strategies which are capable of treading new input-output relations, new classes of data sets and provide new types of output information. The general aim of the proposed project is the development of reconstruction strategies for tomography-type problems which allow for the specification of the type of information which should be extracted from the data (geometric or functional) and which are flexible enough to be easily adapted to different data acquisition devices (CT, SPECT). In this project we propose the complementary use of geometric and functional variables for the reconstruction. Geometrical variables are capable of describing the shapes of objects, the location of cracks or the course of fibers, whereas functional variables describe physical quantities at given points in space. We will investigate different Mumford - Shah type models that allow a simultaneous reconstruction of a piecewise smooth functional variable and the belonging singularity set (the shape of the object). This requires an optimization of the functional with respect to both types of variables. We aim to combine shape sensitivity analysis, the level-set machinery and ideas from nonlinear programming to design new algorithms for the reconstruction of the minimizers. As a consequence, the functional and geometric information is extracted directly from the data.

Within the last three decades, mathematical and algorithmic machinery has been applied most successfully to the problem of reconstructing the density distributions within a body from radiation data. The now classical algorithms provide an extremely fast and reliable way to solve the problem of inversion of tomography data. On the other hand, the development of new scanning devices and the increasing demand for quantitative and functional information rather that mere schematic information calls for the development of new mathematical inversion strategies which are capable of treading new input-output relations, new classes of data sets and provide new types of output information. The general aim of the proposed project is the development of reconstruction strategies for tomography-type problems which allow for the specification of the type of information which should be extracted from the data (geometric or functional) and which are flexible enough to be easily adapted to different data acquisition devices (CT, SPECT). In this project we propose the complementary use of geometric and functional variables for the reconstruction. Geometrical variables are capable of describing the shapes of objects, the location of cracks or the course of fibers, whereas functional variables describe physical quantities at given points in space. We will investigate different Mumford - Shah type models that allow a simultaneous reconstruction of a piecewise smooth functional variable and the belonging singularity set (the shape of the object). This requires an optimization of the functional with respect to both types of variables. We aim to combine shape sensitivity analysis, the level-set machinery and ideas from nonlinear programming to design new algorithms for the reconstruction of the minimizers. As a consequence, the functional and geometric information is extracted directly from the data.