Fast algorithms for free-discontinuity problems on high-dimensional biomedical data

• Applicants: Professor Dr. Martin Storath; Professor Dr. Andreas Weinmann
• Subject Area: Mathematics
• Term: from 2016 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 318064553

Functions with discontinuities are ubiquitous in our everyday life and in almost all types of biomedical data. The discontinuities encode significant information: for instance, they represent the boundaries of cellular structures in microscopic images, they correspond to change points in microarray data, and they define tissue layers in tomographic images. Since classical methods destroy this important information, discontinuity preserving models such as the Mumford-Shah model have been developed. Such free-discontinuity problems are algorithmically challenging as they lead to nonsmooth and nonconvex problems. Even for low-dimensional data, the currently used algorithms are computationally demanding. Since the dimensionality of the acquired data increases tremendously, there is urgent need for new algorithms that scale reasonably with the high-dimensionality in terms of trade-off between complexity and accuracy. In this project we aim at developing new efficient algorithms for free-discontinuity problems for high-dimensional biomedical data. We deal with high-dimensional linear data spaces (magnetic particle imaging, feature images) and manifold-valued data spaces (diffusion tensor imaging, shape spaces) that might be defined on higher dimensional lattices. On the one hand, we evaluate the developed methods on real life data from biomedical problems, and on the other hand, we provide mathematical analysis and foundation. Extending our present software toolbox, we will make the algorithms publicly available to help practitioners to process their data.

• DFG Programme: Research Grants