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

具有不连续的函数在我们的日常生活和几乎所有类型的生物医学数据中无处不在。不连续性编码了重要的信息：例如，它们在显微镜图像中代表细胞结构的边界，它们对应于微阵列数据中的变化点，它们在断层扫描图像中定义组织层。由于经典方法破坏了这一重要信息，因此开发了诸如Mumford-Shah模型之类的保留不连续的模型。这种自由不连续问题在算法上具有挑战性，因为它们会导致非光滑和非凸问题。即使对于低维数据，目前使用的算法对计算量的要求也很高。由于采集数据的维数急剧增加，迫切需要新的算法在复杂度和精度之间进行权衡，以适应高维数。在这个项目中，我们的目标是为高维生物医学数据的自由不连续问题开发新的有效算法。我们处理高维线性数据空间（磁粒子成像，特征图像）和流形值数据空间（扩散张量成像，形状空间），这些数据空间可能在高维晶格上定义。一方面，我们根据生物医学问题的实际数据对开发的方法进行评估，另一方面，我们提供数学分析和基础。扩展我们现有的软件工具箱，我们将使算法公开可用，以帮助从业者处理他们的数据。

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.