A Calculus for Non-Smooth Shape Optimization with Applications to Geometric Inverse Problems

该项目的总体目标是对非光滑形状优化问题的理论和数值进行严格的数学处理。目标函数具有几何性质，即，它们强调最佳形状的某些所需特性。在这个过程中，我们将研究不同类别的泛函，它们分别适用于表面平滑或几何分割任务。这些泛函的非光滑性，最终都是基于表面的法向量场，是它们在每种情况下的重要功能。上述考虑的主要动机是所谓的几何逆问题，其中一个未知的几何是从数据中重建。在非侵入式感测领域（例如用于检测夹杂物）以及医学成像中可以找到用于此的许多应用示例。我们正在研究的新的几何泛函允许对要识别的几何形状的预期或期望的属性进行详细的控制。在项目的第一阶段，我们主要考虑了法向量场的总表面变化，它具有边缘保持的性质。在第二阶段，另一方面，我们首先考虑可用于几何分割的泛函，即根据某些特征对表面进行分类。例如，这些泛函可以帮助表达对表面片段的特定取向的偏好。这使得有可能带来结晶学，地质学和材料科学领域的专业知识，例如。此外，我们认为泛函的基础上的广义，二阶全变差的表面法线。作为应用实例，在每种情况下，电阻抗断层成像（EIT）作为一个经典的成像方式的问题是结合到一个几何逆问题的新的几何泛函。在与理论性质的研究相同的水平上，总是一种有效和鲁棒的数值实现。为了实现这一点，我们将开发一个ADMM方法，这将不得不纳入微分几何的工具，由于表面法线的内在属性。

The overall goal of the project is a mathematically rigorous approach to the theory and numerics of non-smooth shape optimization problems. The objective functions are of a geometric nature, i.e., they emphasize certain desired properties of optimal shapes. In the process, we will examine different classes of functionals, which are, as they stand, suitable for surface smoothing or geometry segmentation tasks, respectively. The non-smoothness of these functionals, which ultimately are all based on the normal vector field of the surface, is a crucial feature in each case for their functioning.The main motivation for the above considerations are so-called geometric inverse problems, in which an unknown geometry is to be reconstructed from data. Numerous examples of applications for this can be found in the field of non-invasive sensing, for example for the detection of inclusions, but also in medical imaging. The novel geometric functionals we are examining allow a detailed control over expected or desired properties of the geometries to be identified. In the first phase of the project, we mainly considered the total surface variation of the normal vector field in this context, which has the property of being edge preserving.In the second phase, on the other hand, we first look at functionals that can be used for geometry segmentation, that is, the classification of a surface according to certain features. These functionals can, for instance, help to express a preference for specific orientations of the surface segments. This makes it possible to bring in expert knowledge in the field of crystallography, geology and material science, for example. Furthermore, we consider functionals based on the generalized, second-order total variation of the surface normals. Thereby, a preference for certain curvature properties of the surface can be expressed.As application examples, in each case problems of electrical impedance tomography (EIT) as a classical imaging modality are to be combined with the new geometric functionals into a geometric inverse problem. On an equal level with the investigation of the theoretical properties is always an efficient and robust numerical realization. To achieve this, we will develop an ADMM method, which will have to incorporate tools of differential geometry due to the intrinsic properties of the surface normal.