Non-Smooth Variational Models with Second Order and Local Anisotropy Priors for Restoring Cyclic and Manifold-Valued Images

成像中的变分方法目前正朝着一种相当通用和灵活的工具发展，使各种成像任务的方法非常成功。许多有用的技术依赖于非光滑的凸函数。正则化函数中的一阶和二阶导数的组合或由图像的局部结构控制的各向异性的结合导致了非常强大的图像恢复技术。分离算法和原始对偶优化方法是最小化这些函数的最先进的技术。它们的优势在于将原始问题分解为一系列可以有效计算的近端映射。在图像处理和计算机视觉的各种应用中，感兴趣的函数在圆或流形上取值。尽管流形长期以来在这些领域中发挥着重要作用，但将近年来广泛应用于实值图像处理的非光滑优化结果与流形值设置相结合的论文很少。这为未来的研究留下了很大的潜力。在我们的项目中，我们想推广凸模型用于恢复实值图像到循环和流形值图像。我们想关注对称空间在图像处理中的应用。对于Hadamard空间，模型仍然是凸的，这在一般情况下是凸的，例如，对于球体，情况并非如此。我们模型的一个具体特征是它们的正则化项将包含一阶和二阶差异或方向各向异性。我们项目的挑战包括为流形值信号和图像构建适当的恢复模型，分析模型，以及开发有效的最小化算法，包括收敛结果。在我们的项目中开发的方法有很大的应用潜力。除此之外，我们将使用我们的模型来分析脑电图数据和电子背散射衍射数据。还计划了一个公开可用的软件包。

Variational methods in imaging are nowadays developing towards a quite universal and flexible tool, allowing for highly successful approaches on various imaging tasks. Many useful techniques rely on non-smooth, convex functionals. Combinations of first and second order derivatives in regularization functionals or the incorporation of anisotropies steered by the local structures of the image have led to very powerful image restoration techniques. Splitting algorithms together with primal-dual optimization methods are the state-of-the-art techniques for minimizing these functionals. Their strength consists in the splitting of the original problem into a sequence of proximal mappings which can be computed efficiently.In various applications in image processing and computer vision the functions of interest take values on the circle or in manifolds. Although manifolds play an important role in these fields for a long time, there are only few papers which combine results on non-smooth optimization which were recently extensively exploited in real-valued image processing with manifold-valued settings. This leaves high potential for future research.In our project we want to generalize convex models for the restoration of real-valued images to cyclic and manifold-valued images. We want to focus on symmetric spaces having applications in image processing. For Hadamard spaces the models are still convex which is in general, e.g., for spheres, not the case. A specific feature of our models is that their regularization terms will incorporate first and second order differences or directional anisotropies. The challenges of our project include the appropriate construction of restoration models for manifold-valued signals and images, the analysis of the models, and the development of efficient minimization algorithms, including convergence results. There is a rich potential for applications of the methods which will be developed within our project. Among others we will use our models for the analysis of Electroencephalographical data and of Electron Backscattered Diffraction data. A publicly available software package is planed as well.