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

如今，成像中的变分方法正朝着一种相当通用和灵活的工具发展，允许在各种成像任务中使用非常成功的方法。许多有用的技巧都依赖于非光滑的凸泛函。一阶和二阶导数在正则化泛函中的组合或由图像的局部结构引导的各向异性的结合导致了非常强大的图像恢复技术。分裂算法和原始-对偶优化方法是最小化这些泛函的最新技术。它们的优点在于将原始问题分解成一系列可以有效计算的近似映射。在图像处理和计算机视觉中的各种应用中，感兴趣的函数取值于圆或流形。虽然流形在这些领域中扮演着重要的角色很长一段时间，但将非光滑优化的结果结合在一起的论文很少，这些结果最近在流形环境下的实值图像处理中得到了广泛的应用。这为未来的研究留下了很大的潜力。在我们的项目中，我们希望将实值图像恢复的凸模型推广到循环图像和流形图像。我们希望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，这通常是，例如，对于球面，不是这种情况。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分或方向各向异性。我们项目的挑战包括为多值信号和图像建立适当的恢复模型，分析模型，以及开发高效的最小化算法，包括收敛结果。将在我们的项目中开发的方法有很大的应用潜力。在其他方面，我们将使用我们的模型来分析脑电数据和电子背向散射衍射数据。此外，还计划推出一个可公开使用的软件包。

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.