Abstract We present a family of non-local variational regularization methods for solving tomographic problems, where the solutions are functions with range in a closed subset of the Euclidean space, for example if the solution only attains values in an embedded sub-manifold. Recently, in [R. Ciak, M. Melching and O. Scherzer, Regularization with metric double integrals of functions with values in a set of vectors, J. Math. Imaging Vision 61 2019, 6, 824–848], such regularization methods have been investigated analytically and their efficiency has been tested for basic imaging tasks such as denoising and inpainting. In this paper we investigate solving complex vector tomography problems with non-local variational methods both analytically and numerically.

中文翻译：

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矢量断层扫描的度量双积分正则化

摘要 我们提出了一系列用于解决断层扫描问题的非局部变分正则化方法，其中的解决方案是在欧几里德空间的封闭子集中具有范围的函数，例如，如果解决方案仅获得嵌入子流形中的值。最近，在 [R. Ciak、M. Melching 和 O. Scherzer，使用一组向量中的值的函数的度量双积分进行正则化，J. Math。Imaging Vision 61 2019, 6, 824–848]，已经对这种正则化方法进行了分析研究，并且已经针对基本成像任务（例如去噪和修复）测试了它们的效率。在本文中，我们研究了使用非局部变分方法从分析和数值上解决复杂的矢量断层扫描问题。