Image modelling, inpainting, decomposition and restoration by redundant representations and variational calculus

该项目将研究框架（Banach空间中的稳定冗余非正交展开）的使用及其在逆问题（图像处理中的修复，恢复和分解任务）领域的数值应用。在反问题中，特别是在图像处理中，最重要的任务之一是识别将问题嵌入正确设置的底层模型函数空间。当处理（特殊）有界变差函数空间时-这是图像处理中非常受欢迎的空间-问题通常是用于近似解的相关PDE（Euler拉格朗日）方案在数值上非常密集且耗时。最好能避开这个缺点，用一些数值上节省的方法来求得解。此外，在某些应用中，通常假设解具有（超）稀疏展开，或者需要以非常节省空间的方式表达解。为此，我们的目标是在这里回答这样一个问题，即我们如何表征所考虑的函数空间和变分问题，以及/或者我们如何用其他更容易处理的框架（例如，将SBV函数嵌入Besov和振荡空间）来代替它们。特别是，我们要求适当的特征的基础功能空间（或其替代品）的框架，允许适当的离散化/分解。一旦实现了这一点，我们可以重写变分问题（我们希望不会太远离原来的问题），并致力于为框架系数构造方案，这些方案可以轻松地提供数值求解的工具。在这里，我们主要集中在两个应用程序：首先，解决非线性问题的变分形式的解决方案是假设有（超）稀疏帧扩展，其次，调查使用框架建模自由不连续性问题，如芒福德沙阿一样的泛函，因为它们出现在图像修复/恢复领域。

The project will address investigations on the usage of frames (stable redundant nonorthogonal expansions in Banach spaces) and their use in numerical applications in the field of inverse problems (inpainting, restoration and decomposition tasks in image processing). In inverse problems, and in particular in image processing, one of the very first tasks is to identify the underlying model function spaces that embed the problem into the right setting. When dealing with spaces of (special) bounded variation functions - which is a very favoured space in image processing - the problem is often that the associated PDE (Euler Lagrange) schemes to approximate the solution are numerically very intensive and time consuming. It would be desirable to bypass this drawback and to derive the solution in some numerically thrifty way. Moreover, in certain applications the solution is often assumed to have a (super) sparse expansion, or it is required to express the solution in a very space saving way. To this end, we aim to answer here the question on how we can characterize the function spaces and variational problems under consideration and/or how we can replace them by other easier to handle frameworks (e.g. embeddings of SBV functions into Besov and oscillation spaces). In particular, we ask for proper characterizations of the underlying function spaces (or its replacements) by means of frames that allow an adequate discretization/decomposition. Once this is achieved, we may rewrite the variational problems (where we hope being not too far off the original problem) and aim to construct schemes for the frame coefficients that easily provide tools to derive the solution numerically. Here we essentially focus on two applications: firstly, solving nonlinear problems in its variational form where the solution is assumed to have (super) sparse frame expansion and, secondly, investigating the use of frames for modelling free discontinuity problems such as Mumford Shah like functionals as they appear in the field of image inpainting/restoration.