Singular Feature Extraction and Artefact Reduction in Dynamic Imaging

成像模态涉及被研究对象的一些特征功能的非侵入性恢复，因此代表了逆问题理论的一个众所周知的应用。对于它们中的大多数，所寻求的函数被假定为与时间无关。然而，在许多医疗和工业应用中，由于患者和器官的运动或成像引擎处于工作阶段，这一假设被违反了。在这种情况下，标准的重建技术会导致计算机图像中的运动伪影，从而严重阻碍可靠的诊断。补偿运动意味着在与静态情况相关的反问题中纳入所研究对象的时间依赖性。将时间维度添加到搜索量中不仅会导致未确定问题，还会改变静态问题的性质，例如不适定性的程度、空间分辨率或导致有限的数据问题。本项目打算通过发展动态成像的正则化理论来解决这些问题。为此，本课题分为两部分：首先，我们提出对已知运动的动力学问题进行研究和求解。特别是，我们将分析运动对病态性的影响，处理由局部变形引起的有限数据问题，开发有效的正则化反演方案，然后研究方法对运动模型参数的敏感性。第二部分致力于从运动损坏的数据中直接估计运动，从而将理论从第一步扩展到未知变形。由于对运动和求量的忽略，使得动态逆问题处于高度欠确定状态。为此，我们建议利用精心选择的特征的稀疏性，例如小波或分段常数函数的轮廓，这将从本质上减少所考虑问题的不确定性。总之，该项目将产生一个联合运动估计和图像重建程序，减少图像中的运动伪影，从而有助于诊断。该项目致力于显著提高受物体相关运动影响的层析成像应用的重建质量，并实现比目前更快的时间演变过程的非侵入性可视化，例如在流体流动研究中。

Imaging modalities are concerned with the non-invasive recovery of some characteristic functions of an object under investigation, and hence represent a well-known application of the theory of inverse problems. For most of them, the sought-for functions are assumed to be independent of time. However, this assumption is violated in many medical and industrial applications, e.g. due to patient and organ motion or while imaging engines at working stage. In this case, the standard reconstruction techniques lead to motion artefacts in the computed images which can significantly impede a reliable diagnostics. To compensate for the motion implies to incorporate the time-dependency of the investigated object in the inverse problem associated to the static case. Adding the time dimension to the searched-for quantity does not only lead to an underdetermined problem, it also alters the nature of the static problem such as the degree of ill-posedness, the spatial resolution or lead to limited data issues. This project intends to address these points by the development of a regularization theory for dynamic imaging.For this purpose, the project is divided in two parts: First, we propose to study and solve the dynamic problem for known motion. In particular, we shall analyse the effect of the motion on the ill-posedness, deal with limited data problems arising from local deformations, develop efficient and regularized inversion schemes and then study the sensitivity of the methods to the parameters of the motion model. The second part is devoted to estimate the motion directly from the motion-corrupted data and thus to extend the theory from the first step to unknown deformations. The ignorance of both motion and searched-for-quantity brings the dynamic inverse problem to be highly underdetermined. At this end, we propose to exploit the sparsity of well chosen features, for instance wavelets or contours for piecewise constant functions, which will inherently reduce the underdeterminancy of the considered problem. Altogether, the project will result in a joint motion estimation and image reconstruction procedure which reduces the motion artefacts in the image and hence helps for the diagnosis.The project is dedicated to significantly improve the quality of reconstruction in tomographic applications affected by object related motion and to enable the non-invasive visualization of faster time-evolving processes than at present, for instance in fluid flow studies.