Singular Feature Extraction and Artefact Reduction in Dynamic Imaging

成像模态涉及到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众所周知的应用。对于其中的大多数，所寻求的功能被假定为与时间无关。然而，在许多医疗和工业应用中，这一假设被违反，例如，由于患者和器官运动或成像引擎处于工作阶段。在这种情况下，标准重建技术导致计算图像中的运动伪影，这会显著妨碍可靠的诊断。为了补偿的运动意味着将时间依赖性的研究对象在与静态的情况下的逆问题。将时间维添加到搜索量中不仅会导致欠定问题，还会改变静态问题的性质，例如不适定性程度，空间分辨率或导致有限的数据问题。本计画将针对这些问题，发展一套动态影像的正则化理论。为此，本计画分为两个部分：第一，我们提出研究并解决已知运动的动态问题。特别是，我们将分析的不适定性的运动的影响，处理有限的数据问题所产生的局部变形，开发高效和正则化的反演方案，然后研究的方法的运动模型的参数的敏感性。第二部分致力于直接从运动损坏的数据中估计运动，从而将理论从第一步扩展到未知变形。运动和搜索量的忽略使得动力学反问题是高度欠定的。在这方面，我们建议利用稀疏的精心挑选的功能，例如小波或轮廓分段常数函数，这将从根本上减少所考虑的问题的不确定性。总而言之，该项目将导致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。该项目致力于显着提高重建质量的断层扫描应用受对象相关的运动，并使非侵入性可视化的速度比目前的时间演变过程，例如在流体流动的研究。

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.