Singular Feature Extraction and Artefact Reduction in Dynamic Imaging

成像模式涉及对被研究对象的某些特征函数的非侵入性恢复，因此代表了反问题理论的众所周知的应用。对于它们中的大多数，所寻找的函数被假定为与时间无关。然而，这一假设在许多医疗和工业应用中是违反的，例如，由于病人和器官的运动，或者当发动机在工作阶段成像时。在这种情况下，标准的重建技术会导致计算图像中的运动伪影，这会显著阻碍可靠的诊断。补偿运动意味着将被研究对象的时间依赖性合并到与静态情况相关联的逆问题中。在搜索量上增加时间维度不仅会导致问题的欠定，还会改变静态问题的性质，如不适定程度、空间分辨率或导致有限的数据问题。本项目旨在通过发展动态成像的正则化理论来解决这些问题。为此，本项目分为两个部分：首先，我们提出了研究和解决已知运动的动力学问题。特别是，我们将分析运动对不适定性的影响，处理局部变形引起的有限数据问题，开发高效的正则化反演方案，然后研究这些方法对运动模型参数的敏感性。第二部分致力于直接从被运动破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。对运动和搜索量的忽视使动力学反问题变得非常不确定。在这一点上，我们建议利用精心选择的特征的稀疏性，例如用于分段常数函数的小波或轮廓，这将内在地减少所考虑问题的不确定性性。总之，该项目将导致联合运动估计和图像重建过程，减少图像中的运动伪影，从而帮助诊断。该项目致力于显著提高受对象相关运动影响的断层成像应用中的重建质量，并能够非侵入性地可视化比目前更快的时间演变过程，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.