Nonlinear mass-preserving registration for magnetic resonance imaging (MRI) and positron emission tomography (PET)

图像配准是医学成像中的一项重要任务。问题是在不同时间、使用不同设备或从不同角度感测到的对象之间自动建立点对点对应。图像配准对于运动校正或数据融合等应用是必不可少的。由于图像配准是一个不适定的逆问题，因此在需要的变换域上进行适当的建模和附加约束是不可避免的。质量保持(MP)是校正磁共振成像(MRI)中场不均匀引起的图像失真和正电子发射断层扫描(PET)中呼吸和心脏运动校正的关键因素之一。然而，MP技术在这些医疗应用中的广泛应用需要解决这个项目中要解决的具有挑战性的数学问题。因此，现有的MP技术将在数值效率和实现方面得到优化。特定问题的数据保真度将提高对噪声的稳健性，甚至对算法的简化。基于非线性弹性的正则化能量是MP背景下产生解存在性的一个关键成分。因此，这一优秀的正则化及其数值离散化将在本项目中得到进一步的研究。通过使用反问题的正则化理论，将对所开发的模型有更深刻的数学理解。

Image registration is an important task in medical imaging. The problem is to automatically establish point-to-point correspondences between objects sensed at different times, with different devices, or from different perspectives. Image registration is essential for applications such as motion correction or data fusion. As image registration is an ill-posed inverse problem, proper modeling and additional constraints on the wanted transformation field become inevitable. Mass-preservation (MP) is one of those constraints that proved to be crucial for the correction of image distortions due to field inhomogeneities in Magnetic Resonance Imaging (MRI) and the respiratory and cardiac motion correction in thoracic Positron Emission Tomography (PET). A wide utilization of MP techniques in those medical applications requires, however, the solution of challenging mathematical questions to be addressed in this project. Existing MP techniques will hence be optimized with respect to numerical efficiency and implementation. Problem specific data fidelities will yield an improved robustness against noise and even simplifications of the algorithms. A key component yielding existence of solutions in MP setting is a regularization energy based on nonlinear elasticity. This outstanding regularizer and its numerical discretization will hence be further investigated in this project. A deep mathematical understanding of the developed models will be gained by using regularization theory for inverse problems.