Many medical imaging applications often require relatively long image acquisition times to form high-SNR images. However, long scan times can lead to motion artifacts. Conventional acquisition and reconstruction methods must sacrifice one for the other, i.e., enough measurements for less motion artifacts or vice versa.;Motion-compensated image reconstruction (MCIR) methods use all collected measurements, but reduce motion artifacts by incorporating motion information into the image reconstruction framework. Several motion incorporation schemes in MCIR have been proposed and showed superior performance over image reconstruction methods without motion information. However, there has been little research that emphasizes the motion aspects of MCIR. This dissertation addresses a few issues of MCIR methods in motion aspects.;First of all, we investigated methods for motion regularization. The usual choice for a motion regularizer in MCIR has been an elastic regularizer. Recently, there has been a lot of research on regularizing nonrigid deformations with two different motion priors. One prior is that deformations are invertible, and the other is that deformations are rigid on rigid tissues such as bones.;Conventional methods that enforce deformations to be locally invertible require high computational complexity and large memory. We developed a sufficient condition that guarantees the local invertibility and proposed a simple regularizer based on that sufficient condition. Our proposed regularizer encourages the local invertibility of motion estimates in a fast and memory-efficient way.;Using both motion invertibility and rigid motion priors may cause conflicts near the sliding area of the diaphragm and the rib cage. We relaxed our motion invertibility regularizer by using a Geman-type function. This relaxation reduces undesirable bone warping yet better matches the image intensities between deformed and target images and permits discontinuous motion fields near the sliding area.;Secondly, we studied the statistical properties of MCIR, showing that all MCIR methods are closely related to one another. This study also showed how motion affects the spatial resolution and noise properties of MCIR. We also designed spatial regularizers to provide approximately uniform spatial resolution for MCIR. These regularizers enabled different MCIR methods to approximately have the same resolution. Noise properties were compared based on these regularizers.;Lastly, we investigated joint image reconstruction and nonrigid motion estimation with different spatial and motion regularizers and regularization parameters. We performed a 4D PET simulation with the XCAT phantom with lesions. Most MCIR methods produced better-quality images with better SNR and less motion blur. The proposed motion invertibility regularizer allowed more flexibility of deformation estimates compared to a conventional quadratic motion regularizer.
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