| This dissertation considers the reconstruction of images (or, in general, two-dimensional space-limited functions) from certain collections of samples of their Fourier transforms. This problem arises in magnetic resonance imaging (MRI). Specifically, this work focuses on MRI techniques that seek to reduce significantly the data acquisition time.;Part of this study is devoted to a type of sampling strategy proposed for rapid MRI, namely, sampling along spiral scans. Initially, some theoretical aspects of spiral sampling are addressed. For linear spiral scans an exact reconstruction procedure is shown to exist, and formulated as a linear spiral sampling theorem. For nonlinear, expanding spirals it is shown that, in general, there is no exact reconstruction procedure. Hence an approximated procedure is furnished and analyzed. Subsequently, two practical image reconstruction methods are derived from the aforementioned theory, and assessed in comprehensive experiments using synthetic images.;One source of distortions in MRI reconstructions is the inhomogeneities of the static magnetic field. The effect of inhomogeneities on spiral sampling MRI is studied and compared to the effect of inhomogeneities on an appropriate (rapid) Cartesian scan MRI method. It is found that the spiral scan method is more sensitive to inhomogeneities than the Cartesian scan method. The presented analysis also provides a means of determining the acceptable level of inhomogeneity in a system that includes rapid spiral scan imaging.;A second sampling strategy considered in this work is suitable for a multislice MRI procedure. In multislice MRI, a volume of interest is represented by a sequence of two-dimensional images, each one corresponding to a different "slice" of the volume. A method is proposed to reduce the acquisition time of a multislice imaging procedure, by acquiring partial data sets in some of the slices (e.g., every other slice). The "partial data slices" are reconstructed using the method of projections onto convex sets (POCS). In a POCS algorithm it is possible to include any number of convex constraints. The constraints parameters are obtained, in this case, from the side information existing in the "full data slices". The performance of the proposed procedure is demonstrated in extensive experiments using head images from an actual multislice MRI session. |
