Magnetic Resonance Imaging (MRI) is a non-invasive imaging modality. Unlike Computed Tomography (CT), MRI does not use ionizing radiation. In addition, MRI provides a large number of flexible contrast parameters. These provide excellent soft tissue contrast. Over the years, MRI has improved dramatically in both imaging quality and imaging speed. This revolutionized the field of diagnostic medicine. However, imaging speed, which is essential to many of the MRI applications remains a major challenge.

Imaging speed can be improved by faster collection of data. This can be achieved by using sophisticated non-Cartesian k-space trajectories. One of the design challenges is to minimize the gradient waveform duration, subject to both hardware and sequence constraints. Existing methods provide solutions limited to specific trajectories, or solutions which are sub-optimal. Based on optimal control theory, a method for designing gradient waveforms for arbitrary k-space trajectories is developed. It is non-iterative, computationally efficient and provides the time-optimal waveforms that traces k-space trajectories as fast as possible within the hardware limits.

With current hardware and sequence design methods, a point has nearly been reached in which fundamental physical and physiological effects limits the ability to simply encode data more quickly. This fundamental limit has led many researchers to look for methods to reduce the amount of acquired data without degrading image quality. MR image data are often highly redundant, which can be exploited to reduce the amount of acquired data, and hence the scan time.

In this work, a method that exploits the inherent compressibility of MR images is developed. It is based on the recent theory of compressed sensing (CS). In compressed sensing, the data are implicitly compressed in the acquisition process by obtaining fewer, so called, incoherent measurements. This is achieved by various randomized k-space sampling schemes. Images are accurately reconstructed from these measurements using a non-linear recovery processes that enforces the compressibility of the image. As a result, for some applications the scan time can be accelerated up to an order of magnitude.

SPIR-iT is an iTerative Self-consistent Parallel Imaging Reconstruction method. It is auto-calibrating and does not require explicit estimates of the coil sensitivity maps. SPIR-iT formulates the parallel imaging reconstruction through data consistency constraints. It is a general, optimal solution for coil-by-coil parallel imaging from arbitrary k-space trajectories. It is also a general framework for easily incorporating additional image priors, and in particular sparsity/compressibility constraints for combining parallel imaging with compressed sensing.