Non-Cartesian MRI Scan-Time Reduction through Sparse Sampling Magnetic resonance imaging (MRI) signals are measured in the Fourier domain, also called k-space. Samples of the MRI signal can not be taken at will, but lie along k-space trajectories determined by the magnetic field gradients. MRI measurements are usually Cartesian, where the trajectories are parallel and equidistant and sampling along the trajectories is also equidistant. This allows fast reconstruction using the inverse Fast Fourier transform (IFFT). However, this thesis focuses on non-Cartesian MRI. Typical trajectories in this case are radial and spiral, but there exists a multitude of other possibilities. Chapter 2 a basic introduction on MRI relevant for this thesis. Image reconstruction in the non-Cartesian case can not be accomplished by IFFT. In certain cases, however, dedicated reconstruction algorithms are available. For example, for radial scanning there exists the Filtered Back Projection algorithm. Another possibility, aiming at maintaining the IFFT for the transformation to the image domain, is the gridding algorithm. This algorithm, which is capable of image reconstruction from a k-space sampled along arbitrary trajectories, is given extensive attention in this thesis. A major complication in non-Cartesian sampling is the compensation for the non-uniformity of the sampling density. In case the trajectories are rather regular, an analytical expression for the density may be derived or Voronoi triangulation can be applied. Another more recent approach is the Pipe-Menon algorithm. However, all these approaches fail in more irregular cases. The above-mentioned image reconstruction algorithms are described in chapter 3. They are based on the inverse Fourier transform and therefore require k-space sampling to obey the Nyquist criterion. These algorithms cannot cope with k-space undersampling. However, in certain cases there may not be enough time to fully sample k-space; or scan-time is deliberately reduced by omitting trajectories, the total scan-time being proportional to the number of trajectories. Under these circumstances we still want to be able to reconstruct an image. Chapter 4 presents two algorithms that are able to cope with undersampled k-space data, and still reconstruct artefact free images. The first, based on work by G.J. Marseille, who worked on Cartesian scans, aims on estimating values for the missing data. In this approach the missing data are estimated iteratively by shuttling back and forth between image and k-space, while smoothing the image with an edge-preserving filter and resetting the measured data to their original values. This algorithm has a major drawback in that it requires density compensation. This means that this algorithm is only applicable if the trajectories are regular. Moreover, the algorithm requires user input on which k-space points are missing. In certain cases, especially when sampling is irregular, this may be impossible or not desirable. Note that this difficulty is absent in undersampled Cartesian scans. The second algorithm, also based on work by G.J. Marseille, directly estimates the image from the available k-space data. It is based on Bayes theorem, which allows both incorporation of consistency with the measured data, as in maximum likelihood estimation, and prior knowledge about the image. If k-space is undersampled, the k-space data alone give insufficient information to reconstruct satisfactory images. The prior knowledge gives the required additional information. In the Cartesian case, one dimension is always completely measured. Along this dimension IFFT can already be applied. This effectively reduces the image reconstruction problem to one dimension, meaning that different columns in the image can be treated separately. Consequently, the used prior knowledge, i.e. the Lorentzian edge distribution model, is a function taking only into account edges in one direction. In the non-Cartesian case, no dimension is completely sampled. Therefore the image can not be treated column-wise. Moreover, the prior has to take into account edges in both directions, since there is no preferred direction. In this thesis the prior used in Cartesian work is extended to include edges in more than one direction. In contrast to the first algorithm, the Bayesian approach allows one to obviate density compensation. Therefore, this algorithm can handle any type of sampling, to whatever degree of irregularity of sampling. In addition, since the image is directly estimated there is no need for the user to input which k-space points are missing. Chapter 5 discusses image quality measures. The measures are necessary for evaluation of the developed reconstruction algorithms. The performance of the mentioned reconstruction algorithms, in case of deliberate omission of trajectories to reduce the scan-time, depends on which trajectories are omitted. Chapter 6 is devoted to the matter of how to omit trajectories. Ideally, the raw data satisfy Hermitian symmetry. This property can be exploited when omitting trajectories. This chapter also describes two ways of finding optimal omission of trajectories from a full measurement. Finally, one of these methods is applied to an in vivo spiral scan. Optimal distributions seem to omit trajectories irregularly, without clustering too many omitted trajectories together so as to keep local undersampling to a minimum. Finally, chapter 7 deals with applications. The methods alluded to above are tested on simulations and real-world data. The Bayesian estimator appears particularly suited for pseudo-random sample positions. Frank Wajer, Delft University of Technology