Encoding and reconstruction in parallel MRI

The advent of parallel MRI over recent years has prompted a variety of concepts and techniques for performing parallel imaging. A main distinguishing feature among these is the specific way of posing and solving the problem of image reconstruction from undersampled multiple-coil data. The clearest distinction in this respect is that between k-space and image-domain methods. The present paper reviews the basic reconstruction approaches, aiming to emphasize common principles along with actual differences. To this end the treatment starts with an elaboration of the encoding mechanisms and sampling strategies that define the reconstruction task. Based on these considerations a formal framework is developed that permits the various methods to be viewed as different solutions of one common problem. Besides the distinction between k-space and image-domain approaches, special attention is given to the implications of general vs lattice sampling patterns. The paper closes with remarks concerning noise propagation and control in parallel imaging and an outlook upon key issues to be addressed in the future.