NATURAL K-PLANE COORDINATE RECONSTRUCTION METHOD FOR MAGNETIC-RESONANCE-IMAGING - MATHEMATICAL FOUNDATIONS

The mathematical basis and motivation for a new, noninterpolative, dir ect method for reconstructing magnetic resonance imaging (MRI) data is presented. The reconstruction method is called the natural k-plane co ordinate reconstruction method (NKPCRM) and can be used for reconstruc ting MRI data collected in the presence of continuously varying gradie nt magnetic fields. A continuous, theoretically useful NKPCRM is prese nted along with a practical discrete NKPCRM both for single-and multip le-shot data acquisitions. The continuous method gives rise to continu ous operators on function spaces that can be characterized as integrab le curve band-pass operators. The discrete reconstruction method reduc es to a Fourier summation weighted by the Jacobian of a ''naturally'' chosen coordinate system. In the case of a Cartesian coordinate system , the new method reduces to the discrete Fourier transform normally us ed for MRI reconstruction. The NKPCRM is rigorously analyzed from a ma thematical point of view, and specific implementations such as Lissajo us, spiral, and rose scans are discussed. (C) 1997 John Wiley & Sons, Inc.