The k‐trajectory formulation of the NMR imaging process with applications in analysis and synthesis of imaging methods

The fundamental operations of nuclear magnetic resonance (NMR) imaging can be formulated, for a large number of methods, as sampling the object distribution in the Fourier spatial-frequency domain, followed by processing the digitized data (often simply by Fourier transformation) to produce a digital image. In these methods, which include reconstruction from projections, Fourier imaging, spin-warp imaging, and echo-planar imaging, controllable gradient fields determine the points in the spatial-frequency domain which are sampled at any given time during the acquisition of data (the free induction decay, or FID). The detailed time dependence of the resulting trajectory of sample points (the k trajectory) determines the relative weight and accuracy with which image information at each spatial frequency is measured, establishing theoretical limitations on image quality achievable with a given imaging method. We demonstrate here that these considerations may be used to compare the theoretical capabilities of NMR imaging methods, and to derive new imaging methods with optimal theoretical imaging properties.