Constrained reconstruction: A superresolution, optimal signal‐to‐noise alternative to the Fourier transform in magnetic resonance imaging

Many problems in physics involve imaging objects with high spatial frequency content in a limited amount of time. The limitation of available experimental data leads to the infamous problem of diffraction limited data which manifests itself by causing ringing in the image. This ringing is due to the interference phenomena in optics and is known as the Gibbs phenomenon in engineering. Present techniques to cope with this problem include filtering and regularization schemes based on minimum norm or maximum entropy constraints. In this paper, a new technique based on object modeling and estimation is developed to achieve superresolution reconstruction from partial Fourier transform data. The nonlinear parameters of the object model are obtained using the singular value decomposition (SVD)-based all-pole model framework, and the linear parameters are determined using a standard least squares estimation method. This technique is capable, in principle, of unlimited resolution and is robust with respect to Gaussian white noise perturbation to the measured data and with respect to systematic modeling errors. Reconstruction results from simulated data and real magnetic resonance data are presented to illustrate the performance of the proposed method.