Image reconstruction using projection onto convex sets, model constraints, and linear prediction theory for the removal of phase, motion, and Gibbs artifacts in magnetic resonance and ultrasound imaging

The Fourier transform is the standard image reconstruction technique used in magnetic resonance imaging (MRI), and it is an integral part of the inverse scattering formalism in ultrasound (US) imaging. Unfortunately, artifacts such as Gibbs ringing induced by a finite sampling window or systematic errors in phase may significantly impede the interpretation of the resulting Fourier transform images. Further, when only a few parameters are needed to characterize the object function, it is likely not to be the best technique for optimal signal-to-noise. In this paper, the application of a parameter estimation reconstruction scheme using a priori constraints to remove Gibbs ringing and improve resolution and signal-to-noise is presented for MRI and US. Projection onto convex set theory is also used to regenerate uncollected data in partial Fourier imaging, and model constraints are used to correct motion artifacts in MRI. These methods are found not to require unrealistic values of the signal-to-noise ratio and are likely to prove practical in future applications.