MR imaging with a reduced number of encoding steps

Several methods have been used to obtain complete MR images from a reduced set of measured encodes 1,2,3,4,5,6. All of these techniques use some sort of a priori information to complete the incomplete data set. These techniques have had varying degrees of success in a variety of applications: imaging contrast uptake, interventional procedures, spectroscopy, fMRI contrast changes, and routine scans of a given population. In this paper we make three points: First, significant effort has been made toward finding the set of phase encodes that minimizes the expected L2 norm of the encoding error. We show that in experiments many sets of phase encodes come close to attaining the Karhunen-Loéve (K-L) limit. However, second, we show that other image quality metrics result in much more pleasing images that have much better detail than images where the L2 norm is optimized. For this purpose we use best basis algorithms to find a local trigonometric and wavelet packet bases which optimizes several cost functions including the Lp norms and entropy; the L0.25 norm gave the most pleasing results. Lastly, we show that although ringing produced by undersampling can not be eliminated it can be effectively controlled when the general shape of the object is known. Ringing renders techniques like SLIM1 impractical. We are able to control ringing by fitting the reconstructed signal to a piecewise monotonic function 7. The larger peaks are kept and the fast oscillations characteristic of ringing are eliminated. The signal can be pulled out of noise or fast oscillations when the signal energy is only one quarter of the noise energy.