Theoretical comparison of Fourier and wavelet encoding in magnetic resonance imaging

A theoretical comparison of the wavelet and Fourier encoding methods is made with respect to resolution, sensitivity to artifact, and signal-to-noise ratios (SNR´s). A general mathematical description is developed in which magnetic resonance (MR) image encoding is represented by a projection of the function representing the “MR signal density” onto an approximation subspace of the finite energy functions. Characteristics of the subspace are used to define a “generalized” point-spread function for space-variant systems. Using the formal model of MR image encoding it is shown that wavelet encoding approaches the resolution limit defined by Fourier encoding. Artifact is treated according to whether or not the source of the variation in the measured data is stationary. Nonstationary imperfections perturb the projection operation and result in encoding method-dependent effects which can be modeled by a distortion matrix suitable for treating shift variant systems. A framework is developed in which to derive expressions for SNR´s applicable to a general class of MR encoding methods