Abstract
Fundamental operations of Magnetic Resonance Imaging (MRI) 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 to produce a digital image. In these methods, controllable gradient fields determine the points in the spatial-frequency domain which are sampled at any given time during the acquisition of the Free Induction Decay (FID) signal. Unlike the constant gradient case in which equally spaced samples of the FID signal in time correspond to uniform samples in the Fourier domain, for time-varying gradients, linear sampling in time corresponds to nonlinear sampling in the Fourier domain, and therefore straightforward inverse Fourier transformation is not sufficient for obtaining samples of the object distribution. MRI methods using time-varying gradients, such as sinusoids, are particularly important from a practical point of view, since they require considerably shorter data acquisition times. In this paper, we derive the optimum continuous time filter and its various discrete time implementations for FID signals resulting from sinusoidal gradients. In doing so we find that the estimation error associated with implementation based on linear temporal sampling, or equivalently nonlinear spatial frequency sampling, is smaller than that of nonlinear temporal sampling. In addition, we will show that that the optimal maximum likelihood estimator for sinusoidal gradients has higher error variance than that of constant gradients.
