Constrained imaging: overcoming the limitations of the Fourier series

Magnetic resonance imaging (MRI) is usually implemented as a Fourier transform-based technique. During data acquisition, spatially resolved information relating to spin density, relaxation rates, chemical shifts, and other parameters is phase and frequency encoded in the measured data. Image reconstruction is accomplished through the use of the Fourier series model, which can be evaluated efficiently using a fast Fourier transform (FFT) algorithm. Theoretically, the Fourier series is capable of producing perfect images if the data space (often called k-space) is sufficiently covered. In practice, several problems arise with this model due to finite sampling. Specifically, finite sampling leads to a truncation or the Fourier series, which results in image blurring and ringing. Image blurring is attributed to a loss of spatial resolution. In fact, with the Fourier series model, the resulting image resolution is limited to roughly the reciprocal of the frequency interval over which the data are sampled. The ringing artifact is due to the well-known Gibbs phenomenon, which is more pronounced for images with sharp edges. In order to overcome these limitations associated with the direct application of the Fourier series model, many alternatives have been proposed in the past decade to incorporate a priori information into the imaging process. This article discusses the constrained imaging concept. Specifically, the authors review 3 model-based imaging techniques that the authors have developed in the past few years. An essential feature of these methods is that a parametric model in the form of a generalized series is superimposed on the underlying measured data or image.