A new class of sampling theorems for Fourier imaging of multiple regions

Abstract

Traditional Fourier imaging utilizes the Whittaker-Kotel'nikov-Shannon (WKS) sampling theorem. This specifies the spatial frequency components which need to be measured in order to reconstruct an image completely contained within a known field of view (FOV). Here, the authors generalize this result in order to find the optimal k-space sampling for images that vanish except in multiple, possibly non-adjacent regions within the FOV. This provides the basis for region Fourier imaging, a method of producing such images from a fraction of the k-space samples required by the WKS theorem. Sampling is optimal in the sense that it is minimal and does not lead to noise amplification during image reconstruction, just as for WKS sampling. Image reconstruction is computationally cheap because it is performed with small fast Fourier transforms. The new technique can also be used to reconstruct images that have low spatial frequency components throughout the entire FOV and high spatial frequencies (i.e. edges) confined to multiple small regions. The method's greater sampling efficiency can be parlayed into increased temporal or spatial resolution whenever the imaged objects have signal or edge intensity confined to multiple small portions of the FOV. The method may be applicable to various types of magnetic resonance (MR) imaging, as well as to other Fourier imaging modalities and multi-band telecommunications. The technique is demonstrated by using it to reconstruct MR angiographic images of the carotid arteries of a volunteer.