Reconstruction of Non-Cartesian k-Space Data

MRI is an unusual medical imaging modality in that the raw data used to form an image can be collected in an infinite number of ways. While conventional Cartesian k-space scanning is by far the most common scanning method, non-Cartesian approaches have shown promise for rapid imaging, motion robustness, ultrashort TE imaging, rapid spectroscopic imaging, and other applications. A key component of implementing any nonCartesian scanning technique is of course a method for reconstructing an image from arbitrary k-space positions. This lecture will cover fast methods for non-Cartesian image reconstruction. It is straightforward to reconstruct data from arbitrary k-space samples using a technique known as conjugate phase reconstruction (CPR) [1, 2]. CPR is performed by multiplying each k-space point by the conjugate of the k-space encoding exponential and by a density compensation factor and then summing all of the data to arrive at a reconstructed pixel. However, this process must be repeated for each reconstructed pixel, and thus CPR is very slow. There have been a variety of methods published for fast nonCartesian image reconstruction. Here we will focus on a technique known as gridding, which is fast, simple and widely used [3–7]. Other methods have advantages in certain situations. The basic idea of gridding is simply to distribute the data from a non-Cartesian k-space trajectory onto a rectilinear grid, which is then be followed by an inverse fast Fourier transform (FFT) to transform to image space. The distribution is performed using a convolution operation, which can be thought of as an interpolation. The speed of gridding varies depending upon the details of the particular implementation, but roughly speaking about half of the reconstruction time is taken up by the FFT, so a simple non-Cartesian reconstruction takes about twice as long as a conventional Cartesian reconstruction. Gridding consists of the following steps: