In the radial sampling scheme, the k-space data is acquired along intersecting spokes instead of parallel rows. Corresponding measurement sequences can be derived rather easily from existing sequences with a Cartesian acquisition. However, due to the non-equidistant sampling positions a special reconstruction strategy is required, and the filtered backprojection method or the regridding technique are commonly used. Both approaches are closely related and differ only in the interpolation technique used, which is needed to map the acquired information from the spoke geometry onto a grid. Because the regridding technique offers a slightly higher flexibility, it is more frequently used in current practice.
Radial sampling offers several advantages over the Cartesian scheme that include a lower sensitivity to object motion and the ability to perform readout oversampling in all directions without additional acquisition steps, which eliminates any aliasing effects.
Further, the k-space center is oversampled, which yields an interesting undersampling behavior. Although the reduction of the spokes leads to spurious streaking artifacts, a high degree of the object information remains visible even for stronger reduction factors, which is not the case for Cartesian sampling. Moreover, each sampled spoke captures an equal amount of low and high spatial frequency information, which offers more homogeneous image updates in dynamic MRI applications. Finally, it is possible to apply a number of consistency criteria for the detection and first-order correction of inaccuracies in the measured spokes. On the other hand, the number of k-space lines for a fully sampled data set is 57% higher than in the Cartesian scheme, which prolongs the
examination time. Radial sampling is also more sensitive to deviations of the gradient time courses, which, however, is a manageable problem with modern MRI systems.
Further, the technique suffers from a pronounced sensitivity to off-resonance effects, which arises from the varying readout direction in k-space as well as the intersection of all spokes in the k-space center. Therefore, radial sampling is less suited to obtain images with T?2 contrast, and practical applications should employ short echo times or RF refocusing pulses.
Chapter 5
Iterative Reconstruction from Incomplete Radial Data
In this chapter, the problem of the streaking artifacts is addressed, which appear when only a low number of spokes is acquired. A novel reconstruction technique is presented that compensates for the missing information by incorporating prior object knowledge.
The feasibility of the method is demonstrated with experimental data from studies of phantoms and the human brain in vivo acquired using a radial spin-echo sequence.
Finally, the relation to the recently proposed compressed-sensing concept is outlined.
5.1 Reconstruction from Undersampled Data
As discussed in Section 4.1, the radial acquisition scheme demands a higher number of sampled lines to fully cover all areas of k-space. In fact, π/2·n spokes have to be acquired for an image with a base resolution of n pixels in order to ensure that the outmost samples of two neighboring spokes have a maximum distance of ∆k = 1/FOV, which corresponds to the well-known Nyquist condition for conventional Fourier imag-ing. While this requirement enables to obtain high-quality images with conventional reconstruction methods like regridding, it prolongs the data acquisition by about 57%
relative to that of a corresponding fully-sampled Cartesian data set. This factor is, of course, highly undesirable as it increases the total duration of the examination and hampers the use of radial techniques in clinical settings.
If the number of acquired spokes is reduced to a value far below the recommended value, the reconstructed image presents with two characteristic features: while most object information remains visible at good spatial resolution, the use of a regridding (or filtered back projection) approach results in streaking artifacts. This property has already been addressed in Section 4.3.1 on the PSF, and the artifacts are demonstrated
Figure 5.1: Regridding reconstructions (Shepp-Logan phantom, 256×256 matrix) using simulated data from 402, 64, and 24 spokes (256 data samples). The lower right panel shows the Fourier transform of the image reconstructed from 24 spokes. It reveals unmeasured gaps in the k-space in-between spokes (arrows). The reconstructions from 64 and 24 spokes suffer from streaking artifacts caused by undersampling.
again in Figure 5.1 for a reduction of the spokes from 402 to 64 and 24. Although the strength of the streaking artifacts increases with the extent of undersampling, it is remarkable how much information about the object can still be seen in an undersampled image from only 24 spokes.
The origin of the streaking artifacts may be best understood when considering the Fourier transform of an undersampled regridding image, which is shown in the bottom right of Figure 5.1. It can be seen that the resulting k-space pattern matches the acquired data at the spoke positions, but in-between the spokes the Fourier transform is zero (except for a small surrounding of the spokes resulting from the convolution with the interpolation kernel). Obviously, this solution with many gaps and jumps is not an accurate representation of the Fourier transform of the true object, which explains the failure of the regridding method for undersampled data sets. Therefore, the conventional reconstruction approaches are not appropriate for strongly undersampled acquisitions.
To obtain improved reconstructions from such data, it is necessary to employ a dedicated technique that takes the undersampling into account, so that distracting streakings are removed while the visible object information is preserved.