Synopsis
Cartesian
k-space sampling on a regular grid provides optimal conditioning for image
reconstruction. Yet, there are several reasons why it can be beneficial to
deviate from the regular Cartesian sampling scheme. It may for example be to
achieve faster coverage of k-space, to make use of self-navigating properties,
to shape the point-spread function or to reduce the echo time. The most
commonly used non-Cartesian acquisitions are radial and spiral sampling, but a
large range of advanced sampling schemes have been explored. This presentation
will cover basic considerations related to arbitrary sampling, from gradient
waveform design to image reconstruction.
Introduction
Cartesian k-space
sampling on a regular grid provides optimal conditioning for inversion of the
Fourier encoding, in order to reconstruct an image from the acquired MR signal.
Yet, there are several reasons why it can be beneficial to deviate from the
regular Cartesian sampling scheme for different applications. It may for
example be to achieve faster coverage of an area of k-space, to make use of
self-navigating properties, to shape the point-spread function (PSF) or to
reduce the echo time. There are innumerable versions of non-Cartesian sampling,
but among the most commonly used are radial and spiral sampling. Radial
trajectories sample the center of k-space repeatedly, which can be utilized for
real-time imaging and sliding window techniques (1). Spiral sampling can provide fast coverage of k-space by efficient
use of the gradients, and a very short echo time can be achieved (2).Gradient waveform design
The exact
sampling pattern in k-space is determined by the gradient time-courses. However,
given a specific desired k-space sampling pattern, the corresponding gradient
waveforms are not uniquely determined. As the k-space path is produced by the
time-integral of the gradient, different gradient waveforms can trace out the
same path by scaling both the gradient amplitude and the duration. The optimal
gradient waveforms to achieve specific desired sampling properties can be
determined by an optimization approach. The optimization can be geared towards different
objectives, such as minimizing the gradient currents (3), minimizing the deviation from a pre-defined trajectory (4), yielding time-optimal gradient waveforms that follow a certain
k-space path (5) or pass a specific set of points in k-space (6). Due to physical limitations of the gradient chain, the
optimization must incorporate amplitude and slew rate constraints on the
gradient waveforms.Image reconstruction
To retrieve an
image from the acquired k-space samples, the reconstruction algorithm has to
take into account that the sampling points for arbitrary trajectories do not
fall on a regular Cartesian grid. Most commonly used is a gridding approach (7,8), where a local convolution kernel in k-space is used to estimate
the values at sampling points on a regular Cartesian grid from the arbitrary
set of k-space samples. With the new set of estimated Cartesian k-space
samples, a standard Fast Fourier Transform can be utilized. To compensate for
the uneven distribution of sampling points, the k-space data should be
multiplied with a density compensation filter before taking the Fourier
Transform.
In a more
general formulation of the reconstruction problem, the encoding process can be
regarded as a linear operator acting on the magnetization:
$$Ex=m$$
where $$$E$$$ is the
encoding matrix, $$$x$$$ is the magnetization vector and $$$m$$$ is the measured signal. In
the special case of Cartesian sampling, the conjugate transpose of $$$E$$$ is also
the inverse operator (i.e. the inverse Fourier Transform). However, in the non-Cartesian
case, this no longer holds true. A least-squares solution to the problem can be
obtained by the pseudo-inverse of $$$E$$$.
As the matrix
size is typically too large for direct inversion, the solution can be
approximated by an iterative optimization algorithm (9). The conditioning of the problem can be improved by regularization,
for example using Tikhonov regularization which penalizes the L2-norm of the
solution vector.
Artefacts
The encoding
process can be perturbed by disturbances such as gradient waveform deviations,
B0 inhomogeneity, motion, etc. How this translates into image
artefacts differs from the Cartesian case, and depends both on the given sampling
trajectory, and the chosen reconstruction algorithm. In the case of spiral
sampling, B0 offsets and gradient imperfections generally result in
blurring of the image. Radial sampling is sensitive to slight
direction-dependent gradient delays, leading to inconsistent sampling of the
k-space center. Numerous methods have been proposed to address gradient
imperfections, including gradient delay correction (2,10), direct gradient measurements (11,12) or gradient estimation based on a system characterization (13,14).Acknowledgements
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