Arbitrary Trajectories: ACQ, Gradients, Reconstruction, Artifacts
S. Johanna Vannesjo1

1Wellcome Centre for Integrative Neuroimaging, FMRIB, University of Oxford, United Kingdom

### 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).

### 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

No acknowledgement found.

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Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)