S. Johanna Vannesjo^{1}

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.

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.

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