• Cartesian k-space sampling provides many simplifying features for image reconstruction, but non-Cartesian sampling offers many advantages for dynamic and undersampled imaging

• Non-Cartesian data is typically interpolated to a Cartesian dataset and then inverted by FFT, but interpolation has important consequences for the final image, so it must be chosen carefully and properly compensated.

• Iterative techniques that cycle between Cartesian images and non-Cartesian k-space open new avenues for undersampling and dynamic imaging

Cartesian or rectilinear sampling has dominated MR imaging for many reasons. Cartesian sampling is minimally affected by many experimental imperfections such as off-resonance spins and timing errors. Furthermore, as long as Nyquist sampling requirements are fulfilled, reconstruction is both analytically tidy and computationally efficient. However, many exciting studies have shown important advantages of nonlinear trajectories in a wide range of research and clinical applications. They have advantages for highly undersampled techniques, continuous imaging applications, motion tracking, and even for new paradigms in contrast generation. [1-4]

However, non-Cartesian acquisition schemes require more sophisticated reconstruction methods. To understand the steps in reconstructing an image from non-Cartesian data, it is useful to review the basis for the reconstruction of Cartesian data. Recall that in a discrete Fourier transform (FT), the sampling function in k-space creates aliasing in image space via convolution with the FT of the sampling function. When the sampling function is a comb function with equal spacing δk, the image is convolved with another comb function with spikes separated by 1/δk. Thus, as long as 1/δk, is a reasonable FOV for the object, these copies do not overlap when we reconstruct a single period or image, and the discrete FT of Cartesian data creates no aliasing artifacts. [5] Furthermore, the harmonic basis associated with Cartesian sampling lends itself to fast calculations via Fast Fourier Transform algorithm (FFT). [6]

In principle, non-Cartesian k-space data could be reconstructed by adding the spatial functions associated with the collected k-space points and weighting these by the sampling density in k-space. However, even on modern computers, this is generally considered too slow for clinical applications. Instead, the most widely used approach is to use the collected k-space to estimate the underlying continuous data and then resample onto a Cartesian grid. In practice, these steps are performed simultaneously to estimate the data at the Cartesian locations. The FFT algorithm can then be applied to the equally spaced data, yielding an image. [7-9]

While this seems straightforward, the chosen means of estimating the Cartesian k-space data alters the sampling function, which has significant consequences for the resulting image. Avoiding and compensating for these effects accounts for the additional steps in a non-Cartesian reconstruction, though several packages are available that yield very good results with standard parameters [10,11]. The steps in gridding include: density compensation, interpolation onto a grid, 2D FFT, and then deapodization to remove the roll-off effects of interpolation.

Initial density compensation is needed because non-Cartesian trajectories acquire k-space with uneven sampling. A simple blurring of the data would overrepresent regions where k-space is densely sampled. To achieve density compensation, the data is pre-multiplied by a weighting function that normalizes the sampling density across k-space. Multiple density compensation techniques exist, but widely used approaches include Jacobian forms for analytical trajectories and numerical methods. [12,13]

Much effort has focused on characterizing the best choice of gridding interpolator, since the transform of the interpolator introduces rolloff over the FOV as well as sidelobes. Resampling onto a Cartesian grid then introduces aliased copies of the image centered at 1/δk, so the sidelobes of the aliased copies will interfere at the edges of the reconstruction. Rolloff could be compensated by a simple magnitude correction, but aliasing is addressed by two complementary strategies:

1) Choose a convolution kernel with minimal sidelobes: An infinite sinc is an ideal convolution kernel because its transform has no roll-off and no sidelobes. However, finite approximations to the infinite sinc introduce significant apodization and sidelobes. Single lobe kernels, such as the Kaiser-Bessel function, can lower the aliased energy, though it is important to consider the amount of aliased energy expected after deapodization. [7-9,14]

2) Oversample to further separate aliased copies: Another approach to reducing aliasing is to estimate the data on a finer Cartesian grid than that needed for the reconstruction FOV. This increases separation between aliased copies, reducing overlap between sidelobes, but there is a computational expense, particularly for 3D applications. There are excellent discussions of these tradeoffs in the literature, but with judicious choice of convolution kernel, even a small rate of oversampling (e.g. 1.25x) can yield very fast and accurate images. [14] Combining these steps produces the standard gridding reconstruction pipeline:

1) Multiply data by a function that compensates for uneven sampling density in k-space

2) Convolve data with a kernel

3) Resample onto a (generally larger) rectilinear grid

4) Inverse 2D FFT

5) Deapodize to compensate for roll-off due to convolution

While this process will generate a very good reconstruction from perfect non-Cartesian data, many reconstruction problems require an iterative transformation between data and image space. These include procedures to reconstruct from undersampled data, as well as methods to correct for experimental imperfections [15-18]. The inverse gridding procedure needed for these approaches is basically the reverse of the reconstruction, skipping density compensation since the image is rectilinear.

Finally, for more arbitrary encoding schemes, iterative solutions can be sought entirely in the spatial domain. [19] These procedures do not have the computational benefits of the FFT, but are entirely flexible in their ability to model alternate forms of image encoding including coil-sensitivity, off-resonance maps, gradient delays, and even non-Fourier based image encodings. [20] In matrix based reconstruction, the image encoding process is written out as a system of linear equations (one for each datapoint) in a matrix known as the encoding matrix, E, such that: Em=Data, where m is the reconstructed image and Data is the measured data. There are a number of iterative algorithm that can be used to seek a numerical solution for the m that minimizes ||Data-Em||².