Non-cartesian sampling methods used in MRI and other modalities continue to play an important part in imaging. Although many techniques have been proposed to overcome the difficulty that arises from nonuniform data, many approaches (including gridding) require a form of data weighting that accounts for the changes in sampling density across k-space. The various methods that have been proposed to calculate appropriate weights for sampling density correction can be grouped into three families of techniques. These basic forms are analytical, algebraic, and convolution approaches.

The analytical methods attempt to define the corresponding area of each sample point which is inversely proportional to the weight. With spiral and radial trajectories, for example, the weight can be calculated by taking the absolute value of the Jacobian determinant of the trajectory with respect to the trajectory's parameters (1, 2). The Jacobian is the result of a continuous coordinate transformations, which can degenerate due to the discrete nature of sampling (1). For a general case the area can be approximated by taking a Voronoi diagram of the trajectory (3). These techniques lack a definition for optimality and their rationale can be purely conceptual.

Most of the algebraic methods represent gridding in an algebraic form, and deduce a solution for the appropriate sampling density weights that minimizes error in the gridded k-space (4–6). Despite the optimality constraints, these techniques do not perform as well when considering the error in the final reconstructed image, at least when not regularized (7). This is often attributed to the system being ill conditioned. It may also be because the optimality is defined as the gridded error in k-space. Hybrid techniques such as (6, 7) combine this form of algebraic gridding and the convolution techniques into a single concept. An additional algebraic method finds appropriate weights that minimize total sidelobe energy in the point spread function (PSF) (8).

Several methods for calculating the compensating weights using a convolution have been proposed, including the earliest form discussed by Jackson (9). Subsequently other methods have improved upon this by using iterative methods. This includes a geometric algorithm proposed in (10), an algorithm that is a first order difference filter (11) and the previously stated algebraic methods of convolution (6, 7).

Each of the convolution schemes attempt to find a solution for the weights such that, when convolved with the kernel, it will provide unity and hence an ideal Modulation Transfer Function (MTF). Stated mathematically each scheme attempts to minimize the error

where *W* is the set of weighted sampling points, *P* is the kernel, and ☆ is the convolution operator. This is equivalent to minimizing the error in the PSF:

where *e*, *w*, and *p* are the Fourier Transforms of *E*, *W*, and *P* respectively. The ideal point spread function is represented by the delta Dirac function, δ. The actual PSF of the given reconstruction is given by *w*. Error outside the region of interest is effectively ignored since it is modulated by a profile, *p*, which tends to zero with radii larger than the field of view (FOV).

In this study, sampling density correction is considered to be completely independent from the gridding. The gridding algorithm (implemented by a convolution interpolation, FFT and deapodization) is an approximation to the direct fourier transform (DFT) (also known as conjugate phase reconstruction in the MRI community). The DFT is also dependent on sampling density correction. The error from the gridding step has been characterized and minimized for even modest oversampling ratios and kernel widths (9, 12). Consequently analysis of error from the different sampling density schemes in this article are calculated by the slower DFT, to eliminate the small, but confounding error introduced by gridding.

Rather than attempting to minimize error in gridded k-space or in the raw PSF, this study attempts to minimize error in the final reconstructed image. While minimizing error in the gridded MTF, PSF and final image seem to accomplish the same goal, the subtle difference between the techniques may result in the discrepancies found in practice.

The goals of this study are to design an appropriate kernel for convolution methods and provide a relatively fast implementation that will enable this process for clinical use. The accuracy of the results from the designed kernel will be compared to state of the art methods for sampling density correction. As an additional criterion, the efficient implementation must be feasible for general 3D imaging trajectories.