## Three Dimensional Resampling of Slices

There are a variety of reasons for resampling (or "reslicing") the MR volume after the data have been acquired. For instance, to compare the image from one study with the results of another or with a reference image it might be necessary to create slices in locations or orientations that differ from where they were originally acquired. In cases where noise has been introduced into the data by head motion through the slice plane, interpolation between slices may be required for adequate motion correction.

A three-dimensional MR data volume is composed of an aggregate of two-dimensional slices, when using common acquisition methods. When resampling the MR volume between slices there are some important issues to consider. It is well known from Fourier theory (see, e.g., [1]) that, given a data volume composed of slices collected perpendicular to the z axis, the interslice distance determines the maximum spatial frequency, Kz, that can be recorded without aliasing. However in Noll et al. [13] it is further observed that important objectives that govern slice selection in two-dimensional data acquisition can ultimately exacerbate the problem of interpolating the data at some later stage. In particular, it is noted there that generating rectangular slice profiles with narrow transition regions, in order to achieve precise localization with minimal slice-to-slice interaction, has the effect of increasing the content of high spatial frequencies in the data.

Later, during data processing, these same design criteria that were deemed desirable in acquiring the data present challenges if the data need to be interpolated.

For effective interpolation it is helpful if the data that are to be interpolated follow a smooth function with a band-limited spectral character — i.e., one that tapers smoothly to zero at a frequency below Nyquist. Then an interpolation operator can be chosen whose spectral shape encompasses the frequencies in the data. However, the sharp boundaries in a typical rectangular slice profile produce frequencies in fc-space at or above the Nyquist frequency. The combination of significant spectral energy right at Nyquist and some aliased data in the z dimension make it difficult to interpolate the data effectively between slices.

Practical guidelines for two-dimensional data acquisition to allow for more effective between-slice interpolation are offered in Noll et al. [13]. First, it is suggested that the use of a slice-profile excitation pulse that achieves the desired slice localization yet limits energy at high spatial frequencies will be helpful. After evaluating several types of pulses, they concluded that a Gaussian shape pulse exhibits the desired properties.

Second, Noll etal. [13] suggests eliminating the gap between slices or even the step of overlapping adjacent slices. This facilitates through-plane interpolation by increasing the density of information in the z dimension. Of course, this type of profile would use an interleaved ordering of slices during acquisition in order to keep crosstalk and signal loss at a minimum in the areas of overlap (see, e.g., [4]).

It is also possible to approach the solution of this problem from the other side — viz., by designing more effective interpolation operators. The goal is to select an interpolator whose spectral characteristics best accommodate the behavior of the Kz data. The sinc function is widely accepted as an ideal interpolating function under the right conditions (see, e.g., [1], p. 62; [12], p. 79). In particular, the sinc operator has a flat spectrum which effectively captures the spectral content of the data up to its cutoff frequency. (As noted in Section 3, there can be undesirable artifacts that accompany sinc interpolation, depending on its implementation.)

In Noll et al. [13] the performance of the (Hamming) windowed sinc operator is compared with a linear one. The results of their simulation studies indicated that the optimal way to handle two-dimensional MR data seemed to be, first, to taper the spectral content of the data by using the Gaussian-type excitation pulse for slice selection; second, to use a sinc-type interpolator that can perform optimally on the band-limited data.

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