In this chapter, we discuss some of the important issues pertaining to the resampling of MR data from one grid of pixels to another grid of pixels. In particular, we note that the way in which resampling is done can impact the information content, accuracy, and statistical properties of the data. In fMRI studies this means that the resampling procedure used in data processing can influence whether areas of activation are properly detected. Throughout most of this chapter we focus on two-dimensional registration (a) because of its primary importance for most current methods of fMRI data collection and (b) because most of the important issues in three dimensions already arise in two. We use the term resampling to mean the interpolation of the data to other than the original pixel locations. We also use the term transformation rather than interpolation when we are focusing on the information in the data rather than the images themselves. The subject of image registration for MR images is much broader than we could possibly cover here and, of course, is much broader than MR itself. A good overview of the general problem of image registration is given in Hajnal et al. [3].

Since fMRI BOLD effects are small, typically only 1-5% changes in the signal, small motions of the brain have significant impact on the data. Adjacent voxels within the brain can have signal values that vary by 10% or more and signal changes on the order of 100% or more occur at the edge of the brain. Thus, very small motions can have highly significant effects. For example, stimulus-correlated head motion has been shown to generate false positive effects [8].

Every method that adjusts fMRI data for head motion (registration) involves two steps: (a) determination (i.e, estimation) of the amount of motion and (b) correction of the data for that amount ofmotion. In this chapter we focus on the correction part of the registration method. Furthermore, we focus on correction for rigid motion of the brain as that is the most important effect.

Motion correction is one of many steps that should be performed during the routine processing of fMRI data. As such, any motion correction method should remove as much of the effect of motion as possible without otherwise affecting the data. If the motions of the brain are essentially random, they will appear to be simply an increase in the underlying noise level. Correction of these motions will reduce the apparent noise level and hence increase the amount of detected activation. On the other hand, if the motions of the brain are correlated with the stimulus, correction of the motion may ( properly) reduce the amount of detected activation.

A critical part of motion correction is the production of an image relocated to the pixels of the fixed target image: that is, the resampling of the data onto the appropriate pixel grid. That is our focus here. This chapter has three sections. In Section 2 we discuss the idea of conservation of information in the context of resampling of the data. We consider the effects of transforming (resampling) the data by various procedures: e.g., nearest-neighbor, (bi- or tri-) linear, polynomial (cubic,

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quintic,...), windowed sine, and Fourier. The choice of interpolator determines the degree to which the original information is conserved during resampling. We show empirically that not all the available (and popular) methods are equally desirable in terms of their impact on the data. Relative to the others, application of the full Fourier technique, in k-space, appears most optimal, among the methods considered, in conserving information. This is for two reasons. First, from a sampling theory point of view, other forms of interpolation exhibit a lower order of accuracy than the full Fourier method. Second, from the point of view of statistical theory, less-accurate interpolators tend to smear information during resampling, which has the effect of introducing a false level of correlation in the data. This can lead to misinterpretation of the functional information contained in the data.

In Section 3 we discuss three distinct implementations of Fourier-based methods for moving (translating and rotating) MR data on a grid: the discrete Fourier transformation, the chirp-z transformation, and rotation by successive shearing. We are interested in both the comparative accuracy and the relative computational efficiency of these different approaches.

The principal messages of this chapter are contained in the argument that runs through the first three sections. The final section expands the scope of the discussion by turning to specific issues pertaining to 3D sampling and resampling. Given the way in which MR data are acquired, what are the implications for 3D processing and imaging? In particular, as we synthesize a 3D data volume from an inherently 2D acquisition process (via resampling and interpolation), what is the impact on the conservation of information and the preservation of statistical properties?

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