"The 64 x 64 pixel image of a brain was rotated k/64 radians, translated 1/4 pixel on the diagonal, translated back 1/4 pixel, and rotated back k/64 radians. All statistics were computed over a 40 x 40 central subimage. The mean squared difference (MSD) was calculated between the input (unmoved) image and the one that was moved. Also, the minimum, maximum, mean, and variance are compared for input and output images, as well as their zerolag covariance. Two different implementations were compared for both Fourier and linear methods of resampling.

included some summary information on the final image: maximum value (Max), minimum value (Min), and average value (Mean). A number of the implementations had minor artifacts at the edges of the final image. In order to downplay the influence of these edge effects, we computed the statistics over a 40 x 40 central subimage.

Two observations that can be made from this table are (a) only the full Fourier technique appears to move and then restore the position of an image accurately; (b) implementations differ; i.e., the same basic method implemented in different ways can yield results with different accuracy. These results are consistent both with our understanding of sampling theory and with the discussion in Section 3 concerning the fidelity of the full Fourier method in conserving information. The windowed sinc methods improve with increasing operator length (or bandwidth). The Fourier method is superior because it does not suffer from the edge effects (Gibbs phenomenon) of the windowed sinc methods.

Results from the same experiment are also portrayed in a statistical light. We give the variance across the final image and the covariance between the input and original and final image. It is clear that, for most methods, one of the consequences of performing motion correction is a reduction in variance — that is, a falsely elevated degree of correlation — among the pixels in the image.

Figure 1 reinforces the message of Table 1 in a pictorial fashion. Three different methods, full Fourier, 4-point windowed sinc (WS4), and linear, were used to move an image in the "round trip" described for Table 1. Figure 1A shows obvious degradation of the image moved by the linear approach, whereas both the Fourier and WS4 methods appear to produce high-fidelity images. Figure 1B shows the result of subtracting each of the moved images from the input image. Here it becomes evident that the full Fourier method has conserved the information in the original image but the other two methods have introduced discrepancies.

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