In practice, there are a finite number of samples of the residual field that are available. This takes the form of a regular 4D array of samples, including the three spatial dimensions and the temporal dimension. These samples (post-normalisation) shall be denoted as: where refers to the time index and takes discrete values. The number of samples in each dimension is by by by , with being a shorthand for the number of voxels.
Now, each sample point in the 4D field is a random variable, with expectation
given by equation 30. Therefore, due to the linearity of
the expectation, the results can be averaged over all the sample points
to achieve a more accurate estimate. That is, for a single point in time only:
(33) |
Note that , and is the same notation used in [2].