# reference volume and Euclidean norm

Should the choice of reference volume when motion correcting have any impact on how enorm is computed?

If you are asking, “Should the Enorm values be the same regardless of what reference volume is used during motion correction?” then my initial thought would be: No.

Consider two similar, good quality reference volumes, A and B, and trying to align other good quality data to either. Because the two reference volumes look quite similar, the result of aligning to either should be quite similar, meaning that the alignment parameters would be very similar in each case. However, they are not the exact same, so we wouldn’t expect exactly the same cost function outcomes; and alignment is a stochastic process (it involves randomness, whilst searching through the vast emptiness of parameter space), so the alignment parameters will likely not be exactly the same even if the volumes were exactly the same. Therefore, the enorm values that are derived from these alignment parameters will not be the same either-- but, the results would likely be quite similar if A and B are quite similar (particularly if using, for example, a nice, stable cost function like “lpc+zz”). Note that trying to align spiky, noisy data to even good reference volumes might lead to pretty different alignment parameter (and therefore enorm) outcomes, because of the bigger variability in cost function values.

However, the more different that reference volumes look, the less similar one would expect alignment parameters to be-- how quickly they diverge, I don’t really know; it would be highly data/example dependent. But then the enorm values could easily be pretty different.

–pt

Thanks.That’s helpful. I guess I’m also asking whether it should be computed the same way regardless of reference volume. Should the enorm value of zero be associated with the first volume even if the volume with minimum outliers was the reference volume? Or does it matter?

Hi-

Consider at time series of length N with zero-based counting, so the time index goes from [0, N-1]. “enorm” is calculated by using the difference of neighboring time point values. Therefore, for N time points, one has “N-1” difference values: at time [i], the value of enorm depends on the value at times [i] and [i-1], for i=1,…,N-1. The th value of the enorm time series is always 0—basically, at time , the subject didn’t move with respect to their previous time point (because there wasn’t want to calculate a difference from yet).

NB: the value of enorm might also be 0 at the start of each concatenated time series, if you have multiple time series entered into processing, for the same reasoning.

–pt