# Choice of axes in 1D cross-correlation of a signal in a 2D space

I am working on a research project in which we present an observer with a target moving in a random walk in a plane in front of them, and record the eye movements they make to track it. We cross-correlate the movements of the stimulus and the eye movements in the response to estimate how well the observers are tracking the target and how long the latency is between its movements and their eye movements.

We have been using normalized 1D cross-correlation in each of the X and Y directions (normalized based on the overlap between the two data vectors, so that all correlations are between +1 and -1). However, there have been questions raised about doing the cross-correlation along different axes or by doing a circular cross-correlation in angular space as described on pages 2 and 8 of this text.

I have a couple questions:

1. Do we get different information by cross-correlating along different pairs of independent axes? If we were only comparing the raw signals, it would be clear that changing axes in this way is a linear transformation that just represents the same information in different coordinates. But when we're performing the presumably non-linear operation of normalized cross-correlation, that's not obvious to me anymore.

2. What about circular cross-correlation as described in that link? My intuition suggests that it tells us something about angular movement that is not preserved if we collapse down to the X and Y directions individually. Is that right at all?