Comment by ColinWright

Ah! Now I understand.

I was stepping out with my wife for a day out and had read your reply very cursorily. That reading had left me quite puzzled -- "I would have done exponentially weighted moving average (EWMA) over time for trails. Why is \phi important here in any form. Is \phi the weight of the EWMA ?".

Now I get it, decrementing the pixels were quite peripheral to the main story.

The main story is that of finding a scan sequence that (a) cycles through a set of points without repetition and (b) without obvious patterns discernible to the eye.

In this, the use \phi is indeed neat. I don't think it would have occurred to me. I would have gone with some shift register sequence with cycle length 1024 * 1024 or a space filling curve on such a grid.

This becomes even more interesting if you include the desiderata that the minimum distance between any two temporally adjacent pixels must not be small (to avoid temporal hot spots).

Finding MiniMax, min over temporal adjacency, max over all 1024* 1024! sequences, might be intractable.

Another interesting formulation could be, that for any fixed kxk sized disc that could be drawn on the grid, the temporal interval between any two "revisit" events need to be independent of the disk's position on the grid.

I think this is the road to small discrepancy sequences of quasi Monte Carlo.