Stack SNR  Impact of resampling during alignment
Looking at Ray's imaging design spreadsheet I was reminded of some thoughts I had on stack SNR calculations, and wondered if this analysis sounds plausible.
There's no real practical benefit in this, it doesn't improve the stack SNR achieved! It just means that the stack SNR will often be better than might be expected.
The "normal" SNR calculation goes as follows:
Signal in one sub = 50e
Total noise in one sub = 10e
Hence, SNR for one sub = 5.0
But the SNR for a 4 sub stack is better, it's SQRT(4) times better, so = 10.0
In general:
Stack SNR = Individual sub SNR x SQRT(No of subs)
But, my thinking is that, in practice:
Stack SNR = Individual sub SNR x SQRT(No of subs) x 1.5
This is where I get the extra 1.5 factor from...
When stacking, individual subs are aligned. For example, a sub might need to be moved by 2.5px in x and 4.5px in y to align it with the reference sub.
But, in order to do this alignment, the subs must be resampled (let's say by bilinear interpolation). In this example, the resampled pixel that is mapped to (100,100) in the reference sub would be resampled by taking the mean value of the original (102,96), (103,96), (102, 95) and (103,95) pixels.
However, in taking the mean of these 4 pixels, by normal error combination theory, the noise (or error) in each resampled pixel is reduced from the original noise. In this example, the original noise is reduced by SQRT(4) = 2.
Of course, if the alignment offset was a whole number of pixels in x and y (and no rotation was necessary), resampling would not reduce the noise per pixel. In this case, the pixels, with all their original noise, would just be shifted to be directly on top of reference pixel positions.
On the other hand, if the offset was something else, like 0.25 px in x and 0.25px in y, resampling would reduce the noise by between 1 and 2. In fact, a bit of arithmetic shows it to be reduced by 1.6.
The fractional part of the offset in pixels between subs in practice are random, e.g. 0.3 when the offset was 2.3 pixels. It can be shown that for a random offset in both x and y, the average noise reduction introduced by resampling during stacking is 1.5 (assuming bilinear interpolation when resampling).
