When scaling an image down it is common to first convolve the image with a Gaussian filter. Given a scale factor $0 < s < 1,$ what is an appropriate $\sigma$ to use?
For example, if $s = 0.5$ (halving the width of the image) it seems reasonable to choose $\sigma = 1$ since it will weight a value in the middle by 40%, the two neighbors by 25% each, and their neighbors by 5% each, i.e. 1-D filter looks like this approximately:
[0.05, 0.25, 0.40, 0.25, 0.05]
This seems reasonable since you will choose every other value when you subsample and the the neighboring values will have about 50% influence.
If $s = 0.25$ (a quarter of the width) it makes sense to use $\sigma = 2$ which yields a 1-D filter that looks like the following (weight 22% for center pixel):
[0.03, 0.07, 0.13, 0.19, 0.22, 0.19, 0.13, 0.07, 0.03]
Thus the filter's radial influence seems appropriate when choosing every 4th value when downsampling.
So my intuitive rule of thumb is $\sigma = 1/(2s).$
Following this, if $s = 0.75$ then $\sigma = 2/3$ and my filter looks like this
[0.007, 0.194, 0.598, 0.194, 0.007]
This was purely an intuitive approach to choosing filter parameters. I couldn't find a more theoretical choice (I am sure it will involve Fourier Transforms). What is a good "rule" for picking $\sigma$ given $s?$