# Gaussian Pyramid - How is Subsampling Rate Related to Sigma?

I found a gaussian pyramid implementation in a MOPS paper (feature detection). They use sampling rate $s=2$ and $\sigma=1$ - i.e. to generate a new level of the pyramid, the current level is smoothed with Gaussian blur of that sigma and then subsampled. The same parameters are used to build each new level.

However, I need to use smaller sampling rates, e.g. $s=1.5$, to get more pyramid levels (non-integer sampling rates would be achieved by interpolation).

How to choose appropriate sigma, knowing the sampling rate?

My first guess is to use $\sigma=\sqrt{s/2}$, since the variance of the gaussian filter is half the sampling rate (radius) and sigma (standard deviation) is square root of that quantity. But I am not sure if that's correct.

In another words: Given a sampling rate, I need to pick gaussian blur sigma preventing aliasing.

• Why more pyramid levels? – geometrikal May 24 '16 at 23:50
• @geometrikal Because the feature detector is very sensitive to scale change so I need many scales to make it effectively scale-invariant. It would be better to use something like SIFT, which find maxima in both space and scale, but for now I need something simple. – Libor May 26 '16 at 7:38

So one would choose $\sigma$ to keep the amount of energy in the aliased bands to an acceptable level. Using $\sigma = 0.25 f_s$ would give 68%, $2\sigma = 0.25 f_s$ would give 95%, and so on.