When using a gaussian kernel with sigma as it's parameter you actually using:
Now, the minimum value quantized is 1(gray level), the gaussian tail beyond it is irrelevant.
Our goal is to stop before we reach the edge of one gray level, let us denote this edge as r
, and the we get to solve:
the +1 is because we want the radius to be non-inclusive. taking log of the above, and reordering gives us:
Which is what you searched for.