Repeated Gaussian Blur in Image Processing

Question

I have two questions relating to repeated Gaussian blur.

What happens when we repeatedly apply gaussian blur to an image keeping the sigma and the radius same ?

And is it possible that after n iterations of repeatedly applying gaussian blur (sigma = s1) the image becomes the same as it would be on applying gaussian blur ( of sigma = s2; s1 < s2 ) 1 time on the original image. And if so what is the general formula for deriving that n number of times we have to apply gaussian blur with s1, given s1 and s2 (s1 < s2).

Answer 1

Successively applying multiple gaussian blurs to an image has the same effect as applying a single, larger gaussian blur, whose radius is the square root of the sum of the squares of the blur radii that were actually applied. In your case, s2 = sqrt(n*s1^2), and the blur radii is approximated as 3*si where i = 1, 2, which means pixels at a distance of more than 3si are small enough to be considered effectively zero during the blurring process.