Gaussian re-blur in image processing

Question

I have two questions related to Gaussian re-blurring.

What happens when we reapply the Gaussian blur to the image, keeping the sigma and radius the same?

And is it possible that after n iterations of repeated use of gaussian blur ( sigma = s1 ), the image will become the same as when applying gawussian blur (of sigma = s2; s1 < s2 ) 1 time in the original image. And if so, what is the general formula for obtaining this n number of times, we must apply the Gaussian blur with s1 given by s1 and s2 ( s1 < s2 ).

Answer 1

Applying several Gaussian blurring to an image successively, it has the same effect as applying one 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 approach 3*si , where i = 1, 2 , which means that pixels at a distance of more than 3si are small enough to be considered equal to zero during the blur process.