# What is the equivalent smoothing function to running the same Gaussian 8 times?

Suppose there is an image which is to be smoothed by convolving it with a Gaussian kernel with standard deviation $\sigma$. If the image is then smoothed with this kernel 8 consecutive times, is the total smoothing still Gaussian with some calculable standard deviation $\hat{\sigma}$? Or is it some other distribution instead?

So in your case, that would be equivalent to blurring the image with a single Gaussian with $\hat{\sigma}$: $$\hat{\sigma} = \sqrt{8\sigma^2} = 2\sqrt{2} \sigma$$
• Of course, for objects with appropriate size (with respect to $\sigma$) away from the borders. Just mentioning for the record – Laurent Duval Nov 13 '15 at 15:14