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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?

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From Wikipedia:

Applying multiple, successive 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.

So in your case, that would be equivalent to blurring the image with a single Gaussian with $\hat{\sigma}$: $$ \hat{\sigma} = \sqrt{8\sigma^2} = 2\sqrt{2} \sigma $$

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    $\begingroup$ In the continuous and infinite support limit $\endgroup$ – Laurent Duval Nov 13 '15 at 14:59
  • $\begingroup$ Well, yes, there will be discretization effects that will mean the image processing version won't quite match that, but it's a close enough approximation for most purposes. $\endgroup$ – Peter K. Nov 13 '15 at 15:00
  • $\begingroup$ Of course, for objects with appropriate size (with respect to $\sigma$) away from the borders. Just mentioning for the record $\endgroup$ – Laurent Duval Nov 13 '15 at 15:14

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