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I was taught in my image processing class that when a mean filter is applied infinite times on a given image, the intensity of each pixel reaches the same value. I understood this that time entirely on intuition. But looking back, I tried to come up with the formal proof of this result but couldn't make any progress. So is there a way to actually proof this notion. Any help would be appreciated.

gkgkgkkgkgkgk
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  • The Central Limit Theorem says that the convolution of a probability distribution with itself $n$ times, as $n\to\infty$, tends to the Normal Distribution with a variance of $n\sigma^2$, where $\sigma^2$ is the variance of the original distribution. – robjohn Sep 02 '23 at 13:35

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The convolution operation is associative.
So the operation is:

$$ \boldsymbol{H} \ast \boldsymbol{H} \ast \dots \ast \boldsymbol{H} \ast \boldsymbol{I} = \left( \boldsymbol{H} \ast \boldsymbol{H} \ast \dots \ast \boldsymbol{H} \right) \ast \boldsymbol{I} $$

The equivalent kernel of infinite convolutions has infinite support.
Hence, for any shift over the image, it actually sums the same pixels.
Hence the output has the same value for pixels locations.

Royi
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