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I am having some trouble working through verifying linearity and shift invariance when the transformation is under a summation.

The given transformation is as follows: $$ y(m,n)=\sum_{i=-1}^{i=1}\sum_{j=-1}^{j=1}x(m+i,n+j) $$

I understand the basics of verifying linearity ($T[a_1f_1(x,y)+a_2f_2(x,y)]=a_1T[f_1(x,y)]+a_2T[f_2(x,y)]$) and shift invariance ($g(x-h,y-k)=T[f(x-h,y-k)]$), but I am unsure how to apply them to the given transformation. I could not find any other questions that covered this situation. It has been a good while since I have had to do this math, so I am just getting back into the subject.

Any help would be much appreciated.

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  • $\begingroup$ Hint: let $x(m,n)$ be the $m$th row and $n$th column of an $M \times N$ image. Then, $y(m,n)$ is the 2D convolution of $x(m,n)$ with a 2D window of ones. $\endgroup$
    – mhdadk
    Commented May 31, 2023 at 18:32
  • $\begingroup$ Funny, as this is related to image processing. Could you explain a bit more about what you mean by the 2D window of ones? Assuming it is discrete, then I could assume the summation is -1,0,1 for i and j? $\endgroup$
    – pflykyle
    Commented May 31, 2023 at 19:44
  • $\begingroup$ See here: ai.stackexchange.com/a/20062/41856 and set $h[m,n] = 1$ for all $m,n$. $\endgroup$
    – mhdadk
    Commented May 31, 2023 at 22:06

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