How to convert infinite intergral to sum

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How to convert Wiener filter formulas from integral to sum? They are for images therefore it must be possible to convert them to sums. Any help will be appreciated:

I could not find much info on this online as my integral is from $\infty$ to $-\infty$. For example how would you convert the following?

$\int\limits_{-\infty}^{\infty}f(r)[x(t-r)+ y(t - r)]\;dr$

And is it basically changing all the occurrences of $r$ or each conversion requires extensive computation?

How would you go about double integrals? Is it just another summation added? (with it's limit)