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Here is an unknown luminosity function $f(x,y)$ and its integration results: $$p_{i,j}= \frac{\iint\limits_{D_{i,j}} \! f(x,y) \, dx \, dy}{\iint\limits_{D_{i,j}} \,dx\,dy}$$ I need to express the result of the following transformation in terms of $p_{i,j}$ and $g(x,y)=\frac{1}{2 \pi \sigma^2} e^{-\frac{x^2+y^2}{2\sigma^2}}$? $$q_{i,j}=\frac{\iint\limits_{D_{i,j}} \! f(x,y) g(x,y) \, dx \, dy}{\iint\limits_{D_{i,j}} \,dx\,dy}$$ How can I do this?

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You can't. You only know the integral of $f$ over each region, and the $q_{i,j}$ aren't determined by those integrals. If the regions are small enough that $g$ doesn't vary too much over a region, you can approximate $g$ by its average over the region to get $q_{i,j}\approx g_{i,j}p_{i,j}$, where

$$ g_{i,j}= \frac{\iint\limits_{D_{i,j}} \! g(x,y) \, \mathrm dx \, \mathrm dy}{\iint\limits_{D_{i,j}} \,\mathrm dx\,\mathrm dy}\;.$$

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