I need to compute one specific value in the original domain from the 2D frequency domain data I have. I can't just use IFFT for a whole set for performance reasons.
I know how to do this in 1D by computing the sum of cosines, but 2D case is a bit confusing.
If what I wrote isn't clear enough I will put it in some context as suggested by Hrodelbert. The frequency domain in question is ocean waves spectrum. What I ask how to compute is actual water height at a choosen point.
Just evaluate the formula for that point. That is, if $F(n,m)$ is the frequency domain data and you want the backwards transform at the point $(x_0,y_0)$, just calculate
$$ f(x_0,y_0) = \sum_{n=0}^{N-1} \sum_{m=0}^{M-1} F(n,m) e^{j2\pi nx_0/N} e^{j2\pi my_0/M}. $$
It's that simple. The difficulty arises when you want to find $f(x,y)$ for large ranges of $x$ and $y$. In that case, the above quickly becomes computationally burdensome. However, if you are interested only in a single point, all you have to do is weight each sample and sum once.