In these slides, specifically page 11, the following formula is reported:

$$ \frac{1}{N} \sum_{k=1}^N \frac{L_i(l_k)f(l_k,v)\cos(\theta_{l_k})}{p(l_k,v)} \approx \left( \frac{1}{N} \sum_{k=1}^N L_i(l_k) \right) \left( \frac{1}{N} \sum_{k=1}^N \frac{f(l_k,v)\cos(\theta_{l_k})}{p(l_k,v)} \right) $$

The formula above should be an approximation of the rendering equations. I'm looking for a paper that explains how that formula is derived, I cannot manage to find neither the title of a paper (other than the name "Dimitar" mentioned in the same page I pointed out).

I've done some research about some possible method that would justify. This is the closest thing I managed to find, however there's some hypothesis that doesn't seem to me fits.

  • $\begingroup$ "Dimitar" refers to the presentation by Dimitar Lazarov, earlier in the same SIGGRAPH course. See page 21 there for the "split approximation". $\endgroup$ Commented Jan 11, 2018 at 22:30

1 Answer 1


I do not have a derivation but i can explain the reasoning behind its usage.

On the left side is a (monte carlo) estimator for the rendering equation. It states the very same thing that the rendering equation states but on a finite number of discrete samples.

on the right side we have the $L_i(l_k)$ factor separated and "integrated" on its own. This approximation holds true when $L_i(l_k)$ is a constant, since $\sum_{k}f(k)\times n = n \times\sum_{k}f(k)$ - and, likewise, $\int_{k}f(k)\times n\space dk = n\times\int_{k}f(k)dk$ - when $n$ is a constant.

$L_i(l_k)$ can be though of the colour of the incoming light and $\int_{k}f(l_k,v)dk$ can be thought of as the intensity of the reflected light. multiplying those together you get that now you can precompute $\int_{k}L_i(l_k)dk$, saving a lot of time in either texture lookups (in the case of sky / environment lighting / screen space ray marching) or raytracing calculations.

However, since $L_i(l_k)$ is usually not a constant, this integral is not separable. Yet, in some cases (like when dealing with diffuse lighting or, more specifically, diffuse ambient lighting), the information that $f(l_k,v)$ carries is low-frequency enough that it can reasonably be approximated as a constant with respect to the distribution of $L_i(l_k)$. Experimentally, it can be shown that the final result is not too different from what you would expect.

  • $\begingroup$ I'm missing the extra 1/N in your explanation. $\endgroup$ Commented Jan 12, 2018 at 0:23
  • $\begingroup$ @user8469759 the 1/N is implicit in the integrals. it comes from "averaging" samples in numerical integration $\endgroup$ Commented Jan 12, 2018 at 3:14
  • $\begingroup$ Still, there's one more factor 1/N. Also I agree the BRDF is almost constant in realistic cases, but I wouldn't be sure of the $cos$ factor and the probability $p$ due to the importance sampling process. $\endgroup$ Commented Jan 12, 2018 at 10:30
  • $\begingroup$ those are meant to be "distributed" into the sigma notation sums, that is why there are parentheses. they should really be interpreted as integrals. $\endgroup$ Commented Jan 14, 2018 at 2:00
  • $\begingroup$ Hi, what do you mean by "distributed"? could you elaborate that? $\endgroup$ Commented Feb 6, 2018 at 15:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.