Recently I feel a bit confused about one-sample model MIS. One sample model MIS can be found here in Veach 1997. 9.2.4 The one-sample model, and this one-sample model is widely used in the opensource renderers, for example: the combination of BSDF sampling and emitter sampling in mitsuba2/3 and pbrt, path connection in BDPT for mitsuba and pbrt, etc. Basically, the estimator is:
$$ \frac{w_I(X)f(X)}{c_Ip_I(X)} $$ $c_I$ is the discrete PMF for choosing $I$th sampling strategy, $p_I(·)$ is the PDF for the $I$th sampling strategy. This is reasonable, since I can easily prove that with several unbiased estimator, the one-sample model is unbiased. Here I present a simple case where balance heuristic is used:
$$ \mathbb{E}\left(\frac{w_I(X)f(X)}{c_I p_I(X)}\right) = \int\frac{\frac{p_1(X)}{p_1(X) + p_2(X)}f(X)}{p_{c_1} p_{1}(X)}\times \left(p_{c_1} p_1(X)\right) + \frac{\frac{p_2(X)}{p_1(X) + p_2(X)}f(X)}{p_{c_2} p_{2}(X)}\times \left(p_{c_2} p_2(X)\right)dX = \int \frac{p_1(X) + p_2(X)}{p_1(X) + p_2(X)}f(X) dX= \int f(X) dX $$
My problem is:
From the opensource renderers, I find that the one-sample model used in them only samples via one strategy (other strategies have zero probability to get sampled). For example:
- In path tracer NEE, we only use emitter sampling (and we don't actually sample BSDF when doing NEE). This can be found in pbrt-v3 and mitsuba2/3.
- In BDPT path connection, we only sample "once" and calculate the weight for that exact path.
Is this correct for one specific strategy to have zero probability to be sampled? I know that $p_{c_I}$ in the nominator and denominator will cancel each other out, but it still doesn't feel right.
If I use balance heuristic, the estimator can actually be simplified when we only use one strategy to sample (the corresponding discrete PMF is 1):
$$ \frac{\frac{p_1(X)}{p_1(X) + p_2(X)}f(X)}{p_{1}} = \frac{f(X)}{p_1(X) + p_2(X)} < f(X) / p_1(X) $$ My problem for the eqn above is: Can I say that using one-sample MIS with heuristic will produce smaller mean-value than the original Monte Carlo estimator? Since I know that the original estimator is unbiased, doesn't this indicate that the one-sample MIS which samples via only one strategy is biased? Did I misunderstand anything here?