p.272 Twice brighter sample

There is an example of importance sampling based Light source and BRDF in p.270, Figure 9.6. Here is the figure. In this case, to avoid biasing, importance weight is compensated by probability. This leads the following interesting equation.

where, \hat{w_2} = p_2/(p_1 + p_2). Notice that the values of p_1, p_2 here. When p_1 is non-zero, usually p_2 is smaller than p_1 (p_1 >> p_2). However, the Equation is the case of p_2 sample happens in p_1 's non-zero region as shown as a red line in Figure1. In general, since the region of p_2 is larger than p_1, this usually doesn't happen. But if it happens, the condition is f/(p_1 + p_2) -> f/p_1, and this is almost equal to p_1 's contribution because p_1 >> p_2 . Veach explains this is seen as a bright spots in Figure 9.5(c). This is interesting. Of course this problem is when the number of sample is quite low and this kind of bad luck is visible.

When the number of samples increases, this kind of bad luck situation becomes less effect since the difference of density distribution p_1 and p_2 . This is an example the variance is large when there are less samples.

p.282 Figure 9.12(a)

Figure 9.12(a) shows the dark and noisy region near the light source. But this method is light source sampling, therefore, the darkness isn't caused by sampling miss. So why is neat the light region dark?

The reason is sample points, that is perpendicular to the area light, are too close to the light source. In this setting, the cos term becomes almost zero.

Also if the sample point is too close to the area light source, geometry term is unstable shown in Figure 2. If there are some distance between sample point and an area light, when the sample happens at upper region and lower region in the light source, the distance is rather stable. On the other hand, the sample point is too close to the light source, the distance can be almost zero to the other side of the light source. This causes high variance. This is a kind of paradoxical situation, near the light source points becomes dark and noisy.

Acknowledgements

Thanks to Matthias, Carsten, Daniel.

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