Multiple importance sampling characterization by weighted mean invariance

Abstract

In this paper, we examine the linear combination of techniques and multiple importance sampling for Monte Carlo integration from a new perspective of quasi-arithmetic weighted means. The invariance property of these means allows us to define a new family of heuristics. We illustrate our results with several rendering examples, including environment mapping and path tracing.

Appendix

Further justification for the heuristics $\alpha _k \propto \frac{1}{c_k v_k}$ used in Sect. 5.3 comes from the following. Observe that for $\alpha _k \propto \frac{1}{c_k v_k}$, the second term of Eq. 13 becomes

$$\begin{aligned} \frac{H(\{c_k v_k\})^2}{H(\{c_k\}) H(\{v_k\})}, \end{aligned}$$

(15)

while for $\alpha _k = \frac{1}{M}$ the second term of Eq. 13 becomes

$$\begin{aligned} A(\{c_k\}) A(\{v_k\}), \end{aligned}$$

(16)

where $A(\{v_k\})$ stands for arithmetic mean of $\{v_k\}$. We want to see that the expression in Eq. 15 is less than the expression in Eq. 16. A necessarily unfavorable case is when $c_i \propto 1/v_i$. Let us consider without loss of generality that the proportionality constant is 1, i.e., $c_i = 1/v_i$. In that case $H(\{v_k\}) = A(\{c_k\})^{-1}$, $H(\{c_k v_k\}) = 1$, and both expressions in Eqs. 15 and 16 are equal. In the necessarily favorable case $c_i = v_i$, and the expression in Eq. 15 becomes $\frac{H(\{v_k^2\})^2}{H(\{v_k\})^2}$, but $\frac{H(\{v_k^2\})}{H(\{v_k\}) } \le H(\{v_k\})$, as $H(\{v_k\})$ corresponds to a power mean with exponent $-1$ while $\sqrt{H(\{v_k^2\})}$ to exponent − 2. Thus $\frac{H(\{v_k^2\})^2}{H(\{v_k\})^2} \le H(\{v_k\})^2 \le A(\{v_k\})^2$.