ABSTRACT
Splitting the domain of integration [sa, sb] into n non-overlapping intervals along the ray such that sa = s0 < s1 < . . . < sn−1 < sn = sb, the integral form of the RTE in participating465 media may be reformulated as440
L(~sa, ωˆ) (11.113)466
T (~sa, ~s)κt(~s)Lu(~s, ωˆ)ds+ T (~sa, ~sb)L(~sb, ωˆ)
T (~s0, ~sj−1) ∫ sj sj−1
T (~sj−1, ~s)κt(~s)Lu(~s, ωˆ)ds+ T (~s0, ~sn)L(~sb, ωˆ)
T (~si−1, ~si)
Lm(~sj−1, ~sj , ωˆ) + (
T (~si−1, ~si) ) L(~sb, ωˆ) (15.1)
Although the most obvious approach consists in using a fixed step size, the sampling process can be made more effective by coarsely evaluating the portions of the integration domain that are likely to have a smaller impact on the final estimate. Without any a priori knowledge about the radiance term, the size of the intervals may be optimally defined such that the cumulated opacity increases in constant steps. Doing so uniformly distributes the samples in the domain of the associated cumulative distribution, effectively leading to coarser132 steps as the cumulated transmittance factor of the intervals decreases.
