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Bi-directional Estimators for Light Transport

Veach proposed a bi-directional ray-tracing algorithm that is similar to the previous one, but it uses combined sampling techniques [VG94]. In this algorithm the eye path and the light path are only connected at the end points by a deterministic shadow ray. The pixel intensities are estimated according to the Monte Carlo method:

Figure: A complete transport path

sum=0 for n=1 to N for k=1 to Max-length x=CHOOSE-PATH(P,k) sum=sum+f(x)/p(x) return sum/N

CHOOSE-PATH(P,k) generates a path x of length k which potentially contributes to the intensity of pixel P. f(x) is the differential contribution to P of light flowing along x,and p(x) is the probability density with which CHOOSE-PATH generates x.

Veach proposed combining sampling techniques for generating random paths, where the distribution of the samples is constructed from several tex2html_wrap_inline981 importance sampling distributions. The relative number of samples taken from tex2html_wrap_inline983 is denoted tex2html_wrap_inline985 , where tex2html_wrap_inline987 . The tex2html_wrap_inline985 values are fixed in advance before any samples are taken. The resulting distribution can be written as:



Using the combining sampling techniques unbiased and low variance estimators can be constructed that allow samples to be weighted differently, depending on which distribution tex2html_wrap_inline983 they where chosen from. Each estimator is parametrised by a set of weighting functions tex2html_wrap_inline993 , where tex2html_wrap_inline995 gives the weight associated with a sample x drawn from tex2html_wrap_inline983 . The combined estimator is given by


where the tex2html_wrap_inline1001 is the jth sample from distribution tex2html_wrap_inline983 . For the unbiased estimation tex2html_wrap_inline1007 for all x.

Csébfalvi Balázs
Tue Apr 15 18:39:13 METDST 1997