Master Thesis - Computer Graphics and Visualization - TU Delft

2.7 BiDirectional Path Tracing Unbiased Rendering
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light transport path. However, it is not always evident which importance sampling strategy
is most optimal in a certain situation. For highly glossy materials, sampling the last vertex
according to the local BSDF often results in less variance than when making an explicit
connection. This is because the glossy BSDF is highly irregular, but explicit connections
do not take the shape of the BSDF in consideration.
By dropping the path space partition restriction, multiple sampling techniques may be
combined to form more robust samplers. Multiple Importance Sampling (MIS) seeks to
combine different importance sampling strategies in one optimal but unbiased estimate [51].
A sampler may generate paths according to one of n sampling strategies, where pi(X) is the
probability of sampling path X using sampling strategy i. Per sampling strategy i, a sampler
generates ni paths Xi,1 ···Xi,ni . Hence, path Xi, j is sampled with probability pi(Xi, j). These
samples are then combined into a single estimate using
n 1
Ik = ∑
i=1
ni
ni
∑ wi(Xi, j)
j=1
f (Xi, j)
pi(Xi, j)
(2.11)
In this formulation, wi is a weight factor used in the combination. For this estimate to
be unbiased, it is enough that for each path X ∈ Ω with f (X) > 0, pi(X) > 0 whenever
wi(X) > 0. Furthermore, whenever f (X) > 0, the weight function must satisfy
n
∑
i=1
wi(X) = 1 (2.12)
In other words, there must be at least one strategy to sample each contributing path and
when some path may be sampled with multiple strategies, the weights for these strategies
must sum to one. A fairly straightforward weight function satisfying these conditions is
wi(X) =
nipi(X)
∑ n j=1 n j p j(X)
(2.13)
This weight function is called the balance heuristic. Veach showed that the balance heuristic
is the best possible combination in the absence of further information [50]. In particular,
they proved that no other combination strategy can significantly improve over the balance
heuristic. The general form of the balance heuristic, called the power heuristic, is given by
wi(X) =
2.7 BiDirectional Path Tracing
nipi (X) β
∑ n j=1 n j p j (X) β
(2.14)
In the PT sampler, all but the last path vertex are sampled by tracing a path backwards from
the eye into the scene. This is not always the most effective sampling strategy. In scenes
with mostly indirect light, it is often hard to find valid paths by sampling backwards from
the eye. Sampling a part of the path forward, starting at a light source and tracing forward
into the scene, can solve this problem.