ABSTRACT
Le(~p→ ~o) + ∫ S fr(~o, ~p, ~q)G(~p, ~q)Lo(~q → ~p)dA(~q) (13.8)
= Le(~p→ ~o) + (TLLo)(~p→ ~o) = (SLLe)(~p→ ~o) (13.9)
where the radiance transport operator reads
(TLLo)(~p→ ~o) , ∫ S fr(~o, ~p, ~q)G(~p, ~q)Lo(~q → ~p)dA(~q) (13.10)
while the solution operator SL (4.128) 150
= 1 + TLSL = (1− TL)−1 is defined in terms of the global reflectance distribution function (GRDF) fg [Dutre´ et al., 1994, Lafortune and Willems, 1994a, Lafortune, 1996] as
(SLLe)(~p→ ~o) , ∫ S
∫ S fg(~o, ~p, ~q, ~r)G(~q, ~r)Le(~r → ~q)dA(~q)dA(~r) (13.11)
13.3.2 Importance Transport
Considering adjoint photons, called importons, conceptually emanating from the sensor and propagating into the scene towards the light sources, radiance transport may alternatively 564 be mathematically described in terms of an adjoint quantity known as importance. Given the emitted importance We(~p, ωˆi) of the sensor’s surface at point ~p (such as the local response defining the exposure of a film or the radiosity of a surface patch in the radiosity method), 657
593the exitant importance is given by the importance equation [Pattanaik and Mudur, 1993c, Pattanaik, 1993a, Ch. 6] as
Wo(~p, ωˆi) , We(~p, ωˆi) +Wr(~p, ωˆi) (13.12)
where, by exploiting the adjoint conservation of importance in free space to reformulate the incident importance Wi(~p, ωˆo) , Wo
( v(~p, ωˆo),−ωˆo
) , the reflected importance reads
Wr(~p, ωˆi) , ∫
Ω Wi(~p, ωˆo)fr(~p, ωˆo, ωˆi)〈nˆp, ωˆo〉dωˆo (13.13)
∫ S Wo(~o,−p̂o)fr(~p, p̂o, ωˆi)G(~o, ~p)dA(~o) (13.14)
Defining a surface point ~q such that p̂q = ωˆi as well as the shorthand notations Wo|e(~p→ ~q) , Wo|e(~p, p̂q), the importance equation may then be expressed as a Fredholm integral equation of the second kind known as the three-point form 150
Wo(~p→ ~q) (13.12) 565
We(~p→ ~q) + ∫ S Wo(~o→ ~p)G(~o, ~p)fr(~o, ~p, ~q)dA(~o) (13.15)
= We(~p→ ~q) + (TWWo)(~p→ ~q) = (SWWe)(~p→ ~q) (13.16)
(TWWo)(~p→ ~q) , ∫ S Wo(~o→ ~p)G(~o, ~p)fr(~o, ~p, ~q)dA(~o) (13.17)
while the solution operator SW (4.128)150
= 1 + TWSW = (1 − TW )−1 is defined in terms of the GRDF as
(SWWe)(~q → ~r) , ∫ S
∫ S We(~o→ ~p)G(~o, ~p)fg(~o, ~p, ~q, ~r)dA(~p)dA(~o) (13.18)
13.3.3 Measurement Equation
Similarly to incident power or the exposure of an optical system, a measurement is defined657 by the so-called measurement equation as the integral of incident radiance modulated by the sensor’s importance function
M , ∫ S
∫ Ω Li(~p, ωˆi)We(~p, ωˆi)〈nˆp, ωˆi〉dωˆidA(~p) (13.19)
∫ S We(~p, p̂q)G(~p, ~q)Lo(~q,−p̂q)dA(~q)dA(~p) (13.20)
∫ S We(~p→ ~q)G(~p, ~q)Lo(~q → ~p)dA(~q)dA(~p) (13.21)
Recursively substituting the three-point form of the rendering equation into the measurement equation then yields
∫ S We(~p0 → ~p1)G(~p0, ~p1)Lo(~p1 → ~p0)dA(~p1)dA(~p0)
∫ S We(~p0 → ~p1)G(~p0, ~p1)
× ( Le(~p1 → ~p0) +
∫ S fr(~p0, ~p1, ~p2)G(~p1, ~p2)Lo(~p2 → ~p1)dA(~p2)
) dA(~p1)dA(~p0)
∫ S We(~p0 → ~p1)G(~p0, ~p1)Le(~p1 → ~p0)dA(~p1)dA(~p0)
∫ S We(~p0 → ~p1)F (~p0, . . . , ~p2)Lo(~p2 → ~p1)dA(~p2)dA(~p1)dA(~p0)
= ...
