Real-Time Rendering, Third Edition, 3rd Edition

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Chapter 9

Global Illumination

“If it looks like computer graphics,

it is not good computer graphics.”

—Jeremy Birn

Radiance is the final quantity computed by the rendering process. So

far, we have been using the reflectance equation to compute it:

L

o

(p, v)=



Ω

f(l, v) ⊗L

i

(p, l)cosθ

i

dω

i

, (9.1)

where L

o

(p, v) is the outgoing radiance from the surface location p in the

view direction v; Ω is the hemisphere of directions above p; f(l, v)isthe

BRDF evaluated for v and the current incoming direction l; L

i

(p, l)isthe

incoming radiance into p from l; ⊗ is the piecewise vector multiplication

operator (used because both f (l, v)andL

i

(p, l) vary with wavelength, so

are represented as RGB vectors); and θ

i

is the angle between l and the

surface normal n. The integration is over all possible l in Ω.

The reflectance equation is a restricted special case of the full rendering

equation, presented by Kajiya in 1986 [619]. Different forms have been used

for the rendering equation. We will use the following one:

L

o

(p, v)=L

e

(p, v)+



Ω

f(l, v) ⊗ L

o

(r(p, l), −l)cosθ

i

dω

i

, (9.2)

where the new elements are L

e

(p, v) (the emitted radiance from the surface

location p in direction v), and the following replacement:

L

i

(p, l)=L

o

(r(p, l), −l). (9.3)

This replacement means that the incoming radiance into location p from

direction l is equal to the outgoing radiance from some other point in the

opposite direction −l. In this case, the “other point” is defined by the

ray casting function r(p, l). This function returns the location of the first

surface point hit by a ray cast from p in direction l (see Figure 9.1).

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328 9. Global Illumination

L

i

(p,l)

L

o

(r(p,l),-l)

r(p,l)

p

l

-l

Figure 9.1. The shaded surface location p, lighting direction l, ray casting function

r(p, l), and incoming radiance L

i

(p, l), also represented as L

o

(r(p, l), −l).

The meaning of the rendering equation is straightforward. To shade

a surface location p, we need to know the outgoing radiance L

o

leaving

p in the view direction v. This is equal to the emitted radiance L

e

plus

the reflected radiance. Emission from light sources has been studied in

previous chapters, as has reflectance. Even the ray casting operator is not

as unfamiliar as it may seem. The Z-buffer computes it for rays cast from

the eye into the scene.

The only new term is L

o

(r(p, l), −l), which makes explicit the fact

that the incoming radiance into one point must be outgoing from an-

other point. Unfortunately, this is a recursive term. That is, it is com-

puted by yet another summation over outgoing radiance from locations

r(r(p, l), l



). These in turn need to compute the outgoing radiance from

locations r(r(r(p, l), l



), l



), ad infinitum (and it is amazing that the real

world can compute all this in real-time).

We know this intuitively, that lights illuminate a scene, and the photons

bounce around and at each collision are absorbed, reflected, and refracted

in a variety of ways. The rendering equation is significant in that it sums

up all possible paths in a simple (looking) equation.

In real-time rendering, using just a local lighting model is the default.

That is, only the surface data at the visible point is needed to compute

the lighting. This is a strength of the GPU pipeline, that primitives can

be generated, processed, and then be discarded. Transparency, reflections,

and shadows are examples of global illumination algorithms, in that they

use information from other objects than the one being illuminated. These

effects contribute greatly to increasing the realism in a rendered image, and

also provide cues that help the viewer to understand spatial relationships.

One way to think of the problem of illumination is by the paths the

photons take. In the local lighting model, photons travel from the light

to a surface (ignoring intervening objects), then to the eye. Shadowing

techniques take into account these intervening objects’ direct effects. With

environment mapping, illumination travels from light sources to distant ob-

jects, then to local shiny objects, which mirror-reflect this light to the eye.

Irradiance maps simulate the photons again, which first travel to distant