Bi-Directional Reflectance Distribution Functions

(BRDF’s)

Matthew McCrory

What is a BRDF?

Must know something about light and how it interacts with matterWhen light interacts with matter:

Complicated light-matter dynamic occursDependent on characteristics of both the light and the matterExample, sandpaper vs. a mirror

What is a BRDF?

Typical light-matter interaction scenario:Incoming Light

Transmitted Light

Reflected Light

Scattering and Emission Internal Reflection Absorption

3 types of interaction: transmission, reflection, and absorbtionLight incident at surface = reflected + absorbed + transmittedBRDF describes how much light is reflected

What is a BRDF?

Viewer/light position dependency (incoming/outgoing rays of light)

Example – Shiny plastic teapot with point lightDifferent wavelengths (colors) of light may be absorbed, reflected, transmitted differentlyPositional variance – light interacts differently with different regions of a surface, e.g. woodBRDF must capture this view and light dependent nature of reflected light

What is a BRDF?In functional notation:

Or

For position invariant BRDF’s

vuooii ,,,,,BRDF

ooii ,,,BRDF

Differential Solid AnglesMore appropriate to speak of light in terms of quantity of light arriving at or passing through a certain area of spaceLight doesn’t come from a single direction

More appropriate to consider a small region of directions

Small surface element

Normal

Small area

Incoming light direction

wi

Neighborhood of directions

Differential Solid Angles

Patch formed at intersection of pyramid and unit sphereDifferential Solid Angle defined as surface area of path

sin

d

d

sphere of radius 1

dddwdddwwidthheightdw

sin))(sin(

))((

Definition of a BRDFGiven:

Incoming light direction wi, and an outgoing reflected direction wo, each defined relative to a small surface element

BRDF defined as: the ratio of the quantity of reflected light in direction wo, to the amount of light that reaches the surface from direction wi.

n wi

light source

Small surface element

Differential solid angle dwi

θii

o

EL

BRDF

Definition of a BRDFLight arriving from direction wi proportional to the amount arriving at the differential solid angle.Given light source Li, total light arriving through the region is Li*dw

Incoming light must be projected onto surface element. Accomplished by modulating by (= N.wi)

BRDF given by:

icos

iii

o

dwLL

BRDFcos

Classes and Properties of BRDF’s

2 classesIsotropicAnistropic

2 important propertiesReciprocity

Conservation of energy

=

Surface

Incoming lightReflected light

1cos,,,BRDF

ooooii dw

The BRDF Lighting EquationGoal:

Define a general lighting equation that expresses how to use BRDF’s for computing the illumination produced at a surface point

Light arrives from > 1 point

Surface

Outgoing light

Incoming lightE

The BRDF Lighting EquationAmount of light reflected in outgoing direction is the integral of the amount of light reflected in the outgoing direction from each incoming direction

More convenient to think discreetly

ioiioo dwwwLL ),( todue

in

oiioo wwLL ),( todue

The BRDF Lighting EquationFor each incoming direction, the amount of reflected light in the outgoing direction is defined in terms of the BRDF.Given:

Li is the light intensity from direction wi. Ei is the amount of light arriving from direction wi:

Ei must take into account surface area intensity instead of differential solid angle

iooiiio EBRDFL ),,,( todue

iiii dwLE cosiiooiiio LBRDFL cos),,,( todue

in

iiooiio LBRDFL cos),,,(

The BRDF Lighting EquationFinally! The general BRDF lighting equation for a single point light source is:

For multiple light sources, each light must be used in the equation and the sum is the amount of outgoing light

iiooiio LBRDFL cos),,,(

Analytical Models and Acquired BRDF Data

How can we compute BRDFs for use in the general BRDF lighting equation?

Evaluate mathematical functions derived from analytical modelsResample BRDF data acquired by empirical measurements of real-world surfaces

Some ExamplesFrom the National Institute of Standards and TechnologyCourse and fine metallic paint on vases

Some ExamplesTwo tiles rendered using BRDFs obtained from the measured surface topology of actual tile samples

Improvements on BRDFBRDF assumes light enters and leaves a surface at the same point, which isn’t true in real lifeLight scatters beneath a surface and leaves at different places than where it enteredBi-direction Scattering Surface Reflection Distribution Functions (BSSRDFs) account for just that

Comparison

BRDF vs BSSRDF

Real-time BRDFSome hardware vendors like Nvidia are making BRDF lighting doable in real-time

That’s it!

Questions?