1. BRDFs
Randy Rauwendaal

2. Acronyms BRDF – Bidirectional Reflectance Distribution Function
SPDF – Scattering Probability Density Function
BSDF – Bidirectional Scattering Distribution Function
BTDF – Bidirectional Transmission Distribution Function
BSSRDF – Bidirectional Scattering Surface Reflectance Distribution Function

3. What is a BRDF?
We’ll get to that later, but first we must understand how light interacts with matter

4. Light & Matter
Interaction depends on the physical characteristics of the light as well as the physical composition and characteristics of the matter
Incoming Light
Transmitted Light
Reflected Light
Scattering and Emission
Internal Reflection
Absorption

5. Light & Matter
Three types of interactions when light first hits a material
Reflection
Absorption
Transmittance

6. Conservation of Energy
A BRDF describes how much light is reflected when light makes contact with a certain material
The degree to which light is reflected depends on the viewer and light position relative to the surface normal and tangent
BRDF is a function of incoming (light) direction and outgoing (view) direction relative to a local orientation at the light interaction point

7. Wavelength
When light interacts with a surface, different wavelengths of light may be absorbed, reflected, and transmitted to varying degrees depending upon the physical properties of the material itself
This means that a BRDF is also a function of wavelength

8. Positional Variance
Light interacts differently with different regions of the surface, this is know as positional variance
Noticeable in materials that reflect light in manner that produces surface detail (wood, marble, etc.)

9. Notation  is wavelength
I and i, represent the incoming light direction in spherical coordinates
o and o represent the outgoing reflected direction in spherical coordinates
u and v represent the surface position parameterized in texture space

10. Position-Invariant BRDFs
When the spatial position is not included as a parameter to the function the reflectance properties of the material do not vary with spatial position
Only valid for homogenous materials
A spatially variant BRDF can be approximated by adding or modulating the result of a BRDF lookup with a texture

11. Spherical Coordinatesx y z   v
1.0
Cartesian coordinates (vx,vy,vz)
Spherical coordinates (, )

12. Differential Solid Angles
Since BRDFs measure how light reflects off a surface when viewed we must know how much light arrives at a surface element from a particular direction
Light is measured as energy per-unit surface area (i.e. Watt/meter2)
Units are in radians squared or steradians
We must consider flow through a neighborhood of directions when determining the amount of light that arrives at or leaves a surface
Small surface element
Normal
Small area
Incoming light direction wi
Neighborhood of directions

13. Differential Solid Angles
The differential solid angle is defined to be the area of the small blue patch
Given spherical coordinates (,) and small differential angular changes denoted d, d, the differential solid angle, dw, is defined to be
sin   d d
sphere of radius 1
The area quantity has units of radians squared, or steradians

14. Definition of a BRDF wi=incoming light direction
wo=outgoing reflected direction
Lo=quantity of light reflected in direction wo
Ei=quantity of light arriving from direction wi

15. Definition of a BRDF n wi
light source
Small surface element
Differential solid angle dwi θi
The amount of light arriving from direction wi is proportional to the amount of light arriving at the differential solid angle
The region is uniformly illuminated as the same quantity of light, Li, arrives for each position on the differential solid angle, meaning the total amount of incoming light arriving through the region is Li*dwi
To determine the amount of light with respect to the surface element, the incoming light must be projected onto the surface element by modulating by
This projection is similar to that which happens with diffuse Lambertian lighting and is accomplished by modulating that amount by cosθi=N·wi.
This means Ei=Licosθidwi, as a result, the BRDF is given by cosθi=N·wi

16. Classes and Properties of BRDFs
There are two classes of BRDFs, isotropic BRDFs and anisotropic BRDFs
The important properties of BRDFs are reciprocity and conservation of energy
BRDFs that have these properties are considered to be physically plausible

17. Isotropic BRDFs
The term isotropic is used to describe BRDFs that represent reflectance properties that are invariant with respect to rotation of the surface around the surface normal vector.
Smooth plastics

18. Anisotropic BRDFs
BRDFs that describes reflectance properties that do exhibit change with respect to rotation of the surface around the surface normal vector
Most real world surfaces

19. Reciprocity
If the incoming and outgoing directions are swapped, the value of the BRDF does not change =

20. Conservation of Energy
Surface
Incoming light
Reflected light
The quantity of light reflected must be less than or equal to the quantity of incident light
The sum of all outgoing directions of the BRDF times the project solid angle must be less than one

21. The BRDF Lighting Equation
Surface
Outgoing light
Incoming light E
The amount of light reflected in the outgoing direction is the integral of the amount of light reflected in the outgoing direction from each incoming direction
Where Lo due to i(wi,wo) represents the amount of light reflected in direction wo

22. The BRDF Lighting Equation
The intensity of the light reflected in the outgoing direction is defined by modulating the intensity of the light arriving at the surface point with the corresponding BRDF
Where Ei is the amount of light arriving from direction wi
Again we project the light by modulating with cosθi=N·wi
This means Ei=Licosθidwi, but in practice, in the discrete case, the dwi term indicates that all incoming directions are equally weight and Ei=Licosθi
And the light reflected in the outgoing direction is

23. The BRDF Lighting Equation
Often, rather than computing a sum of light contributions from many incoming light directions and summing these results to determine the final output color, a small number of individual point light sources define the set of incoming directions and light intensities to use in the evaluation of the lighting equation
Where Lij is the intensity of the jth light source and wij=(θijθij) is the direction of the jth light source
For a single point light source, the light reflected in the direction of an observer is

24. BRDFs and the Phong Model
Notice that the final expression is very similar to the general BRDF lighting equation
The Phong lighting model can be considered a special case of the BRDF based lighting
The Phong model is convenient to use for materials with reflectance properties approximated by

25. Acquiring BRDF Data Many simple analytical models have been developed
Cook-Torrance model
Modified Phong model
Ward’s model
BRDFs can be obtained through physical measurement with a device called a gonioreflectometer
By using real data there is no need to try to determine which analytical model to use to achieve a certain visual lighting effect

27. References
Chris Wynn. An Introduction to BRDF-Based Lighting, NVIDIA Corporation
Peter Shirley, Helen Hu, Brian Smits, and Eric Lafortune. A Practitioners’ Assessment of Light Reflection Models. In Pacific Graphics, pages 40-49, October 1997
M. Ashikhmin and P. Shirley. An Anisotropic Phong BRDF Model. Journal of Graphics Tools: JGT, 5(2), pages 25-32, 2000