BRDFs

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?

• Well 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

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 Coordinates

• 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

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

• The area quantity has units of radians squared,
  or steradians

14
Definition of a BRDF

• wiincoming light direction
• wooutgoing reflected direction
• Loquantity of light reflected in direction wo
• Eiquantity of light arriving from direction wi

15
Definition of a BRDF

• 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 Lidwi
• 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
  cos?iNwi.
• This means EiLicos?idwi, as a result, the BRDF
  is given by cos?iNwi

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

• 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

• 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
  cos?iNwi
• This means EiLicos?idwi, but in practice, in the
  discrete case, the dwi term indicates that all
  incoming directions are equally weight and
  EiLicos?i

• 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(?ij?ij) 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
• Wards 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

26
(No Transcript)
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