BRDFs Randy Rauwendaal

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

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

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

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

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

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

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.)

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

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

Spherical Coordinates x y z   v 1.0 Cartesian coordinates (vx,vy,vz) Spherical coordinates (, )

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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