Rendering General BSDFs and BSSDFs

Randy Rauwendaal

BSDFs

• Bidirectional Scattering Distribution Function

• Gives a mathematical description of the way that light is scattered by a surface

• Is a generalization of a BRDF and a BTDF

The Bidirectional Scattering Distribution Function• The BSDF is a mathematical description of the light-scattering properties of a

surface• Let Lo(ωo) denote the radiance leaving x in direction ωo

• The light strikes the surfaces and generates an irradiance

• It can be observed experimentally that as dE(ωi) is increased there is a proportional increase in the observed radiance dLo(ωo)

• The BSDF is now defined to be this constant of proportionality

The Scattering Equation

By integrating the relationship

Over all directions, we can now predict Lo(ωo)

This is summarized by the (surface) scattering equation,

This equation can be used to predict the appearance of the surface, given a description of the incident illumination

The BRDF and BTDF

• Usually scattered light is subdivided reflected and transmitted components, which are treated separately

– Bidirectional Reflectance Distribution Function (BRDF), denoted fr

– Bidirectional Transmittance Distribution Function (BTDF), denoted ft

• The BRDF is obtained by simply restricting fs to the domain

• The BTDF is similarly obtained by restricting fs to the domain

• Thus the BSDF is the union of two BRDF’s (one for each side of the surface), and two BTDF’s (one for light transmitted in each direction).

– More convenient, 1 function instead of 4

– Purely reflective or transmissive surfaces are special case of this formulation

Properties of the BRDF• BRDF’s that describe real surfaces have basic properties

– Symmetry

– Energy Conservation

• These properties are unique to reflection

– It cannot be assumed that other distribution functions satisfy these properties

The BSDF for Specular Reflection

• For a perfect mirror, the desired relationship between Li and Lo is that

• Where MN(ωo) is the mirror direction, obtained by reflecting around the normal N

• Now we define this BSDF in terms of a special Dirac distribution δσ┴, we is defined by the property that

• The Dirac distribution (or delta function) also has the properties that– δ(x) = 0 for all x ≠ 0

– ∫R δ(x)dx = 1

• Which implies the useful identity

The BSDF for Specular Reflection• Now we can write the equation for a BSDF for a perfect mirror

• By expanding the definition of the projected solid angle, we can write the scattering distribution function in the form

• Which allows us to write the mirror BSDF as

• Expressions containing Dirac distributions must be evaluated with great care, particularly when the measure functions associated with the Dirac distribution is different than the measure function used for integration

The BSDF for Refraction• We define a mapping

• Such that R(ωi) is the transmitted direction corresponding to the incident direction ωi

• Given this mapping, the relation between Li and Lt due to refraction can be expressed as

• The corresponding BSDF is thus

• This functions expresses the relationship between ωi and ωt, and also the fact that the radiance is scale by a factor of (ηt/ηi)2

Reciprocity and Conservation Laws for General BSDF’s

• The BSDF of any physically plausible material must satisfy

• Where ηi and ηo are the refractive indices of the materials containing ωi and ωo respectively

• This is a generalization of the BRDF property of symmetry

• We also investigate how light scattering is constrained by the law of conservation of energy, and we derive a simple condition that must be satisfied by any BSDF that is energy conserving

A Reciprocity Principle for General BSDF’s

• To prove a reciprocity condition for general BSDF’s, we consider the light energy scattered between to directions ωi and ωo at a point x in an isothermal enclosure

• By the principle of detailed balance, the rates of scattering from ωi to ωo and from ωo to ωi are equal (dΦ1 = dΦ2), while by Kirchoff’s equilibrium radiance law, the incident radiance from each direction is proportional to the refractive index squared

• Putting these facts together we get the desired reciprocity condition

Conservation of Energy• Theorem If fs is the BSDF for a physically valid surface, which is either the boundary of an

opaque object or the interface between two non-absorbing media, then

• Proof

• Where E denotes the irradiance, and M denotes the radiant exitance

• Considering the radiance distribution, we let the incident power be concentrated in a single direction ωi

• From which the requirement E ≤ M gives the desired result

BSSDFs

• Bidirectional Surface Scattering Distribution Function

• There not a whole lot material on general BSSDFs, so instead we’ll focus on…

BSSRDFs

• Bidirectional Surface Scattering Reflectance Distribution Function

• The BSSRDF relates the outgoing radiance to the incident flux

• The BRDF is an approximation of the BSSRDF for which it is assumes that light enters and leaves at the same point

• The outgoing radiance is computed by integrating the incident radiance over incoming directions and area, A

Symbol Reference

The Diffusion Approximation• The diffusion approximation is based on the observation that the light distribution in highly

scattering media tends to become isotropic

• The volumetric source distribution can be approximated using the dipole method

• The dipole method consists of positioning two point sources near the surface in such a way as to satisfy the required boundary condition

• The diffuse reflectance due to the dipole source can be computed (with much hand waving) as

• Taking into account the Fresnel reflection at the boundary for both the incoming light and the outgoing radiance

• Where Sd is the diffusion term of the BSSRDF, which represents multiple scattering

The Diffusion Approximation

An incoming ray is transformed into a dipole source for the diffusion approximation

Single Scattering Term• The total outgoing radiance, due to single scattering is computed by integrating the incident

radiance along the refracted outgoing ray

• The single scattering BSSRDF is defined implicitly by the second line of this equation

Single scattering occurs only when the refracted incoming and outgoing rays intersect, and is computed as an integral over path length s along the refracted outgoing ray

The BSSRDF Model

• The complete BSSRDF model is a sum of the diffusion approximation and the single scattering term

• This model accounts for light transport between different locations on the surface, and it simulates both the directional component (due to single scattering) as well as the diffuse component (due to multiple scattering)

BRDF Approximation• We can approximate the BSSRDF with a BRDF by assuming that the incident illumination is

uniform

• By integrating the diffusion term we find the total diffuse reflectance of the material

• The integration of the single scattering term for a semi-infinite medium gives

• The complete BRDF model is the sum of the diffuse reflectance scaled by the Fresnel term and the single scattering approximation

• This model has the same parameters as the BSSRDF

• Note that the amount of light is computed from the intrinsic material parameters

• The BRDF approximation is useful for opaque materials, which have a very short mean free path

Rendering Using the BSSRDF

• The BSSRDF model derived only applies to semi-infinite homogeneous media, for a practical model we must consider

– Efficient integration of the BSSRDF (importance sampling)

– Single scattering evaluation for arbitrary geometry

– Diffusion approximation for arbitrary geometry

– Texture (spatial variation on the object surface)

BRDF vs BSSRDF

BRDF vs BSSRDF

BRDF vs BSSRDF

References

Eric Veach, Robust monte carlo methods for light transport simulation, 1997

Jensen, H. W., Marschner, S. R., Levoy, M., and Hanrahan, P. 2001 A practical model for subsurface light transport. In Proceeding of SIGGRAPH 2001, 511-518

Henrik Wann Jensen and Juan Buhler. A rapid hierarchical rendering technique for translucent materials. ACM Transactions on Graphics, 21(3):576.581, July 2002.