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
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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
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It can be observed experimentally that as dE(ωi) is increased there is a proportional
increase in the observed radiance dLo(ωo)
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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
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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
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The BTDF is similarly obtained by restricting fs to the domain
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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
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BRDF’s that describe real surfaces have basic properties
– Symmetry
– Energy Conservation
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These properties are unique to reflection
– It cannot be assumed that other distribution functions satisfy these properties
The BSDF for Specular Reflection
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For a perfect mirror, the desired relationship between Li and Lo is that
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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
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The Dirac distribution (or delta function) also has the properties that
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δ(x) = 0 for all x ≠ 0
∫R δ(x)dx = 1
Which implies the useful identity
The BSDF for Specular Reflection
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Now we can write the equation for a BSDF for a perfect mirror
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By expanding the definition of the projected solid angle, we can write the scattering
distribution function in the form
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Which allows us to write the mirror BSDF as
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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
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We define a mapping
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Such that R(ωi) is the transmitted direction corresponding to the incident direction ωi
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Given this mapping, the relation between Li and Lt due to refraction can be expressed as
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The corresponding BSDF is thus
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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
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The BSDF of any physically plausible material must satisfy
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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
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A Reciprocity Principle for General
BSDF’s
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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
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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
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Proof
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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
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From which the requirement E ≤ M gives the desired result
BSSDFs
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Bidirectional Surface Scattering Distribution Function
There not a whole lot material on general BSSDFs, so instead we’ll focus on…
BSSRDFs
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Bidirectional Surface Scattering Reflectance Distribution Function
The BSSRDF relates the outgoing radiance to the incident flux
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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
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Symbol Reference
The Diffusion Approximation
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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
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Taking into account the Fresnel reflection at the boundary for both the incoming light and the
outgoing radiance
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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
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The total outgoing radiance, due to single scattering is computed by integrating the incident
radiance along the refracted outgoing ray
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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
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The complete BSSRDF model is a sum of the diffusion approximation and the
single scattering term
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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
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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
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The integration of the single scattering term for a semi-infinite medium gives
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The complete BRDF model is the sum of the diffuse reflectance scaled by the Fresnel term
and the single scattering approximation
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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
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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.