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## Rendering General BSDFs and BSSDFs

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Title: Rendering General BSDFs and BSSDFs

1
Rendering General BSDFs and BSSDFs
• Randy Rauwendaal

2
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

3
The Bidirectional Scattering Distribution Function
• The BSDF is a mathematical description of the
light-scattering properties of a surface
• Let Lo(?o) denote the radiance leaving x in
direction ?o
• The light strikes the surfaces and generates an
• It can be observed experimentally that as dE(?i)
is increased there is a proportional increase in
• The BSDF is now defined to be this constant of
proportionality

4
The Scattering Equation
• By integrating the relationship

Over all directions, we can now predict
Lo(?o) 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
5
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 BRDFs (one for
each side of the surface), and two BTDFs (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

6
Properties of the BRDF
• BRDFs 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

7
The BSDF for Specular Reflection
• For a perfect mirror, the desired relationship
between Li and Lo is that
• Where MN(?o) is the mirror direction, obtained by
reflecting around the normal N
• Now we define this BSDF in terms of a special
Dirac distribution ds-, we is defined by the
property that
• The Dirac distribution (or delta function) also
has the properties that
• d(x) 0 for all x ? 0
• ?R d(x)dx 1
• Which implies the useful identity

8
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

9
The BSDF for Refraction
• We define a mapping
• Such that R(?i) is the transmitted direction
corresponding to the incident direction ?i
• 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
?i and ?t, and also the fact that the radiance is
scale by a factor of (?t/?i)2

10
Reciprocity and Conservation Laws for General
BSDFs
• The BSDF of any physically plausible material
must satisfy
• Where ?i and ?o are the refractive indices of the
materials containing ?i and ?o 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

11
A Reciprocity Principle for General BSDFs
• To prove a reciprocity condition for general
BSDFs, we consider the light energy scattered
between to directions ?i and ?o at a point x in
an isothermal enclosure
• By the principle of detailed balance, the rates
of scattering from ?i to ?o and from ?o to ?i are
equal (dF1 dF2), while by Kirchoffs
from each direction is proportional to the
refractive index squared
• Putting these facts together we get the desired
reciprocity condition

12
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
• Considering the radiance distribution, we let the
incident power be concentrated in a single
direction ?i
• From which the requirement E M gives the
desired result

13
BSSDFs
• Bidirectional Surface Scattering Distribution
Function
• There not a whole lot material on general BSSDFs,

14
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

15
Symbol Reference
16
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
• Where Sd is the diffusion term of the BSSRDF,
which represents multiple scattering

17
The Diffusion Approximation
An incoming ray is transformed into a dipole
source for the diffusion approximation
18
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
19
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)

20
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

21
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)

22
BRDF vs BSSRDF
23
BRDF vs BSSRDF
24
BRDF vs BSSRDF
25
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.