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Rendering General BSDFs and BSSDFs


It cannot be assumed that other distribution functions satisfy these properties ... The BSDF of any physically plausible material must satisfy ... – PowerPoint PPT presentation

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Title: Rendering General BSDFs and BSSDFs

Rendering General BSDFs and BSSDFs
  • Randy Rauwendaal

  • 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(?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 observed radiance dLo(?o)
  • The BSDF is now defined to be this constant of

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

Properties of the BRDF
  • BRDFs that describe real surfaces have basic
  • 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(?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

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(?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

Reciprocity and Conservation Laws for General
  • 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
  • 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 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
    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 ?i
  • From which the requirement E M gives the
    desired result

  • Bidirectional Surface Scattering Distribution
  • There not a whole lot material on general BSSDFs,
    so instead well focus on

  • 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
  • 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
  • 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
  • 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
  • Single scattering evaluation for arbitrary
  • Diffusion approximation for arbitrary geometry
  • Texture (spatial variation on the object surface)

  • 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.