PBR Chapter15 Light Transport II: Volume Rendering

• The Equation of Transfer(LTE)
• Generalized Path Space
• Sampling Volume Scattering
• Volumetric Light Transport
• Sampling Subsurface Reflection Functions
• Subsurface Scattering Using the Diffusion Profile

The Equation of Transfer(LTE)

• Two effects that contribute to radiance along the ray
  • The emitted and reflected radiance from the surface. This radiance may be attenuated by the participating media; the beam transmittance from the ray origin to the point accounts for this.
  • the added radiance along the ray due to volume scattering and emission but only up to the point where the ray intersects the surface

Generalized Path Space

• path integration
• An integral that can consider an arbitrary sequence of both 2D surface locations A and 3D positions in a participating medium V .
• a sum of many integrals considering all possible sequences of surface and volume scattering events
  • f (BSDF or phase function)
  • G

Sampling Volume Scattering

• Without loss of generality, the following discussion assumes that there is always a surface at some distance.

• two cases when path tracing

  • particle intersection
  • medium interaction
    • probability

• Homogeneous Medium

  • attenuation coefficients varied by wavelength
    • a uniform sample is first used to select a spectral channel i
      • sample distance with transmittance pdf
      • interaction lighting calculation
      • weighting factor

• Heterogeneous Medium

  • regular tracking becomes costly when there are many voxels
  • ray marching introduces systemic statistical bias which generally won’t converge to the right result
  • delta tracking （unbiased)
    • filling the medium with additional (virtual) particles until its attenuation coefficient is constant everywhere
    • however whenever an interaction with a particle occurs, it is still necessary to determine if it involved a “real” or a “virtual” particle
    • the scattering and absorption coefficients are still permitted to vary with respect to wavelength — however, their sum must be uniform
    • steps
      • precompute the inverse of the maximum density scale factor over the entire medium delta tracking 会把media的 density “填充” 成 maximum density
      • transforming the ray into the medium coordinate system and normalizing the ray direction
      • computes the parametric range of the ray’s overlap with the medium’s bounds
      • each delta-tracking iteration performs a standard exponential step through the uniform medium 实际是根据当前位置的density / maximum density 来判断是否是real particles
        • compute the transmittance along a ray segment

• Sampling Phase Function

  • The PDF for the Henyey–Greenstein phase function is separable into theta and phi components
  • p(phi) = 1 / (2 * pi)
  • pdf of theta

Volumetric Light Transport

• VolPathIntegrator’s main responsibility is to implement the Li() method.
• Usually, the ray is first intersected with the surfaces in the scene to find the closest surface intersection, if any. Next, participating media are accounted.
• In scenes with very dense scattering media, a more efficient implementation would be to first sample a medium interaction.

Sampling Subsurface Reflection Functions

• sample points p_i on the surface and to compute the incident radiance at these points
• an efficient way to compute the specific value of the BSSRDF S for each sampled point p_i and incident direction
• VolPathIntegrator path tracer integration to evaluate the BSSRDF
• BSSRDF sampling
• sampling the SeparableBSSRDF
  • assume that the BSSRDF is only sampled for rays that are transmitted through the surface boundary, so 1 - F_r(cos theta_o) has nothing needs to sample
  • S_w(w_i)
    • defined as a diffuse-like term scaled by the normalized Fresnel transmission
    • just uses the default cosine-weighted sampling routine to calculate pdf
  • S_p sampling an out position
    • need a way of mapping a 2D distribution function onto an arbitrary surface using a parameterization of the surface in the neighborhood of the outgoing position
    • a simpler approach
      • difficulties
        • first two problems can be addressed with introducing additional tailored sampling distributions and combining them using multiple importance sampling
        • per wavelength
        • additionally replicated three times with different projection axes given by the basis vectors of a local frame
        • stages
          • choosing a projection axis calc all based on local coordinated of p_o allocate a fairly large portion (50%) of the sample budget to perpendicular rays. The other half is equally shared between tangential projections
            • uniformly choose a spectral channel and re-scale u1 once more
            • sampling S_r
              • in order to reduce the computational expense of the ray-tracing step, the probe ray is clamped to a sphere of radius r_max around p_o
              • calc probe trace start
              • calc intersection chain and choose one from it uniformly
              • calc the pdf and the S_p
• sampling the TabulatedBSSRDF
• Subsurface Scattering in Path Tracer

Subsurface Scattering Using the Diffusion Profile

• initialize the TabulatedBSSRDF with a radial profile function S_r that accurately describes subsurface scattering for given properties of the scattering medium (theta_a, theta_s, the phase function asymmetry parameter g, and the relative index of refraction).
• photon beam diffusion (PBD)
  • assumption
    • the distribution of light in the translucent medium is modeled with the diffusion approximation
    • homogeneous scattering properties throughout the "semi-infinite" medium
    • builds upon the separable BSSRDF approximation of Equation
      • precompute S_r for a range of radii and albedo values and use the results to populate the BSSRDFTable of a TabulatedBSSRDF
• principle of similarity
  • for an anisotropically scattering medium with a high albedo, the medium can instead be modeled as having an isotropic phase function with appropriately modified scattering and attenuation coefficients
  • isotropization due to multiple scattering events from the Henyey–Greenstein phase function. As n grows large, this converges to the isotropic phase function 1/ (4 * pi).
  • reduced scattering coefficient
  • reduced albedo 因为假设了isotropic phase function，所以需要改变scattering coefficients来弥补假设带来的artifacts， theta_s probability of scattering
• diffusion theory