A Practical Model for Subsurface Light Transport

In computer graphics, a bidirectional reflectance distribution function (BRDF) is used to model light reflectance properties at a surface, and is defined as the ratio of the radiance (incident light) to the irradiance (reflected light) per unit surface area. All BRDF models operate on surface scattering, i.e., the assumption that “light scatters at one surface point” and that light enters and exits a material at the same position, and do not model subsurface transport of incident light. Although this assumption stands valid for metals, translucent surfaces modeled using BRDF exhibit a distinct hard and computer-generated appearance and poor blending of local color and geometry features. While there have been works to model subsurface transport of light, the existing methods are either slow or inefficient for anisotropic or highly scattering translucent media (such as skin and milk). This papers attempts to address this shortcoming by proposing a model for subsurface light transport in translucent materials using bidirectional surface scattering reflectance distribution function (BSSRDF). BSSRDFs are a generalization of BRDFs and, unlike the latter, can model light transport between any two rays that hit a surface. Since the exact BSSRDF derivation is quite involved, we only present a brief summary here, followed by its extension to a model for rendering computer graphics.

The BSSRDF model comprises of a sum of diffusion approximation and single scattering terms and is parameterized by these medium properties: $\sigma_a$ (absorption coefficient), $\sigma’_s$ (reduced scattering coefficient), $\eta$ (the relative refractive index), and $p(\cdot)$ (phase function, which is non-zero for anisotropic scattering). The diffusion approximation relies upon the fact that light distribution tends to be isotropic in a highly scattering medium. To model sub-surface reflection, the authors use a dipole approach (i.e., 2 point sources) to approximate the volumetric source distribution. They position the real light source one mean free path beneath the surface, and a negative virtual ’light source’ above the surface, resulting in a dipole. The single scattering term in the BSSRDF can be obtained by extending the BRDF to account for local variations in lighting over the surface, and can be computed by integrating the incident radiance over all the incoming and the outgoing rays. Since BSSRDF is a generalization of BRDF, the latter can be computed from the former by assuming that the incident illumination is uniform. The authors then propose a measurement apparatus to verify their BSSRDF model and measure the model parameters for rendering several materials, including translucent materials for which the scattering coefficient is much smaller than the absorption coefficient. The surface of a sample is illuminated with ‘a tightly focused beam of white light’ and the radiant exitance is captured using a 3-CCD camera. A series of different exposure times is used to capture a HDR (high dynamic range) image to account for the exponential signal intensity fall-off away from the illuminant source. The absorption and scattering coefficients can be computed by measuring the total diffuse reluctance using least squares fit subject to the the constraint that the integral of the point-wise diffuse reflectance over the entire surface is equal to the total reflectance ($\int R_d \ dA = R$). Comparing the computed measurements for a white marble sample for the camera’s green channel shows almost perfect agreement with those computed using a Monte Carlo simulation, confirming the former’s correctness. Finally, the authors extend this model to obtain a practical rendering model by considering the following in a ray-tracing context: (a) using Monte Carlo sampling to efficiently sample locations on the surface to integrate the incoming light over the entire surface area, (b) reparameterizing the single scattering for arbitrary geometry by using a shadow ray that does not refract at the surface, (c) modifying the diffusion approximation using a dipole source by always evaluating with a minimum distance to avoid singularities associated with highly curved surfaces and sharp edges, and (d) only considering texture variation at the surface. Comparative evaluation of BSSRDF model’s rendering of a marble bust matches the appearance of a full Monte Carlo simulation and is $250\times$ faster on the same machine (5 minutes vs 1250 minutes). Unlike BRDF which yields a ‘hard’ appearance to the rendering with even very small bumps on the surface being visible, BSSRDF outputs a smooth appearance which is more realistic. A similar observation is made for rendering a glass of milk where BSSRDF models capture subtle details in the milk’s appearance, including Rayleigh scattering which makes the milk appear more realistic. Finally, BSSRDF modelling of skin, a highly scattering, translucent, and anisotropic material, using just one layer shows the superiority of the approach with a overall soft appearance and subtle details such as color bleeding in the shadow regions and light absorption by blood.

This paper proposes a novel BSSRDF model for rendering computer graphics which combines dipole diffusion approximation with single scattering computation, and the experiments show the qualitative and quantitative superiority of the method. The proposed approach is well-grounded in theoretical methods and has had a tremendous impact in rendering computer-generated imagery, acknowledged by a Technical Achievement Oscar Award 3 years after its publication. The paper, although quite well written, is quite dense and assumes a lot background knowledge on part of the reader, making it challenging for someone not familiar with light transport models. It would have been really helpful to point the reader to appropriate sources for the background material. Although superior to BRDF, BSSRDF models are at least twice as slower to render.