BSDF

The Bidirectional Scattering Distribution Function (BSDF) is a fundamental concept in optics and computer graphics that describes how light scatters at a surface point. It captures the relationship between incoming radiance from a given direction and outgoing radiance in another direction, encompassing both reflection and transmission.

Mathematically, the BSDF is a function f(x, ωi, ωo) that relates differential outgoing radiance to differential incoming irradiance. In differential form, dLo(x, ωo) = f(x, ωi, ωo) Li(x, ωi) cos θi dωi, where ωi and ωo are the incident and outgoing directions, θi is the angle between ωi and the surface normal, and Li is the incoming radiance. The BSDF must be non-negative, and it often satisfies reciprocity, f(ωi, ωo) cos θi = f(ωo, ωi) cos θo, under homogeneous, physically plausible conditions. Energy conservation requires that, for any fixed ωo, the integral of f(ωi, ωo) cos θi over the hemisphere does not exceed 1.

The BSDF generalizes both the BRDF (reflection) and the BTDF (transmission). The BRDF describes how light reflects at a surface, while the BTDF describes how light transmits through it. In rendering, the BSDF is used to model a surface’s scattering behavior by combining or selecting appropriate models, such as Lambertian diffuse reflectance, microfacet-based specular reflection (e.g., Cook-Torrance), Oren-Nayar rough diffuse, and various transmission models. It is a central component of physically based rendering, enabling accurate shading computations and light transport simulations.