OrenNayar

The Oren–Nayar model is a diffuse reflectance model for rough surfaces, introduced by Zvi Oren and Michael Nayar in 1994. It generalizes Lambertian reflectance by accounting for surface microstructure, providing a more accurate description of how light is scattered from rough, matte materials such as plaster, paint, and fabric.

In this model, the bidirectional reflectance distribution function (BRDF) for diffuse reflection depends on the incident and viewing directions. Let θi and θr be the angles between the surface normal and the incoming and outgoing directions, respectively, and let φ be the relative azimuth angle between these directions. Define α = max(θi, θr) and β = min(θi, θr). The roughness is described by σ, the standard deviation of the microfacet slopes. Two coefficients A and B, which depend on σ, are defined as:

A = 1 − (σ^2 / (2(σ^2 + 1)))

B = 0.45 σ^2 / (σ^2 + 0.57)

The Oren–Nayar BRDF for diffuse reflection is:

f_r(ωi, ωr) = (ρd / π) [ A + B max(0, cos(φ) sin α tan β) ]

where ρd is the diffuse reflectance (albedo) of the surface. When σ → 0, A → 1 and B → 0, reducing the model to Lambertian reflectance, f_r = ρd / π.

The model captures the dependence of reflected light on both incident and viewing geometry and provides a physically motivated interpolation between perfectly smooth and very rough surfaces. It is widely used in computer graphics and physically based rendering to improve realism for rough diffuse materials.