Bidirectional reflectance distribution function

The bidirectional reflectance distribution function (BRDF), symbol $f_r(w_i,w_r)$ , is a function of four real variables that defines how light from a source is reflected off an opaque surface. It is employed in the optics of real-world light, in computer graphics algorithms, and in computer vision algorithms. The function takes an incoming light direction, $w_i$ , and outgoing direction, $w_r$ (taken in a coordinate system where the surface normal $n$ lies along the z-axis), and returns the ratio of reflected radiance exiting along $w_r$ to the irradiance incident on the surface from direction $w_i$. Each direction $w$ is itself parameterized by azimuth angle $\varphi$ and zenith angle $\theta$ , therefore the BRDF as a whole is a function of 4 variables. The BRDF has units sr⁻¹, with steradians (sr) being a unit of solid angle.

Diagram showing vectors used to define the BRDF. All vectors are unit length. $w_i$ points toward the light source. $w_r$ points toward the viewer (camera). $n$ is the surface normal.

Applications

The BRDF is a fundamental radiometric concept, and accordingly is used in computer graphics for photorealistic rendering of synthetic scenes (see the rendering equation), as well as in computer vision for many inverse problems such as object recognition. BRDF has also been used for modeling light trapping in solar cells (e.g. using the OPTOS formalism) or low concentration solar photovoltaic systems.

In the context of satellite remote sensing, NASA uses a BRDF model to characterise surface reflectance anisotropy. For a given land area, the BRDF is established based on selected multiangular observations of surface reflectance. While single observations depend on view geometry and solar angle, the MODIS BRDF/Albedo product describes intrinsic surface properties in several spectral bands, at a resolution of 500 meters. The BRDF/Albedo product can be used to model surface albedo depending on atmospheric scattering.

Some examples

• Lambertian model, representing perfectly diffuse (matte) surfaces by a constant BRDF.
• Lommel–Seeliger, lunar and Martian reflection.
• Hapke scattering model, physically motivated approximation of the radiative transfer solution for a porous, irregular, and particulate surface. Often used in astronomy for planet/small body surface reflection simulations. Multiple versions and modifications exist.
• Phong reflectance model, a phenomenological model akin to plastic-like specularity.
• Blinn–Phong model, resembling Phong, but allowing for certain quantities to be interpolated, reducing computational overhead.
• Torrance–Sparrow model, a general model representing surfaces as distributions of perfectly specular microfacets.
• Cook–Torrance model, a specular-microfacet model (Torrance–Sparrow) accounting for wavelength and thus color shifting.
• Ward model, a specular-microfacet model with an elliptical-Gaussian distribution function dependent on surface tangent orientation (in addition to surface normal).
• Oren–Nayar model, a "directed-diffuse" microfacet model, with perfectly diffuse (rather than specular) microfacets.
• Ashikhmin–Shirley model, allowing for anisotropic reflectance, along with a diffuse substrate under a specular surface.
• Trowbridge–Reitz model, a specular-microfacet model that models surfaces as consisting of microfacets of not only random orientation but also random curvature.
• HTSG (He, Torrance, Sillion, Greenberg), a comprehensive physically based model.