where h is the Planck’s constant, c is the speed of light in vacuum and λ is the wavelength of the incident photon. The energy flowing through a surface per unit time is called Radiant Flux, indicated with Φ and measured in Watts [W]:
The flux flowing per unit of surface area is called radiant flux area density, measured [W/m2] and indicated by u:
Two different terms are used in order to distinguish the flow of energy toward a surface from the flow leaving a surface: in the first case we refer to irradiance (E); in the second one the term used is radiosity (B). If instead of referring to the ratio of flux per unit of surface area we take into account a solid angle, we can define the intensity I, the radiant energy leaving a point in the direction Φ per unit solid angle, measured in [W/sr]:
Finally, the radiant flux per unit solid angle and per unit projected area is called radiance:
In the following we indicate the radiance arriving at a surface with Li and the radiance leaving a surface with Lr.
We are now ready to define the BRDF as the ratio of the reflected radiance Lr to incident irradiance E:
where vi and vr are vectors describing the incident (i) and exitant (r) directions. By taking into account the incident radiance Li instead of Ei, thus considering the solid angle around the incident lighting direction and the cosine of the angle between the latter and the surface normal, we can write the Equation (2.7) in a different form, which allows understanding how the units of a BRDF are inverse steradian [1/sr]:
Researchers have measured hundreds of BRDFs, suggested implementation techniques and utilized user input to edit and enhance materials. Recent implementations have expanded material libraries but have not improved significantly upon material representation efficiency. However, the uptake of acquired models has not been widespread across rendering packages due to their data and storage requirements.
To understand the way the BRDF is parameterized, let’s take into consideration a point p on a surface and the surface normal n at that specific location on the surface; on the plane tangent to the surface in p we fix a reference direction t, called tangent direction, and its perpendicular direction b on the plane: n × t × b defines a local reference frame. Once we set the incoming light direction and the outgoing direction (viewing direction), the angle between the surface normal and the viewing direction is called θi; similarly the angle between the surface normal and the outgoing direction is called θr. If we take the projection of the viewing direction on the tangent plane, the angles between the tangent direction and the projection of the incoming direction are called respectively ϕi, and ϕr.
Figure 2.3 shows the geometry of the BRDF and the vectors used for parameterizations:
• n is the normal at a specific point p on the surface.
• t is the tangent vector. It is perpendicular to the normal n and hence it is tangent to the surface at p.
• b is the bi-tangent vector, defined as b = n × t. In literature it is also named as binormal vector.
• h is the halfway vector [Rus98], defined as:
Another very common way to parameterize the BRDF is the halfway h vector shown in Figure 2.4, defined by the normalized vector sum of the incoming and outgoing directions. In this case we are taking into account the angle between the surface normal n and the halfway vector h. This has important implications in the way the measured data can be stored, compressed and can speed up computation of specific models. The use of the halfway vector enables another possibility to define a local reference frame, in which one of the axes is aligned with h and the other two are given by
There exist other coordinate systems and parameterizations especially suited for dimensionality reduction of some isotropic BRDF models, for instance the barycentric coordinate system with respect to a triangular support proposed by Stark et al. [SAS05], or the hybrid model described by Barla et al. which could lead to a better repartition of samples to cover most of the effects of materials [BBP15].
Figure 2.3: Geometry of BRDF.
2.2 BASIC BRDF MODELS
There are many reflectance models that are simplified subsets of the BRDF function. One of the simplest reflectance models is the Lambertian model, which represents the perfect diffuse reflectance and is often used in many interactive applications, since it requires no recalculation with the change of viewing direction. The model simply assumes that the surface reflects light uniformly in all directions with the same radiance (see Figure 2.5, in yellow), constant with vr, unlike other BRDF models: fr(vi, vr) = ρd/π, where ρd is the diffuse albedo.
In the case of a pure specular BRDF all the light is reflected in a single direction for a given incident direction (see Figure 2.5, in light blue). In fact, light that is incident within a differential solid angle dωi from direction (θi, ϕi) is reflected in a differential solid angle ωr in direction (θi, ϕi + π), hence the pure specular BRDF can be formalized with a double Dirac delta function:
Surfaces not perfectly smooth, which have some roughness at the micro-geometry level, have a glossy appearance and show broader highlights, other than specular reflections (see Figure 2.5).
Some materials, like the surface of the moon or some biological tissues, show a phenomenon called retro-reflection in which light is scattered not only in the forward direction but also in the direction of the illuminant. Real material tends to display a mixture of the aforementioned basic reflectance types, giving rise to very complex reflection properties.