• 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.
Figure 2.4: Halfway vector parameterization.
BRDFs can be classified by taking into account the characteristics of the reflection, whether they change by rotating the surface around its normal direction:
• Isotropic BRDFs are able to represent materials whose reflection does not depend on the orientation of the surface, since the reflectance properties are invariant to rotations of the surface around n.
• Anisotropic BRDFs can describe materials whose reflection changes with respect to rotation of the surface around n; this class includes materials like brushed metal, satin, velvet and hair.
The Fresnel effect predicts the fraction of power that is reflected and transmitted and has a great impact on the appearance (Figure 2.7). Many basic BRDF models have lost importance in the context of physically based modeling because they do not account for a Fresnel term. For conductive materials, like metals, the fraction of light reflected by pure specular reflection is roughly constant for all angles of incidence, whereas for non-conductive materials (dielectrics), the amount of light reflected increases at grazing angles; see Figure 2.6 for a comparative example of the behavior of metals and dielectrics. The fraction of light reflected is called Fresnel reflectance, which can be obtained from the solution of Maxwell’s equations and depends also on the polarization state of the incident light. For unpolarized light, the Fresnel reflectance F at the interface between the surface and the air is given by
Figure 2.5: Basic reflectance models of the incoming light (in orange): perfect diffuse (yellow), glossy (purple) and perfect specular (light blue). Renderings of diffuse, glossy and specular spheres are shown, placed inside a Cornell box [GTGB84].
where η is the index of refraction of the surface and is the angle of transmission. In computer graphics, it is very common to use Schlick’s approximation of the Fresnel reflectance [Sch94]:
where F(0) is the Fresnel reflectance at normal incidence; in the following chapters we will generally use the symbol F to refer either to the exact Fresnel reflectance or one of its approximations.
A BRDF should respect some basic physical properties, namely non-negativity, reciprocity and energy conservation:
• non-negativity: the BRDF is a non-negative function, hence for any pair of incident and outgoing direction fr (vr, vi) ≥ 0;
• the Helmholtz reciprocity principle states that the light path is reversible for any pair of incident and outgoing direction: fr (vi, vr) = fr (vr, vi). This principle holds only for corresponding states of polarization for incident and emerging fluxes, whereas large discrepancies might occur for non-corresponding states of polarization [CP85]. In designing a rendering system possible non-reciprocity should be taken into account [Vea97].
