LITMIR.BIZ

      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 and t’ = b’ × h.

      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(v_i, v_r) = ρ_d/π, where ρ_d 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: where is the specular albedo. Perfect specularity is valid only for highly polished mirrors and metals.

      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.

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