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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 (v_r, v_i) ≥ 0;

      • the Helmholtz reciprocity principle states that the light path is reversible for any pair of incident and outgoing direction: f_r (v_i, v_r) = fr (v_r, v_i). 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].

      Figure 2.6: Fresnel reflectance for metals (a) and dielectrics (b).

      Figure 2.7: A dielectric sphere rendered without accounting for the Fresnel reflectance (a) and accounting for it (b).

      • Energy conservation assumes that the energy reflected cannot exceed incident energy [DRS07]: Lr ≤ Ei, hence over the unit hemisphere Ω+ above the surface

      CHAPTER 3

       Models of BRDF

      In the last 40 years many material models have been proposed and some BRDFs are able to describe a wide subset of a material properties we have mentioned. Generally they can be classified into three big families that have different purposes and different ways of calculating interaction with light:

      • Phenomenological

      • Physically based

      • Data driven

      Phenomenological models, also known as Empirical models, are entirely based on reflectance data, which is fitted to analytical formulas, thus approximating the reflectance and reproducing characteristics of real world materials, but they will not necessarily appear realistic unless placed in an accurately simulated environment for such a model.

      Physically based models are based on Physics and Optics, with the assumption that the surface is rough at a fine scale, therefore described by a collection of micro facets with some distribution D of size and direction; these models are also referred to as “first principle” models. Usually they are represented by accurate and adjustable formulae; however the most common mathematical model has the form:

      which also takes into account the Fresnel term F. Effects like masking and self-shadowing (see Figure 3.1) [AMHH08] depend on the projected area of the microfacets and hence on the distribution D, generally described by the geometrical attenuation term G; for a review of common masking functions and a derivation of the exact form of the masking function from the microsurface profile, see the work by Heitz [Hei14]. This class of models can represent unique properties of the material and may include subsurface structure, generally resulting in complex calculations due to the interaction of the light with the surface structure.

      Data-Driven models approximates measured BRDFs with a suitable function space, for example, spherical harmonics or wavelets, weighted sum of separable functions or product of functions. Measured BRDF data, produced by most of the setups described in Chapter 4, can be stored in a table or a grid and then interpolated to produce a large look-up table when data is needed. This method is simple but inefficient in terms of storage. Moreover, the measured raw data is often noisy, hence the noise is likely to appear in the rendered material. A measured BRDF can be fitted to analytic models and employed to reconstruct the BRDF, thus significantly reducing storage size. The downside of this strategy is related to the inflexibility of many models, which are hard to edit and able to represent only limited classes of materials. A different solution is to approximate measured BRDFs with a suitable function space, for example, spherical harmonics or wavelets, weighted sum of separable functions, or product of functions. We refer to this class of models as Data-Driven models, described in Section 3.3.

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