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CHAPTER 2

Reflectance Functions

Human perception of a material depends on how the light that reflected, transmitted, absorbed by an object and reaches the viewer [DR05]. Hence, the appearance of materials may vary significantly depending on a wide range of properties such as color, smoothness, geometry, roughness, reflectance and angle from which the material is viewed and lighting directions.

Clearly not all materials interact with light in the same way: some let part of the light go through the surface, even to the extent of being transparent or semi-transparent; some scatter the light back, toward the light source itself; others show a mirror-like behavior. A ray of light can hit a surface at particular point of an object surface and possibly “travel” under its surface in different directions before leaving the surface in a different spot, after some time, with a totally different direction than initially. The most general reflectance function hence would need to take into account a number of variables (namely 16), which includes the wavelength of the incident ray of light (1 variable) and its direction (2 variables), the time when the ray of light hits the surface (1 variable) and the location on the surface (3 variables), the wavelength (1 variable) and direction (2 variables) of the ray of light leaving the surface at a (possibly) different location on the surface (3 variables) after some time (1 variable), and it must also account for the transmittance direction though the surface (2 variables). These parameters and the parametrization of such a general reflectance function RF are summarized in the following Table 2.1 and Equation 2.1:

The reflectance function RF needs to be measured for each wavelength in the visible spectrum or, at the very least, for the color channels of the RGB color space. Such a reflectance function, schematized in Figure 2.1, can fully describe a material appearance; however, due to its high dimensionality it is currently unfeasible to measure and would produce a vast amount of data, which cannot be handled by current computer graphics and virtual reality applications, considering also that the most general definition of a RF includes the dependence on the polarization state of the incident and reflected light.

For these reasons, in computer graphics and related fields it is customary to rely on several classes of simplified reflectance functions, obtained by discarding some dimensions, more suited for a practical use; in Figure 2.2 we report the taxonomy of the reflectance functions introduced in this section, along with their parameterizations.

The aforementioned simplifications are obtained by assuming the radiance to be constant along the rays of light, which allows discarding the z coordinate of the points on the surface under consideration. By dropping the dependency on the time (ti and t₀, hence assuming that the light transport does not take a measurable time), on the wavelength (λi and λ₀), thus assuming that the interaction with the surface does not change the wavelength of the light and restricting our attention to the RGB color bands) and assuming no transmittance (θt = ϕt = 0), we obtain the Bidirectional Surface Scattering Reflectance Distribution Function (BSSRDF), the most complex reported in Figure 2.2, which has 8 dimensions. The BSSRDF is able to represent a ray of light incident at a point on the surface, traveling under the surface where it gets scattered in different directions before leaving the surface from a different point and in a different direction. Many common translucent materials like milk, skin and alabaster are characterized by their subsurface scattering behavior that smooth the appearance of surface details, with the light shining through them. Thanks to its properties the BSSRDF is able to describe phenomena like translucency, self shadowing, self occlusions and inter-reflections. Unfortunately, it is still a very complex function to measure and often simpler representations are preferred over it.

Table 2.1: Parameters of a general reflectance function

Figure 2.1: Schematic representation of the general reflectance function RF.

Figure 2.2: Reflectance functions.

If we assume that the ray o light leaves the surface at the same location where it was incident (hence xi = xr = x, yi = yr = y), we obtain the Bidirectional Texture Function (BTF), a 6-dimensional representation able to describe not only the local variations in reflectance but also the mesoscopic effects due to small-scale geometry, like self-shadowing, self-occlusions and inter-reflections: BTF(x, y, θi, ϕi, θr, ϕr). The term BTF was first introduced by Dana et al. [DVGNK99] as an image-based representation that can describe the fine-scale appearance of a rough surface. The mesoscopic effects are difficult to quantify and separate from the measured data, hence BTF acquisition generally needs a large number of samples of the surface as well as high-end hardware support due to lengthy acquisition times and storage demands [HF13]. Nevertheless, there exist low cost acquisition setups, like the kaleidoscopic device by Han and Perlin [HP03] or the more recent mechanical gantry with rotating arms by Filip et al. [FVK14], built using a toy construction set. BTFs generally result in very realistic material appearance. The first BTF database, described in [DVGNK99], contains 61 real-world surfaces, each observed under 205 different combinations of lighting and viewing illuminations (plus 205 additional measurements for anisotropic surfaces), consisting of over 14.000 images.

A similar parameterizations is used to represent the Spatially Varying Bidirectional Reflectance Distribution Function (SVBRDF), used to describe opaque, smooth materials that can have different reflectance at each point of the surface (non-homogeneous materials) [HF13]. The SVBRDF parameterization hence must takes into account the location over the surface: SVBRDF (x, y, θi, ϕi, θr, ϕr). Capturing the SVBRDF sometimes requires long measurements and processing times as well as large, specialized and expensive hardware rigs, although under certain assumptions approximate measurements can be performed even with cellphone or tablet cameras [AWL15, RPG15]. The SVBRDF cannot describe subsurface scattering and mesoscopic effects.

For a homogeneous material that can reflect light but also transmit it through its surface we need to reintroduce the transmittance angles (θt, ϕt), thus obtaining the Bidirectional Scattering Distribution Function (BSDF), comprising scattering effects for both reflection and transmission: BSDF(θi, ϕi, θr, ϕr, θt, ϕt). The BSDF can describe both transparent and opaque materials.

If we take into account only the transmittance of homogeneous material it is possible to describe it with the Bidirectional Transmittance Distribution Function (BTDF), suitable to model how the light passes through transparent or semi-transparent surfaces [WMLT07], [HF13]: BTDF(θi, ϕi, θt, ϕt).

An opaque, smooth, homogeneous material can be represented with the Bidirectional Reflectance Distribution Function (BRDF): BRDF(θi, ϕi, θr, ϕr). By looking at Figure 2.2 it is possible to note how the BRDF can be considered a special case of the more complex functions described above [ASMS01]. In fact, the BRDF can be considered as a special case of the BTF and the SVBRDF when the position on the sample surface is fixed; any BTF datasets can be approximated as a sparse linear combination of rotated analytical BRDFs [WDR11] and the SVBRDF parameterization includes extra parameters with respect to the BRDF simply to take into account the location over the surface, but it must fulfill the BRDF reciprocity and energy conservation properties, which will be described in the next sections. Finally, a BSDF can be modeled as a sum of a BRDF (for the reflection component) and a BTDF (for the transmittance component).

2.1 DEFINITION OF THE BRDF

As discussed in the previous section, one of the possible ways to represent the way an opaque, homogeneous material interacts with the light is through the BRDF (Bidirectional Reflectance Distribution Function), a radiometric function currently used to varying levels of accuracy in photorealistic rendering systems. It describes, in the general case, how incident energy redirects in all directions across a hemisphere above the surface. Historically, the BRDF was defined and suggested over the more generalized BSSRDF (Bidirectional Scattering Surface Reflectance Distribution function) [JMLH01] by Nicodemus [NRH*77], as a simplified reflectance representation for opaque surfaces: the BRDF assumes that light entering a material leaves the material at the same position, whereas the BSSRDF can describe light transport between any two incident rays on a surface. Many common translucent materials like milk, skin and alabaster cannot be represented by a BRDF since they are characterized by their subsurface scattering behavior that smooths the surface details, with the light shining through them [GLL*04]. These materials are expensive to measure and render. However many techniques have been proposed JMLH01], [DS03], [HBV03], [DWd*08], [DI11], [KRP*15].

Before defining the reflectance we provide a brief introduction to some important radiometric terms. Radiant Energy, the basic unit of energy, is measured in Joules [J] and indicated with the symbol Q:

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 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.

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 (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