Assuming the geometric model of optics, light interacts in three possible ways after contact with any matter, i.e. reflection, refraction and absorption. Different ma- terials reflect, refract and absorbs different amount of light which can be as simple as pure reflections in a mirrored surface to much more complex reflections which adds components of diffraction and scattering. Ignoring the complex interactions for brevity, reflection happens in broadly three ways for most materials; diffused, specular and a mixture of these two, termed asglossy in Computer Graphics termi- nology. The following sections will describe the mathematical representation of the light interaction.
2.2.1 Bi-directional Reflection Distribution Function (BRDF)
The different types of reflections can be estimated by a function which, given a specific radiance amount Li in a particular direction !i, calculates the radiance Lo
the Bi-directional Scattering Surface Reflection Function (BSSRDF). However, as- suming that the light incident on a point x on a surface reflects from the same pointxsimplifies the relation by ignoring the more complex, sub-surface scattering phenomenon. The new simplified function is called Bi-directional Reflectance Dis- tribution Function (BRDF) [Bartell et al., 1981]. The BRDF at a point x is given by the function fr(x, ! ⇥) where and ⇥ are the incident and reflected ray
direction respectively. In the following equation, the ! is the solid angle between light source and the area similar to the diagram in Figure 2.2.
fr(x, !⇥) = dL(x!⇥)
dE(x ) =
dL(x!⇥)
dL(x )cos✓d! (2.2) BRDF is the ratio of the outgoing radiance L(x ! ⇥) and the incoming irradianceE(x )at the pointx. Equation 2.2 further decomposes the irradiance in terms of incoming radiance at an angle✓with the surface normal at pointx. The incoming radiance is calculated over the entire hemisphere around the point x. In practice, the hemisphere is discretised into blocks of solid angles d! [Dutre et al., 2006]. The total radiance is computed by integrating the discreet radiances over the full hemisphere.
2.2.2 BRDF Properties
There are four main properties of a BRDF as described below.
2.2.2.1 Anisotropy
Anisotropy is one of the main properties of BRDF. It is a seven dimensional function of 3D position x, outgoing ray angle ⇥, and incoming ray angle . fr(x,⇥, ) =
fr((x, y),(✓i, i),(✓o, o)). A more detailed representation uses the wavelength ( )
of the incoming light but it is ignored here for brevity. Changing any of these di- mensions naturally changes the BRDF too. For example, if the surface containing pointxis rotated around the surface normal onx, as the incident angle changes, the BRDF will also change. However there are a few materials that are isotropic and this kind of rotation has no effect on their reflective properties.
2.2.2.2 Helmholtz Reciprocity
The Helmholtz reciprocity dictates that reversing the direction of light does not change the amount of reflected light [Von Helmholtz, 1867].
Therefore, both the direction in the BRDF function can be used interchangeably.
2.2.2.3 Linearity
Linearity is one of the primary properties of BRDF which states that the value of the BRDF for a particular incident direction is independent of any number of irradiance along other incident directions on the same point. BRDF acts as a linear function with respect to all possible simultaneous incident directions and thus is integrable over the hemisphere surrounding the pointx [Bartell et al., 1981].
dL(x!⇥) =fr(x, !⇥)dE(x )
Integrating over hemisphere (⌦) around x, L(x!⇥) =
Z
⌦
fr(x, !⇥)dE(x ) (2.4)
This is especially useful while deriving the “Rendering Equation” as described in section 2.3.
2.2.2.4 Energy conservation
The final property is conservation of energy which is central to the formulation of Physically-Based BRDF. It ensures that the incoming energy is always exactly same to the sum of total outgoing energy and the absorbed energy. Expanding equation 2.2, the ratio of incoming and outgoing radiance is always 1 [Dutre et al., 2006][Bartell et al., 1981].
dL(x!⇥)
dL(x ) =fr(x, !⇥)cos✓d!1 (2.5) The next three sections briefly describe the most commonly occurred BRDF types, diffused, specular and the glossy, respectively.
2.2.3 Diffused and Specular BRDF
Diffused BRDF:Diffused surfaces are made of materials that reflect light uniformly over the entire hemisphere around any given pointxon the surface. Regardless of the incident angle of light, a point in a purely diffused surface looks exactly the same from all directions around it. An Ideal diffused BRDF is given by fr(x, $⇥) = ⇢d/⇡
where ⇢d denotes the reflectance, i.e. the fraction of incident energy reflected on
Specular BRDF:Incident light at a given point on Specular surfaces reflects and refracts light only in one particular direction. Following the geometric model of optics [Hamilton, 1828], given the incident angle ( i) and the surface normal (Nˆx)
on the point of incident, the reflection direction (R) can be calculated as:
R= 2( ˆNx· i)N i (2.6)
Light can pass through mediums such as air, glass and water. In the process, the direction ⇥r in the refractive medium of the light changes from the incident
direction ( i) (in the incident medium) on the point of contact between the two
mediums. Snell’s law [Feynman et al., 2011] gives us the relation as:
⌘rsin⇥r=⌘isin i (2.7)
Here, the constants⌘r and ⌘i are called the refractive index and they belong to the
refractive medium and the incident medium respectively. Usually ⌘ is same for a given medium but in some cases it can vary throughout the medium causing complex phenomenons such as sub-surface scattering [Hanrahan and Krueger, 1993].
2.2.4 Glossy BRDF
Both diffused and specular BRDFs are mostly theoretical because real-world materi- als show a degree of ideal specular or ideal diffused BRDFs. Often materials have a combination of diffused and specular BRDFs. Such kind of surfaces are called glossy surfaces. Due to the apparent complexity of modeling BRDF for real-world objects, there are a number of shading models proposed that can replicate different types of glossy surfaces. This section provides a discussion on the 5 most relevant BRDF models out of them.
2.2.4.1 Lambertian model
The Lambertian model [Lambert, 1760] is the simplest and oldest of all models. Constructed around 1760 by Swiss mathematician Johann H. Lambert, it models the diffused BRDF where the value of BRDF is constant (fr(x, $ ⇥) =kd =⇢d/⇡)
over any direction of the hemisphere overx[Bartell et al., 1981; Dutre et al., 2006].
2.2.4.2 Phong model
Phong is the most popular shading model to date partly because it is slightly less complicated than some of the other models and it is hardware accelerated. The
BRDF is given by:
fr(x, $⇥) =ks
R·⇥n ˆ
N· +kd (2.8)
Here, reflected ray direction R can be found with Equation 2.6 and kd is found
from the lambertian constant BRDF⇢d/⇡. All the other symbols bear usual refer-
ences. However, Phong model does not support energy conservation or Helmholtz reciprocity as a tradeoff for its simplicity and the less computational requirements [Phong, 1975].
2.2.4.3 Modified Blinn-Phong model
Modified Blinn-Phong model [Blinn, 1977] solves the issues with Phong model and uses the half vector (H) between incident direction and eye direction ⇥.
fr(x, $⇥) =ks( ˆN·H)n+kd (2.9)
All the symbols used have the same references as the previously described models. Modified Blinn-Phong is a fairly popular model in Computer Graphics.
2.2.4.4 Cook-Torrance model
The Modified Blinn-Phong model was further augmented by Cook and Torrance to introduce a micro-faceted model [Cook and Torrance, 1982]. It assumes that the surface is made of randomly placed and oriented micro-sized smooth planer surfaces. Given a distribution of the facets, this model can model real-world objects fairly closely. fr(x, $⇥) = F( ) ⇡ D(✓h)G ( ˆN · )( ˆN ·⇥)+kd (2.10) Here “F” is the unpolarised Fresnel reflectance calculated over the angle [Kenyon, 2008], D is the micro-facet distribution and the G denotes a geometry term for shad- owing between those micro-facets.
2.2.4.5 Ward model
Gregory Ward took a new approach based on empirical data [Ward, 1992]. Ward and other comparable models such as LaFortune [Marschner et al., 1999] are the closest to the BRDF of real-world objects because of the intuitive parameterisation for the amount of diffused reflectance (⇢d), specular reflectance (⇢s) and the measure
will still hold the energy conservation law. fr(x, $⇥) = ⇢d ⇡ +⇢s e tan 2✓ h ↵2 4⇡↵2 q ( ˆN · )( ˆN·⇥) (2.11) A number of other empirical models with many different approaches exist, such as models that fit empirical data obtained from real-world materials to known distributions and they all have their own set of advantages. However, assuming the general notion of the radiometric quantities and how light reacts with the materials can be used to model the equilibrium of light energy in a scene. Next Section 2.3 describes the “rendering equation” as a mathematical formulation of the energy equilibrium in a scene.