08 | Materials and Appearances

• Instructor: Lingqi Yan | UCSB
• Bilibili: https://www.bilibili.com/video/BV1X7411F744
• Lectures covered: Lecture 17 (Materials and Appearances)

Note: the previous post (#7) drove all the way to path tracing and solved the rendering equation. This one turns around and stares at the one term in that equation that is “known” yet endlessly complicated — $f_r$, the BRDF. Materials, appearances, glossiness, Fresnel, microfacets: they all answer the same question. When light hits this surface, how much energy goes where? The advanced rendering topics of Lecture 18 (BDPT, MLT, photon mapping, VCM, plus non-surface models such as participating media and hair) get their own post.

1 Materials and Appearances

Take one mesh and assign different materials to different parts, and the rendering changes completely: glazed ceramic shows a partial specular highlight, clay is diffuse, and the coffee on top is textured.

In graphics, a material is a BRDF. Different BRDFs define different material appearances — the BRDF decides how light gets reflected.

The family of materials we walk through looks roughly like this:

2 Diffuse / Lambertian Material

Light is reflected uniformly into all directions.

BRDF:

where $\rho$ is the albedo — the colour of the surface.

Derivation: assume the incoming light is uniform ($L_i$ constant). By energy conservation (the point emits no light and absorbs none, so whatever it receives it must reflect):

Since $\int_{H^2} \cos\theta_i \, d\omega_i = \pi$ over the hemisphere, if $L_o = L_i$ (perfect reflection, no absorption at all) then $f_r = \frac{1}{\pi}$. Multiply by the albedo $\rho$ and you get the diffuse BRDF.

3 Glossy Material

Somewhere between diffuse and a perfect mirror: reflected light concentrates in a small range around the specular direction instead of spreading out uniformly or collapsing to a single ray. Copper and aluminium surfaces behave like this.

4 Ideal Reflective / Refractive Material

When light travels from one medium (say air) towards another (say glass), the change in propagation speed splits it: part is specularly reflected, part enters the new medium and bends.

Such materials handle reflection and refraction together, which is why they are also called a BSDF (Bidirectional Scattering Distribution Function).

Perfect Specular Reflection

Azimuth is the horizontal angle measured from a reference direction (usually due north) clockwise to the target direction.

Angle of incidence equals angle of reflection: $\theta_i = \theta_o = \theta$

Azimuth relation: $\phi_o = (\phi_i + \pi) \mod 2\pi$

Computing the reflected direction:

The BRDF of perfect specular reflection can be written as a Dirac $\delta$ — it is non-zero only along the mirrored direction, and zero everywhere else.

Specular Refraction

Light can also pass through the surface. Entering a new medium makes it refract.

Snell’s Law: the refracted angle depends on the index of refraction (IOR) of the media on both sides. Different materials have different IORs.

where $\eta_i$ and $\eta_t$ are the IORs of the incident and transmitting media.

Common indices of refraction:

Computing the refracted angle:

Total Internal Reflection: when light travels from a denser to a rarer medium ($\eta_i / \eta_t > 1$) at a sufficiently large angle of incidence:

The value under the square root goes negative, so $\cos\theta_t$ has no real solution and refraction simply does not exist. Look at the condition: $1 - \cos^2\theta_i$ lies in $[0, 1]$, so only $\frac{\eta_i}{\eta_t} > 1$ (denser to rarer) can make the radicand negative. The light cannot escape and is entirely reflected back into the original medium.

Snell’s Window / Circle: the visible consequence of total internal reflection — looking up at the water surface from below, you only see the world above through a circular patch overhead; outside that circle, total internal reflection shows you a mirror image of the water below.

5 The Fresnel Term

The phenomenon: reflectance depends on the angle of incidence — the more grazing the view, the stronger the reflection. That is why distant water surfaces and tabletops always look “reflective”.

The exact formulas (accounting for polarisation; for reference only):

The Schlick approximation (faster, and what renderers actually use):

• $R(\theta)$: reflectance at angle of incidence $\theta$
• $R_0$: the base reflectance at normal incidence ($\theta = 0$)
• $\theta$: the angle of incidence (between ray and normal)
• $\cos\theta$: the dot product of incident direction and normal (for unit vectors)

where $R_0$ follows from the two IORs:

// cosTheta = dot(normal, viewDir), both unit vectors
float R0 = pow((n1 - n2) / (n1 + n2), 2.0);
float fresnel = R0 + (1.0 - R0) * pow(1.0 - cosTheta, 5.0);

The Schlick approximation uses two terms, $R_0$ and $(1 - \cos\theta)^5$, to turn the complicated Fresnel equations into a fast interpolation: it returns $R_0$ at normal incidence, approaches 1 at grazing angles, and transitions smoothly in between.

6 Microfacet Material

The Earth is not a smooth sphere — it has mountains and basins — yet photographs from space still show a specular highlight.

Microfacet theory

• Macroscale: the surface is flat and rough
• Microscale: the surface is bumpy, and each microfacet behaves like a tiny mirror

The microfacet BRDF

The key is the distribution of microfacet normals.

• Normals concentrated $\Rightarrow$ glossy
• Normals spread out $\Rightarrow$ diffuse

Microfacet materials show up everywhere: motorcycle dashboards, rough cushions, smooth leather, near-mirror metals, wooden furniture.

7 Isotropic and Anisotropic Materials

In materials and rendering, isotropic and anisotropic describe whether a surface’s properties depend on direction.

The one-line difference: an isotropic material’s reflection looks the same as you rotate the object; an anisotropic material’s reflection stretches and deforms with direction.

An anisotropic material’s reflection varies with azimuth because of directional microstructure on the surface — the grain direction of brushed metal, for instance.

8 Properties of BRDFs

A BRDF must satisfy three basic physical constraints — non-negativity, reciprocity and energy conservation. Beyond that it can be classified as isotropic or anisotropic, and it is linear.

9 BRDF Measurement

A BRDF is a theoretical model, but real materials reflect light in very complicated ways:

• The microstructure differs wildly from material to material
• The same material behaves completely differently at different angles
• Theoretical models (Cook-Torrance, for instance) are only approximations whose parameters need real data to be calibrated

So trustworthy BRDF data has to come from actual measurement.

Motivation: skip developing and deriving a model, and render directly with real material data.

Method (using a gonioreflectometer):

for each outgoing direction wo:
    move the light source to illuminate the surface from wo
    for each incident direction wi:
        move the sensor to direction wi
        measure the incident radiance

Efficiency optimisations:

• For isotropic surfaces, the dimensionality drops from 4D to 3D
• Reciprocity halves the number of measurements
• Cleverer optical system design

Image-based measurement: Marschner et al. 1999 replaced point-by-point mechanical measurement with a camera.

Challenges in measurement:

• Accurate measurement at grazing angles (important because of the Fresnel effect)
• High-frequency specular peaks need dense enough sampling to capture
• Retro-reflection
• Spatially varying reflectance

Representing measured data: desirable properties are compactness, faithful reproduction of the measurements, efficient evaluation for arbitrary direction pairs, and suitability for importance sampling. A common choice is tabular representation: store regularly sampled values over $(\theta_i, \theta_o, |\phi_i - \phi_o|)$ (reparameterising when needed to match specular peaks better). Storage requirements are heavy — the MERL BRDF database [Matusik et al. 2004], for example, holds 90 × 90 × 180 sets of measurements.

Summary

This chapter is the gateway to the “materials” side, taking apart the known term $f_r$ in the rendering equation:

This post is note #8 in the GAMES101 - Modern Computer Graphics learning series.