GAMES101 Lecture 17 - Materials and Appearances

GAMES101_Lecture_17.pdf

I. Materials and Appearances

Textures and appearances are closely related:

• Under different lighting conditions textures appears to be different.

Some of the features from natural materials:

• Water

• Scattering

• Hair/Fur

• Clothes

• Subsurface Scattering (SSS)

• ...

The term material is equivalent to BSDF.

Bidirectional Scattering Distribution Function, BSDF

• The generalization of BRDF and BTDF (Bidirectional Transmittance Distribution Function), which takes both refraction and reflection into consideration.

Diffuse/Lambertian Material

From [Mitsuba render, Wenzel Jakob, 2010

Light is equally reflected in each output direction.

Suppose the incident lighting is uniform in radiance, and without self-emission we have:

If the material absorbs no light, then $f_r=1/\pi$$f_r = 1/\pi$.

On lambertian material we have

in which $\rho$$\rho$ is called albedo, or color.

Glossy Material

From [Mitsuba render, Wenzel Jakob, 2010

Ideal Reflective/Refractive Material

From [Mitsuba render, Wenzel Jakob, 2010

• Part of the spectrum is absorbed by the underlying material.

Perfect Specular Reflection

From PBRT

$\theta$$\theta$ and $\varphi$$\phi$ are obtained from the local coordinate system.

BRDFs for the perfect specular reflection are difficult to write.

• Related to the $\delta$$\delta$ function

Specular Refraction

Light refracts when it enters a new medium.

Snell's Law

Transmitted angle depends on

• index of refraction (IOR) for incident ray

• IOR for exiting ray

Index of frefraction is wavelength dependent. These are averages.

Law of Refraction

Definition: Total internal reflection: When light is moving from a more optically dense medium to a less optically dense medium, i.e.,

then light incident on boundary from large enough angle will not exit the medium. The critical angle can be computed from equation $\text{1}$$\ref{total_internal_reflection}$ by substituting ${\theta }_t=\pi /2$$\theta_t = \pi / 2$ into the equation.

• Snell's Window/Circle

  

Fresnel Reflection/Term

Reflectance depends on incident angle (and polarization of light).

This example: reflectance increases with grazing angle [Lafortune et al. 1997]

Fresnel Term

• Polarization: The component of the electric field parallel to the incidence plane is termed p-like (parallel) and the component perpendicular to this plane is termed s-like (from senkrecht, German for perpendicular). (Wikipedia)

Dielectric, $\eta =1.5$$\eta = 1.5$:

Conductor:

• Conductors have negative indices of refraction.

Formulae

Accurate: polarization taken into consideration

$n$$n$ is related to $\eta$$\eta$, the intrinsic property of the material.

Approximate: Schlick's approximation

Microfacet Material

State of art.

Microfacet Theory

Rough surface:

• Macroscale: flat & rough

• Microscale: bumpy & specular

Microfacet: individual elements of surface act like mirrors

Key: The distribution of their normals. Each microfacet has its own normal.

• Concentrated <==> Glossy

  

  

• Spread <==> Diffuse

  

  

Microfacet BRDF:

In which:

• $\text{F}$$\text{F}$ is the Fresnel term

• $\mathbf{\text{G}}$$\textbf{G}$ is the shadowing-masking term

  • Microfacets may block each other.

  • Happens often when lights are near the grazing angle.

• $\mathbf{\text{D}}$$\textbf{D}$ is the distribution of normals

  • How many normals are there that match the direction of $\mathbf{\text{h}}$$\textbf{h}$?

Isotropic/Anisotropic Materials (BRDFs)

Key: Directionality of the underlying surface.

Anisotropic: Reflection depends on azimuthal angle $\varphi$$\phi$

• Results from oriented microstructure of the surface, e.g.

  • Brushed Metal

  • Nylon

  • Velvet

II. Further on BRDFs

Properties of BRDFs

• Non-negativity: On any point, $f_r$$f_r$ is always non-negative.

  $$f_r({\omega }_i,{\omega }_r)\ge 0$$

• Linearity: BRDFs can be directly summed.

  $$L_r(\text{p},{\omega }_r)={\int }_{H^2}{f}_r(\text{p},{\omega }_i,{\omega }_r)L_i(\text{p},{\omega }_i)\cos {\theta }_i\mathrm{d}{\omega }_i$$

  The nature of integration make BRDFs addable.

  

• Reciprocity Principle

  $$f_r({\omega }_r,{\omega }_i)=f_r({\omega }_i,{\omega }_r)$$

  

• Energy Conservation

  $$\mathrm{\forall }{\omega }_r,{\int }_{H^2}{f}_r({\omega }_i,{\omega }_r)\cos {\theta }_i\mathrm{d}{\omega }_i\le 1$$

• Isotropic/Anisotropic

  • If isotropic: $f_r({\theta }_i,{\varphi }_i;{\theta }_r,{\varphi }_r)=f_r({\theta }_i,{\theta }_r,{\varphi }_r-{\varphi }_i)$$f_r (\theta_i, \phi_i; \theta_r, \phi_r) = f_r (\theta_i, \theta_r, \phi_r - \phi_i)$, which essentially means that the dimension of BRDF is reduced by 1.

    Then from reciprocity we have

    $$f_r({\theta }_i,{\theta }_r,{\varphi }_r-{\varphi }_i)=f_r({\theta }_r,{\theta }_i,{\varphi }_i-{\varphi }_r)=f_r({\theta }_i,{\theta }_r,|{\varphi }_r-{\varphi }_i|)$$

  

Measuring BRDFs

Target:

• Avoid need to develop/derive models

  • Automatically includes all of the scattering effects present

• Can accurately render with real-world materials

  • Useful for product design, special effects, ...

General Approach

• For each outgoing direction ${\omega }_o$$\omega_o$

  • move light to illuminate surface with a thin beam from ${\omega }_o$$\omega_o$

  • For each incoming direction ${\omega }_i$$\omega_i$

    • move sensor to be at direction ${\omega }_i$$\omega_i$ from surface

    • measure incident radiance

Gonioreflectometer:

Spherical gantry at UCSD

Improving Efficiency

• Isotropic surfaces reduce dimensionality from 4D to 3D

• Reciprocity reduces # of measurements by half

• Clever optical systems

Challenges

• Accurate measurements at grazing angles

  • Important due to Fresnel effects

• Measuring with dens enough sampling to capture high frequency specularities

• Retro-reflection

• Spatially-varying reflectance

• ...

Representing Measured BRDFs

Desirable qualities:

• Compact representation

• Accurate representation of measured data

• Efficient evaluation for arbitrary pairs of directions

• Good distributions available for importance sampling

Tabular Representation

MERL BRDF Database [Matusik et al. 2004], $90\times 90\times 180$$90 \times 90 \times 180$ measurements

Store regularly-spaced samples in $({\theta }_i,{\theta }_o,|{\varphi }_i-{\varphi }_o|)$$(\theta_i, \theta_o, \abs{\phi_i - \phi_o})$

• Better: reparameterize angles to better match specularities

• Generally need to resample measured values to table

• Very high storage requirements

Appendix A: Microfacet Models

Reference: Microfacet Models (pbr-book.org)

Introduction

Many geometric-optics-based approaches to modeling surface reflection and transmission are based on the idea that rough surfaces can be modeled as a collection of small microfacets. They are often modeled as heightfields, where the distribution of facet orientations is described statistically.

• Microsurface is used to describe microfacet surfaces

• Macrosurface is used to describe the underlying smooth surface (as represented by a Shape, or other Object in our framework for homework).

The microfacet-based BRDF models work by statistically modeling the scattering of light from a large collection of microfacets.

• If we assume that the differential area $\mathrm{d}A$$\dd{A}$ is relatively large compared to the size of individual microfacets, then a large number of microfacets are illuminated,

• and it is the aggregate behavior of these microfacets that determines the observed scattering.

The two main components of microfacet models are:

• A representation of the distribution of facets, and

• A BRDF for individual microfacets.

  • Perfect Mirror Reflection

    • most commonly used

  • Specular Transmission

    • useful for modeling many translucent materials

  • The Oren-Nayar Model

    • treats microfacets as Lambertian reflectors

Three important geometric effects to consider with Microfacet Reflection Models:

• Masking

  • Microfacet is occluded by another facet

• Shadowing

  • Microfacet may lie in the shadow of a neighboring microfacet

• Interreflection

  • Cause a microfacet to reflect more light than predicted by the amount of direct illumination and the low-level microfacet BRDF

Oren-Nayar Diffuse Reflection

Idea: Real-world objects do not exhibit perfect Lambertian reflection.

• Rough surfaces generally appear brighter as the illumination direction approaches the viewing direction.

• Describe rough surfaces by V-shaped microfacets, which is

  • Described by a spherical Gaussian distribution with a single parameter $\sigma$$\sigma$, the standard deviation of the microfacet orientation angle

  • Interreflections: Only consider the neighboring microfacet

Approximation:

where if $\sigma$$\sigma$ is in radians,

Microfacet Distribution Functions

One important characteristics of a microfacet surface is represented by the distribution function $D({\omega }_h)$$D(\omega_h)$, which gives the differential area of microfacets with the surface normal ${\omega }_h$$\omega_h$. In pbrt, microfacet distribution functions are defined in the same BSDF coordinate system as BxDFs. As such,

• a perfectly smooth surface could be describe by a delta distribution that was non-zero only when ${\omega }_h$$\omega_h$ was equal to the surface normal: $D({\omega }_h)=\delta ({\omega }_h-(0,0,1))$$D(\omega_h) = \delta (\omega_h - (0, 0, 1))$

Microfacet distribution functions must be

• Normalized: Given a differential area of the microsurface, $\mathrm{d}A$$\dd{A}$, then the projected area of the microfacet faces above that area must be equal to $\mathrm{d}A$$\dd{A}$.

  $${\int }_{H^2(\mathbf{\text{n}})}D({\omega }_h)\cos {\theta }_h\mathrm{d}{\omega }_h=1$$

  

Beckmann Distribution

A widely used microfacet distribution function based on a Gaussian distribution of microfacet slops is due to Beckmann and Spizzichino.

The traditional definition of the Beckmann-Spizzichino model is

where if $\sigma$$\sigma$ is the RMS slope of the microfacets, then $\alpha =\sqrt{2}\sigma$$\alpha = \sqrt{2}\sigma$.

• RMS: Root Mean Square

The anisotropic microfacet distribution function is

• Note that the original isotropic variant falls out when ${\alpha }_x={\alpha }_y$$\alpha_x = \alpha_y$

When programming, the algorithm directly translates the above equation, but pay special attention to the following issues:

• Infinity value of $\tan \theta$$\tan\theta$ corresponds to viewing at perfectly grazing directions, and is thus valid. Zero should be explicitly returned.

Trowbridge-Reitz Distribution

Anisotropic variant given by

In comparison to the Beckmann-Spizzichino model,

• It has higher tails - it falls off to zero more slowly for directions far from the surface normal.

  • This characteristic matches the properties of many real-world surfaces well

Usually, we specify ${\alpha }_x$$\alpha_x$ and ${\alpha }_y$$\alpha_y$ by roughness, which is between $[0,1]$$[0, 1]$, and value close to zero correspond to near-perfect specular reflection.

Masking and Shadowing

Smith's Masking-Shadowing Function: Some microfacets will be invisible from a given viewing or illumination direction because,

1. They are back-facing

2. Some of the forward-facing microfacet area will be hidden due to being shadowed by back-facing microfacets.

This is described by

which gives the fraction of microfacets with normal ${\omega }_h$$\omega_h$ that are visible from direction $\omega$$\omega$.

• Note that $0\le G_1(\omega ,{\omega }_h)\le 1$$0 \leq G_1(\omega, \omega_h) \leq 1$.

Normalization Constraint: A differential area $\mathrm{d}A$$\dd{A}$, as shown in Figure 8.17, has area $\mathrm{d}A\cos \theta$$\dd{A} \cos\theta$ when viewed from a direction $\omega$$\omega$ that makes an angle $\theta$$\theta$ with the surface normal. The area of visible microfacets seen from this direction must also be equal to $\mathrm{d}A\cos \theta$$\dd{A} \cos\theta$, which leads to a normalization constraint for $G_1$$G_1$:

Compute $G_1$$G_1$ using $\mathrm{\Lambda }(\omega )$$\Lambda(\omega)$: Because the microfacets form a heightfield, every back-facing microfacet shadows a forward-facing microfacet of equal projected area in the direction $\omega$$\omega$. If $A^{+}(\omega )$$A^+(\omega)$ is the projected area of forward-facing microfacets as seen from the direction $\omega$$\omega$ and $A^{-}(\omega )$$A^-(\omega)$ is the projected area of backward-facing microfacets, then $\cos \theta =A^{+}(\omega )-A^{-}(\omega )$$\cos\theta = A^+(\omega) - A^-(\omega)$.

We can thus alternatively write the masking-shadowing function as the ratio of visible microfacets area to total forward-facing microfacet area:

Shadowing-masking functions are traditionally expressed in terms of an auxiliary function $\mathrm{\Lambda }(\omega )$$\Lambda(\omega)$, which measures invisible masked microfacet area per visible microfacet area.

After some algebra we have

Specifying $\mathrm{\Lambda }(\omega )$$\Lambda(\omega)$: The microfacet distribution alone doesn't impose enough conditions to imply a specific $\mathrm{\Lambda }(\omega )$$\Lambda(\omega)$ function.

• If we assume that there is no correlation between the heights of nearby points on the microsurface, then it's possible to find a unique $\mathrm{\Lambda }(\omega )$$\Lambda(\omega)$ given $D({\omega }_h)$$D(\omega_h)$.

  • For many microfacet models, a closed-form expression can be found.

  • Although this isn't true in reality, the resulting $\mathrm{\Lambda }(\omega )$$\Lambda(\omega)$ functions turned out to be fairly accurate when compared to measure reflection from actual surfaces.

  • Beckmann-Spizzichino: Under the assumption of no such correlation, we have

    $$\mathrm{\Lambda }(\omega )=\frac{1}{2}(\mathrm{erf}\left(a\right)-1+\frac{e^{-a^2}}{a\sqrt{\pi }})$$

    where $\alpha =1/(\alpha \tan \theta )$$\alpha = 1/(\alpha \tan\theta)$ and $\mathrm{erf}$$\erf$ is the error function, $\mathrm{erf}\left(x\right)=2/\sqrt{\pi }{\int }_0^x{e}^{-u^2}\mathrm{d}u$$\erf(x) = 2/\sqrt{\pi}\int_{0}^{x}e^{-u^2}\dd{u}$.

    In pbrt, a rational polynomial approximation is used, to avoid calling std::erf() and std::exp() which are fairly expensive to evaluate.

Computing the interpolated $\alpha$$\alpha$ for anisotropic distributions: It is the easiest to compute $\mathrm{\Lambda }(\omega )$$\Lambda(\omega)$ by taking their corresponding isotropic function and stretching the underlying surface according to the ${\alpha }_x$$\alpha_x$ and ${\alpha }_y$$\alpha_y$ values. Equivalently, one can compute an interpolated $\alpha$$\alpha$ value for the direction of interest and use that with the isotropic function.

• The function is close to one over much of the domain, but falls to zero at grazing angles.

• Increasing surface roughness causes the function to fall off more quickly.

Computing $G({\omega }_o,{\omega }_i)$$G(\omega_o, \omega_i)$: This function gives the fraction of microfacets in a differential area that are visible from both directions ${\omega }_o$$\omega_o$ and ${\omega }_i$$\omega_i$. Defining $G$$G$ requires some additional assumptions. For starters,

• If we assume that the probability of a microfacet being visible from both directions is the probability that it is visible from each direction independently, then we have

  $$G({\omega }_o,{\omega }_i)=G_1({\omega }_o)G_1({\omega }_i).$$

  In practice,

  • This often underestimates $G$$G$.

  • The closer together the ${\omega }_o$$\omega_o$ and ${\omega }_i$$\omega_i$ are, the more correlation there is between $G_1({\omega }_o)$$G_1(\omega_o)$ and $G_1({\omega }_i)$$G_1(\omega_i)$.

• A more accurate model can be derived, assuming that microfacet visibility is more likely the higher up a given point on a microfacet is. This assumption leads to the model

  $$G({\omega }_o,{\omega }_i)=\frac{1}{1+\mathrm{\Lambda }({\omega }_o)+\mathrm{\Lambda }({\omega }_i)}.$$

The Torrance-Sparrow Model

For very detailed derivation, refer to the PBR book. Basically, we assume that individual microfacets are perfectly specular, and therefore only those that have the direction of their normals matched with the orientation of ${\omega }_h$$\omega_h$ will reflect light.