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The Comprehensive PBR Guide

by Allegorithmic — vol. 1

Light and Matter : The theory of Physically—Based Rendering and Shading

Cover by Gaëtan Lassagne, written by Wes McDermott

vol. 1 — The theory of Physically—Based Rendering and Shading

• Light Rays — 2

• Absorption and Scattering (Transparency and Translucency) — 3

• Diffuse and Specular Reflection — 4

Microfacet Theory — 5

• Color — 6

• BRDF — 6

• Energy Conservation — 7

• Fresnel Effect — 7

F0 (Fresnel Reflectance at 0 Degrees) — 8

• Conductors and Insulators (Metals and Non Metal) — 9

Metals — 9

Non-Metals — 10

• Linear Space Rendering — 11

• The Key Factors — 11

• References — 12

Technical edit by: Cyrille Damez and Nicolas Wirrmann

vol.1 — The theory of Physically—Based Rendering and Shading

Light and Matter

The theory of Physically—Based Rendering and Shading

The Light Ray Model states that a light ray has the

trajectory of a straight line in homogeneous transparent

media such as air. The Light Ray Model also says that the

ray will behave in a predictable manner when

encountering surfaces such as opaque objects or passing

through a different medium such as air to water. This

makes it possible to visualize the path the light ray will

follow as it moves from a starting point to where it

eventually changes into another form of energy such as

heat.

The light ray that hits a surface is called the Incident Ray

and the angle that at which it hits is called the Angle of

Incidence as shown in figure 01.

A light ray is incident on a plane interface between two

media.

When a light ray hits a surface, either or possibly both of

these things can happen:

1. The light ray is reflected off the surface and travels

in a different direction. It follows the Law of Reflection,

which states that the Angle of Reflection is equal to the

Angle of Incidence (Reflected Light).

2. The light ray passes from one medium to another in

the trajectory of a straight line (Refracted Light).

At this point, we can state that light rays split into two

directions: reflection and refraction. At the surface, the

ray is either reflected or refracted and it can be

eventually absorbed by either medium. However,

absorption doesn’t occur at the surface.

Light is a complex phenomenon as it can exhibit properties of both a wave and a particle. As a result, different models

have been created to describe the behavior of light. As texture artists, we are interested in the Light Ray Model as it

describes the interaction of light and matter. It’s important for us to understand how light rays interact with surface

matter because our job is to create textures that describe a surface. The textures and materials we author interact

with light in our virtual worlds and the more we understand about how light behaves, the better our textures will look.

In this guide, we will discuss the theory behind the physics through which physically-based rendering models are

based upon. We will start with a light ray and work up to defining the key factors for PBR.

vol. 1 — The theory of Physically—Based Rendering and Shading

When traveling in an inhomogeneous medium or

translucent material, light can be absorbed or scattered:

1. With absorption, the light intensity decreases as it is

changed into another form of energy (usually heat), and

its color changes as the amount of light absorbed

depends on the wavelength, but the direction of the ray

doesn’t change.

2. With scattering, the ray direction is changed randomly,

the amount of deviation depending on the material.

Scattering randomizes light direction but the intensity

doesn’t change. An ear is a good example. The ear is thin

(absorption is low), so you can see the scattered light

penetrating out of the back of the ear. If there is no

scattering and the absorption is low, rays can pass

directly through the surface such as with glass. For

example, if you are swimming in a pool, which is

hopefully clean, you can open your eyes and see at a

fairly good distance through the clear water. However,

let’s imagine that same pool hasn’t been cleaned in a

while and the water is dirty. The dirt particles scatter the

light and thus make the clarity of the water much lower.

The further light travels in such a medium/material, the

more it is absorbed and/or scattered. Therefore, object

thickness plays a large role in how much the light is

absorbed or scattered. A thickness map can be used to

describe object thickness to the shader as shown in figure

02.

Absorption and Scattering (Transparency and Translucency)

Specular reflection is light that has been reflected at the

surface, as we discussed above in the Light Ray section.

The light ray is reflected off the surface and travels in a

different direction. It follows the Law of Reflection, which

states that on a perfectly planar surface the Angle of

Reflection is equal to the Angle of Incidence. However, it

is important to note that most surfaces are irregular and

that the reflected direction will therefore vary randomly

based on the surface roughness. This changes light

direction, but the light intensity remains constant.

Rougher surfaces will have larger and dimmer looking

highlights. Smoother surfaces will keep specular

reflections focused, which can appear to look brighter or

more intense when looked at from the proper angle.

However, the same total amount of light is reflected in

both cases as shown in figure 03.

Diffuse reflection is light that has been refracted. The

light ray passes from one medium to another and is

scattered multiple times inside the object. Then it is

Diffuse and Specular Reflection

Object thickness plays a large role

in how much the light is absorbed

or scattered

vol.1 — The theory of Physically—Based Rendering and Shading

refracted again out of the object making its way back to

the original medium at approximately the same point

where it went through the first time as shown in figure

04.

Diffuse materials are fairly absorbent, meaning that if the

refracted light travels for too long in that material, it has

a good chance of being completely absorbed. This means

that if the light ever comes out of that material, it has

probably not traveled very far from the point of entry.

That’s why the distance between the entry and exit

points can be neglected. The Lambertian model, which is

usually used for diffuse reflection in a traditional shading

sense, does not take surface roughness into account, but

there are diffuse reflection models that do such as Oren-

Nayar.

Materials that have both high scattering but low

absorption are sometimes referred to as «participating

media» or «translucent materials». Examples of these are

smoke, milk, skin, jade and marble. Rendering of the

latter three may be possible with the additional modeling

of subsurface scattering where the difference between

the ingoing and outgoing point of the light ray is no

longer neglected. Accurate rendering of medium with

highly varying and very low scattering and absorption like

smoke or fog may require even more

expensive methods such as Monte Carlo

simulations.

Rougher surfaces will

have larger and dimmer

looking highlights

vol. 1 — The theory of Physically—Based Rendering and Shading

In theory, both diffuse and

specular reflection are dependent

on the surface irregularities

where the light rays intersect. In

practice though, the effect of

roughness on diffuse reflection is

much less visible because of the

scattering happening inside the

material. As a result, the

outgoing direction of the ray is

fairly independent of surface

roughness and the incident

direction. The most common

model for diffuse reflection

(Lambertian) completely neglects

it.

In this document, we have

referred to these surface

irregularities as surface

roughness. Actually, it is often

referred to by several names

such as roughness, smoothness, glossiness or micro-

surface, depending on the PBR workflow in use, but they

describe the same aspect of a surface, which is sub-texel

geometric detail.

These surface irregularities are authored in the roughness

or glossiness map depending on the workflow you are

using. A physically-based BRDF is based on the

microfacet theory which supposes that a surface is

composed of small-scaled planar detail surfaces of

varying orientation called microfacets. Each of these

small planes reflects light in a single direction based on

its normal as shown in figure 05.

Micro-facets whose surface normal is oriented exactly

halfway between the light direction and view direction will

reflect visible light. However, not all microfacets where

the microsurface normal and the half normal are equal

will contribute as some will be blocked by shadowing

(light direction) or masking (view direction) as is

illustrated in figure 05.

The surface irregularities at a microscopic level cause

light diffusion. For example, blurred reflections are due

to scattered light rays. The rays are not reflected in

parallel so we perceive the specular reflection as blurred

as shown in figure 06.

The surface irregularities at a

microscopic level cause light

diffusion

vol.1 — The theory of Physically—Based Rendering and Shading

The color of a surface

(which is to say the color

that we see) is due to

which wavelengths are

emitted by the light source,

which are absorbed by the

object and which others

are reflected both

specularly and diffusely.

The remaining reflected

wavelengths are what we

see as color.

For example, the skin of an

apple mostly reflects red

light. Only the red

wavelengths are scattered

back outside the apple skin

and the others are

absorbed by it as shown in figure 07.

It also has bright specular highlights the same color as

the light source because with materials like the skin of an

apple that are not electrical conductors (dielectrics),

specular reflection is almost independent of wavelength.

Therefore, for such materials the specular reflection is

never colored. We will discuss more about the different

type of materials (metals and dielectrics) in later sections.

A Bidirectional Reflectance Distribution Function (BRDF)

simply put is a function that describes the reflectance

properties of a surface. In computer graphics, there are

different BRDF models some of which are not physically

plausible. For a BRDF to be physically plausible, it must

be energy conserving and exhibit reciprocity. For

reciprocity, I am referring to the Helmholtz Reciprocity

principle, which states that incoming and outgoing light

rays can be considered as reversals of each other without

affecting the outcome of the BRDF.

The BRDF used by Substance’s PBR shaders is based on

Disney’s «principled» reflectance model, which is based on

the GGX microfacet distribution. GGX provides one of the

better solutions in terms of specular distribution in that it

has a shorter peak in the highlight and a longer tail in the

falloff, which is to say that it looks

more realistic as shown in figure 08.

Object GGX provides

one of the better

solutions in terms of

specular distribution

Substance PBR shaders use the

GGX microfacet distribution

vol. 1 — The theory of Physically—Based Rendering and Shading

Energy Conservation plays a vital role in physically-based rendering solutions. It states that the total amount of light

re-emitted by a surface (reflected and scattered back) is less than the total amount it received. In other words, the

light reflected off the surface will never be more intense than it was before it hit the surface. As artists, we don’t have

to worry about controlling Energy Conservation. This is one of the nice aspects of PBR in that energy conservation is

always enforced by the shader. It’s part of the physically-based model and it allows us to focus more on art rather

than physics.

The Fresnel reflection factor also plays a vital role in

physically-based shading as a coefficient of the BRDF.

The Fresnel Effect as observed by French physicist

Augustin-Jean Fresnel states that the amount of light you

see reflected from a surface depends on the viewing

angle at which you perceive it.

For example, think of a pool of water. If you look straight

down, perpendicular to the water surface, you can see

down to the bottom. Viewing the water surface in this

manner would be at zero degrees or normal incidence,

normal being the surface normal. Now, if you look at the

pool of water at a grazing incidence, more parallel to the

water surface, you will see that the specular reflections

on the water surface become more intense and you may

not be able to see below the surface of the water at all.

Fresnel is not something that we control in PBR as we did

in traditional shading. Again, this is another physics

aspect that is handled for us by the PBR shader. When it

comes to viewing a surface at a grazing incidence, all

smoothed surfaces will become a nearly 100% reflector

at a 90 degree angle of incidence.

For rough surfaces, reflectance will become increasingly

specular but we won’t approach 100% specular

reflection. What matters then is the angle between the

normal of each microfacet and the light, not the angle

between the normal of the «macrosurface» and the light.

Because the light rays are dispersed into different

directions, the reflection appears softer or dimmer. What

you get at a macroscopic level is a bit like the average of

all the Fresnel effect you would have for the microfacets.

For rough surfaces, reflectance will become increasingly

specular but we won’t approach 100% specular reflection

vol.1 — The theory of Physically—Based Rendering and Shading

F0 (Fresnel Reflectance at 0 Degrees)

When light hits a surface straight on or perpendicularly (0

degree angle), there is a percentage of that light that is

reflected back as specular. Using the Index of Refraction

(IOR) for a surface, you can derive the amount that is

reflected back and this is referred to as

F0 (Fresnel 0) as shown in figure 09.

The amount of light that is refracted

into the surface is referred to a 1-F0.

The F0 range for most common

dielectrics will be from 0.02 — 0.05 and

for conductors the F0 range will be

0.5-1.0. Thus, the reflectivity of a

surface is determined by the refractive

index as shown in the following

equation from Sebastien Lagarde’s

«Feeding a Physically-based Shading

Model» blog post as shown in figure 10.

It is the F0 reflectance value that we

are concerned with in regards to

authoring our textures. Non-metals

(dielectrics/insulators) will have a

greyscale value and metals (conductors)

will have an RGB value. With regards to

PBR and from an artistic interpretation

of reflectance, we can state that for a

common smooth dielectric surface, F0 will reflect

between 2% and 5% of light and 100% at grazing angles

as was shown in figure 09.

The dielectric (non-metal) reflectance values don’t

actually change very drastically. In fact, when altered by

roughness the actual changes in value can be hard to

see. However, there is a difference in the values. In

figure 11, you can see a chart that shows the F0 ranges

for both metal and non-metal materials.

Notice that the ranges for non-metals do not deviate

from each other drastically. Gemstones are an exception

as they have higher values. We will discuss F0 as it

specifically relates to conductors and insulators a bit later.

vol. 1 — The theory of Physically—Based Rendering and Shading

When creating materials for PBR, I find it helpful to think in terms of metal or non-metal. I simply ask myself if the

surface is metal or not. If it is, I follow one set of guidelines and if it’s not, I follow another. This can be a rather

simplistic approach as some materials may not fall into these categories such as metalloids, but in the overall process

of creating materials, distinguishing between metal and non-metal is a good approach and metalloids are an exception.

To set up guidelines for materials, we first must understand what we are trying to create. With PBR, we can look at

the properties of metals (conductors) and non-metals (insulators) to derive this set of guidelines.

Conductors and Insulators (Metals and Non-Metals)

Metals (conductors) are good conductors of heat and

electricity. Simply put, the electric field in conducting

metals is zero and when an incoming light wave made of

electric and magnetic fields hits the surface, it is partially

reflected and all the refracted

light is absorbed. The reflectance

value for polished metal is going

to be high at a range of about

70-100% reflective as shown in

figure 12.

Some metals absorb light at

different wavelengths. For

example, gold absorbs blue light

at the high-frequency end of the

visible spectrum so it appears

yellow as a result. However,

since the refracted light is

absorbed, the color tint of

metals come from the reflected

light and thus in our maps, we

don’t give metals a diffuse color.

For example, in the specular/

gloss workflow, raw metal is set

to black in the diffuse map and

the reflectance value is a tinted

color value in the specular map.

With metals, the reflectance

value will be RGB and can be tinted. Since we are

working within a physically-based model, we need to use

real-world measured values for the metal reflectance in

our maps.

Another important aspect with metals in terms of

texturing is that metal can corrode. This means that

weathering elements can play a large role in the

reflective state of metal. If the metal rusts for example,

this changes the reflective state of the metal and the

corroded areas are then treated as a dielectric material as

shown in figure 13.

Also, metal that is painted is not treated like a metal but

rather a dielectric as well. The paint acts as a layer on

top of the raw metal. Only the raw metal exposed from

chipped away paint is treated as metal. The same goes

for dirt on the metal or any matter that obscures the raw

metal.

I stated above that I always ask myself if a material is a

metal or not. However, to be more precise, the question

should also inquire the state of the metal such as is it

painted, rusted or covered in dirt/grease. The material

will be treated as dielectric if it is not raw metal and there

could be some blending between metal and non-metal

depending on the weathering.

Refracted light is absorbed, the color tint of metals come from

the reflected light and thus in our maps, we don’t give metals a

diffuse color

Weathering elements can play a

large role in the reflective state of

metal

vol.1 — The theory of Physically—Based Rendering and Shading

Non-metals (insulators/dielectrics)

are poor conductors of electricity.

The refracted light is scattered and/

or absorbed (often re-emerging from

the surface) and thus they reflect a

much smaller amount of light than

metals and will have an albedo color.

We stated earlier that the value for

common dielectrics would be around

2-5% based on the F0 as computed

by the index of refraction. These

values are contained within the

linear range of 0.017-0.067 (40-75

sRGB) as shown in figure 14. With

the exception of gemstones, most

dielectrics will not be greater than

4%.

Just as with metals, we need to use real-world measured

values, but it can be difficult to find an IOR for other

materials that are not transparent. However, the value

between most common dielectric materials doesn’t

change drastically, so we can utilize a few guidelines to

follow in terms of reflectance values, which we will cover

in volume two.

The value for common dielectrics is

around 2-5% based on the F0 as

computed by the Index of

Refraction (IOR)

vol. 1 — The theory of Physically—Based Rendering and Shading

Linear space rendering can take up an entire article all on its own. So, we won’t go in-depth into the specifics.

However, the important takeaway is that computations are calculated in linear space.

Simply put, linear space rendering provides correct math for lighting calculations. It’s about creating an environment

that allows light to behave as it does in the real world. In linear space, the gamma is 1.0. However, for this to look

correct to our eyes, the linear gamma needs to be shifted. Gamma-encoded space (sRGB) compensates for images

that are displayed on a computer screen. The value of the image is adjusted for display.

When computing color values and performing operations on colors, all computations should be performed in linear

space. A simple way to look at it is that if an image is to be displayed in the render such as base color or diffuse, then

these maps need to be set as sRGB. What happens in Substance is that if the image is tagged as sRGB, it will be

converted to linear for calculations and then set back to sRGB for display. However, when you store mathematical

values that purely denote surface attributes in a texture such as roughness or metallic, then these maps must be set

as linear.

Substance handles the conversion between linear/sRGB space for inputs automatically as well gamma-correction on

the computed result in the rendered viewport. As the artist, you don’t need to worry about the internal working of

linear-space computations and conversions in the Substance pipeline. When using Substance materials via the

Substance Integration plugin, the conversions for linear space are also handled automatically.

However, it’s important to understand the process, as when Substance maps are utilized as exported bitmaps and not

Substance materials, you may need to manually handle the conversions depending on the renderer you are using. You

need to know that base color/diffuse maps are sRGB and the rest are linear.

Now that we have explored the basic theory behind the physics, we can derive some key factors for PBR.

1. Energy Conservation. A reflected ray is never brighter than the value it had when it first hit

the surface. Energy Conservation is handled by the shader.

2. Fresnel. The BRDF is handled by the shader. The F0 reflectance value has minimal change

for most common dielectrics and falls within a range of 2% — 5%. The F0 for metals is a high

value ranging from 70-100%.

3. Specular intensity is controlled through the BRDF, roughness or glossiness map and the F0

reflectance value.

4. Lighting calculations are computed in linear space. All maps that have gamma-encoded

values such as base color or diffuse are usually converted by the shader to linear, but you may

have to make sure that the conversion is properly done by checking the appropriate option

when importing the image in your game engine or renderer. Maps that describe surface

attributes such as roughness, glossiness, metallic and height should be set to be interpreted as

linear.

When using Substance materials via the Substance Integration

plugin, the conversions for linear space are also handled

automatically.

vol.1 — The theory of Physically—Based Rendering and Shading

1. Physically-Based Shading at Disney Brent Burley, Walt Disney Animation Studios.

https://disney-animation.s3.amazonaws.com/library/s2012_pbs_disney_brdf_notes_v2.pdf

2. Microfacet Models for Refraction through Rough Surfaces

http://www.cs.cornell.edu/~srm/publications/EGSR07-btdf.pdf

3. Feeding a Physically-Based Shading Model by Sebastien Lagarde

Feeding a physically based shading model

4. An Introduction to BRDF Models by Daniël Jimenez Kwast

http://hmi.ewi.utwente.nl/verslagen/capita-selecta/CS-Jimenez-Kwast-Daniel.pdf

Allegorithmic develops the new generation of 3D texturing software: Substance Painter, Substance

Designer and Bitmap2Material. With most AAA game studios using these tools, Substance has become the

standard for creating next-generation PBR (Physically Based Rendering) assets.

For more information on Substance, please visit our website at

www.allegorithmic.com