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INFOMAGR Advanced Graphics Jacco Bikker - November 2016 - February 2017 Lecture 13 - BRDFs Welcome! , = (, ) , + , , ,

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SLIDE 1

𝑱 π’š, π’šβ€² = 𝒉(π’š, π’šβ€²) 𝝑 π’š, π’šβ€² +

𝑻

𝝇 π’š, π’šβ€², π’šβ€²β€² 𝑱 π’šβ€², π’šβ€²β€² π’†π’šβ€²β€²

INFOMAGR – Advanced Graphics

Jacco Bikker - November 2016 - February 2017

Lecture 13 - β€œBRDFs”

Welcome!

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SLIDE 2

Today’s Agenda:

  • Phong BRDF
  • Microfacets
  • The Lobe
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SLIDE 3

Phong

Advanced Graphics – BRDFs 3

BRDFs, Recap

Recall that a BRDF defines the relation between incoming and outgoing radiance for directions and a surface point: 𝑔

𝑠 𝑦, πœ„π‘—, πœ„π‘ =

𝑒𝑀𝑝(𝑦, πœ„π‘) 𝑀𝑗 𝑦, πœ„π‘— cos πœ„π‘—π‘’πœ•π‘—

π‘œ

πœ„π‘ πœ„π‘—

𝑦 π‘œ π‘œ What about materials that are not purely specular, nor diffuse?

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SLIDE 4

Phong

Advanced Graphics – BRDFs 4

BRDFs, Recap

We already know how to do materials that are diffuse and shiny. But that gets us good looking marble floors, not glossy reflections.

50% diffuse, 50% specular, 50% diffuse, 50% glossy (or: 100% glossy)

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SLIDE 5

Phong

Advanced Graphics – BRDFs 5

Glossy Reflection

Glossy reflections: sending out rays in random directions close to the reflected vector.

Simple solution: 𝑆 = π‘ π‘“π‘”π‘šπ‘“π‘‘π‘’(𝐹, 𝑂); 𝑄 = 𝐽 + 𝑆 + π‘‘π‘‘π‘π‘šπ‘“(π‘ π‘π‘œπ‘’π‘π‘›π‘„π‘π‘—π‘œπ‘’π½π‘œπ‘‡π‘žβ„Žπ‘“π‘ π‘“(), π‘‘π‘žπ‘“π‘‘π‘£π‘šπ‘π‘ π‘—π‘’π‘§) 𝑆 = π‘œπ‘π‘ π‘›π‘π‘šπ‘—π‘¨π‘“(𝑄 βˆ’ 𝐽); Or: 𝑆 = π‘ π‘“π‘”π‘šπ‘“π‘‘π‘’(𝐹, 𝑂); 𝑄 = 𝐽 + 𝑆 + π‘‘π‘‘π‘π‘šπ‘“(π‘ π‘π‘œπ‘’π‘π‘›πΈπ‘—π‘ π‘“π‘‘π‘’π‘—π‘π‘œπ½π‘œπΌπ‘“π‘›π‘—π‘‘π‘žβ„Žπ‘“π‘ π‘“π·π‘π‘‘π‘—π‘œπ‘“π‘‹π‘“π‘—π‘•β„Žπ‘’π‘“π‘’(𝑆), π‘‘π‘žπ‘“π‘‘π‘£π‘šπ‘π‘ π‘—π‘’π‘§) 𝑆 = π‘œπ‘π‘ π‘›π‘π‘šπ‘—π‘¨π‘“(𝑄 βˆ’ 𝐽);

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SLIDE 6

Phong

Advanced Graphics – BRDFs 6

Glossy Reflection

In OpenGL, shading is defined as follows*: 𝐽𝑦 = π‘™π‘π‘›π‘π‘—π‘“π‘œπ‘’π‘—π‘π‘›π‘π‘—π‘“π‘œπ‘’ +

π‘›βˆˆπ‘šπ‘—π‘•β„Žπ‘’π‘‘

𝑙𝑒𝑗𝑔𝑔𝑣𝑑𝑓 𝑂 βˆ™ 𝑀𝑛 + π‘™π‘‘π‘žπ‘“π‘‘π‘£π‘šπ‘π‘  𝑆𝑛 βˆ™ π‘Š

π‘“π‘¦π‘žπ‘π‘œπ‘“π‘œπ‘’

𝐽𝑛

*: Bui-Tuong Phong, Illumination for Computer Generated Images, 1975.

𝑆𝑛 𝑀𝑛 π‘Š

𝑦 where

  • π‘™π‘π‘›π‘π‘—π‘“π‘œπ‘’, 𝑙𝑒𝑗𝑔𝑔𝑣𝑑𝑓 π‘π‘œπ‘’ π‘™π‘‘π‘žπ‘“π‘‘π‘£π‘šπ‘π‘  are

material properties (typically: rgb);

  • 𝑆𝑛 is the vector 𝑀𝑛 reflected in the

normal π‘œ;

  • 𝐽𝑛 is the illumination from light 𝑛.

π‘œ

𝛻

cos πœ„ = 𝜌

𝛻

π‘‘π‘π‘‘π‘“π‘¦π‘žπœ„ =?

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SLIDE 7

Phong

Advanced Graphics – BRDFs 7

Blinn-Phong BRDF (images: Disney BRDF Explorer)

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SLIDE 8

Phong

Advanced Graphics – BRDFs 8

Modified Phong BRDF

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SLIDE 9

Phong

Advanced Graphics – BRDFs 9

Fixing Phong

Recall the requirements for a proper BRDF:

  • Should be positive: 𝑔

𝑠 πœ•π‘, πœ•π‘— β‰₯ 0

  • Helmholtz reciprocity should be obeyed: 𝑔

𝑠 πœ•π‘, πœ•π‘— = 𝑔 𝑠 πœ•π‘—, πœ•π‘

  • Energy should be conserved:

𝛻 𝑔 𝑠 πœ•π‘, πœ•π‘— cosπœ„π‘ π‘’πœ•π‘ ≀ 1

BRDFs obeying these rules are called physically plausible. For a path tracer, we have additional requirements:

  • 1. It β€˜would be nice’ if we could generate a random direction proportional to the BRDF (IS)
  • 2. We need to be able to calculate the probability density (importance) for a given direction

(for MIS).

The BRDF relates irradiance to

  • utgoing radiance. Irradiance is

measured per unit area; the reflected energy cannot exceed 1 per unit area, hence the cos: we essentially convert outgoing radiance into irradiance as well. Alternatively, this also follows from the Helmholtz reciprocity principle: we can exchange πœ•π‘ and πœ•π‘—.

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SLIDE 10

Phong

Advanced Graphics – BRDFs 10

Fixing Phong

  • 1. Sampling the specular lobe proportional to the BRDF:

Sampling proportional to 𝑂 βˆ™ 𝑀, according to the G.I.C. : 𝑦 = cos 2πœŒπ‘ 

1

1 βˆ’ 𝑠2 𝑧 = sin 2πœŒπ‘ 

1

1 βˆ’ 𝑠2 𝑨 = 𝑠2

𝑆 βˆ™ π‘Š 1 𝑆 βˆ™ π‘Š 2 𝑆 βˆ™ π‘Š 10 𝑆 βˆ™ π‘Š 50

Sampling proportional to 𝑆 βˆ™ π‘Š 𝛽, according to the G.I.C. : 𝑒 = 𝑠

2

2 𝛽+1, 𝑦 = cos 2πœŒπ‘ 

1

1 βˆ’ 𝑒 𝑧 = sin 2πœŒπ‘ 

1

1 βˆ’ 𝑒 𝑨 = 𝑒

𝑨 𝑦 𝑧

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SLIDE 11

Phong

Advanced Graphics – BRDFs 11

Tangent / Local Space

Setting up a local coordinate system in 2D:

  • First axis is the normal;
  • Second axis is perpendicular to normal.

𝑂 = π‘œ π‘ˆ = βˆ’π‘œ. 𝑧 π‘œ. 𝑦 Setting up a local coordinate system in 3D: 𝑂 = π‘œ π‘ˆ = π‘œπ‘π‘ π‘›π‘π‘šπ‘—π‘¨π‘“(𝑂 Γ— 𝑋) 𝐢 = π‘ˆ Γ— 𝑂 where 𝑋 is a random unit vector; 𝑋 β‰  𝑂.

𝑨 𝑦 𝑧

Good choice for 𝑋: 𝑋 = (0,1,0) if abs(π‘œπ‘¦) > 0.99; 𝑋 = 1,0,0 otherwise.

𝑢 𝑼 π‘ͺ

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SLIDE 12

Phong

Advanced Graphics – BRDFs 12

Tangent / Local Space

Converting a vector from world space to local space: 𝑄

𝑦

𝑄

𝑧

𝑄

𝑨

= 𝑄 βˆ™ π‘ˆ 𝑄 βˆ™ 𝐢 𝑄 βˆ™ 𝑂 Local space to world space: 𝑄π‘₯π‘π‘ π‘šπ‘’ = 𝑁 Γ— π‘„π‘šπ‘π‘‘π‘π‘š = 𝑄

π‘¦π‘ˆ + 𝑄 𝑧𝐢 + 𝑄 𝑨𝑂

𝑢 𝑼 π‘ͺ

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SLIDE 13

Phong

Advanced Graphics – BRDFs 13

Normalizing the Lobe

A material cannot reflect more energy than it receives:  We thus scale the BRDF by the inverse of its integral over the hemisphere. For the Lambertian BRDF: π‘‘π‘‘π‘π‘šπ‘“ =

1 𝜌 (because cos πœ„ integrates to 𝜌)

For the cosine lobe, the scale is

𝛽+1 2𝜌 (*). However, there is a problem:

*: Physically Based Rendering, page 969; also see: http://www.farbrausch.de/~fg/stuff/phong.pdf

π‘œ π‘œ π‘œ

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SLIDE 14

Phong

Advanced Graphics – BRDFs 14

The Modified Phong BRDF

The Phong BRDF has problems:

  • 1. it does not doesn’t obey the Helmholtz reciprocity rule.
  • 2. We can only estimate the integral of the lobe, since it

intersects the surface. Compensating for the intersection: Modified Phong BRDF. 𝑔

𝑠 𝑦, πœ„π‘—, πœ„π‘ = 𝑙𝑒

1 𝜌 + 𝑙𝑑 𝛽 + 2 2𝜌 π‘‘π‘π‘‘π›½πœ’ where πœ’ is the angle between the view vector and the reflected light vector. For 𝑙𝑒 + 𝑙𝑑 < 1, this guarantees energy preservation*.

*: Lafortune & Willems, Using the Modified Phong Reflectance Model for Physically Based Rendering, 1994.

𝑔

𝑠 𝑦, πœ„π‘—, πœ„π‘ = 𝑙𝑒 1 𝜌 + 𝑙𝑑 𝛽+2 2𝜌 π‘‘π‘π‘‘π›½πœ’ :

Total hemispherical reflection (note: integrating over

  • utgoing directions, since these cannot exceed 1):

𝜍 𝑦, πœ„π‘ =

Ξ©

𝑔

𝑠 𝑦, πœ„π‘—, πœ„π‘ cos ΞΈπ‘π‘’πœ•π‘

=

Ξ©

(𝑙𝑒 1 𝜌 + 𝑙𝑑 𝛽 + 2 2𝜌 π‘‘π‘π‘‘π›½πœ’) cos ΞΈπ‘π‘’πœ•π‘ = 𝑙𝑒 + 𝑙𝑑 𝛽 + 2 2𝜌

Ξ©

π‘‘π‘π‘‘π›½πœ’ cos ΞΈπ‘π‘’πœ•π‘ The last integral is maximal when the incoming light arrives along the normal; in this case πœ’ = πœ„π‘, and the integral is 2𝜌/ (π‘œ + 2). Therefore, in this case: 𝜍 𝑦, πœ„π‘ = 𝑙𝑒 + 𝑙𝑑 and conservation of energy is guaranteed iff: 𝑙𝑒 + 𝑙_𝑑 ≀ 1.

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SLIDE 15

Phong

Advanced Graphics – BRDFs 15

The Modified Phong BRDF

Despite the kludges, we now have a decent BRDF for glossy materials. We can sample it: 𝑒 = 𝑠

2

2 𝛽+1, 𝑦 = cos 2πœŒπ‘ 

1

1 βˆ’ 𝑒 𝑧 = sin 2πœŒπ‘ 

1

1 βˆ’ 𝑒 𝑨 = 𝑒 Normalize it: 𝛽 + 2 2𝜌 π‘‘π‘π‘‘π›½πœ’ We have a PDF: 𝛽 + 2 2𝜌 π‘‘π‘π‘‘π›½πœ’

And finally, we can blend it with the Lambertian BRDF:

  • Define a probability π‘ž of sampling Phong;
  • Draw a random number 𝑠

0;

  • Sample Phong if 𝑠

0 < π‘ž, Lambert otherwise;

  • Combine PDFs: 𝑄𝐸𝐺 = π‘ž 𝑄𝐸𝐺

π‘žβ„Žπ‘π‘œπ‘• + 1 βˆ’ π‘ž 𝑄𝐸𝐺 𝑒𝑗𝑔𝑔

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SLIDE 16

Today’s Agenda:

  • Phong BRDF
  • Microfacets
  • The Lobe
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SLIDE 17

Microfacet

Advanced Graphics – BRDFs 17

BRDFs Without Issues

We now have two BRDFs without problems:

  • 1. The Lambertian BRDF
  • 2. The pure specular BRDF

These are plausible and can be sampled. The PDF is also clear. And, we have the somewhat kludged modified Phong BRDF.

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SLIDE 18

Microfacet

Advanced Graphics – BRDFs 18

Microfacet BRDFs*

We can simulate a broad range of materials if we assume: at a microscopic level, the material consists of tiny specular fragments.

  • If the fragment orientations are chaotic, the material appears diffuse.
  • If the fragment orientations are all the same, the material appears specular.
  • Different but similar orientations yield glossy materials.

*: Torrance & Sparrow, Theory for Off-Specular Reflection from Roughened Surfaces. 1967.

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SLIDE 19

Microfacet

Advanced Graphics – BRDFs 19

Microfacet BRDFs*

The Microfacet BRDF: 𝑔

𝑠 𝑀, π‘Š = 𝐺 𝑀, π‘Š 𝐻 𝑀, π‘Š, 𝐼 𝐸 𝐼

4 𝑂 βˆ™ 𝑀 𝑂 βˆ™ π‘Š Ingredients:

  • 1. Normal distribution D
  • 2. Geometry term G
  • 3. Fresnel term F
  • 4. Normalization
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SLIDE 20

Microfacet

Advanced Graphics – BRDFs 20

Normal Distribution

Microfacet BRDF, ingredient 1: 𝐸(𝐼)  the normal distribution function. Parameter 𝐼: the halfway vector: 𝑔

𝑠 π‘Š, 𝑀 = β‹―

A microfacet that reflects 𝑀 towards π‘Š must have a normal halfway π‘Š and 𝑀:

H = normalize(V + L).

𝑔

𝑠 𝑦, πœ„π‘—, πœ„π‘ =

𝑒𝑀𝑝(𝑦, πœ„π‘) 𝑀𝑗 𝑦, πœ„π‘— cos πœ„π‘—π‘’πœ•π‘—

𝑾 𝑴

𝑰

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SLIDE 21

Microfacet

Advanced Graphics – BRDFs 21

Normal Distribution

Intuitive choices for D: 𝐸 𝐼 = 𝐷: microfacet normals are equally distributed  diffuse material. 𝐸 𝐼 = ∞, 𝑔𝑝𝑠𝐼 = 0,0,1 0, π‘π‘’β„Žπ‘“π‘ π‘₯𝑗𝑑𝑓 : all microfacet normals are (0,0,1)  pure specular. Good practical choice for D: the Blinn-Phong distribution; 𝐸 𝐼 = 𝛽 + 2 2𝜌 𝑂 βˆ™ 𝐼

𝛽

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SLIDE 22

Microfacet

Advanced Graphics – BRDFs 22

Geometry Term

Microfacet BRDF, ingredient 2: 𝐻(π‘Š, 𝑀, 𝐼)  the geometry term.

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SLIDE 23

Microfacet

Advanced Graphics – BRDFs 23

Geometry Term

Intuitive choice for G: 𝐻(π‘Š, 𝑀, 𝐼) = 1: no occlusion. Good practical choice for G*: 𝐻 π‘Š, 𝑀, 𝐼 = min(1, min 2 𝑂 βˆ™ 𝐼 𝑂 βˆ™ π‘Š π‘Š βˆ™ 𝐼 , 2 𝑂 βˆ™ 𝐼 𝑂 βˆ™ 𝑀 π‘Š βˆ™ 𝐼

*: Physically Based Rendering, page 455

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SLIDE 24

Microfacet

Advanced Graphics – BRDFs 24

Fresnel Term

Microfacet BRDF, ingredient 3: 𝐺(𝑀, 𝐼)  the Fresnel term. So far, we assumed that the light reflected by a specular surface is only modulated by the material color. This is not true for dielectrics: here we use the Fresnel equations to determine reflection. In nature, Fresnel does not just apply to dielectrics.

𝒐 𝝏𝒑 𝝏𝒋

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SLIDE 25

Microfacet

Advanced Graphics – BRDFs 25

From β€œReal-time Rendering, 3rd edition, A. K. Peters.

Fresnel Term

Iron is specular, but reflectivity differs depending on incident angle. Aluminum is even more interesting: reflectivity depends on wavelength. The three lines in the graph: Top: blue, middle: green, bottom: red. Copper takes this to extremes: at grazing angles, it appears white. The lines in the graph: Top: red, middle: green, bottom: blue.

(hence its reddish appearance)

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SLIDE 26

Microfacet

Advanced Graphics – BRDFs 26

Fresnel Term

For Fresnel, we once again use Schlick’s approximation: 𝐺

𝑠 = π‘™π‘‘π‘žπ‘“π‘‘π‘£π‘šπ‘π‘  + (1 βˆ’ π‘™π‘‘π‘žπ‘“π‘‘π‘£π‘šπ‘π‘ )(1 βˆ’ 𝑀 βˆ™ 𝐼 )5

Note that this is calculated per color channel (π‘™π‘‘π‘žπ‘“π‘‘π‘£π‘šπ‘π‘  is an rgb triplet). Values for π‘™π‘‘π‘žπ‘“π‘‘π‘£π‘šπ‘π‘  for various materials: Iron 0.56, 0.57, 0.58 Copper 0.95, 0.64, 0.54 Gold 1.00, 0.71, 0.29 Aluminum 0.91, 0.92, 0.92 Silver 0.95, 0.93, 0.88

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SLIDE 27

Microfacet

Advanced Graphics – BRDFs 27

Bringing it All Together

The Microfacet BRDF: 𝑔

𝑠 𝑀, π‘Š = 𝐺 𝑀, π‘Š 𝐻 𝑀, π‘Š, 𝐼 𝐸 𝐼

4 𝑂 βˆ™ 𝑀 𝑂 βˆ™ π‘Š 𝐺

𝑠 = π‘™π‘‘π‘žπ‘“π‘‘π‘£π‘šπ‘π‘  + (1 βˆ’ π‘™π‘‘π‘žπ‘“π‘‘π‘£π‘šπ‘π‘ )(1 βˆ’ 𝑀 βˆ™ 𝐼 )5

𝐻 π‘Š, 𝑀, 𝐼 = min(1, min 2 𝑂 βˆ™ 𝐼 𝑂 βˆ™ π‘Š π‘Š βˆ™ 𝐼 , 2 𝑂 βˆ™ 𝐼 𝑂 βˆ™ 𝑀 π‘Š βˆ™ 𝐼 𝐸 𝐼 = 𝛽 + 2 2𝜌 𝑂 βˆ™ 𝐼

𝛽

For a full derivation of the denominator of the BRDF, see Physically Based Rendering, section 8.4.2.

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SLIDE 28

Microfacet

Advanced Graphics – BRDFs 28

Lambertian BRDF

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SLIDE 29

Microfacet

Advanced Graphics – BRDFs 29

Blinn-Phong Microfacet BRDF, Ξ±=1

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SLIDE 30

Microfacet

Advanced Graphics – BRDFs 30

Blinn-Phong Microfacet BRDF, Ξ±=10

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SLIDE 31

Microfacet

Advanced Graphics – BRDFs 31

Blinn-Phong Microfacet BRDF, Ξ±=50

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SLIDE 32

Microfacet

Advanced Graphics – BRDFs 32

Blinn-Phong Microfacet BRDF, Ξ±=500

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SLIDE 33

Microfacet

Advanced Graphics – BRDFs 33

Blinn-Phong Microfacet BRDF, Ξ±=50000

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SLIDE 34

Microfacet

Advanced Graphics – BRDFs 34

Specular BRDF

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SLIDE 35

Today’s Agenda:

  • Phong BRDF
  • Microfacets
  • The Lobe
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SLIDE 36

The Lobe

Advanced Graphics – BRDFs 36

Sampling the Microfacet BRDF

A BRDF merely tells us how much energy we can expect to leave in a certain direction, given an incoming direction. We would like to pick a direction proportional to importance (even though any direction on the hemisphere is β€˜correct’). Lambertian: sample proportional to cos πœ„ (𝑄𝐸𝐺 =

π‘‚βˆ™π‘€ 𝜌 ).

Phong: sample proportional to specular lobe (𝑄𝐸𝐺 =

𝛽+2 𝜌

π‘Š βˆ™ 𝑆

𝛽, where 𝑆 = π‘ π‘“π‘”π‘šπ‘“π‘‘π‘’(𝑀, 𝑂).

Microfacet: sample proportional to 𝐸 (𝑄𝐸𝐺 =

𝛽+2 2𝜌

𝑂 βˆ™ 𝐼

𝛽).

Sampling proportional to 𝑆 βˆ™ π‘Š, according to the Global Illumination Compendium:

𝑦 = cos 2πœŒπ‘ 

1

1 βˆ’ 𝑠

2

𝑧 = sin 2πœŒπ‘ 

1

1 βˆ’ 𝑠

2

𝑨 = 𝑠

2

Sampling proportional to 𝑆 βˆ™ π‘Š 𝛽, according to the Global Illumination Compendium:

𝑒 = 𝑠

2

2 𝛽+1, 𝑦 = cos 2πœŒπ‘ 

1

1 βˆ’ 𝑒 𝑧 = sin 2πœŒπ‘ 

1

1 βˆ’ 𝑒 𝑨 = 𝑒

slide-37
SLIDE 37

The Lobe

Advanced Graphics – BRDFs 37

Sampling the Microfacet BRDF

Sampling proportional to D: 𝐸 𝐼 = 𝛽 + 2 2𝜌 𝑂 βˆ™ 𝐼

𝛽

𝑒 = 𝑠

2

2 𝛽+1, 𝑦 = cos 2πœŒπ‘ 

1

1 βˆ’ 𝑒 𝑧 = sin 2πœŒπ‘ 

1

1 βˆ’ 𝑒 𝑨 = 𝑒 Vector 𝑦 𝑧 𝑨 is the halfway vector, but we where looking for 𝑀… Luckily, π‘Š is 𝑀 reflected in 𝐼, and therefore: 𝑀 = 2 π‘Š βˆ™ 𝐼 𝐼 βˆ’ π‘Š

slide-38
SLIDE 38

The Lobe

Advanced Graphics – BRDFs 38

Sampling the Microfacet BRDF

Problem: we are going to sample directions into the surface. Formal solution: these rays are occluded and return no energy. Better solution: use RIS.

  • 1. Randomly select 𝑂 directions using the PDF;
  • 2. Assign a weight 𝑋

𝑗 (>0) to each direction;

  • 3. Pick a direction 𝑗 proportional to 𝑋;
  • 4. Scale PDF by

𝑋𝑗𝑂 𝑋𝑗.

Note: using stratification, N=2 is sufficient (why?).

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SLIDE 39

The Lobe

Advanced Graphics – BRDFs 39

Microfacet BRDF, 1spp, no RIS

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SLIDE 40

The Lobe

Advanced Graphics – BRDFs 40

Microfacet BRDF, 1spp, RIS enabled

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SLIDE 41

The Lobe

Advanced Graphics – BRDFs 41

Quo Vadis

Alternative normal distributions: Beckmann: 𝐸 𝐼 = 𝐼 βˆ™ 𝑂 πœŒπ›½π‘π‘‘π‘π‘‘4πœ„β„Ž 𝑓

βˆ’π‘’π‘π‘œ2πœ„β„Ž 𝛽𝑐

2

GGX: 𝐸 𝐼 = 𝛽2 𝜌 𝐼 βˆ™ 𝑂 2 𝛽2 βˆ’ 1 + 1

2

…and many others.

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SLIDE 42

The Lobe

Advanced Graphics – BRDFs 42

Quo Vadis

Alternative geometric shadowing terms: Neumann: 𝐻 𝑀, π‘Š, 𝐼 = (𝑂 βˆ™ 𝑀)(𝑂 βˆ™ π‘Š) max(𝑂 βˆ™ 𝑀, 𝑂 βˆ™ π‘Š) GGX: 𝐻 𝐼 = 2(𝑂 βˆ™ π‘Š) 𝑂 βˆ™ π‘Š + 𝛽2 + 1 βˆ’ 𝛽2 𝑂 βˆ™ π‘Š 2 …and many others.

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SLIDE 43

The Lobe

Advanced Graphics – BRDFs 43

Quo Vadis

Anisotropic materials:

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SLIDE 44

The Lobe

Advanced Graphics – BRDFs 44

Quo Vadis

The Other Side: BSSRDF*

*: Jensen et al., 2001. A Practical Model for Subsurface Light Transport.

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SLIDE 45

The Lobe

Advanced Graphics – BRDFs 45

Quo Vadis

Variance Reduction for BSSRDFs:

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SLIDE 46

The Lobe

Advanced Graphics – BRDFs 46

Quo Vadis

Measured BRDFs:

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SLIDE 47

Today’s Agenda:

  • Phong BRDF
  • Microfacets
  • The Lobe
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SLIDE 48

INFOMAGR – Advanced Graphics

Jacco Bikker - November 2016 - February 2017

END of β€œBRDFs”

next lecture: β€œGrand Recap”