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Three Papers on Illumination
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Who am I?
• Engineer at Two Sigma working on distributed systems problems
• Studied computer graphics in school
• Just got a kitten, she is adorable
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The views expressed herein are not necessarily the views of Two Sigma Investments, LP or
any of its affiliates (collectively, “Two Sigma”). The information presented herein is only for
informational and educational purposes and is not an offer to sell or the solicitation of an
offer to buy any securities or other instruments. Additionally, the information is not intended
to provide, and should not be relied upon for investment, accounting, legal or tax advice. Two
Sigma makes no representations, express or implied, regarding the accuracy or completeness
of this information, and you accept all risks in relying on the above information for any
purpose whatsoever.
Important Legal Information
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What we are trying to do
By Gilles Tran - http://www.oyonale.com/modeles.php?lang=en&page=40, Public
Domain, https://commons.wikimedia.org/w/index.php?curid=745313 4
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Lighting effects – diffuse
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Lighting effects – specular
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Lighting effects – (soft) shadows
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Lighting effects – refraction and reflection
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Lighting effects – caustics
http://madebyevan.com/webgl-water
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Lighting effects – color bleed
http://madebyevan.com/webgl-path-tracing/
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What we are going to talk about
• Background
• Continuous shading of curved surfaces – Henri Gouraud
• An Improved Illumination Model for Shaded Display – Turner Whitted
• The Rendering equation – James T. Kajiya
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The model
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3d scenes
Mesh of triangles Functions
https://www.shadertoy.com/view/4sS3zG
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Rendering
Items first
for each item
for each pixel
if pixel between eye and item
draw something here
Pixels first
for each pixel
for each item
if ray from eye through pixel hits item
draw something here
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Some math, Vectors and Rays
Direction and magnitude
Starting point and direction
Vector
Ray
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Some math, Normals
N
N
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Lighting models
(, %)
X’
x
L
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Lighting models – Lambert cosine law
Incoming light Outgoing light
∗ = cos ( ℎ ) Constant (viewer independent)
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Machbanding
By The original uploader was Aliwiki at French Wikipedia -
Transferred from fr.wikipedia to Commons by Korrigan using
CommonsHelper., FAL,
https://commons.wikimedia.org/w/index.php?curid=4770182
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Mach bands
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Its 1971…
We are here
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Put it in hardware!
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for each triangle
for each row in image
for each col in row
if pixel between eye and item
draw something here
Hidden surface removal problem
1-100
~400
~600 O(T*H*W)
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Hidden surface removal (simple)
*Project triangles into screen space*
for each row in image
for each triangle
if row intersects triangle line segments
for col between intersected line segments
draw something here*
* Consider depth
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Gouraud Illumination model
= <
+ ?
@ A B
B∈EFGHIJ
N
L
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The one cool trick…
A
B
C
row
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How well does it work?
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Reflection
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Refraction
By JrPol - Own work, CC BY-SA 4.0,
https://commons.wikimedia.org/w/i
ndex.php?curid=38535231
By Josell7 - Own work, CC BY-SA 3.0,
https://commons.wikimedia.org/w/index.php?curi
d=21670922
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The model
= <
+ ?
@ A B
B∈EFGHIJ
+ J
+ I
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The model
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Ray tracing
By Henrik - Own work, GFDL,
https://commons.wikimedia.org/w/index.php?curid=3869326
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The rendering equation
, % = , % , % + N , %, %% %, %% ′′
Q
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The rendering equation
, % = , % , % + N , %, %% %, %% ′′
Q
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, % = , % , % + N , %, %% %, %% ′′
Q
The rendering equation
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, % = , % , % + N , %, %% %, %% ′′
Q
The rendering equation
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The rendering equation
, % = , % , % + N , %, %% %, %% ′′
Q
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The rendering equation
, % = , % , % + N , %, %% %, %% ′′
Q
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Approximation
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Approximation
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Previous methods as approximation
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Previous methods as approximation
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Integration
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Expectation
= N
() = N
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Expectation
= N ≈
1
@ F
W
FXY
() = N ≈
1
@ (F
)
W
FXY
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Monte Carlo integration
N
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Monte Carlo integration
N = N
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Monte Carlo integration
N = N
=
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Monte Carlo integration
N = N
=
() = N
Where
=
()
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Monte Carlo integration
N = N
=
() = N
Where
=
()
≈
1
@
F
(F
)
W
FXY
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Monte Carlo integration example
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Monte Carlo Integration + Rendering Equation
N , %, %% %, %% ′′
Q
X’
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Path Tracing
X’
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Variance reduction
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Path tracing example
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Summary
1. Continuous shading of curved surfaces – Henri Gouraud
2. An Improved Illumination Model for Shaded Display – Turner Whitted
3. The Rendering equation – James T. Kajiya
= <
+ ?
@ A B
B∈EFGHIJ
, % = , % , % + N , %, %% %, %% ′′
Q
= <
+ ?
@ A B
+ J
+ I
B∈EFGHIJ
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Thank you
Andy van Dam John “Spike” Hughes
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