SOME THEORY BEHIND REAL-TIME RENDERING

Jaroslav Křivánek Charles University in Prague

Off-line realistic rendering (not yet in rea-time)

Ray tracing

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Image created by Bertrand Benoit Rendered in Corona Renderer

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Image created by Jeff Patton Rendered in Corona Renderer

ŠKODA Rapid Catalogue 7

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Image created by Weta Digital © 20th Century Fox

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Image created by Weta Digital © 20th Century Fox

vimeo >> “The Great Gatsby VFX” 11

vimeo >> “The Great Gatsby VFX” 12

Global illumination

Global illumination – Color bleeding

Image credit: Michael Bunnell 14 14

Global illumination – Caustics

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Global illumination in games

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Basic light transport theory

Image generation

Pixel value = average radiance reflected from surfaces visible through the pixel

Generating an image involves (some sort of) light transport simulation

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Radiance L (x, ω)

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Radiance

Radiometric quantity measuring “amount of light energy” along a ray

Denoted L (x, ω) x … position ω … direction

Units [W / m2 sr ]

Essential properties of radiance

Proportional to perceived brightness xxxxx

Constant along a ray xxxxxx

x

ω

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Light reflection

x 20 20

Light reflection

BRDF

Bi-directional Reflectance Distribution Function

Obr. Wojciech Matusik

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Bidirectional Reflectance Distribution Function

dωi Lr(ωo)

θo

n Li(ωi)

θi

]sr[dcos)(

)(d)( 1

iiii

oroi

−

⋅⋅=→

ωθωωωω

LLfr

BRDF – Formal definition

„incoming“

„outgoing“

„reflected“

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Surface appearance and the BRDF

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Appearance BRDF lobe (for four different viewing directions)

Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/

Surface appearance and the BRDF

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Appearance BRDF lobe (for four different viewing directions)

Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/

Surface appearance and the BRDF

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Appearance BRDF lobe (for four different viewing directions)

Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/

Surface appearance and the BRDF

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Appearance BRDF lobe (for four different viewing directions)

Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/

Reflection equation

Total reflected radiance: integrate contributions of incident radiance, weighted by the BRDF, over the hemisphere

∫ ⋅→⋅=)(

iioiiior dcos),(),(),(x

xxxH

rfLL ωθωωωω

dωi θo

n Li(x, ωi)

θi Lr(x, ωo)

upper hemisphere over x

CG III (NPGR010) - J. Křivánek 2015

But where does the incident radiance come from? Other surfaces in the scene!

Rendering Equation

x

L(r(x,ωi), −ωi)

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ωi

ωo Li(x,ωi)

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∫ ⋅→⋅−+

=

)(iioiiio

oeoo

dcos),()),,(r(

),(),(

x

xx

xx

HrfL

LL

ωθωωωω

ωω

Recursive calculation of rendering equation

To calculate L(x, ωo) I need to calculate L(r(x, ω’), −ω’)

for all directions ω’ around the point x. For the calculation of each L(r(x, ω’), −ω’) I need to do

the same thing recursively At each step, apply Monte Carlo integration to sample

the hemisphere etc.

x

ω’ r(x, ω’)

ω’’

ωo

r( r(x,ω’), ω’)

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Monte Carlo integration

General approach to numerical evaluation of integrals

x1

f(x)

0 1

p(x)

x2 x3 x4 x5 x6

Integral:

Monte Carlo estimate of I:

Correct „on average“:

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Slide credit: Iwan Kawrakov

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L: Equilibrium radiance Le: Emitted radiance T: Transport & scattering operator

LTLL += e

Rendering equation – Operator form

LTLL += e

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Progressive approximation

Each application of T corresponds to one step of reflection & light propagation

...e3

e2

ee ++++= LTLTTLLL

emission

direct illumination

one-bounce indirect illumination

two-bounce indirect illumination

OpenGL shading 33

Progressive approximation

eL

eL

eLT eLTT eLTTT

ee LTL + e2

ee LTTLL ++ e3

e ... LTL ++

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Idea: Distributed ray tracing (Cook ‘84)

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Path tracing

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Path tracing, Naive

getLi (x, ω): y = traceRay(x, ω) return Le(y, –ω) + // emitted radiance Lr (y, –ω) // reflected radiance Lr(y, ω): ω′ = genUniformRandomDir( n(y) ) return getLi (y, ω′) * brdf(y, ω, ω′) * dot(ω′, n(y)) * 2π

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Path tracing, Naive

getLi (x, ω): y = traceRay(x, ω) return Le(y, –ω) + // emitted radiance Lr (y, –ω) // reflected radiance Lr(y, ω): ω′ = genUniformRandomDir( n(y) ) return getLi (y, ω′) * brdf(y, ω, ω′) * dot(ω′, n(y)) * 2π

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Practical formulas for light transport

P. Dutré: Global Illumination Compendium, http://people.cs.kuleuven.be/~philip.dutre/GI/

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ROBUST GI CALCULATION

Path sampling techniques

Path tracing Light tracing Bidirectional path tracing

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All possible bidirectional techniques vertex on a light sub-path vertex on en eye subpath

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All possible bidirectional techniques vertex on a light sub-path vertex on en eye subpath

VPLs

no single techniques importance samples all the terms

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Bidirectional path tracing

Use all of the above sampling techniques

Combine using Multiple Importance Sampling

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BPT Implementation

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Results

BPT, 25 samples per pixel PT, 56 samples per pixel

Imag

es: E

ric

Vea

ch

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Insufficient path sampling techniques

Reference solution Bidirectional path tracing 47

Insufficient path sampling techniques

In BPT, Specular-Diffuse-Specular paths sampled with zero (or very small) probability

diffuse – D

specular – S

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PHOTON MAPPING

(DENSITY ESTIMATION)

Photon mapping (Density estimation)

1. Many fwd walks + store collisions (“photon map”) 2. Radiance estimate: (Kernel) density estimation

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Photon mapping

1. Photon tracing „Photon“ emission Particle tracing Storage in the „photon map“

2. Rendering with photon map Like path tracing Photon map query instead

of recursion

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Photon mapping – SDS paths

© Wojciech Jarosz

© H.W.Jensen

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Our work: Vertex Connection and Merging BPT + PM combination

Bidirectional path tracing (30 min) 54

Photon mapping (Density estimation) (30 min) 55

Vertex connection and merging (30 min) 56

THANK YOU!

Questions?

[cgg.mff.cuni.cz/~jaroslav]