10 bidirectional methodsvda.univie.ac.at/teaching/imagesynthesis/17w/lecture... · 2018-01-09 ·...

052213 VU Image Synthesis

Light Transport III:Bidirectional Methods

Alireza Ghane + Torsten Möller

[email protected] [email protected]

vda.cs.univie.ac.at/Teaching/ImageSynthesis/17w/

1

© Torsten Möller

Reading• Chapter 16 of “Physically Based Rendering” by Pharr, Jakob,

Humphreys • Chapter 19, 20 in “Principles of Digital Image Synthesis,” by A.

Glassner • “Realistic Image Synthesis Using Photon Mapping,”

by H. W. Jensen

2

The path-space measurement equation

3

© Torsten Möller

• Expressing LTE in terms of geometry within the scene

• Replacing the integrand (dωi)with an area integrator over the whole scene geometry and remembering:

• - visibility term (either one or zero)

LTE - Geometric Formulation

4

€

dω i =cos ʹ ́ θ

ʹ p − ʹ ́ p 2 dA ʹ ́ p ( )

€

ωo

€

ω i

€

ʹ p €

ʹ ́ p

€

p

© Torsten Möller

LTE - Geometric Formulation• Geometry coupling term

• New (geometric) formulation of the Light Transport Equation (LTE)

• Randomly pick points in the scene and create a path • vs. (previously) randomly pick directions over a sphere

5

€

ωo

€

ω i

€

ʹ p €

ʹ ́ p

€

p

© Torsten Möller

• Recursive evaluation

€

+...

LTE - Geometric Formulation

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€

p1€

p2

€

p0

€

p3

© Torsten Möller

LTE - Geometric Formulation• compact formulation:

• For a path • Where p0 is the camera and pi is a light source

7

€

p i = p0p1...pi

€

p1€

p2

€

p0

€

p3

© Torsten Möller

LTE - Geometric Formulation• with:

• Where

• Is called the throughput • Special case:

8

€

p1€

p2

€

p0

€

p3

© Torsten Möller

Measuring over a pixel• Integrating over pixel j:

• Where We is the filter function around a pixel:

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© Torsten Möller

Integrating over a pixel• And hence the integral:

• A - light bounces around until it hits the camera • B - a quantity from the sensor bounces around and only makes a

contribution if it hits a light

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© Torsten Möller

Symmetry• Concept of measurement (We) and emission (Le) is symmetrical:

• Swapping the role of camera and light creates the idea of particle tracing

• We also known as the importance of ray between p0 and p1

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© Torsten Möller

Non-symmetry: Refraction• When light passes from one medium into another whose index of

refraction is lower the transmitted angle is larger than the incident angle

• at one point the light gets reflected - called total internal reflection • For water ηw = 4/3 • Radiance gets compressed

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Li =⌘2i⌘2t

Lt

© Torsten Möller

Non-symmetry: Refraction• Radiance gets compressed

• Choose a proper adjoint BTDF:

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Li =⌘2i⌘2t

Lt

f?(p,!o,!i) = f(p,!i,!o) =⌘2t⌘2i

f(p,!o,!i)

© Torsten Möller

Non-symmetry: shading• Sometime replace proper normal ng with Interpolated normal ns:

• We can achieve this by a different BTDF:

• Hence, the adjoint would be:

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Lo(p,!o) = Le(p,!o) +

Z

S2

f(p,!o,!i)Li(p,!i)|ng · !i|d!i

fshade(p,!o,!i) =|ns · !i||ng · !i|

f(p,!o,!i)

f?shade(p,!o,!i) =

|ns · !o||ng · !o|

f?(p,!o,!i)

Photon Mapping

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© Torsten Möller

Theoretical foundation for particle tracing

• Particle tracing algorithm creates N samples of illumination at points pj, on surfaces in the scene

(pj, ωj, βj) • Recording

• Incident illumination from direction ωj • With throughput weight βj

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© Torsten Möller


• Particle tracing algorithm creates N samples of illumination at points pj, on surfaces in the scene

(pj, ωj, βj) • Recording

• Incident illumination from direction ωj • With throughput weight βj

• What are good values for ωj and βj

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© Torsten Möller


• Given an importance function We(p, ω), ideally, we want

• E.g. if we want to compute the total flux:

• We would use an importance function of We(p, ω)=1 if p is on the wall surface and ω.n > 0

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E

2

4 1

N

NX

j=1

�jWe(pj ,!j)

3

5 =

Z

a

Z

S2

We(p,!)Li(p,!)|cos✓|dAd!

⌦ =

Z

Awall

Z

H2(n)Li(p,!)|cos✓|dAd!

© Torsten Möller



19

E

2

4 1

N

NX

j=1

�jWe(pj ,!j)

3

5 =

Z

a

Z

S2


© Torsten Möller



• hence, good weights could be:

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E

2

4 1

N

NX

j=1

�jWe(pj ,!j)

3

5 =

Z

a

Z

S2


�i,j =Le(pi,ni ! pi,ni�1)

p(pi,ni)

ni�1Y

j=1

1

1� qi,j

f(pi,j+1 ! pi,j ! pi,j�1)G(pi,j+1 $ pi,j)

p(pi,j)

© Torsten Möller

Photon mapping• In order to compute reflected radiance at a point, need to estimate

exitant radiance at a point p in a direction ωo:

• Hence, we have as a measuring function:

• Not too useful • Spread out contribution instead and us a smooth interpolator • This reuse of photons is a big boost to the efficiency of photon

mapping

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Z

S2

f(p,!o,!i)Li(p,!i)| cos ✓i|d!i =

Z

A

Z

S2

�(p� p0)f(p0,!o,!i)Li(p0,!i)| cos ✓i|d!idA(p

0)

We(p0,!) = �(p0 � p)f(p,!o,!i)

© Torsten Möller

Photon Mapping Example• Pinhole camera • Point light • Only transmissive glass • Tracing ray through glass hits surface through refractive behaviour • No direct light strategies are possible / will hit the light • Photon mapping:

• trace particles leaving the light • Deposit them on the surface and • now collect their contributions

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Glass

Diffuse

© Torsten Möller

Photon Mapping• Kernel density estimation collects the photons

• dN — radius is chosen adaptively such that N photons are enclosed

• Big problem — storage of photons • Usually using a kd-tree

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Lo(p,!o) ⇡1

NpdN (p)2

NpX

j

k

✓p� pjdN (p)

◆�jf(p,!o,!i)

© Torsten Möller

Photon Mapping• Basic idea : Density estimation with a discrete density of photons • 2-step algorithm

• Photon trace: • Simulates the transport of individual photons • Photons emitted from source • Photons deposited on surfaces • Photons reflected from surfaces to other surfaces

• Rendering • Photons collected by rendering

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Progressive Photon Mapping

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© Torsten Möller

Progressive Photon Mapping• Trace rays from camera, find a VisiblePoint at each pixel • Store VisiblePoint in a 3D spatial structure that allows fast lookups • Trace photons from lights; at intersections search for nearby

VisiblePoints and contribute illumination • Decrease acceptance radius as more photons contribute

• When we hit a specular surface trace one bounce further

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© Torsten Möller



• LTE broken into two parts:

• Le is easy and direct term Ld can be handled in a standard fashion

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L(p,!o) = Le(p,!o) +

Z

S2

f(p,!o,!i)Ld(p,!i)| cos ✓i|d!i +

Z

S2

f(p,!o,!i)Li(p,!i)| cos ✓i|d!i

© Torsten Möller



• LTE broken into two parts:

• Li can be handled either by: • Sample next bounce according to BSDF (like a path tracer) • Save current point to collect illumination from photons

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L(p,!o) = Le(p,!o) +

Z

S2

f(p,!o,!i)Ld(p,!i)| cos ✓i|d!i +

Z

S2

f(p,!o,!i)Li(p,!i)| cos ✓i|d!i

© Torsten Möller



• adv: • No photon storage used • Arbitrary number of photons can be traced

• Cons: • Storage limited by number of VisiblePoints

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Stochastic Progressive Photon Mapping (SPPM)

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© Torsten Möller

SPPM• Three modifications:

• Same as PPM but with one VisiblePoint per pixel • Use a constant radius for all visible points:

• Update the radius as more photons are traced

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Lo(p,!o) ⇡1

Np⇡r2

NpX

j

k

✓p� pjdN (p)

◆�jf(p,!o,!i)

© Torsten Möller

SPPM — updating the radius

• Where • Ni is the number of points that have contributed to the point

after the its tiertion • Mi is the number of photons that contributed during the current

iteration • ri is the search radius for the its iteration • 𝜏 maintains the sum of products of photons with BSDF values

• 𝛾 is usually 2/3

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Ni+1 = Ni + �Mi

rI+1 = ri

rNi+1

Ni +Mi

⌧i+1 = (⌧i + �i)r2i+1

r2i

�i =MiX

j

�jf(p,!o,!j)

33http://graphics.stanford.edu/courses/cs348b/lecture/bidir

Bidirectional Path Tracing

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Metropolis Light Transport

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10 bidirectional methodsvda.univie.ac.at/teaching/imagesynthesis/17w/lecture... · 2018-01-09 ·...

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