[PPT] - PATH CONSTRUCTION Iliyan Georgiev Solid Angle MONTE CARLO METHODS PowerPoint Presentation

SLIDE 1

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

PATH CONSTRUCTION

Iliyan Georgiev

Solid Angle

SLIDE 2

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

PATH INTEGRAL FRAMEWORK

2 — PATH CONSTRUCTION

Ij = Z

P

fj(x)dx

Pixel value

Z

P

hIji = 1 N

N

X

i=1

fj(xi)

Pixel estimator

p(xi)

SLIDE 3

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

PATH INTEGRAL FRAMEWORK

3 — PATH CONSTRUCTION

Ij = Z

P

fj(x)dx

Pixel value Pixel estimator

p(xi)

path pdf

Z

P

hIji = 1 N

N

X

i=1

fj(xi)

path contribution

SLIDE 4

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

PATH INTEGRAL FRAMEWORK

4 — PATH CONSTRUCTION

Path contribution

fj(x) = Wj(x0, x1) "Y

i

fs(xi)G(xi, xi+1)T(xi, xi+1) # Le(xk, xk−1) x0

0 x1 1 xk 1 xk−1

Ij = Z

P

fj(x)dx

Pixel value Pixel estimator

p(xi)

Z

P

hIji = 1 N

N

X

i=1

fj(xi)

SLIDE 5

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

PATH INTEGRAL FRAMEWORK

5 — PATH CONSTRUCTION

Path contribution

fj(x) = Wj(x0, x1) "Y

i

fs(xi)G(xi, xi+1)T(xi, xi+1) # Le(xk, xk−1)

camera response

x0

0 x1 1 xk 1 xk−1

Ij = Z

P

fj(x)dx

Pixel value Pixel estimator

p(xi)

Z

P

hIji = 1 N

N

X

i=1

fj(xi)

SLIDE 6

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

PATH INTEGRAL FRAMEWORK

6 — PATH CONSTRUCTION

Path contribution

fj(x) = Wj(x0, x1) "Y

i

fs(xi)G(xi, xi+1)T(xi, xi+1) # Le(xk, xk−1) x0

0 x1 1 xk 1 xk−1

BSDF/ phase

Ij = Z

P

fj(x)dx

Pixel value Pixel estimator

p(xi)

Z

P

hIji = 1 N

N

X

i=1

fj(xi)

camera response

SLIDE 7

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

PATH INTEGRAL FRAMEWORK

7 — PATH CONSTRUCTION

Path contribution

fj(x) = Wj(x0, x1) "Y

i

fs(xi)G(xi, xi+1)T(xi, xi+1) # Le(xk, xk−1) x0

0 x1 1 xk 1 xk−1

geometry

Ij = Z

P

fj(x)dx

Pixel value Pixel estimator

p(xi)

Z

P

hIji = 1 N

N

X

i=1

fj(xi)

BSDF/ phase camera response

SLIDE 8

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

PATH INTEGRAL FRAMEWORK

8 — PATH CONSTRUCTION

Path contribution

fj(x) = Wj(x0, x1) "Y

i

fs(xi)G(xi, xi+1)T(xi, xi+1) # Le(xk, xk−1) x0

0 x1 1 xk 1 xk−1

transmittance

Ij = Z

P

fj(x)dx

Pixel value Pixel estimator

p(xi)

Z

P

hIji = 1 N

N

X

i=1

fj(xi)

geometry BSDF/ phase camera response

SLIDE 9

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

PATH INTEGRAL FRAMEWORK

9 — PATH CONSTRUCTION

Path contribution

fj(x) = Wj(x0, x1) "Y

i

fs(xi)G(xi, xi+1)T(xi, xi+1) # Le(xk, xk−1) x0

0 x1 1 xk 1 xk−1

emitted radiance

Ij = Z

P

fj(x)dx

Pixel value Pixel estimator

p(xi)

Z

P

hIji = 1 N

N

X

i=1

fj(xi)

transmittance geometry BSDF/ phase camera response

SLIDE 10

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

PATH INTEGRAL FRAMEWORK

10 — PATH CONSTRUCTION

Path contribution

fj(x) = Wj(x0, x1) "Y

i

fs(xi)G(xi, xi+1)T(xi, xi+1) # Le(xk, xk−1) x0

0 x1 1 xk 1 xk−1

Ij = Z

P

fj(x)dx

Pixel value Pixel estimator

p(xi)

Z

P

hIji = 1 N

N

X

i=1

fj(xi)

emitted radiance transmittance geometry BSDF/ phase camera response

SLIDE 11

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

Z

P

hIji = 1 N

N

X

i=1

fj(xi)

Ij = Z

P

fj(x)dx

Pixel value Pixel estimator

p(xi)

PATH INTEGRAL FRAMEWORK

11 — PATH CONSTRUCTION

Path contribution

fj(x) = Wj(x0, x1) "Y

i

fs(xi)G(xi, xi+1)T(xi, xi+1) # Le(xk, xk−1) x0

0 x1 1 xk 1 xk−1

) ∝

ideally proportional emitted radiance transmittance geometry BSDF/ phase camera response

SLIDE 12

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

UNIDIRECTIONAL PATH SAMPLING

12 — PATH CONSTRUCTION

ω1 x0

SLIDE 13

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

UNIDIRECTIONAL PATH SAMPLING

13 — PATH CONSTRUCTION

x0

0 x1

p(t1|x0, ω1) ∝ T(x0, x1)

distance sampling

ω1

p(t1|

SLIDE 14

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

UNIDIRECTIONAL PATH SAMPLING

14 — PATH CONSTRUCTION

x0

0 x1

direction sampling

p(ω2|x1) ∝ fs(x1)

ω2

SLIDE 15

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

UNIDIRECTIONAL PATH SAMPLING

15 — PATH CONSTRUCTION

x0

0 x1

distance sampling

x2 ω2

p(t2|x1, ω2) ∝ T(x1, x2) p(t2|

SLIDE 16

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

UNIDIRECTIONAL PATH SAMPLING

16 — PATH CONSTRUCTION

x0

0 x1 2 x3 3 x4 4 x5

x2

A series of distance and direction sampling decisions

SLIDE 17

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

UNIDIRECTIONAL PATH SAMPLING

17 — PATH CONSTRUCTION

p(x) ∝ Wj(x0, x1) "Y

i

fs(xi)G(xi, xi+1)T(xi, xi+1) # ) # Le(xk, xk−1)

not importance sampled high variance when light sources are small cannot render illumination from point light sources

x0

0 x1

2 x3

3 x4 4 x5

x2

A series of distance and direction sampling decisions

SLIDE 18

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

EXPLICIT LIGHT SAMPLING

18 — PATH CONSTRUCTION

x0 x2

SLIDE 19

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

EXPLICIT LIGHT SAMPLING

19 — PATH CONSTRUCTION

x0

0 x1

x2

SLIDE 20

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

EXPLICIT: TRANSMITTANCE

20 — PATH CONSTRUCTION

x0 x2

p(t1|x0) ∝ T(x0, x1)

0 x1

SLIDE 21

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

EXPLICIT: TRANSMITTANCE

21 — PATH CONSTRUCTION

x0 x2

T(x0, x1) ∈ [0, 1]

/ G(x1, x2) = 1 kx1, x2k2 2 [0, 1]

0 x1

SLIDE 22

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

EXPLICIT: EQUIANGULAR

22 — PATH CONSTRUCTION

x0 x2

p(t1|x0) / G(x1, x2) = 1 kx1, x2k2

SLIDE 23

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

EXPLICIT: EQUIANGULAR

23 — PATH CONSTRUCTION

x0 x2

p(t1|x0) / G(x1, x2) = 1 kx1, x2k2

uniform angular distribution

SLIDE 24

Transmittance sampling, 16 spp Equiangular sampling, 16 spp

SLIDE 25

MIS combination Transmittance sampling Equiangular sampling

SLIDE 26

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

UNIDIRECTIONAL + NEXT EVENT

26 — PATH CONSTRUCTION

x0

0 x1 2 x3

x2

SLIDE 27

Equiangular connections Transmittance connections

SLIDE 28

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

angular singularity

UNIDIRECTIONAL + NEXT EVENT

28 — PATH CONSTRUCTION

x0

0 x1 2 x3

x2

SLIDE 29

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

JOINT PATH SAMPLING

29 — PATH CONSTRUCTION

x0

2 x3

SLIDE 30

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

JOINT PATH SAMPLING

30 — PATH CONSTRUCTION

x0

2 x3 0 x1

x2

SLIDE 31

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

JOINT PATH SAMPLING

31 — PATH CONSTRUCTION

x0

2 x3 0 x1

x2

J

i

n t p a t h s a m p l i n g : 1 ) P r e s c r i b e j

i

n t p d f   2 ) D e r i v e c

n

d i t i

n

a l p d f s v i a   s u c c e s s i v e j

i

n t p d f m a r g i n a l i z a t i

n

  3 ) C

n

d i t i

n

a l s a r e

b

t a i n e d i n   r e v e r s e

r

d e r

TRADITIONAL: prescribes conditional pdfs, no explicit control over joint pdf JOINT SAMPLING: prescribe joint pdf, conditional pdfs derived from it

SLIDE 32

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

JOINT PATH SAMPLING

32 — PATH CONSTRUCTION

p(x1, x2) ∝ G(x0, x1)G(x1, x2)G(x2, x3)

joint pdf

x0

2 x3 0 x1

x2

SLIDE 33

37 — PATH CONSTRUCTION

x0

2 x3 0 x1

x2

joint pdf via tabulation

SLIDE 38

Equiangular Transmittance Joint sampling

path lengths 1-3

isotropic phase function

SLIDE 39

Equiangular Transmittance Joint sampling

path lengths 1-8

isotropic phase function

SLIDE 40

Joint tabulated path sampling Transmittance connections

path lengths 1-3

anisotropic phase function

SLIDE 41

Joint tabulated path sampling Transmittance connections

path lengths 1-8

anisotropic phase function

50 — PATH CONSTRUCTION

2 x3 3 x4 4 x5

x2

0 x1

x0

SLIDE 51

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

BIDIRECTIONAL PATH TRACING

51 — PATH CONSTRUCTION

4 x5

x2

0 x1

x0

2 x3 3 x4

SLIDE 52

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

BIDIRECTIONAL PATH TRACING

52 — PATH CONSTRUCTION

4 x5

x2

0 x1

x0

2 x3 3 x4

SLIDE 53

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

BIDIRECTIONAL PATH TRACING

53 — PATH CONSTRUCTION

4 x5

x2

0 x1

x0

2 x3 3 x4

SLIDE 54

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

BIDIRECTIONAL PATH TRACING

54 — PATH CONSTRUCTION

4 x5

x2

0 x1

x0

2 x3 3 x4

hIji = X

s≥0

X

t≥0

ws,t(xi,j) fj(xi,j) ps,t(xi,j)

Combined MIS pixel estimator:

# vertices from light

# vertices from eye

SLIDE 55

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

SUMMARY

55 — DISTANCE SAMPLING

UNIDIRECTIONAL SAMPLING

Almost ideal on paper, rarely useful in practice

NEXT EVENT ESTIMATION

Improvement, but singularity in indirect lighting (reduced convergence rate)

JOINT PATH SAMPLING

Substantial improvement in the presence of singularities

BIDIRECTIONAL PATH TRACING

Avoids singularities, more robust thanks to mixing many sampling techniques
Difficult to implement