Deterministic Importance Sampling with Error Diffusion

László Szirmay-Kalos, László Szécsi

Budapest University of Technology

Eurographics Symposium on Rendering, 2009

Numerical integration

0 1

f: integrand

g: target density

samples

Quadrature error

fg

f/g

M2

1best:

Role of

Random sampling

undersampling

oversampling

Role of

Wanted

Previous work• Importance sampling:

– Transformation of uniform samples– Rejection sampling

• Metropolis (Veach97)• Population Monte Carlo (Lai07)• Importance re-sampling (Talbot05), thresholding (Burke05)

• Stratification:

• Low-discrepancy series (Shirley91,Keller95,Kollig02)• Poisson-disk/blue-noise (Cook86,Dunbar06,Kopf06)• Tiling (Ostromukhov05-07, Lagae06) • Sample relaxation (Agarwal03,Kollig03,Wan05,Spencer09)

2D only?

Proposed method

• Simultaneously targets– Importance sampling

• Importance function– Point samples– Cheap

– Stratification• Minimize discrepancy in the target domain

• Simple!

Sample generation: Phase 1

f

I: importance function

I

Tentative samples

Normalizationconstant: b

Sample generation: Phase 2

f

g=I/b

M2

1

G

Frequency modulator

Comparator (quantizer)

-+ Integrator

Tentative samples

Realsamples

g(i) y(i)

Frequency domain analysis

)()()1()( 11 zyznzzzg

Delay Light-blue noise

White noise:

-+ Integrator

Tentative samples

Realsamples

g(i) y(i)

n(i)

Transfer function in the Z-transform domain:

Delta-Sigma modulator:Noise-Shaping Feedback Coder

H(z)

quantizer

+

-

)()())(1()( zyznzHzg

No delayControllable blue noise

Tentative samples

g(i)+

g(i) y(i)+

Realsamples

Noise shaping

filter

Transfer function in the Z-transform domain:

Application in higher dimensions

Importancemap

pixels

Importancemap

Application in higher dimensions

sequence neighborhood

Application in higher dimensions

Importancemap

Equivalence• Deterministic importance sampling allowing

arbitrary importance functions and minimizing the error of distribution

• Delta-Sigma modulation• Error diffusion halftoning (e.g. Floyd-Steinberg)

Environment mapping with light source sampling

v=1

v=1

lighting reflection visibility

Light source sampling = Error diffusion halftoning of the Environment Map

Error diffusion

Random sampling

Similar complexity and running times!

Light source sampling results

Random Error diffusion Reference

Light source sampling results for diffuse objects

Random Error diffusion Reference

Environment mapping with product sampling

lighting reflection visibility

• Separate importance map for every shaded point• Computational cost ???:

– Similar to importance re-sampling– Negligible overhead more complex scenes

Product sampling: Diffuse objectsBRDF sampling Importance resampling Error diffusion

11 sec 13 sec 13 sec

Product sampling: Specular objectsBRDF sampling Importance resampling Error diffusion

11 sec 13 sec 13 sec

Product sampling with occlusions

BRDF sampling

Importanceresampling

Error diffusion

Even higher dimensions• Regular grid: Curse of dimensionality!

• Solution: Low-discrepancy seriescurrentsample

Errordistribution

sequence ofvisiting samples

Elemental interval property

8

12

7

3

4

5

6

9

10

11

1

2

The algorithm in d-dimensions

8

12

7

3

4

5

6

9

10

11

1

2

8,I(u8)

2,I(u2)

5,I(u5)

11,I(u11)

4,I(u4)

10,I(u10)

1,I(u1)

7,I(u7)

12,I(u12)

6,I(u6)

9,I(u9)

3,I(u3)

d-dimensional arrayd-dimensional cube

+ normalization constant b

Virtual point light source method

6D primary sample space

paths

VPLs ofa path

power

Geometryfactor

visibility

BRDF

VPL with error diffusion

6D primary sample space

Approximatevisibility

VPL with error diffusion results (4D, 16 real from 420 tentative)

Classical VPL Importance resampling Error diffusion

8D integration (equal time test)

Error diffusionClassical VPL

Conclusions

• Delta-sigma modulation is a powerful sampling algorithm.

• In lower dimensions sampling is equivalent to the error diffusion halftoning of the importance image.

• In higher dimensions, implicit cell structure of low-discrepancy series can help to fight the curse of dimensionality.

Open question: Optimal error shaping filter

Higher weight for faster changing coordinate