adam goodenoughsampling: strategies and toolsapril 11, 2005 sampling: strategies and tools adam...

Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005

Sampling:Strategies and Tools

Adam Goodenough

DIRSIG Meeting

April 11, 2005


• DIRSIG = Integration• Numerical integration is inefficient• Each sample can lead to many other

samples• Monte Carlo integration uses knowledge

of sample “importance”• Techniques are essential to photon

mapping• Applicable in many other areas

Introduction


Derivation


Outline• Monte Carlo integration

• Monte Carlo integration of an example “scene”

• DIRSIG Tool: CDSampleGen

• Samplers and Projections

• Arbitrary BRDF Representation and Sampling

• Future Applications


Integration• Equation 4.31 in Schott (1997)


Monte Carlo• Monte Carlo integration for the same term

iii dd)sin(


Sky Dome• Picture of the sky taken with a fisheye lens• Data taken from Conesus collect


Parameterized• Sky dome unwrapped for easy sampling• Only the green band was acquired

• Parameterized as θ versus Φ

θ

Φ0

0 2π

½π


Ward BRDF• Models specular lobe + diffuse • Anisotropic (brushed surfaces)• Parameters are “physical”• BRDF accuracy has been validated against

measurements

Lig

htly B

rush

edA

lum

inu

m


Setup• Using sky dome and Ward BRDF• Viewer at 80o zenith and 45o azimuth• Calculate the numerical integral first• Calculate Monte Carlo integrals using different

sampling methods

View Direction


Integral• Regular Numerical Integration

50 SamplesLout = 0.0629




Grid• Monte Carlo Integration with Grid sampling

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 10 20 30 40 50 60 70 80 90 100

Number of Samples

Ab

so

lute

Err

or

Power (Grid)

5000 SamplesLout = 0.0559Num: Lout = 0.0556


Random• Monte Carlo Integration with Random sampling

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 10 20 30 40 50 60 70 80 90 100

Number of Samples

Ab

so

lute

Err

or

Power (Grid)

Power (Random)



Shirley• Monte Carlo Integration with Shirley sampling


0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 10 20 30 40 50 60 70 80 90 100

Number of Samples

Ab

so

lute

Err

or

Power (Grid)

Power (Random)

Power (Shirley)


Cosine• Integration with Cosine Weighted Sampling


0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 10 20 30 40 50 60 70 80 90 100

Number of Samples

Ab

so

lute

Err

or

Power (Grid)

Power (Cosine-Weighted Grid)


Importance• Cosine and BRDF Weighted Sampling


0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 10 20 30 40 50 60 70 80 90 100

Number of Samples

Ab

so

lute

Err

or

Power (Grid)

Power (BRDF-Weighted Grid)


DIRSIG Tool• CDSampleGen

• None• Gaussian• Sphere• Hemisphere• Sphere

Section

• Grid• Random• N-Rooks• Stratified• Shirley• Halton• Hammersley

Samplers

• Cosine• Henyey-

Greenstein• Schlick• Ward BRDF• Factored

Geometry Weighted

Projections


Samplers• Grid • Random


Samplers• N-Rooks (Latin Hypercube)


Samplers• Stratified • Shirley


Samplers• Halton “Sequence” (Quasi-Monte Carlo)


• Set vs. Sequence• Hammersley Set

Samplers

• Sets are defined by “n” and “m” and yield nxm samples

• Sets are randomly shuffled

• All samplers work in either mode


• Quasi-Monte Carlo Samplers

Samplers

• Halton Sequence and Hammersley Set

• Deterministic

• Optimal uniformity and repeatability

• “Leaping” and “Scrambling”


Projections• None • Gaussian


Projections• Gaussian Revisited

PDF Random Shirley


Projections• Sphere • Hemisphere


Projections• Sphere Section


Projections• Henyey-Greenstein SPF (N-Terms)


Projections• Schlick SPF (N-Terms)


Projections• Cosine • Ward BRDF


Arbitrary BRDF• Synthetic or Modeled 4-D Measurements

Measurements

Factorization

Re-Creation


Components• Re-Parameterization • Non-Negative Matrix

Factorization• Most BRDFs consist of

diffuse and specular components

• Variability exists primarily around specular direction

• Re-Parameterize into Half-Angle space

• Factor Y into two matrices F and G

• F and G are non-negative• Iterative algorithm• Gradient descent• Rate of descent picked to

ensure monotonically decreasing RSE or distortion


Approach• Re-Parameterization and Factorization From Lawrence et al. (2004)

Re-Shuffling

Re-Parameterizing Re-Shuffling and NMF

NMF

G

F

G

uv


Example• Significance of u and v From Lawrence et al. (2004)


Compression• Large data sets represented by few terms• Better accuracy than general basis functions

(Zernike, Spherical Harmonics, LaFortune Lobes)• Compression ratios of ~200:1 shown for

measured data (measurements performed by Matusik [2003])

From Lawrence et al. (2004)


Sampling• Compression rates are sub-optimal!• Factorization form maintains each sampling

variable (incident direction) independently• Factored forms are used directly to sample• F- matrix maintains view direction indexing• G (u and v) contains incident direction PDFs


Future Apps• Standard for storing BRDF measurements• Addition of a spectral dimension• Storage of modeled, spectral phase

function data


Questions

adam goodenoughsampling: strategies and toolsapril 11, 2005 sampling: strategies and tools adam...

Documents

adam goodenoughsampling

direction slide

term slide

derivation slide

random monte carlo integration

grid monte carlo integration

cosine integration

brushed aluminum slide