volumetric zero-variance-based path guidingjirka/path-guiding-in...volumetric monte-carlo path...
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S. Herholz: A Unified Manifold Framework for Efficient BRDF Sampling โฆ
Sebastian Herholz1 Yangyang Zhao2 Oskar Elek3
Derek Nowrouzezahrai 2 Hendrik P. A. Lensch1 Jaroslav Kลivรกnek3
Volumetric Zero-Variance-Based
Path Guiding
1University of Tรผbingen 2McGill University Montreal 3Charles University Prague
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MOTIVATION
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding2
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MOTIVATION
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding3
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MOTIVATION
โข A correct physically-based simulation of volumetric effects is crucial for rendering realistic scenes
โข In the recent years, brute-force path tracing these effects startedto become applicable in production environments ([Fong2017], [Novak2018])
โข Increased complexity of the light transport makes it still challenging
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding4
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5
No guiding Guiding
(Our)
10 min10 min
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VOLUMETRIC MONTE-CARLO PATH TRACING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding6
๐ ๐
p ๐
โข The variance is defined by how well we can generate
random paths proportional to the volumetric light transport:
๐2 = ๐๐(๐)
๐(๐)
๐ผ = เถฑ๐ ๐ ๐๐
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VOLUMETRIC MONTE-CARLO PATH TRACING: ZERO-VARIANCE
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding7
๐ ๐
p๐ง๐ฃ ๐ (optimal)
๐ผ = เถฑ๐ ๐ ๐๐
โข If the PDF for all paths is proportional to the light transport function
we would get a perfect zero-variance estimator:
๐2 = ๐๐(๐)
๐๐ง๐ฃ(๐)= 0
We need to know the
shape of ๐ ๐ฟ !
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THE 4 SAMPLING DECISIONS:
1. Scatter: ๐๐ ๐๐ , ๐๐
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding8
3. Direction: ๐๐ ๐๐+1|๐๐+1, ๐๐ 4. Termination: ๐๐ ๐ ๐๐ , ๐๐โ1
2. Distance: ๐๐ ๐๐+1|๐๐ , ๐๐
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VOLUMETRIC RANDOM WALK - DECISIONS
๐ฅ0
๐0
๐ฅ๐
๐๐
๐ฅ๐+1
๐๐+1
๐ฅ๐โ1
๐๐โ1
๐ฅ๐+2
๐๐+2
โข Path PDF :
๐ ๐ฟ = เท
๐=1
๐โ1
๐๐ โฆ โ ๐๐ โฆ โ ๐๐ โฆ โ 1 โ ๐๐ ๐ โฆ
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding9
Source of variance
Path segment PDF
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VOLUME RENDERING EQUATION
โข Incident radiance (volume):
โข In-scattered radiance:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding10
๐ฟ ๐ฅ, ๐ = ๐ โฆ โ ๐ฟ๐(โฆ ) + เถฑ๐ โฆ โ ๐๐ (โฆ ) โ ๐ฟ๐(โฆ )d๐
๐ฟ๐ โฆ = เถฑ๐ โฆ โ ๐ฟ(โฆ )d๐โฒ
Surface contribution Volume contribution
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VOLUME RENDERING EQUATION
โข Incident radiance (volume):
โข In-scattered radiance:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding11
๐ฟ ๐ฅ, ๐ = ๐ โฆ โ ๐ฟ๐(โฆ ) + เถฑ๐ โฆ โ ๐๐ (โฆ ) โ ๐ฟ๐(โฆ )d๐
๐ฟ๐ โฆ = เถฑ๐ โฆ โ ๐ฟ(โฆ )d๐โฒ
Known Local
Quantities
Transmittance
Phase function
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VOLUME RENDERING EQUATION
โข Incident radiance (volume):
โข In-scattered radiance:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding12
๐ฟ ๐ฅ, ๐ = ๐ โฆ โ ๐ฟ๐(โฆ ) + เถฑ๐ โฆ โ ๐๐ (โฆ ) โ ๐ฟ๐(โฆ )d๐
๐ฟ๐ โฆ = เถฑ๐ โฆ โ ๐ฟ(โฆ )d๐โฒ
Unknown Light
Transport Quantities
Incident radiance
In-scattered radiancesurface radiance
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STANDARD SAMPLING
13
๐ฅ๐+1๐ฅ๐ ๐๐ ๐ฅ๐+1 ๐๐๐ฅ๐+1
๐๐+1
1+2 Scatter and Distance:
โข Based on the transmittance
3 Direction:
โข Based on the phase function
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STANDARD SAMPLING
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โข Based on local albedo or throughput
4 Termination:
๐ฅ๐
๐๐๐ฅ๐+1
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CHALLENGES FOR VOLUME SAMPLINGWhy do we need volumetric path guiding?
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LIGHT SHAFTS
โข Light shafts:
- We need to scatter inside the light shaft.
- We need to follow the direction of the light shaft.
- We need to scatter towards the light shaft.
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LIGHT SHAFTS
โข Light shafts:
- We need to scatter inside the light shaft.
- We need to follow the direction of the light shaft.
- We need to scatter towards the light shaft.
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding17
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LIGHT SHAFTS
โข Light shafts:
- We need to scatter inside the light shaft.
- We need to follow the direction of the light shaft.
- We need to scatter towards the light shaft.
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding18
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No guiding (1024 spp) Our guiding (1024 spp)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding19
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SUB-SURFACE-SCATTERING
๐ฅ
๐
โข Sub-Surface-Scattering:
- We โoftenโ need stay close to the surface
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding20
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SUB-SURFACE-SCATTERING
๐ฅ
๐
โข Sub-Surface-Scattering:
- We โoftenโ need to stay close to the surface
- We need to leave the object with the right direction
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding21
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No guiding (256 spp) Our guiding (256 spp)
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DENSE MEDIA
โข Dense media (back illuminated):
- We may need to โavoidโ generating a scattering
event even if the transmittance is low
(e.g. strong light source behind the volume).
๐ฅ
๐
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding23
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No guiding (256 spp) Our guiding (256 spp)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding24
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NON-DENSE MEDIA
โข Non-dense media (no back illumination):
- We may need to โforceโ a scattering event
even if the transmittance is high.
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No guiding (256 spp) Our guiding (256 spp)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding26
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SPECIALIZED SOLUTIONS: SHORTCOMINGS
โข Many individual solutions:
โข Equiangular Sampling: [Kulla2012]
โข Joint-Importance Sampling: [Georgiev2012]
โข Zero-Variance Dwivedi Sampling: [Krivanek2014]
[Meng2016]
โข Directional (illumination-based) guiding: [Pegoraro2008][Bashford2012]
โข Only considering sub-sets or special cases:
โข Surface-bounded volumes
โข Homogenous or isotropic volumes
โข Single scattering
โข None of the current methods importance samples the full volumetric light transport!
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding27
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SPECIALIZED SOLUTIONS: SHORTCOMINGS
โข Many individual solutions:
โข Equiangular Sampling: [Kulla2012]
โข Joint-Importance Sampling: [Georgiev2012]
โข Zero-Variance Dwivedi Sampling: [Krivanek2014]
[Meng2016]
โข Directional (illumination-based) guiding: [Pegoraro2008][Bashford2012]
โข Only considering sub-sets or special cases:
โข Surface-bounded volumes
โข Homogenous or isotropic volumes
โข Single scattering
โข None of the current methods importance samples the full volumetric light transport!
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding28
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SPECIALIZED SOLUTIONS: SHORTCOMINGS
โข Many individual solutions:
โข Equiangular Sampling: [Kulla2012]
โข Joint-Importance Sampling: [Georgiev2012]
โข Zero-Variance Dwivedi Sampling: [Krivanek2014]
[Meng2016]
โข Directional (illumination-based) guiding: [Pegoraro2008][Bashford2012]
โข Only considering sub-sets or special cases:
โข Surface-bounded volumes
โข Homogenous or isotropic volumes
โข Single scattering
โข None of the current methods importance samples the full volumetric light transport!
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding29
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ZERO-VARIANCE-BASEDVOLUMETRIC PATH GUIDING
30
TUE: 30TH JULY TIME: 9:00 AMROOM: 152
TECH TALK:
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[Hoogenboom 2008]
ZV-BASED VOLUMETRIC PATH GUIDING: GOALS
โข Leverage recent success of local surface guiding methods:
โข Extend the concept to volumes
โข Consider the complete volumetric light transport:
โข No prior assumptions or special cases
โข Guide based on the optimal zero-variance decisions
โข Replace unknown quantities by estimates:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding31
๐ฟ ๐, ๐ = เทจ๐ฟ ๐, ๐ ๐ฟ๐ ๐,๐ = เทจ๐ฟ๐ ๐,๐
[Vorba2014],
[Herholz2016],
[Mueller2017]
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ZV-BASED VOLUMETRIC PATH GUIDING: CONTRIBUTIONS
โข Guiding all local sampling decisions:
โข 1+2 Guided product distance sampling:
โข 3 Guided product directional sampling:
โข 4 Guided Russian roulette and Splitting:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding32
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ZV-BASED VOLUMETRIC PATH GUIDING: CONTRIBUTIONS
โข Guiding all local sampling decisions:
โข 1+2 Guided product distance sampling:
โข 3 Guided product directional sampling:
โข 4 Guided Russian roulette and Splitting:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding33
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ZV-BASED VOLUMETRIC PATH GUIDING: CONTRIBUTIONS
โข Guiding all local sampling decisions:
โข 1+2 Guided product distance sampling:
โข 3 Guided product directional sampling:
โข 4 Guided Russian roulette and Splitting:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding34
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VOLUME RADIANCE ESTIMATES
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VOLUME RADIANCE ESTIMATES
โข Spatial caches via BSP-tree: max. 2K photons per node:
โข Similar 3D structure as PPG [Mueller2017]
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VOLUME RADIANCE ESTIMATES
โข Pre-processing step to fit estimates from photons (50M):
โข EM-fitting of von Mises-Fisher mixtures (similar to [Vorba2014]โs GMMs)
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VON MISES-FISHER MIXTURE MODELS
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding38
โข Spherical Distribution:
โข Features (closed-form):
- Sampling
- Convolution
- Product
๐ ๐|ฮ =
๐พ
๐๐๐ฃ(๐|๐๐ , ๐ ๐)
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RADIANCE ESTIMATES
โข Incident Radiance Distribution โข In-Scattered Radiance Distribution
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding39
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INCIDENT RAD. TO IN-SCATTERED RAD. TRANSFORMATION
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding40
โข Convolution between incident radiance ๐ฟ and the phase function ๐
โ
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INCIDENT RADIANCE ESTIMATES
Ground truth (2K spp) Our estimates (VMM)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding41
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IN-SCATTERED RADIANCE ESTIMATES
Ground truth (2K spp) Our estimates (VMM)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding42
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GUIDED SAMPLING DECISIONS
43
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DISTANCE SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding44
2. Scatter distance
1. Volume or surface decision
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DISTANCE SAMPLING
โข Standard distance PDF:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding45
1+2. Event distance
๐๐๐ ๐ก๐ โฆ โ ๐ โฆ โ ๐๐ โฆ
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GUIDED PRODUCT DISTANCE SAMPLING
โข Our guided PDF:
๐๐๐ง๐ฃ โฆ =
๐ โฆ โ ๐๐ โฆ โ เทจ๐ฟ๐(โฆ )
เทจ๐ฟ(โฆ )Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding46
1+2. Event distance
Our estimates
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NAรVE TABULATED APPROACH
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding47
โข Naรฏve tabulated approach:
โข Step through the complete volume and build a tabulated PDF
โข Inefficient (large scenes dense media):
โข we always need to evaluated all bins first
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OUR INCREMENTAL GUIDED PRODUCT DISTANCE SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding48
โข Incremental approach:
โข At each step make a local decision, if we scatter inside the bin
โข We only need to step until the scattering event happens
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Full CDF (30min) Our incremental (30min)
49
Spp: 548
Avg. steps: 18
Spp: 1140
Avg. steps: 4
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45 min
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No guiding
Spp: 960
relMSE: 1.342
45 min
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No guiding Distance guiding
Spp: 960
relMSE: 1.342Spp: 424
relMSE: 0.901
45 min
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No guiding (256 spp) Distance guiding (256 spp)
53
โข Here, distance sampling is not the main cause of variance!!
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DIRECTIONAL SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding54
3. Scatter direction
๐๐๐ฅ๐+1
โข Standard PDF:
๐๐๐ ๐ก๐ โฆ โ แ๐ โฆ
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GUIDED PRODUCT DIRECTIONAL SAMPLING
โข Our guided PDF:
๐๐๐ง๐ฃ โฆ โ แ๐ โฆ โ เทจ๐ฟ(โฆ )
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding55
3. Scatter direction
๐๐๐ฅ๐+1 ๐๐+1
Our estimates
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OUR GUIDED PRODUCT DIRECTIONAL SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding56
Incident radiance VMM
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OUR GUIDED PRODUCT DIRECTIONAL SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding57
Incident radiance VMM Phase function VMM
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OUR GUIDED PRODUCT DIRECTIONAL SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding58
Product sampling VMM
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30 min
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30 min No guiding
Spp: 2212
relMSE: 0.376
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Directional guiding
Spp: 1756
relMSE: 0.048
30 min No guiding
Spp: 2212
relMSE: 0.376
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Directional guiding Dist + Direct
Spp: 1756
relMSE: 0.048Spp: 1228
relMSE: 0.034
30 min No guiding
Spp: 2212
relMSE: 0.376
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No Guiding
(256 spp)
Product Guiding
(256 spp)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding63
Illum Guiding
(256 spp)
IMPORTANCE OF THE PRODUCT FOR DENSE ANISOTROPIC MEDIA
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GUIDED RUSSIAN ROULETTE AND SPLITTING
4a. Termination 4b. Splitting Distance
Directional
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GUIDED RUSSIAN ROULETTE AND SPLITTING
โข Post-sampling compensation strategies:
โข Identify, if we did a sub-optimal sampling decision
โข Terminate: to increase performance
โข Split: bound/reduce sample variance
4b. Splitting
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding65
4a. Termination
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GUIDED RUSSIAN ROULETTE AND SPLITTING
โข Path contribution: ๐ธ[๐]โข The expected contribution
if we continue the path
โข Reference solution: ๐ผโข the final pixel value
๐ =๐ธ ๐
๐ผ
Path contribution
Reference solution
survival prob /
splitting factor
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GUIDED RUSSIAN ROULETTE AND SPLITTING
โข Path contribution: ๐ธ[๐]โข The expected contribution
if we continue the path
โข Reference solution: ๐ผโข the final pixel value
๐ =๐ธ ๐
๐ผ= 1
Path contribution
Reference solution
survival prob /
splitting factor
Zero-Variance
Estimator
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding67
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GUIDED RUSSIAN ROULETTE AND SPLITTING
โข If ๐ โค 1: Russian Roulette
โข Terminates low contributing paths
โข Survival probability: ๐
โข If ๐ > 1: Splitting
โข Splits an under sampled paths with
a potential high contribution (๐ times)
๐ =๐ธ ๐
๐ผ
Path contribution
Reference solution
survival prob /
splitting factor
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding68
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ESTIMATED PATH CONTRIBUTION
โข See course notes or paper for more details
เทจ๐ฟ๐
๐ฟ
๐ธ ๐ฟ = ๐โฒ(๐ฟ) โ เทจ๐ฟ๐ ๐ฅ๐ , ๐๐โ1
Path throughput: ๐(๐ฟ)/๐(๐ฟ) In-scattered radiance estimate
๐ฅ๐
๐๐โ1
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GUIDED RUSSIAN ROULETTE AND SPLITTING: PIXEL ESTIMATE
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding71
โข Ray marched cache to integrate: ๐ โ ๐๐ โ เทจ๐ฟ๐
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No RR
Spp: 468
relMSE: 0.454
45 min
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No RR Guided RR
Spp: 468
relMSE: 0.454Spp: 1500
relMSE: 0.174
45 min
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No RR Guided RR
Spp: 468
relMSE: 0.454Spp: 1500
relMSE: 0.174
45 min
+ Guided splitting
Spp: 1340
relMSE: 0.066
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Guided RR + Guided splitting
Spp: 1500
relMSE: 0.174
Spp: 1340
relMSE: 0.06675
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No guiding
Time: 60 min
Spp: 10644
relMSE: 11.58
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Distance guiding
Time: 60 min
Spp: 4624
relMSE: 3.520
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Distance + directional guiding
Time: 60 min
Spp: 4448
relMSE: 0.468
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Distance + directional guiding + GRRS
Time: 60 min
Spp: 3796
relMSE: 0.321
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OPEN PROBLEMS AND LIMITATIONSOpen Challenges to make it bullet proof
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SPATIAL CACHE STRUCTURE
โข Naive approach to define the resolution:
โข Heuristic based on sample numbers
โข Takes time or many samples to model/separate fine features (e.g. thin shafts or caustics)(PPG by [Mueller2017] has the same problem)
โข Influences the performance of some sampling methods (e.g. distance)
โข Ideal structure should adjust to the light transport characteristics
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INSUFFICIENT CACHE SIZES
โข Shared problem with other 3D caching based guiding approaches(e.g. [Vorba2014], [Mueller2017], โฆ)
โข By merging the samples of a spatial cache we blur the directional distribution
โข Can lead to incorrect estimates of ๐ฟ and ๐ฟ๐
82
Ground truth Our estimate Spatial avg. ground truth
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PRODUCTION CHALLENGESHow can we get good guiding estimates?
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VMM PHASE FUNCTION FITTING: PRE-PROCESSING STEP
โข Pre-processing step:โข Fitting a VMM for each phase function
โข Fitting up to K = 4 componentsโข Details in the course notes
โข Open Challenge:โข Procedural phase functions or
procedural mixtures?
โข How to deal with changing mean cosines (roughening/mollification)?
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VMM PHASE FUNCTION FITTING: PRE-PROCESSING STEP
โข Pre-processing step:โข Fitting a VMM for each phase function
โข Fitting up to K = 4 componentsโข Details in the course notes
โข Open Challenge:โข Procedural phase functions or
procedural mixtures?
โข How to deal with changing mean cosines (roughening/mollification)?
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FITTING/LEARNING THE INCIDENT RADIANCE MIXTURES
Pros:
โข Photons directly represents the light transport
โข Spatial distribution corresponds to important features (light shaft)
โข Number of traced photons can be fixed
โข No additional fitting during rendering
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Photon-based pre-processing ([Herholz2019][Vorba2014])
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FITTING/LEARNING THE INCIDENT RADIANCE MIXTURES
Cons:
โข Pre-processing step:
โข Long time to first render iteration
โข Complex scenes need bidirectional learning:
โข Ping-Pong style [Vorba2014]
โข It is not the ideal solution for artists in production
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Photon-based pre-processing ([Herholz2019][Vorba2014])
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FITTING/LEARNING THE INCIDENT RADIANCE MIXTURES
Pros:
โข First experiments show promising results
โข No pre-processing
โข Refines spatial and directional distributions in each iteration
โข Sample data gets more reliable
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Progressive learning (PPG-style [Mueller2017])
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FITTING/LEARNING THE INCIDENT RADIANCE MIXTURES
Cons (open challenges):
โข Sample count grows exponential:
โข Memory and fitting time increases
โข Shorter update rates ?
โข Online fitting for mixtures?
โข Spatial structure adapts slowly to LT:
โข Important for distance sampling
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Progressive learning (PPG-style [Mueller2017])
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CONCLUSION
โข Even approximate local sampling decisions lead to a good approximation of the globally optimal guiding distribution (and thus significantly reducing MC variance)
โข Converges to optimal ZV estimator in the hypothetical limit (i.e., if the approximations were perfect)
โข Solely based one guiding structure for alldecisions (incident radiance VMMs)
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THANK YOU
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REFERENCES
โข [Fong2017]: โProduction volume renderingโ
โข [Novak2018]: โMonte Carlo methods for volumetric light transport simulationโ
โข [Kulla2012]: โImportance sampling techniques for path tracing in participating mediaโ
โข [Georgiev2012]: โimportance sampling of low-order volumetric scatteringโ
โข [Krivanek2014]: โA zero-variance-based sampling scheme for Monte Carlo subsurface scatteringโ
โข [Meng2016]: โImproving the Dwivedi sampling schemeโ
โข [Vorba2014]: โOnline learning of parametric mixture models for light transport simulationโ
โข [Vorba2016]: โAdjoint-driven Russian roulette and splitting in light transport simulationโ
โข [Herholz2016]: โProduct importance sampling for light transport path guidingโ
โข [Koerner2016]: โSubdivision next-event estimation for path-traced subsurface scatteringโ
โข [Mueller2017]: โPractical path guiding for efficient light-transport simulationโ
โข [Hoogenboom2008]: โZero-varianceMonte Carlo schemes revisitedโ
โข [Pegoraro2008]:โ Sequential Monte Carlo integration in low-anisotropy participating mediaโ
โข [Bashford2012]: โA significance cache for accelerating global illuminationโ
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ADJOINT ESTIMATE ACCURACY
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COMPARING AGAINST EQUIANGULAR SAMPLING
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BUMPY SPHERE
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INCREMENTAL GUIDED DISTANCE SAMPLING
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INCREMENTAL GUIDED DISTANCE SAMPLING
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INCREMENTAL GUIDED DISTANCE SAMPLING
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VMM PHASE FUNCTION FITTING: PRE-PROCESSING STEP
โข Using up to K = 4 components
โข Optimization Problem:
arg mi๐ฮ๐
๐=1
๐
โlog(๐ ๐๐, โฆ , ๐(๐๐, ฮ๐))2
โlog(๐,๐) = log ๐ + ๐ โ log ๐ + ๐
๐ = (1๐ โ 4) โ max(๐1, โฆ , ๐๐)
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VMM PHASE FUNCTION FITTING
โข Manifold representation of the VMM parameters for an HG phase function model for different mean cosines
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