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S. Herholz: A Unified Manifold Framework for Efficient BRDF Sampling β¦
Sebastian Herholz1 Yangyang Zhao2 Oskar Elek3
Derek Nowrouzezahrai 2 Hendrik P. A. Lensch1 Jaroslav KΕivΓ‘nek3
Volumetric Zero-Variance-Based
Path Guiding
1University of TΓΌbingen 2McGill University Montreal 3Charles University Prague
MOTIVATION
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding2
MOTIVATION
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding3
MOTIVATION
β’ A correct physically-based simulation of volumetric effects is crucial for rendering realistic scenes
β’ In the recent years, brute-force path tracing these effects startedto become applicable in production environments ([Fong2017], [Novak2018])
β’ Increased complexity of the light transport makes it still challenging
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding4
5
No guiding Guiding
(Our)
10 min10 min
VOLUMETRIC MONTE-CARLO PATH TRACING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding6
π π
p π
β’ The variance is defined by how well we can generate
random paths proportional to the volumetric light transport:
π2 = ππ(π)
π(π)
πΌ = ΰΆ±π π ππ
VOLUMETRIC MONTE-CARLO PATH TRACING: ZERO-VARIANCE
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding7
π π
pπ§π£ π (optimal)
πΌ = ΰΆ±π π ππ
β’ If the PDF for all paths is proportional to the light transport function
we would get a perfect zero-variance estimator:
π2 = ππ(π)
ππ§π£(π)= 0
We need to know the
shape of π πΏ !
THE 4 SAMPLING DECISIONS:
1. Scatter: ππ ππ , ππ
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding8
3. Direction: ππ ππ+1|ππ+1, ππ 4. Termination: ππ π ππ , ππβ1
2. Distance: ππ ππ+1|ππ , ππ
VOLUMETRIC RANDOM WALK - DECISIONS
π₯0
π0
π₯π
ππ
π₯π+1
ππ+1
π₯πβ1
ππβ1
π₯π+2
ππ+2
β’ Path PDF :
π πΏ = ΰ·
π=1
πβ1
ππ β¦ β ππ β¦ β ππ β¦ β 1 β ππ π β¦
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding9
Source of variance
Path segment PDF
VOLUME RENDERING EQUATION
β’ Incident radiance (volume):
β’ In-scattered radiance:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding10
πΏ π₯, π = π β¦ β πΏπ(β¦ ) + ΰΆ±π β¦ β ππ (β¦ ) β πΏπ(β¦ )dπ
πΏπ β¦ = ΰΆ±π β¦ β πΏ(β¦ )dπβ²
Surface contribution Volume contribution
VOLUME RENDERING EQUATION
β’ Incident radiance (volume):
β’ In-scattered radiance:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding11
πΏ π₯, π = π β¦ β πΏπ(β¦ ) + ΰΆ±π β¦ β ππ (β¦ ) β πΏπ(β¦ )dπ
πΏπ β¦ = ΰΆ±π β¦ β πΏ(β¦ )dπβ²
Known Local
Quantities
Transmittance
Phase function
VOLUME RENDERING EQUATION
β’ Incident radiance (volume):
β’ In-scattered radiance:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding12
πΏ π₯, π = π β¦ β πΏπ(β¦ ) + ΰΆ±π β¦ β ππ (β¦ ) β πΏπ(β¦ )dπ
πΏπ β¦ = ΰΆ±π β¦ β πΏ(β¦ )dπβ²
Unknown Light
Transport Quantities
Incident radiance
In-scattered radiancesurface radiance
STANDARD SAMPLING
13
π₯π+1π₯π ππ π₯π+1 πππ₯π+1
ππ+1
1+2 Scatter and Distance:
β’ Based on the transmittance
3 Direction:
β’ Based on the phase function
STANDARD SAMPLING
14
β’ Based on local albedo or throughput
4 Termination:
π₯π
πππ₯π+1
CHALLENGES FOR VOLUME SAMPLINGWhy do we need volumetric path guiding?
15
LIGHT SHAFTS
β’ 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 Guiding16
LIGHT SHAFTS
β’ 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
LIGHT SHAFTS
β’ 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
No guiding (1024 spp) Our guiding (1024 spp)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding19
SUB-SURFACE-SCATTERING
π₯
π
β’ Sub-Surface-Scattering:
- We βoftenβ need stay close to the surface
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding20
SUB-SURFACE-SCATTERING
π₯
π
β’ Sub-Surface-Scattering:
- We βoftenβ 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
No guiding (256 spp) Our guiding (256 spp)
22
DENSE MEDIA
β’ Dense media (back illuminated):
- We may need to βavoidβ generating a scattering
event even if the transmittance is low
(e.g. strong light source behind the volume).
π₯
π
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding23
No guiding (256 spp) Our guiding (256 spp)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding24
NON-DENSE MEDIA
β’ Non-dense media (no back illumination):
- We may need to βforceβ a scattering event
even if the transmittance is high.
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding25
No guiding (256 spp) Our guiding (256 spp)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding26
SPECIALIZED SOLUTIONS: SHORTCOMINGS
β’ Many individual solutions:
β’ Equiangular Sampling: [Kulla2012]
β’ Joint-Importance Sampling: [Georgiev2012]
β’ Zero-Variance Dwivedi Sampling: [Krivanek2014]
[Meng2016]
β’ Directional (illumination-based) guiding: [Pegoraro2008][Bashford2012]
β’ Only considering sub-sets or special cases:
β’ Surface-bounded volumes
β’ Homogenous or isotropic volumes
β’ Single scattering
β’ None of the current methods importance samples the full volumetric light transport!
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding27
SPECIALIZED SOLUTIONS: SHORTCOMINGS
β’ Many individual solutions:
β’ Equiangular Sampling: [Kulla2012]
β’ Joint-Importance Sampling: [Georgiev2012]
β’ Zero-Variance Dwivedi Sampling: [Krivanek2014]
[Meng2016]
β’ Directional (illumination-based) guiding: [Pegoraro2008][Bashford2012]
β’ Only considering sub-sets or special cases:
β’ Surface-bounded volumes
β’ Homogenous or isotropic volumes
β’ Single scattering
β’ None of the current methods importance samples the full volumetric light transport!
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding28
SPECIALIZED SOLUTIONS: SHORTCOMINGS
β’ Many individual solutions:
β’ Equiangular Sampling: [Kulla2012]
β’ Joint-Importance Sampling: [Georgiev2012]
β’ Zero-Variance Dwivedi Sampling: [Krivanek2014]
[Meng2016]
β’ Directional (illumination-based) guiding: [Pegoraro2008][Bashford2012]
β’ Only considering sub-sets or special cases:
β’ Surface-bounded volumes
β’ Homogenous or isotropic volumes
β’ Single scattering
β’ None of the current methods importance samples the full volumetric light transport!
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding29
ZERO-VARIANCE-BASEDVOLUMETRIC PATH GUIDING
30
TUE: 30TH JULY TIME: 9:00 AMROOM: 152
TECH TALK:
[Hoogenboom 2008]
ZV-BASED VOLUMETRIC PATH GUIDING: GOALS
β’ Leverage recent success of local surface guiding methods:
β’ Extend the concept to volumes
β’ Consider the complete volumetric light transport:
β’ No prior assumptions or special cases
β’ Guide based on the optimal zero-variance decisions
β’ Replace unknown quantities by estimates:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding31
πΏ π, π = ΰ·¨πΏ π, π πΏπ π,π = ΰ·¨πΏπ π,π
[Vorba2014],
[Herholz2016],
[Mueller2017]
ZV-BASED VOLUMETRIC PATH GUIDING: CONTRIBUTIONS
β’ Guiding all local sampling decisions:
β’ 1+2 Guided product distance sampling:
β’ 3 Guided product directional sampling:
β’ 4 Guided Russian roulette and Splitting:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding32
ZV-BASED VOLUMETRIC PATH GUIDING: CONTRIBUTIONS
β’ Guiding all local sampling decisions:
β’ 1+2 Guided product distance sampling:
β’ 3 Guided product directional sampling:
β’ 4 Guided Russian roulette and Splitting:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding33
ZV-BASED VOLUMETRIC PATH GUIDING: CONTRIBUTIONS
β’ Guiding all local sampling decisions:
β’ 1+2 Guided product distance sampling:
β’ 3 Guided product directional sampling:
β’ 4 Guided Russian roulette and Splitting:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding34
VOLUME RADIANCE ESTIMATES
35
VOLUME RADIANCE ESTIMATES
β’ Spatial caches via BSP-tree: max. 2K photons per node:
β’ Similar 3D structure as PPG [Mueller2017]
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding36
VOLUME RADIANCE ESTIMATES
β’ Pre-processing step to fit estimates from photons (50M):
β’ EM-fitting of von Mises-Fisher mixtures (similar to [Vorba2014]βs GMMs)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding37
VON MISES-FISHER MIXTURE MODELS
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding38
β’ Spherical Distribution:
β’ Features (closed-form):
- Sampling
- Convolution
- Product
π π|Ξ =
πΎ
πππ£(π|ππ , π π)
RADIANCE ESTIMATES
β’ Incident Radiance Distribution β’ In-Scattered Radiance Distribution
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding39
INCIDENT RAD. TO IN-SCATTERED RAD. TRANSFORMATION
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding40
β’ Convolution between incident radiance πΏ and the phase function π
β
INCIDENT RADIANCE ESTIMATES
Ground truth (2K spp) Our estimates (VMM)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding41
IN-SCATTERED RADIANCE ESTIMATES
Ground truth (2K spp) Our estimates (VMM)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding42
GUIDED SAMPLING DECISIONS
43
DISTANCE SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding44
2. Scatter distance
1. Volume or surface decision
DISTANCE SAMPLING
β’ Standard distance PDF:
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding45
1+2. Event distance
πππ π‘π β¦ β π β¦ β ππ β¦
GUIDED PRODUCT DISTANCE SAMPLING
β’ Our guided PDF:
πππ§π£ β¦ =
π β¦ β ππ β¦ β ΰ·¨πΏπ(β¦ )
ΰ·¨πΏ(β¦ )Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding46
1+2. Event distance
Our estimates
NAΓVE TABULATED APPROACH
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding47
β’ NaΓ―ve tabulated approach:
β’ Step through the complete volume and build a tabulated PDF
β’ Inefficient (large scenes dense media):
β’ we always need to evaluated all bins first
OUR INCREMENTAL GUIDED PRODUCT DISTANCE SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding48
β’ Incremental approach:
β’ At each step make a local decision, if we scatter inside the bin
β’ We only need to step until the scattering event happens
Full CDF (30min) Our incremental (30min)
49
Spp: 548
Avg. steps: 18
Spp: 1140
Avg. steps: 4
45 min
No guiding
Spp: 960
relMSE: 1.342
45 min
No guiding Distance guiding
Spp: 960
relMSE: 1.342Spp: 424
relMSE: 0.901
45 min
No guiding (256 spp) Distance guiding (256 spp)
53
β’ Here, distance sampling is not the main cause of variance!!
DIRECTIONAL SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding54
3. Scatter direction
πππ₯π+1
β’ Standard PDF:
πππ π‘π β¦ β απ β¦
GUIDED PRODUCT DIRECTIONAL SAMPLING
β’ Our guided PDF:
πππ§π£ β¦ β απ β¦ β ΰ·¨πΏ(β¦ )
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding55
3. Scatter direction
πππ₯π+1 ππ+1
Our estimates
OUR GUIDED PRODUCT DIRECTIONAL SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding56
Incident radiance VMM
OUR GUIDED PRODUCT DIRECTIONAL SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding57
Incident radiance VMM Phase function VMM
OUR GUIDED PRODUCT DIRECTIONAL SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding58
Product sampling VMM
30 min
59
30 min No guiding
Spp: 2212
relMSE: 0.376
Directional guiding
Spp: 1756
relMSE: 0.048
30 min No guiding
Spp: 2212
relMSE: 0.376
Directional guiding Dist + Direct
Spp: 1756
relMSE: 0.048Spp: 1228
relMSE: 0.034
30 min No guiding
Spp: 2212
relMSE: 0.376
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
GUIDED RUSSIAN ROULETTE AND SPLITTING
4a. Termination 4b. Splitting Distance
Directional
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding64
GUIDED RUSSIAN ROULETTE AND SPLITTING
β’ Post-sampling compensation strategies:
β’ Identify, if we did a sub-optimal sampling decision
β’ Terminate: to increase performance
β’ Split: bound/reduce sample variance
4b. Splitting
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding65
4a. Termination
GUIDED RUSSIAN ROULETTE AND SPLITTING
β’ Path contribution: πΈ[π]β’ The expected contribution
if we continue the path
β’ Reference solution: πΌβ’ the final pixel value
π =πΈ π
πΌ
Path contribution
Reference solution
survival prob /
splitting factor
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding66
GUIDED RUSSIAN ROULETTE AND SPLITTING
β’ Path contribution: πΈ[π]β’ The expected contribution
if we continue the path
β’ Reference solution: πΌβ’ the final pixel value
π =πΈ π
πΌ= 1
Path contribution
Reference solution
survival prob /
splitting factor
Zero-Variance
Estimator
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding67
GUIDED RUSSIAN ROULETTE AND SPLITTING
β’ If π β€ 1: Russian Roulette
β’ Terminates low contributing paths
β’ Survival probability: π
β’ If π > 1: Splitting
β’ Splits an under sampled paths with
a potential high contribution (π times)
π =πΈ π
πΌ
Path contribution
Reference solution
survival prob /
splitting factor
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding68
ESTIMATED PATH CONTRIBUTION
β’ See course notes or paper for more details
ΰ·¨πΏπ
πΏ
πΈ πΏ = πβ²(πΏ) β ΰ·¨πΏπ π₯π , ππβ1
Path throughput: π(πΏ)/π(πΏ) In-scattered radiance estimate
π₯π
ππβ1
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding69
GUIDED RUSSIAN ROULETTE AND SPLITTING: PIXEL ESTIMATE
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding71
β’ Ray marched cache to integrate: π β ππ β ΰ·¨πΏπ
No RR
Spp: 468
relMSE: 0.454
45 min
No RR Guided RR
Spp: 468
relMSE: 0.454Spp: 1500
relMSE: 0.174
45 min
No RR Guided RR
Spp: 468
relMSE: 0.454Spp: 1500
relMSE: 0.174
45 min
+ Guided splitting
Spp: 1340
relMSE: 0.066
Guided RR + Guided splitting
Spp: 1500
relMSE: 0.174
Spp: 1340
relMSE: 0.06675
No guiding
Time: 60 min
Spp: 10644
relMSE: 11.58
Distance guiding
Time: 60 min
Spp: 4624
relMSE: 3.520
Distance + directional guiding
Time: 60 min
Spp: 4448
relMSE: 0.468
Distance + directional guiding + GRRS
Time: 60 min
Spp: 3796
relMSE: 0.321
OPEN PROBLEMS AND LIMITATIONSOpen Challenges to make it bullet proof
80
SPATIAL CACHE STRUCTURE
β’ Naive approach to define the resolution:
β’ Heuristic based on sample numbers
β’ Takes time or many samples to model/separate fine features (e.g. thin shafts or caustics)(PPG by [Mueller2017] has the same problem)
β’ Influences the performance of some sampling methods (e.g. distance)
β’ Ideal structure should adjust to the light transport characteristics
81
INSUFFICIENT CACHE SIZES
β’ Shared problem with other 3D caching based guiding approaches(e.g. [Vorba2014], [Mueller2017], β¦)
β’ By merging the samples of a spatial cache we blur the directional distribution
β’ Can lead to incorrect estimates of πΏ and πΏπ
82
Ground truth Our estimate Spatial avg. ground truth
PRODUCTION CHALLENGESHow can we get good guiding estimates?
83
VMM PHASE FUNCTION FITTING: PRE-PROCESSING STEP
β’ Pre-processing step:β’ Fitting a VMM for each phase function
β’ Fitting up to K = 4 componentsβ’ Details in the course notes
β’ Open Challenge:β’ Procedural phase functions or
procedural mixtures?
β’ How to deal with changing mean cosines (roughening/mollification)?
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding84
VMM PHASE FUNCTION FITTING: PRE-PROCESSING STEP
β’ Pre-processing step:β’ Fitting a VMM for each phase function
β’ Fitting up to K = 4 componentsβ’ Details in the course notes
β’ Open Challenge:β’ Procedural phase functions or
procedural mixtures?
β’ How to deal with changing mean cosines (roughening/mollification)?
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding85
FITTING/LEARNING THE INCIDENT RADIANCE MIXTURES
Pros:
β’ Photons directly represents the light transport
β’ Spatial distribution corresponds to important features (light shaft)
β’ Number of traced photons can be fixed
β’ No additional fitting during rendering
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding86
Photon-based pre-processing ([Herholz2019][Vorba2014])
FITTING/LEARNING THE INCIDENT RADIANCE MIXTURES
Cons:
β’ Pre-processing step:
β’ Long time to first render iteration
β’ Complex scenes need bidirectional learning:
β’ Ping-Pong style [Vorba2014]
β’ It is not the ideal solution for artists in production
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding87
Photon-based pre-processing ([Herholz2019][Vorba2014])
FITTING/LEARNING THE INCIDENT RADIANCE MIXTURES
Pros:
β’ First experiments show promising results
β’ No pre-processing
β’ Refines spatial and directional distributions in each iteration
β’ Sample data gets more reliable
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding88
Progressive learning (PPG-style [Mueller2017])
FITTING/LEARNING THE INCIDENT RADIANCE MIXTURES
Cons (open challenges):
β’ Sample count grows exponential:
β’ Memory and fitting time increases
β’ Shorter update rates ?
β’ Online fitting for mixtures?
β’ Spatial structure adapts slowly to LT:
β’ Important for distance sampling
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding89
Progressive learning (PPG-style [Mueller2017])
CONCLUSION
β’ Even approximate local sampling decisions lead to a good approximation of the globally optimal guiding distribution (and thus significantly reducing MC variance)
β’ Converges to optimal ZV estimator in the hypothetical limit (i.e., if the approximations were perfect)
β’ Solely based one guiding structure for alldecisions (incident radiance VMMs)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding90
THANK YOU
91
REFERENCES
β’ [Fong2017]: βProduction volume renderingβ
β’ [Novak2018]: βMonte Carlo methods for volumetric light transport simulationβ
β’ [Kulla2012]: βImportance sampling techniques for path tracing in participating mediaβ
β’ [Georgiev2012]: βimportance sampling of low-order volumetric scatteringβ
β’ [Krivanek2014]: βA zero-variance-based sampling scheme for Monte Carlo subsurface scatteringβ
β’ [Meng2016]: βImproving the Dwivedi sampling schemeβ
β’ [Vorba2014]: βOnline learning of parametric mixture models for light transport simulationβ
β’ [Vorba2016]: βAdjoint-driven Russian roulette and splitting in light transport simulationβ
β’ [Herholz2016]: βProduct importance sampling for light transport path guidingβ
β’ [Koerner2016]: βSubdivision next-event estimation for path-traced subsurface scatteringβ
β’ [Mueller2017]: βPractical path guiding for efficient light-transport simulationβ
β’ [Hoogenboom2008]: βZero-varianceMonte Carlo schemes revisitedβ
β’ [Pegoraro2008]:β Sequential Monte Carlo integration in low-anisotropy participating mediaβ
β’ [Bashford2012]: βA significance cache for accelerating global illuminationβ
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding92
ADJOINT ESTIMATE ACCURACY
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding93
COMPARING AGAINST EQUIANGULAR SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding94
BUMPY SPHERE
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding95
INCREMENTAL GUIDED DISTANCE SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding96
INCREMENTAL GUIDED DISTANCE SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding97
INCREMENTAL GUIDED DISTANCE SAMPLING
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding98
VMM PHASE FUNCTION FITTING: PRE-PROCESSING STEP
β’ Using up to K = 4 components
β’ Optimization Problem:
arg miπΞπ
π=1
π
βlog(π ππ, β¦ , π(ππ, Ξπ))2
βlog(π,π) = log π + π β log π + π
π = (1π β 4) β max(π1, β¦ , ππ)
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding99
VMM PHASE FUNCTION FITTING
β’ Manifold representation of the VMM parameters for an HG phase function model for different mean cosines
Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding100
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