SLIDE 1 Extended Path Integral Formulation for Volumetric Transport
- T. Hachisuka I. Georgiev W. Jarosz J. Křivánek D. Nowrouzezahrai
The University of Tokyo Solid Angle Dartmouth College Charles University in Prague McGill University
SLIDE 2 [Jarosz et al. 2011] [Pauly et al. 2000] [Jensen and Christensen 1998] [Křivánek et al. 2014]
SLIDE 3
SLIDE 4 Bidirectional path tracing [Pauly et al. 2000]
SLIDE 5 Volume photon mapping [Jensen and Christensen 1998]
SLIDE 6 Beam radiance estimate [Jarosz et al. 2008]
SLIDE 7 Photon beams [Jarosz et al. 2011]
SLIDE 8 Comprehensive theory [Jarosz et al. 2011] Point-Point Point-Beam Beam-Point Beam-Beam Point query Beam query Point data Beam data
SLIDE 9 Comprehensive theory [Jarosz et al. 2011] 3D blur 2D blur 1D blur
SLIDE 10 UPBP formulation
- Unified points, beams, and paths as sampling techniques for volumes
[Křivánek et al. 2014]
SLIDE 11 Dimensionality of paths
Path integral: Four vertices Density estimation: Five vertices Same path length
SLIDE 12
x
y
SLIDE 13 x
y ≡
Merge vertices
SLIDE 14 Consider all the paths which result in the same merged path
y0
SLIDE 15 Accept according to the probability of merging
y0 Prob[x ≡ y] = Z p(y0)dy0 y ≡
x
y0 y0
SLIDE 16 Prob[x ≡ y] = Z p(y0)dy0 y ≡
x
y0 y0 y0
Beam-Point 2D
SLIDE 17 Prob[x ≡ y] = Z p(y0)dy0 y ≡
x
y0 y0 y0
Beam-Beam 1D
SLIDE 18 Prob[x ≡ y] = Z p(y0)dy0 y ≡
x
y0 y0 y0
Beam-Beam 1D
UPBP formulation
- Three steps to match with BDPT
- 1. Merge subpaths
- 2. Consider all the paths which result in the same merged path
- 3. Accept the path with the probability of merging
Corresponds to contraction of density estimation path space
SLIDE 19
- Unified path integration and photon density estimation for surfaces
UPS/VCM formulation
[Hachisuka et al. 2012] [Georgiev et al. 2012]
SLIDE 20 Vertex Connection and Merging
- Contract the space of density estimation into the original path space
Path integration Photon density estimation
SLIDE 21 Vertex Connection and Merging
- Contract the space of density estimation into the original path space
Path integration Vertex merging
SLIDE 22 Unified Path Sampling
- Extend the original path space to include photon density estimation
Path integration Photon density estimation
SLIDE 23 Unified Path Sampling
- Extend the original path space to include photon density estimation
Vertex perturbation Photon density estimation
SLIDE 24 Differences
- VCM: precise for path integration, approximate for density estimation
- UPS: precise for density estimation, approximate for path integration
UPS VCM
SLIDE 25 Surfaces Volumes Contraction VCM UPBP Extension UPS Ours
(UVPS)
SLIDE 26
Unified Volumetric Path Sampling
SLIDE 27 Path integral formulation
Vertices are fully connected
SLIDE 28 Extended path integral formulation
Allow disconnected vertices
SLIDE 29 Extended path integral formulation
Blurring kernel as throughput of disconnected vertices
) = K3D(x, y)
SLIDE 30 Point-Point 3D
Precisely models photon density estimation
) = K3D(x, y)
SLIDE 31
3D blur to 2D blur
) = K3D(x, y)
SLIDE 32 3D blur to 2D blur
K2D(x, y) = K3D(x, y)δ(xt − tK)
Flatten a sphere into a disc
SLIDE 33
Beam-Point 2D
) = K2D(x, y)δ
SLIDE 34 Beam-Point 2D
Beam-point 2D = deterministic sampling of one distance
) = K2D(x, y)δ δ(y
t
− t
i n t
)
SLIDE 35
2D blur to 1D blur
) = K2D(x, y)δ
SLIDE 36 2D blur to 1D blur
K1D(x, y) = K2D(x, y)δ(xt − t0
K)
Flatten a disc into a line
SLIDE 37
Beam-Beam 1D
K1D(x, y) =
SLIDE 38 Beam-Beam 1D
δ(y
t
− t
i n t
) δ(xt − tint)
Beam-beam 1D = deterministic sampling of two distances
K1D(x, y) =
SLIDE 39
Beam-Beam 2D
SLIDE 40 Beam-Beam 2D
I n t e r s e c t i
i n t e r v a l
SLIDE 41 Beam-Beam 2D [Jarosz et al. 2011]
Integral over the intersection interval
Z f(x, y)K(x, y)dty
SLIDE 42 Beam-Beam 2D
p ( ty )
Stochastic sampling within the interval
SLIDE 43
Beam-Beam 2D
p ( ty ) δ(tx − t(yproj ))
SLIDE 44 Beam-Beam 2D
p ( ty ) δ(tx − t(yproj )) ) = K2D(x, y)δ
Same 2D kernel as beam-point 2D
SLIDE 45
Beam-Beam 3D
SLIDE 46
Beam-Beam 3D
SLIDE 47 Beam-Beam 3D [Jarosz et al. 2011]
Double integral over the intersection intervals (usually intractable)
dtx Z f(x, y)K(x, y)dty Z
SLIDE 48
Beam-Beam 3D
p ( ty )
SLIDE 49
Beam-Beam 3D
p ( ty )
SLIDE 50 Beam-Beam 3D
) = K3D(x, y)
Same 3D kernel as point-point 3D
p(tx) p ( ty )
SLIDE 51 Beam-Beam 3D
) = K3D(x, y) p ( ty ) p(tx)
Simple Monte Carlo path sampling (no longer intractable)
SLIDE 52 Beam-Beam 3D
Courtesy of Adrien Gruson
SLIDE 53
Beam-Point 3D
SLIDE 54
Beam-Point 3D
SLIDE 55 Beam-Point 3D
) = K3D(x, y) p ( ty )
Same 3D kernel as point-point 3D
SLIDE 56
Bidirectional path tracing
SLIDE 57 Bidirectional path tracing
Duplicate a vertex
δ(x δ(x x − y) p(y)=
SLIDE 58 Bidirectional path tracing
) = K3D(x, y) δ(x δ(x x − y) =
Delta kernel leads to the original path integral formulation
δ(x δ(x x − y) p(y)=
SLIDE 59 Biased bidirectional path tracing
Take disconnected vertices via blurring kernel
) = K3D(x, y) δ(x δ(x x − y) p(y)= δ(x δ(x x − y) =
SLIDE 60 Virtual perturbation
Approximate the implementation of biased BDPT by regular BDPT
p(y) ) = K3D(x, y) δ(x δ(x x − y) ≈ δ(x δ(x x − y) =
SLIDE 61 Conclusion
- Extension of the path space for volumetric light transport
- Better explains density estimation compared to merging
- Formulate beam as Monte Carlo distance sampling
- Enables a practical beam-beam 3D estimator
Fills a theoretical gap in the unified formulation for volumes
