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Radiative Transfer & Volume Path Tracingshz/courses/cs295/slides/15_radiative_transfer.pdfRadiative...
Transcript of Radiative Transfer & Volume Path Tracingshz/courses/cs295/slides/15_radiative_transfer.pdfRadiative...
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Radiative Transfer &Volume Path Tracing
CS295, Spring 2017
Shuang Zhao
Computer Science Department
University of California, Irvine
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Last Lecture
• Refraction & BSDFs• How light interacts with refractive interfaces (e.g.,
glass)
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Today’s Lecture
• Radiative transfer• The mathematical model to simulate light scattering
in participating media (e.g., smoke) and translucent materials (e.g., marble and skin)
• Volume path tracing (VPT)• A Monte Carlo solution to the radiative transfer
problem
• Similar to the normal PT from previous lectures
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Radiative Transfer
CS295: Realistic Image Synthesis
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Participating Media
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[Kutz et al. 2017]
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Translucent Materials
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[Gkioulekas et al. 2013]
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Subsurface Scattering
• Light enters a material and scatters around before eventually leaving or absorbed
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X Absorbed
Participating medium
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Subsurface Scattering
• Light enters a material and scatters around before eventually leaving or absorbed
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Scattered
Participating medium
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Subsurface Scattering
• Light enters a material and scatters around before eventually leaving or absorbed
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Participating medium
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Radiative Transfer
• A mathematical model describing how light interacts with participating media
• Originated in physics
• Now used in many areas• Astrophysics (light transport in space)
• Biomedicine (light transport in human tissue)
• Graphics
• Nuclear science & engineering (neutron transport)
• Remote sensing
• …
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Radiative Transfer Equation (RTE)
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In-scattering Out-scattering
& absorption
EmissionDifferential
radiance
In-scattering Out-scattering
& absorption
Emission
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Radiative Transfer Equation (RTE)
• The RTE is a first-order integro-differential equation
• For a participating medium in a volume with boundary , the RTE governs the radiance values inside this volume (i.e., for all )
• The boundary condition is the radiance field on the boundary (i.e., L(x, ω) for all )
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In-scattering Out-scattering
& absorption
Emission
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Radiative Transfer Equation (RTE)
• Differential radiance
• Scattering coefficient: ,Phase function: , a probability density over given x and ωi
• Extinction coefficient:
• Source term:
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In-scattering Out-scattering
& absorption
Emission
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Radiative Transfer Equation (RTE)
• σt controls how frequently light scatters and is also known as the optical density
• The ratio between σs and σt controls the fraction of radiant energy not being absorbed at each scattering
and is also known as the single-scattering albedo
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In-scattering Out-scattering
& absorption
Emission
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Radiative Transfer Equation (RTE)
• The phase function fp is usually parameterized as a function on the angle between ωi and ω. Namely,
• Example: the Henyey-Greenstein (HG) phase function with parameter -1 < g < 1:
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In-scattering Out-scattering
& absorption
Emission
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The Integral Form of the RTE
• It is desirable to rewrite the RTE as an integral equation• which can then be solved numerically using Monte Carlo
methods
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Integro-differential equation
Integral equation
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Integral Form of the RTE
• For any , let h(x, ω) denotes the minimal distance for the ray (x, -ω) to hit the boundary . In other words,
• When (x, -ω) never hits the boundary,• This can happen when the volume is infinite
• For any with , let
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Integral Form of the RTE
• For any , the attenuation between x and y is
• A line integral between x and y
• for all x and y
• For homogeneous media with ,
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Integral Form of the RTE
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In-scattering EmissionAttenuation
Attenuation Boundary cond.
(The second term vanishes when )
where
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Kernel Form of the RTE
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whereKernel function Source function
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Operator Form of the RTE
• Phase space:
• For any real-valued function g on Γ, define operator K as
where
• Then, the RTE becomes
• Similar to the RE!
• Yield Neumann series
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Volume Path Tracing
CS295: Realistic Image Synthesis
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Solving the RTE
• Given the similarity between the RTE and the RE, Monte Carlo solutions to the RE can be adapted to solve the RTE
• Volume path tracing
• Volume adjoint particle tracing
• Volume bidirectional path tracing
• …
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Volume Path Tracing
• Basic idea• Draw from
• Draw ωi from p(ωi)
• Evaluate L(r, ωi) recursively
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Known
where
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Free Distance Sampling
• is called the “free distance” and is sampled from
where
with λ0 being an arbitrary positive number
• p gives an exponential distribution with varying parameters
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Free Distance Sampling
• For all , it holds that
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Free Distance Sampling
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where
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Free Distance Sampling
• By applying Monte Carlo integration, we have
• Pseudocode:
• Draw from p
• If , return
• Otherwise, return
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Direction Sampling
• One extra integral remains:
• ωi can be sampled based on • In practice, is usually a valid
probability density on ωi, yielding
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Volume Path Tracing
radiance(x, ω):
compute h = h(x, ω) # using ray tracing
draw τ
if τ < h:
r = x – τ*ω
draw ωi
return σs(r)/σt(r)*radiance(r, ωi) + Q(r, ω)/σt(r)
else:
return boundaryRadiance(x – h*ω, ω)
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How to implement this?
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Free Distance Sampling Methods
• How to draw samples from this distribution?
• Homogeneous media•
• Let , then and
• In this case, can be drawn using the inversion method:
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Free Distance Sampling Methods
• Heterogeneous media• varies with x, causing to vary with
• p does not have a close-form expression in general
• Common sampling methods• Ray marching
• Delta tracking
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Ray Marching
• One can apply the inversion method by1. Drawing ξ from U(0, 1)
2. Finding satisfying• This is usually achieved numerically by iteratively
increasing with some fixed step size untilreaches ξ
• The step size is generally picked according to the underlying representation of σt(x) (e.g., voxel size)
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Ray Marching
• Pros• For each sample , can be obtained easily
• Cons• Biased (for any finite step size )
• Resolution dependent• needs to be picked based on the resolution of the
density (σt) field
• Slow for high-resolution density fields
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Delta Tracking
• Also known as Woodcock tracking
• Basic idea• Consider the medium to have homogeneous density
, and use it to draw free distances
• To compensate the fact that “phantom” densities have been introduced, the sampling process continues with probability
at each ri
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Delta Tracking
• Pseudocode:
deltaTracking(x, ω, σtmax)
compute h using ray tracing
τ = 0
while τ < h:
τ += -log(rand())/σtmax
r = x - τ*ω
if rand() < σt(r)/σtmax:
break
return τ
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Delta Tracking
• Pros• Unbiased
• Resolution independent
• Cons• For each sample , is not immediately
available
• Slow for density fields with widely varying σt values (i.e., σt
max >> σt(x) for many x)
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Volume Path Tracing (VPT)
radiance(x, ω):
compute h = h(x, ω)
draw τ
if τ < h:
r = x – τ*ω
draw ωi
return σs(r)/σt(r)*radiance(r, ωi) + Q(r, ω)/σt(r)
else:
return boundaryRadiance(x – h*ω, ω)
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This basic version can be improved
using techniques we have seen earlier:
• Russian roulette
• Next-event estimation
• Multiple importance sampling
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VPT with Next-Event Estimation
• The RTE implies that . Namely,
• By drawing from the aforementioned exponential distribution, we have
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where
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VPT with Next-Event Estimation
• The remaining integral is then split into two:
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Estimate recursively by
drawing ωi based on fp“indirect illumination”
Estimate directly by
area sampling or MIS
“direct illumination”
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VPT with Next-Event Estimation
• Pseudocode:scatteredRadiance(x, ω):
compute h = h(x, ω) # using ray tracing
draw τ
if τ < h:
r = x – τ*ω
rad = directIllumination(r, ω)
draw ωirad += scatteredRadiance(r, ωi)
return σs(r)/σt(r)*rad
else:
return 0
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Direct Illumination for VPT
• Recall that
• For non-emissive materials, Q vanishes and
• In this case,
• The area integral can be further restricted to the subset of where the boundary radiance is non-zero
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Change of measure
Boundary
radiance
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Direct Illumination for VPT
• Phase function sampling:
• Area sampling:
• The two strategies can be combined using MIS
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Draw ωi based on fp
Draw y from
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Next Lecture
• Metropolis light transport (MLT)• Applying the Metropolis-Hasting algorithm to
rendering
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