Path Integration for Light Transport in Volumes.
ABSTRACT Simulating the transport of light in volumes such as clouds or objects with subsurface scattering is computationally expensive. We describe an approximation to such transport using path integration. Unlike the more commonly used diffusion approximation, the path integration approach does not explicitly rely on the assumption that the material within the volume is dense. Instead, it assumes the phase function of the volume material is strongly forward scattering and uniform throughout the medium, an assumption that is often the case in nature. We show that this approach is useful for simulating subsurface scattering and scattering in clouds.

 [Show abstract] [Hide abstract]
ABSTRACT: Central to all Monte Carlobased rendering algorithms is the construction of light transport paths from the light sources to the eye. Existing rendering approaches sample path vertices incrementally when constructing these light transport paths. The resulting probability density is thus a product of the conditional densities of each local sampling step, constructed without explicit control over the form of the final joint distribution of the complete path. We analyze why current incremental construction schemes often lead to high variance in the presence of participating media, and reveal that such approaches are an unnecessary legacy inherited from traditional surfacebased rendering algorithms. We devise joint importance sampling of path vertices in participating media to construct paths that explicitly account for the product of all scattering and geometry terms along a sequence of vertices instead of just locally at a single vertex. This leads to a number of practical importance sampling routines to explicitly construct singleand doublescattering subpaths in anisotropicallyscattering media. We demonstrate the benefit of our new sampling techniques, integrating them into several pathbased rendering algorithms such as path tracing, bidirectional path tracing, and manylight methods. We also use our sampling routines to generalize deterministic shadow connections to connection subpaths consisting of two or three random decisions, to efficiently simulate higherorder multiple scattering. Our algorithms significantly reduce noise and increase performance in renderings with both isotropic and highly anisotropic, loworder scattering.ACM Transactions on Graphics (TOG). 11/2013; 32(6).  SourceAvailable from: isg.cs.tcd.ie
Conference Paper: Accelerated light propagation through participating media
[Show abstract] [Hide abstract]
ABSTRACT: Monte Carlo path tracing is a simple and effective way to solve the volume rendering equation. However, propagating light paths through participating media can be very costly because of the need to simulate potentially many scattering events. This paper presents a simple technique to accelerate path tracing of homogeneous participating media. We use a traditional path tracer for scattering near the surface but switch to a new approach for handling paths that penetrate far enough inside the medium. These paths are determined by sampling from a set of precomputed probability distributions, which avoids the need to simulate individual scattering events or perform ray intersection tests with the environment. We demonstrate cases where our approach leads to accurate and more efficient rendering of participating media, including subsurface scattering in translucent materials.Proceedings of the Sixth Eurographics / Ieee VGTC conference on Volume Graphics; 09/2007
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Eurographics Symposium on Rendering 2003, pp. 1–12
Per Christensen and Daniel CohenOr (Editors)
Path Integration for Light Transport in Volumes
Simon Premože,1Michael Ashikhmin2and Peter Shirley1
1Computer Science Department, University of Utah
2Computer Science Department, SUNY at Stony Brook
Abstract
Simulating thetransport of light involumessuch asclouds orobjectswithsubsurface scatteringiscomputationally
expensive. We describe an approximation tosuch transport using path integration. Unlike the more commonly used
diffusion approximation, the path integration approach does not explicitly rely on the assumption that the material
within the volume is dense. Instead, it assumes the phase function of the volume material is strongly forward
scattering and uniform throughout the medium, an assumption that is often the case in nature. We show that this
approach is useful for simulating subsurface scattering and scattering in clouds.
1. Introduction
The appearance of many materials (e.g., skin, fruits, snow,
clouds) cannot be described by a simple BRDFstyle re
flectance model. The main reason for this is volumetric scat
tering which manifests itself in important lighting effects for
materials4, 5, 19and scenes36. The radiative transfer equation
and propagation of light in a scattering medium have been
both analytically and numerically studied in astrophysics, at
mospheric optics, and more recently medical applications.
For most problems with nontrivial boundary conditions,
phase functions and initial conditions there are no analytic
solutions. Most solutions are based on the diffusion approx
imation which assumes that enough scattering events have
occurred for light to be uniformly scattered in all directions.
This approximation has proven useful for generating images
with subsurface scattering18. Monte Carlo methods are also
often used to compute radiative transport within a medium.
Although simple and powerful, these methods suffer from
slow convergence. Finite element methods are also used, but
they require large amounts of storage to capture discontinu
ities and strong directional light distributions.
Because the diffusion approximation is only appropriate
for dense uniform media19, there is a gap in the computer
graphics literaturewhen accurateapproximations aredesired
for sparse or nonuniform media. This paper attempts to fill
that gap using an alternative to the diffusion approxima
tion based on Feynman’s path integral approach to solving
quantum mechanics problems10. Path integral formulations
of physical processes have been used in physics to solve a
wide variety of problems including energy propagation in
random media38and transfer equation39.
The radiative transfer equation describing light propaga
tion can be viewed as a collection of paths taken by radia
tion as it travels through space. A path integral is an inte
gral over all such possible paths traveled by a photon. Ra
diative transfer is decomposed into a series of smaller prob
lemsformulated by the Green function propagator. Scattered
optical fields are described using the concept of an ensem
ble of effective optical paths of partial contributions. This
physical picture for treating light transport in multiple scat
tering media as a collection of most probable paths pro
vides insight into the light propagation in a medium. Unlike
the randomized approach to using paths of Metropolis Light
Transport45, path integral methods analytically find the most
important paths and develop analytical estimates based on
them. Once the most important path is found, the multiple
scattering contributions are only computed along the most
probable paths and the rest of the paths are dealt withimplic
itlyvia analytic integration of multiplescattering using well
known approximations. We therefore avoid computationally
expensive direct numerical simulation of multiple scattering
in the medium. We provide some intuition behind path in
tegrals and demonstrate that some useful results of the the
ory can be obtained without any heavy mathematical tools.
We discuss solutions to light propagation as a path integral
(a formal sum) and how they can be used for rendering arbi
traryscattering materialsand media. Because manycommon
phenomena cannot be described using only single scattering,
we discuss some observable consequences of multiple scat
c ? The Eurographics Association 2003.
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Premože et al. / Path Integration for Light Transport in Volumes
tering and how they can be exploited for rendering applica
tions.
We show results for inhomogeneous media that cannot
be achieved using the diffusion approximation. The restric
tions on our method are different; while sparse and inho
mogeneous media are allowed, we assume that the den
sity and therefore scattering coefficients vary smoothly, the
phase function is arbitrary but constant within the medium.
This expands the class of problems that can be attacked ef
ficiently. We note that this paper serves mainly as an intro
duction to path integral methods and many improvements in
efficiencyhavenot yetbeen explored inour implementation.
2. Background and Previous Work
Light Transport Approximations
There has been much work in approximating radiative trans
fer in arbitrary media in many fields including computer
graphics. Perez et al.30survey and classify global illumina
tion algorithms in participating media in detail. Pharr and
Hanrahan31and Premože35also provide an extensive list of
existing methods and background. Here we briefly review
recent methods proven practical and robust.
Jensen and Christensen16presented a two pass photon
density estimation method. This method is simple, robust
and efficient but suffers from large memory requirements to
store photons if the extent of the scene is large or the lighting
configuration is very complex. The method is very practical
and it has been used for many phenomena including smoke9,
fire26, stone8, and wet materials17proving its generality. On
the other hand, the photon map becomes rather inefficient in
highly scattering media18.
Veach and Guibas45presented a global illumination algo
rithm that found important paths, and then explored the path
space locally because it was likely that other important paths
would be nearby. Pauly et al.27extended the method for par
ticipating media and proposed suitable mutation strategies
for paths. Although extremely general and robust, as it could
handle any lighting condition and configuration, it still suf
fers from the classical Monte Carlo problems of noise and
slow convergence.
Stam37presented a solution to multiple scattering in non
homogeneous materialsbysolving thediffusionequation us
ing a multigrid method. Jensen et al.18introduced an ana
lytical solution to the diffusion approximation to multiple
scattering, which is especially applicable for translucent ma
terials that exhibit considerable subsurface light transport.
Their method relies on the assumption that the multiply
scattered light is nearly isotropic and cannot be easily ex
tended to inhomogeneous materials. Lensch et al.22imple
mented this method in graphics hardware and Jensen and
Buhler15extended this diffusion approximation to be com
putationally more efficient by precomputing and storing illu
mination in a hierarchical grid. Narasimhan and Nayar25de
scribed a physicallbased multiple scattering model for sim
ulating weather effects such as fog, haze, mist and rain.
An alternative description of light propagation was done
by Pharr and Hanrahan31who described a mathematical
framework for solving the scattering equation in the con
text of a variety of rendering problems and a numerical
solution in terms of Monte Carlo sampling. The scattering
equation describes all scattering events inside the object and
it does not depend on the incoming illumination. Unfortu
nately, there has been no other work exploiting this inter
esting paradigm of scattering objects and interactions be
tween objects on larger scales. Lafortune20also described
the global reflectance distribution function (GRDF) which
corresponds to the scattering equation idea.
Path Integral Methods
Path integral techniques and functional integration have
been widely used in statistical and quantum mechanics to
solve propagators for FokkerPlanck and Schrödinger equa
tions. The concept of photon paths has been wellknown
in the theory of energy propagation in random media38.
Tessendorf39, 40used the path integral approach to study the
propagation of light in weakly scattering media such as wa
ter. Perelman et al.28, 29described energy transport in a tur
bid medium using a quasiparticle Lagrangian, from which
the most probable paths could be found. Wilson and Wang46
constructed a Lagrangian through a turbid medium using
local path descriptors. Miller24also constructed a stochas
tic Lagrangian path integral representation for Green’s evo
lution operator. Constantinou and Demetrescu7showed the
equivalence of the path integral formulation and virtual rays.
Gross12studied multiple scattering of a wave in a system of
random and uncorrelated scattering particles. The path inte
gral methods have also been studied in optical tomography
and medical imaging. Neardiffusive scattering regimes are
very important for obtaining diagnostic information about
multilayer biological tissues where standard diffusion ap
proximation fails32, 33. Jacques and Wang14presented a ba
sic introduction to the path integral description of photon
transport anddiscussed classical pathfor describing themost
probable path of a photon.
3. Mathematical Preliminaries
MEDIUM PROPERTIES AND LIGHT PATHS. In an arbitrary
medium, the underlying optical properties depend on bulk
material properties such as density ρ(x), temperature T(x),
and the particle absorption and scattering crosssections, σa
and σs. Optical properties of the medium are then described
in terms of the scattering coefficient b(x) = σsρ(x), the ab
sorption coefficient a(x)=σaρ(x), the extinction coefficient
c(x) = a(x)+b(x), and the phase function P(x,? ω,? ω?). The
phase function P describes the probability density of light
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Premože et al. / Path Integration for Light Transport in Volumes
spatial spreading
angular spreading
temporal spreading
Figure 1: Scattering in a highly scattering medium. Original
radiance undergoes a series of scattering events that result in
angular, spatial and temporal spreading of the original radi
ance distribution.
x
? ω
a(x)
b(x)
c(x)
g
Q
P(x,? ω,? ω?)
G
? γP(s)
?β(s)
d(s)
s
S
?
?θ2?
Generic location in
Generic direction
Absorption coefficient at a point
Scattering coefficient at a point
Extinction coefficient at a point
Mean cosine of the scattering angle
Volume source distribution
Phase function
Propagator (Green function)
Light point path
Light propagation direction on the path
Displacement along the path
Distance along the path (arclength)
Arclength of the path? γ
Spatial variability
Mean square scattering angle
?
3
Table 1: Notations for important terms used in the paper.
coming from incident direction ? ω scattering into direction
? ω?upon scattering event at point x. The phase function is
normalized so that
cal settings only depends on the phase angle cosθ =? ω·? ω?.
The mean cosine g of the scattering angle is defined:
?4πP(? ω,? ω?)dω?= 1 and in most practi
g =
?
4πP(? ω,? ω?)(? ω·? ω?)dω?.
These optical properties are inherent, because they depend
solely on the medium and not on the structure of the in
coming light field. Upon entering the medium, incoming
light undergoes a series of scattering and absorption events
that modify both the directional structure of the incoming
light field and its intensity. As a result of multiple scattering
events, the original radiance distribution undergoes angular,
spatial and temporal spreading which result in different ra
diance distribution. Figure 1 shows spreading effects in an
arbitrary highly scattering medium. Table 1 summarizes im
portant terms and quantities used in the paper.
Consider aphoton that originatesat point x?withdirection
? ω?and traveling in a medium along a curved arclength pa
ω
←?
ω’
←?
x’
x
γP
←?
path
s’
β(s’)
←?
incident light
viewing direction
Figure 2: Transfer geometry in an enclosed scattering
medium. A photon originating at point x?with direction ? ω?
travels along curved path? γPof length s until it reaches the
final point x with direction ? ω.?β(s?) is the direction of prop
agation at arclength parameter s?on the curved path.
rameterized path? γP(s) until it reaches the final point x with
direction ? ω, (see Figure 2). This path results from an accu
mulated random walk of propagation directions governed by
scattering and absorption events along the path. Because the
path? γPis in general curved, its total length S is greater than
the distance between starting and ending points: x?−x ≤S.
The direction of propagation along the path is defined by
?β(s) =d? γP(s)
ds
. The path therefore satisfies the “twosided”
boundary conditions:
? γP(0) = x?
? γP(S) = x
?β(0) =? ω?
?β(S) =? ω.
(1)
The displacement relative to the point x?at distance s is
obtained by integrating?β:
s
d(s) =
?
0
?β(s?)ds?.
Light undergoes a series of scattering and absorption events
along the path. Note that if we ignore exact backscattering
which returns photons back into the same path, the intensity
of “original” light will be only diminishing because of these
events since inscattering will be due to photons traveling a
different path in the medium. Therefore, if we introduce the
effective attenuation τ which determines how much will the
light intensity be reduced along the length of the path, the
radiance L in the medium will be proportional to
L ∼ ∑
allpaths
e−τ(path).
Thepath integral formulation of light transport isessentially
a mathematically rigorous expression of this simple idea.
RADIATIVE TRANSFER. Light transport in arbitrary me
dia is described by the radiative transport equation2, 13:
(? ω·∇+c(x))L(x,? ω) =
b(x)
?
4πP(? ω,? ω?)L(x,? ω?)dω?+Q(x,? ω),
(2)
where Q(x,? ω) is the source term. In computer graphics, the
source terms Q(x,? ω) is often due to light emitted by the
c ? The Eurographics Association 2003.
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Premože et al. / Path Integration for Light Transport in Volumes
medium itself (Le(x,? ω)). It is often convenient to split the
total radiance within the medium into components and write
it as the sum of unscattered (direct) radiance Lun, the emis
sion Leand the scattered radiance Lsc:
L(x,? ω) = Lun(x,? ω)+Lsc(x,? ω)+Le(x,? ω).
Here Lunis the radiance which intensity has been reduced
due to absorption and outscattering along the length S. Lscis
the radiance that has undergone a series of scattering events
andfinallyscatteredintoasmallcone around theobservation
direction? ω.
PROPAGATOR FOR RADIATIVE TRANSFER. The solu
tion of equation 2 is the limit of the corresponding solution
for thetimedependent problem whereradianceL variesover
time t. It is convenient to express time in units of length s as
t = s/v where v is the speed of light in the medium. With
this notation, the timedependent radiative transfer (TDRT)
equation is
?∂
b(x)
∂s+? ω·∇+c(x)
?
L(s,x,? ω) =
?
4πP(? ω,? ω?)L(s,x,? ω?)dω?+Q(s,x,? ω).
(3)
The solution of the TDRT equation can be formulated in
terms of the Green evolution operator G which is also called
the propagator, the Green function, or the point spread func
tion (PSF). It is defined as the solution of homogeneous
equation
?∂
∂s+? ω·∇+c(x)
?
b(x)
G(s,x,? ω,x?,? ω?) =
?
4πP(? ω,? ω??)G(s,x,? ω??,x?,? ω?)dω??,
(4)
with the initial condition
G(s = 0,x,? ω,x?,? ω?) = δ(x−x?)δ(? ω−? ω?).
Physically, the Green propagator G(s,x,? ω,x?,? ω?) represents
the radiance at point x in direction ? ω at time s due to light
emitted at time zero by a point directional light source lo
cated at x?shining in direction ? ω?. For example, in the ab
sence of scattering (b =0), the solution for the propagator G
is
G(s,x,? ω,x?,? ω?) =
δ(x−? ωs−x?)δ(? ω−? ω?)exp
?
−
?
s
0
a(x−? ω(s−s?))ds?
?
.
Here the light travels in a straight line and is attenuated by
the absorption coefficient a(x). One can see that in this case,
the formulation using the propagator is generally equivalent
to simple raytracing. The Green propagator G(s,x,? ω,x?,? ω?)
represents the angular distribution and density of rays at
point x in direction ? ω generated by point x?in direction ? ω?
and is therefore equivalent to raytracing.
The concept of the Green function has been used in neu
tron transport theory3providing an approach to solving ra
diative transfer problems with arbitrary boundary conditions
by first finding solution G of the elementary transfer prob
lems stated above and then forming the complete solution of
equation 3 by using the superposition principle, i.e. integrat
ing G with the initial radiance distribution:
L(s,x,? ω) =
?
G(s,x−x?,? ω,? ω?)L0(x?,? ω?)dx?dω?,
where G(s,x − x?,? ω,? ω?) is the evolution operator and
L0(x?,? ω?) is the initial distribution. Conceptually, the notion
of theevolution operator isequivalent totheidea of radiative
process introduced by Preisendorfer34. Note that the bound
ary value problem for the Green function is actually ad
joint to the radiative transfer problem, but due to reciprocity
(timereversal invariance) we can solve the light transport
problem by reversing the direction of light propagation1.
PATH INTEGRALS. Consider the problem of finding the
probability that a point particle at the initial position xiand
time tiwill reach a final position xfand time tf, i.e. will
“propagate” (in the quantum mechanical sense) from xito
xf. This quantity can be expressed using quantum mechan
ical propagator G(tf−ti,xf,xi) which for this problem is
the solution of the Schrödinger equation subject to appropri
ate initial conditions. If the problem is broken down into a
series of shorter time steps with propagators G(t,x,x?), the
full propagator is expressed as a superposition of “smaller”
Green function:
G(t,xi,xf) =
lim
N→∞
The object on the righthand side is called a path inte
gral. Feynman formulated quantum mechanics using path
integrals10and showed that with an appropriate definition
of differential measure
space one can write for the quantum mechanical amplitude
of a propagating point particle
?
···
?
G(t/N,xi,x1)···G(t/N,xN−1,xf)dx1···dxN−1.
? x in the infinitedimensional path
?xf,tfxi,ti? =
?
? x(t)eiA[x(t)]/¯ h
where the weight factor contains the classical action
A(xi,xf,tf−ti) for each path. The classical action is the
integral of the Lagrangian over the time the trajectory tra
verses. The path taken by a classical trajectorycould be none
other than the one that minimizes the classical action.
RADIATIVE TRANSFER AS A SUM OVER PATHS. The
path integral (PI) approach provides a particular way to ex
press the propagator G(s,x,? ω,x?,? ω?). It is based on the sim
ple observation that the full process of energy transfer from
one point to another can be thought of as a sum over transfer
events taking place along many different paths connecting
points x and x?, each subjected to boundary conditions re
stricting path directions at these points to ? ω and ? ω?, respec
tively. The full propagator is then just an integral of individ
ual path contribution over all such paths. This object is the
path integral defined above. Note that this is different than
the terminology used in Veach and Guibas45.
c ? The Eurographics Association 2003.
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Premože et al. / Path Integration for Light Transport in Volumes
Because the integration is performed over the infinite
dimensional path space using the not very intuitive differ
ential measure defined for it, the mathematics of path inte
grals is quite complex10, 21. Tessendorf41derived a path inte
gral expression for the propagator G in homogeneous mate
rials. Interested readers are referred to his further work42, 43
for detailed derivations of the path integral formulation. We
present here a more intuitive approach sufficient for our pur
poses and state results from the literature without deriva
tions.
4. Practical Path Integrals
4.1. The Most Probable Path
We first attempt to formulate conditions to find the most
probable path (MPP) among all possible ones which sat
isfy the necessary boundary conditions. We will then as
sume that the full propagator can be sufficiently approx
imated by accounting only for contributions from paths
“close to” this special one. Formally, this approach corre
sponds to evaluating the path integral using WKB (Wentzel
KramersBrillouin) expansion6, which is a wellestablished
method from perturbation theory. In practice, this means that
once the MPP (or its approximation) is identified, we simply
consider its blurred contribution which approximates the ef
fect of surrounding paths. The details of how this is done
are presented in Section 5 below, but one can already see
some potential advantages of this approach over more tra
ditional methods. Compared with MonteCarlo techniques
which perform statistical sampling of random paths, the path
integral approach attempts to find the most important ones
directly and can therefore be considered an extreme form
of variance reduction. The PI formulation also does not ex
plicitly rely on further assumptions about the character of
radiance distribution which are needed, for example, in dif
fusion approximation11. However, if warranted, we can take
direct computational advantage of the fact that the radiance
distribution becomes more and more blurred as one travels
along the MPP.
Consider an inhomogeneous medium with position
dependent scattering and absorption coefficients b(x) and
a(x). Let? γP(s) be some arclength parameterized path from
x?to x. The probability density for a photon to reach x while
traveling exactly along this path and not by any other possi
ble one can be written as a product of two terms: the prob
ability density of experiencing a series of scattering events
which results in this particular path taken and the probability
that the photon will “stay alive” at the end of its journey (i.e.
not be absorbed along the path). We implicitly assume here
that all absorption and scattering events are independent.
Thisallowsthesecond termtobeexpressed directlyfromthe
radiative transfer equation with no scattering which is writ
ten along the path in a trivial form dL(s)/ds = −a(x)L(s)
with initial conditions L(0) = 1 where we use loose nota
tions L(s) for the radiance along the path. In this case, the
fraction of initial radiance which reaches the end of the path
is exactly the probability of a photon not being absorbed.
Therefore, using the solution of the equation above we can
write
p(not absorbed) = exp
?
−
?
S
0
a( ? γP(s?))ds?
?
,
where the argument to the integral is the absorption coeffi
cient along the path and S is total length of the path.
To deal with the scattering term, we adopt an approach
similar to Wilson and Wang46which is inspired by physics,
rather than attempting to follow the more mathematically
rigorous treatment of Tessendorf42. Homogeneous media
have been considered so far in the literature and in many
cases the rigorous mathematical procedures will break down
if scattering/absorption coefficients are allowed to vary
across the medium.†A notable exception is the path con
struction using random walks by Pauly et al.27. We first split
the path into a number of straight line segments connect
ing positions of scattering events that change photon prop
agation direction. Then the probability density of a particu
lar path is a product of probability densities that individual
scattering events will change the propagation direction “just
right” to steer the photon all the way along the path. Using
the phase function P(∆θ) where ∆θ is the change of propa
gation direction, we get for the total probability density by
writing :
p(path shape)
? x = ∏
i∈ ? γP(s)
P(∆θi)dωi,
(5)
where individual factors correspond to the different (ith)
scattering event along the path. Differential solid angles will
eventually become a part of the full differential measure
in the path space and are not of interest for finding the MPP
since they do not affect relative probabilities of different
paths. We now make further assumption that the phase func
tion is strongly peaked in the forward direction which is true
for many important media5. In thiscase, P(θ)can beapprox
imated with the first terms of its Taylor expansion as 1−αθ2
(we drop irrelevant constants here). It can be shown that if
we want to maintain the mean cosine of the scattering an
gle g, α has to be set to α = 1/4(1−g) = 1/(2?θ2?) where
?θ2? is the mean square scattering angle. Note that although
the phase function is strongly forward peaked, this does not
mean that the path itself has to deviate by only a small angle
from its original propagation direction, which is an assump
tion often used to simplify derivations. We would also like
to treat path? γP(s) as a continuous object by taking a limit
in equation 5 as scattering events occur often enough along
the path. Each scattering event changes the propagation di
rection by a small amount δθ. In the case of forward peaked
phase functions, only such scattering events generally have
? x
†For example, the Fourier transformed RT equation will have a
much more complex form in this case, containing convolution over
frequencies.
c ? The Eurographics Association 2003.
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Premože et al. / Path Integration for Light Transport in Volumes
significant probability. The expression of interest is
∏
i∈ ? γP(s)
P(∆θi) ≈ ∏
i∈ ? γP(s)
?1−α(δθi)2?=
∏
i∈ ? γP(s)
?
1−αδsi
??δθi
δsi
?2
δsi
??
,
(6)
where we introduced the lengths of path elements between
scattering events δsi. Note that by construction these seg
ments are physically constrained to be of finite length of the
order of 1/b( ? γP(si)) and can not be simply treated as in
finitely short in the limit since no scattering can physically
occur on an infinitesimal path interval. We also intentionally
rearranged the last expression to highlight its part in square
brackets which indeed can be treated as a full differential of
some function in our approximation. Taking a logarithm of
equation 6, using Taylor series, and replacing one of the δsi
with its physical value we obtain
?
∑
i∈ ? γP(s)
ln1−αδsi
??δθi
δsi
?2
δsi
??
≈
∑
i∈ ? γP(s)
−α
b( ? γP(si))
??δθi
δsi
?2
δsi
?
.
If the scattering scale δs = 1/b is much smaller than the
macroscopic scale of the path, the expression in square
brackets can be taken to give dθ/ds2ds in this limit and the
discrete sum will become an integral along the path. Taking
the exponent, we get:
?
p(path shape) ∼ exp
−
?
s
0
α
b( ? γP(s?))
????
dθ
ds?
????
2
ds?
?
.
We can write this because of a general property of function
limits: if the limit exists, its value does not depend on the
specific way to take the limit and therefore the particular
subdivision of the path we use does not affect the final re
sult. It is comforting to note this expression gives exactly the
result of rigorous treatment of Tessendorf39when applied to
a homogeneous medium. The full expression for path prob
ability density is now
?
p(path) ∼ exp
−
?
s
0
?
a( ? γP(s?))+
α
b( ? γP(s?))
????
dθ
ds?
????
2?
ds?
?
=
exp(−A(path)),
(7)
where A is analogous to classical action along the path.
To find the MPP, we need to determine the path which
minimizes the effective attenuation in equation 7. It is not
possible to write an analytic expression for arbitrary func
tions a(x) and b(x), but some important general trends can
nevertheless be established. Since the expression under the
integral is nonnegative, equation 7 favors shortest paths
with low curvature dθ/ds. For example, if we are interested
intheMPPconnecting twopoints without specifying anyex
tra conditions, this will be the straight line connecting these
points. If a path has to turn to satisfy boundary conditions, it
will tend to curve more in regions with high scattering coef
ficient b(x). Finally, for a homogeneous medium, the MPP
will be completely determined by the applied boundary con
ditions. Explicit expressions can be obtained in some cases
using the standard EulerLagrange minimization procedure
applied to the integral in equation 742and we will use such
results below.
4.2. Spatial Variability
Opticalpropertiesinascatteringmediumcanvaryarbitrarily
spatially. A spatial variability in a medium can be measured
by the number of scattering events that occur along the path
? γP:
s
b?? γP(s?)?ds?.
In uniform media, the spatial variability is just a constant
multiple of the distance s: ? = bs. Given the spatial variabil
ity ? of a path in inhomogeneous media, we can “invert” this
equation and write displacement of the ray from its origin x?
with respect to ? as:
?(s) =
?
0
d(?) =
?
?
0
1
b(x?+d(??))
?β(??)d??.
(8)
This expression suggests a practical way of constructing ac
tual path in inhomogeneous medium by stitching together
straight line segments withlengths given by the local scatter
ing coefficient. The only extra information we need is local
propagation direction which is discussed in the next subsec
tion.
4.3. Finding the MPP
Tessendorf43described the propagation direction?β(?) with
Euler rotation angles and satisfying boundary conditions
(equation 2) using the Fourier series expansion of the an
gles. We follow a simplified version of his formulation to
construct the stationary path?β0which is the path that mini
mizes attenuation along its length. We mentioned before that
due to reciprocity we can construct the path by reversing the
direction of light propagation1. Through the rest of the paper
we take advantage of this property and construct the MPP
starting from the initial (viewing) direction ? ω and ending in
the final (light) direction ? ω?, although the light is actually
moving in the opposite direction.
Let R be a rotation matrix that rotates initial direction ? ω
to the zaxis vector? z = (0 0 1)T. If the final direction ? ω?is
written in spherical coordinates as
cos(θ)
? ω?=
sin(θ)cos(φ)
sin(θ)sin(φ)
,
then the stationary path?β0(s,? z,? ω?) that uniformly rotates? ω?
c ? The Eurographics Association 2003.
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Premože et al. / Path Integration for Light Transport in Volumes
?
d?
x
? ω
? ω?
? ω
? ω
x
? ω
x?
? ω?
θ(?) = 0
θ(?) = π/8
θ(?) = π/4
Figure 3: Construction of the Most Probable Path in a ho
mogeneous medium. The path is constructed by stitching
together piecewise linear path segments and integrating the
propagation direction?β. The propagation direction of path
segments between starting point x and ending point x?uni
formly rotates from the initial direction ? ω to the final direc
tion? ω?, therefore matching the boundary conditions.
into? z over the physical path length S is
?β0(s,? z,? ω?) =
sin(θ(s))cos(φ)
sin(θ(s))sin(φ)
cos(θ(s))
,
(9)
where
θ(s) = θ−
?s
S
?
θ.
Note that we still need to apply rotation R−1that rotates? z
back to? ω such that:
?β0(s,? ω,? ω?) = R−1?β0(s,? z,R? ω?).
One can show that such a “uniformly turning” pathis exactly
the MPP among all paths of fixed length for the homoge
neous medium, i.e. it minimizes effective attenuation given
by equation 7.
(10)
If the stationary path?β0needs to be constructed in inho
mogeneous medium with respect to path spatial variability
? then we will simply replace θ(s) in equation 9 by θ(??)
where the light path? γPis now parameterized according to
the number of scattering events and not the physical path
distance:
θ(??) = θ−
???
?
?
θ.
(11)
Given the starting point x, initial direction ? ω, and final
direction ? ω?, the MPP is constructed by integrating the ve
locity function?β0. The locations on the MPP are found in
terms of displacements d(?) (equation 8) from the starting
point x. The displacements d(?) are defined implicitly and at
first it appears that the spatial variability ? along the entire
path is needed. Note, however, that each displacement d(?)
along the path only depends on the value of ? up to this point
and not on the spatial variability of the entire path, allowing
an “incremental” construction of the path.
The path is constructed by stitching together piecewise
linear path segments. We march along the path in steps of
size d? (step size in spatial variability not in distance). This
step size d? is arbitrary and is analogous to selecting a step
size ds for direct lighting computation when marching along
straight line. A sensible value for d? can also be estimated
from the optical properties and density of the volume. At ev
ery step we first obtain the propagation direction?β(?) along
the existing portion of the path using equations 11 and 10
with the accumulated total spatial variability ?accon the path
so far substituted for ?. We then use equation 8 to compute
the displacement point d(?acc) along the path.Finding the
next point on the path involves increasing the spatial vari
ability of the path so far by d? (?acc= ?acc+d?) and finding
corresponding displacement d(?acc) from the initial point x.
Note that the full path is reconstructed from scratch at each
step. So, for every sampling point on the curved path, the
MPP is reconstructed from the initial point x and not just
from the previous sampling point on the path.
Note that at the starting point x the spatial variability ? is
zero (no scattering events encountered so far) which there
fore causes the first path segment to be in the initial direc
tion ? ω. Similarly, the very last segment of the path will by
construction be in the direction ? ω?, matching the boundary
conditions. The propagation direction of path segments be
tween starting point x and ending point x?uniformly (in ?)
rotates from the initial direction ? ω to the final direction ? ω?.
The segment length in physical space depends on the scat
tering coefficient b at previous displacement point d(?).
The total spatial variability ? along the path is just the
sum of spatial variabilities d? along each segment on the
path. This is again analogous to computing the physical
length s by summing segments ds along the straight line.
As expected, the stationary path?β0is relatively flat in re
gions where scattering coefficient b(x) is small and highly
curved where density is high. Figure 3 illustrates construc
tion of the most important path using the described method.
Quaternions could provide an alternative and more rigorous
approach to uniform rotations.
4.4. Multiple Scattering Phase Function
Tessendorf and Wasson44observe that the width of the phase
function grows with the number of scattering events ?. When
the number of scattering events ? grows large, the probabil
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Premože et al. / Path Integration for Light Transport in Volumes
ity of scattering in any direction is equal and the phase func
tion essentially becomes isotropic. It follows from the WKB
approximation that the average scattering angle ΘMSafter ?
scattering events is:
?ΘMS? =
?
1−exp(−?).
Tessendorf and Wasson44also introduce the idea of the
multiplyscattered phase function which is defined as the
probability of light scattering through an angle θ after ? scat
tering events:
?
PMS(θ,?) =1
NP
θ
?
?
1−exp(−?)
constant
?
,
(12)
where
?4πPMS(θ,?)dω = 1. Intuitively, equation 12 says that
the probability of scattering into an arbitrary angle increases
with the number of scattering events. When the number of
scattering events is large, the phase function PMSbecomes
isotropic. Note that equation 12 holds for arbitrary phase
function.
N
isthenormalizationsuchthat
4.5. WKB Approximation For Multiple Scattering
Tessendorf41derived a path integral expression for the prop
agator G in homogeneous materials:
G(s,x,? ω,x?,? ω?) =
? [d?β][dp]δ(?β(0)−? ω?)δ(?β(s)−? ω)
s
ds??β(s?)
exp(−cs)
?
δ
?
x−
?
0
?
exp
?
b
?
s
0
ds?Z(? p(s))
exp
?
i
?
s
0
ds?? p(s?)·d?β(s?)
ds?
?
,
(13)
where Z(? p) is the Fourier transform of the normalized
Gaussian phase function PGand ? p is the Fourier transform
variable. Interested readers are referred to Tessendorf42, 43
for detailed derivations of the path integral formulation. If
the phase function P is not spatially varying and the sin
gle scattering albedo ω0=b
then the propagator G can be extended to inhomogeneous
materials44, but no details are provided. The WKB approx
imation done by Tessendorf42, 43consists of approximating
the path integral in equation 13 by finding the most probable
(stationary) path and integrating all paths fluctuating around
the MPP. The path integration approximation they obtained
can be expressed in terms of the Green propagator G:
?
The propagator in equation 14 is valid for spatially varying
materials as long as the density of the material is smoothly
varying and the phase function is uniform within the vol
ume. While the WKB approximation was obtained using
the assumption of strongly forward peaked phase functions,
there is no other restriction on the specific phase function
cis smoothly varying in space
G(s,x,? ω,x?,? ω?) ∼ exp
−
?
s
0
ds?c( ? γP(s?))
??
e?−1
?
PMS(θ,?)
(14)
form. In practice, the direct and indirect radiance compo
nents are computed separately and if the phase function al
lows backscattering, it will be computed explicitly in the di
rect radiance component computation.
5. Algorithm
As discussed in section 3, the radiance L received from di
rection ? ω at the observation point x is composed of three
components:
L(x,? ω) = Lun(x,? ω)+Lsc(x,? ω)+Le(x,? ω).
The unscattered component Liun(x,? ω) represents the amount
of unscattered light due to the ithlight source:
Li
un(x,? ω) = Li
light(x,? ω)exp
?
−
?
∞
0
c(x−? ωs)ds
?
,
(15)
where Li
the ithlight source. In practice, the unscattered radiance Lun
and the emitted radiance Leare also the source for the scat
tered radiance13:
light(x,? ω) is the radiant exitance in direction? ω from
QS(x,? ω) = a(x)Le(x,? ω)+b(x)
allNlights
∑
i
P(? ω,? ωi)Li
un(x,? ωi). (16)
To compute the total radiance L in the medium, all exter
nal and internal sources of radiation need to be propagated
through the volume to the point x on the volume that the
camera is looking at. The evolution operator G from equa
tion 4 propagates all energy to a given observation point and
direction. We use the propagator G from equation 14 for
the rendering algorithm described in the next subsection. We
also use results from Tessendorf and Wasson44to develop a
rendering algorithm using the path integration formulation.
Monte Carlo Ray Tracing
Monte Carlo ray tracing is an accurate algorithm for solving
the radiative transfer equation in arbitrary media. We use it
for comparison to evaluate our approximations. We march
through the medium in direction ? ω sampling points along
the ray. The light from previous step is attenuated and the
light that is inscattered into the viewing direction ? ω is gath
ered. The inscattered light is collected recursively for each
inscattered ray:
Ln+1(x,? ω) =
allNlights
∑
l
4π
N
∑
i=1
exp(−c(x)∆x)Ln(x+? ω∆x,? ω)
Lun(x,? ω?
l)P(x,? ω?
l,? ω)b(x)∆x+
M
Lsc(x,? ωi)P(x,? ωi,? ω)b(x)∆x+
where M is the number of directional samples taken. While
Monte Carlo ray tracing is robust and powerful, it is also
slow because of the large number of rays needed.
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Premože et al. / Path Integration for Light Transport in Volumes
5.1. Path Integration Approximation
Our algorithm exploits the WKB approximation from pre
vious section. The WKB approximation computes the
multiplyscattered light by finding the most probable path
and then analytically integrates scattered radiance along this
path and some neighborhood around this path including
quadratic fluctuations around this path (equation 14).
The approximate radiance is then the sum of the singly
scattered (“direct”) and multiply scattered components44:
Lssc(x,? ω) =
∞
?
0
QS(x−s? ω,? ω))exp
?
−
?
s
0
ds?c(x−(s−s?)? ω)
?
ds,
(17)
and
Lmsc(x,? ω) =
nw
∑
i=1
where
?
4πdω??
?
∞
0
dsi{QS(x??,? ω??)×w[x??]×PMS(θ,?)},
(18)
w[x??] = exp(−c(x??)?/b(x??))×(exp(?)−1),
and x??is an intermediate point on the curved path: x??=
x+d(?). The first term in equation 19 is the intensity of the
ray at a sampling point along the path while the second term
is the number of rays that propagate along this path. Note
that the number of rays grows exponentially with the dis
tance. Brute force Monte Carlo ray tracing algorithms would
have trouble with this due to enormous number of poten
tially spawned rays. The path integration formulation does
not have problems with this exponential growth, because the
rays are dealt with via analytic integration.
(19)
As we saw earlier, the MPP we construct is not uniquely
defined by the boundary conditions. The total length of the
path (either arclength S or total spatial variability ? ) remains
a free parameter since it is not possible to find this parameter
without preprocessing or involved analysis. To find the true
MPP one would need to perform a search among paths with
different lengths and choose the one with minimal effective
attenuation. This process can be very expensive for inhomo
geneous media. Fortunately, inpractice it issufficient tocon
struct a small number nwof “quasiMPPs” with different to
tal lengths (spatial variabilities) using the process described
above and sum their contributions, as shown by equation 18.
Based on the density and optical properties of the volume,
spatial variability of the path is estimated. For example, a
raymarching can be performed along three axes and diago
nals to estimate the total length of the path parameter. These
estimates are then used as a free parameter for the MPP gen
eration. Further investigation is needed to estimate the total
path length more robustly and systematically. However, we
found that naïve approach worked well in practice without
spending extra time probing the volume.
The contributions of diffuse (or indirect) outside light
Algorithm 1 Rendering Algorithm Using WKB Approxi
mation
// Preprocessing for efficiency purposes
for each light source S
Compute a light attenuation through the media by ray march
ing
Store light attenuation in a spatial data structure (e.g. octree,
deep shadow map)
end for
// Unscattered (direct) radiance Lssccomputation
Given starting point x and viewing direction ? ω
for each sample point along the ray in direction ? ω
Compute the unscattered radiance contribution from equa
tion 17
end for
// Scattered (indirect) radiance Lmsccomputation
// using WKB approximation
Choose number of important paths nw
Choose the sampling length d?
for each important path?βnw
for each sample point along the path?βnw
Find ds and a point x?on the path from equation 8
for each set of sampling directions Ω
Compute the scattered radiance contribution from equa
tion 18
end for
end for
Add diffuse light source contributions (equation 20)
end for
sources are included as a radiance source at the boundary:
L(x,? ω) = exp{−
?
sboundary
0
ds?c(x−? ω(s−s?))}Ld(? ω)+
+∑
nw
?
4πdω??wb[x?]PMS(θ,?b)Ld(? ω??),
(20)
where Ldis the diffuse radiance in a given direction. Sim
ilarly to the weight w in equation 19 for the collimated
beam13, the weight for the diffuse light source contribution
is:
wb[x?] = exp(−c?b/b)×(exp(?b)−1),
and ?bis the number of scattering events when the path
reached the boundary x?of the volume.
(21)
The rendering algorithm can be implemented using an ex
isting ray marching routine with little difficulty. For every
viewing ray? ω, we first find an intersection with the volume.
This is starting point x. The direct component (equation 17)
is computed using standard ray marching. At every sample
point, the source term (direct radiance and emission) QSis
computed and the intensity is then exponentially attenuated.
To speed up computation we precompute light attenuation in
the volume and store it at all possible depths. For this we use
asimplifiedversionof adeepshadow map23.Fromeachlight
source we compute attenuation along the ray and store it at
every sampling point. This speeds up rendering, because we
c ? The Eurographics Association 2003.
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Premože et al. / Path Integration for Light Transport in Volumes
only pay the price once and the rest is a table lookup. There
fore, the source term QSis just a lookup in the spatial data
structure storing the precomputed attenuation for every light
source. The indirect component (equation 18) is computed
by constructing the most probable path (or some number
of them) using the procedure presented in Section 4.3. We
march along the curved path in steps such that the number of
scattering events is uniform. At every sampling point along
the curved path, we compute the contribution from many di
rections over the entire sphere. Sampling directions are cho
sen based on viewing and lighting directions (i.e. sample
more densely in directions close to the viewing and lighting
directions). We have a fixed number of directions we sample
at every point. One can use a more sophisticated metric to
determine how many directions to sample. In our implemen
tation, the number of sampling directions is a parameter to
the rendering algorithm. Although many directions are sam
pled at every sampling point along the path, this consists just
of a few operations since no rays are actually spawned. This
is unlike in a traditional Monte Carlo algorithm where addi
tional rays are created at every sampling point which causes
exponential growth in number of rays. Similarly to comput
ing the direct component, the source term QScan be just
looked up in the spatial data structure. Once the boundary
x?of the volume is reached, the diffuse light contributions
are explicitly added (equation 20). The rendering algorithm
is summarized in Algorithm 1.
6. Results and Discussion
The algorithm from the previous section was implemented
in C++ (compiled with g++ compiler) as an extension to the
traditional Monte Carlo raytracer. All examples were ren
dered on a Pentium IV 1.7 Ghz with 512 Mb of memory
running Linux. The results of the algorithm for a uniform
density volume is shown in Figure 4. These images used the
scattering parameters for skim milk from Jensen et al.18and
HenyeyGreenstein phase function with g=0.9. The images
are 512 by 384 and used 9 samples per pixel. The runtimes
were 25 minutes for the Monte Carlo method and 8 minutes
for the PI method. Note that the path integration solution
is slightly darker than the bruteforce Monte Carlo solution.
This is because path integration assumes that paths near the
MPP are where all of the light comes from, ignoring con
tributions that are far from the MPP. However, the degree
of darkening is small, showing that the contributions from
those paths is small as well.
Figure 5 shows the results for a nonuniform density. The
images are 512 by 384 and used 9 samples per pixel. The
runtimes were 55 minutes for the PI method and and 580
minutes for the Monte Carlo solution. The cloud densities
were generated using a Perlinstyle noise function, and a
Gaussian phase function was used with µ =0.11. Again, no
tice some slight darkening.
Figure6showsthe resultsfor anonuniform densitycloud
Figure 4: Upper: MC raytracing. Lower: PI Approximation.
with different illumination that includes indirect illumina
tion and skylight contribution. The image is 640 by 480 pix
els and used 9 samples per pixel. The runtime was 95 min
utes.
Most of the theory in this paper is known to other scien
tificcommunities. Wehavedescribed amoreintuitiveway of
constructing path integrals avoiding the heavy mathematics
of functional integration that would ordinarily be required.
We have shown that this theory can be applied to produce
reasonably accurate images. This was accomplished by find
ing the MPP and explicitly computing contributions along
that path. All other paths are dealt with via analytic inte
gration around the MPP. The MPP can be also used in tra
ditional Monte Carlo simulations as a form of variance re
duction. Our formulation can be used as a starting point
for a sophisticated method such as Metropolis Light Trans
port, which effectively attempts to find the MPP by ran
dom walk. We have not pursued a highly efficient imple
mentation; there is much room for the development of algo
rithms based on path integration. For example, a hierarchi
cal implementation analogous to15would surely be bene
ficial. Our images should be viewed as a proofofconcept
that path integration is a valuable tool for dealing with volu
metric lighting. We have not shown that it has advantages
over the dipole approximation18to the diffusion approxi
c ? The Eurographics Association 2003.
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Premože et al. / Path Integration for Light Transport in Volumes
Figure 5: Upper: MC raytracing. Lower: PI Approximation.
Figure 6: A cloud rendered using the Path Integration ap
proximation. Diffuse light source contributions were added
at the boundary of the volume.
mation for dense uniform media, and in fact there is rea
son to prefer diffusion as density increases because path in
tegration ignores backscattering. However, we have shown
that path integration can be applied to sparse and spatially
varying volumes where the diffusion approximation is not
appropriate19. Thus path integration should be viewed as a
complement todiffusion, andshouldbecome part ofthetool
box of dealing with volumebased scatterers. Many classical
rendering problems could be reformulated and solved faster
using thepath integral formulation. Understanding the paral
lels with the classical formulation and thinking of problems
in this framework opens many directions for future research.
7. Acknowledgements
Thanks to Milan Ikits, Emil Praun, and David Weinstein for
reading drafts of the paper. The reviewers provided valuble
suggestions for improving this paper. This work was par
tially supported by NSF grant 9978099.
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