Path Integration for Light Transport in Volumes.

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Eurographics Symposium on Rendering 2003, pp. 1–12
Per Christensen and Daniel Cohen-Or (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 BRDF-style 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 non-trivial 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 non-uniform 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-
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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 physicall-based 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 Fokker-Planck and Schrödinger equa-
tions. The concept of photon paths has been well-known
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 quasi-particle 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. Near-diffusive 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 cross-sections, 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 “two-sided”
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 in-scattering 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
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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 thetime-dependent 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 time-dependent 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
(time-reversal 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 right-hand 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 infinite-dimensional path
?xf,tf|xi,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.
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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-
Kramers-Brillouin) expansion6, which is a well-established
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 Monte-Carlo 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.