# 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.

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**ABSTRACT:**It is a long-standing problem in unbiased Monte Carlo methods for rendering that certain difficult types of light transport paths, particularly those involving viewing and illumination along paths containing specular or glossy surfaces, cause unusably slow convergence. In this paper we introduce Manifold Exploration, a new way of handling specular paths in rendering. It is based on the idea that sets of paths contributing to the image naturally form manifolds in path space, which can be explored locally by a simple equation-solving iteration. This paper shows how to formulate and solve the required equations using only geometric information that is already generally available in ray tracing systems, and how to use this method in in two different Markov Chain Monte Carlo frameworks to accurately compute illumination from general families of paths. The resulting rendering algorithms handle specular, near-specular, glossy, and diffuse surface interactions as well as isotropic or highly anisotropic volume scattering interactions, all using the same fundamental algorithm. An implementation is demonstrated on a range of challenging scenes and evaluated against previous methods.ACM Transactions on Graphics - TOG. 01/2012; - SourceAvailable from: DJEDI NourEddine
##### Conference Paper: Realistic rendering of cumulus clouds

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**ABSTRACT:**Realistic simulation of natural scenes is one of the most challenging problems facing computer graphics, including clouds. The complexity of cloud formation, dynamics and light interaction makes real time cloud rendering a difficult task. In addition, traditional computer graphics methods involve high amounts of memory and computing resources, which currently limits their realism and speed. This paper presents a method for the efficient realistic shading of cumulus clouds taking into account the sky color. Our approach consists in taking advantage of all the a priori knowledge on the characteristics of the cumulus cloud and on the sky light effects. In this paper, we only focus on cloud shading. We do not deal with the modeling and animation of the cloud’s shape.The 2nd International Conference on Software Engineering and New Technologies "ICSENT - 2013", Hammamet, Tunisia.; 12/2013 - [Show abstract] [Hide abstract]

**ABSTRACT:**Central to all Monte Carlo-based 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 surface-based 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 single-and double-scattering subpaths in anisotropically-scattering media. We demonstrate the benefit of our new sampling techniques, integrating them into several path-based rendering algorithms such as path tracing, bidirectional path tracing, and many-light 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 higher-order multiple scattering. Our algorithms significantly reduce noise and increase performance in renderings with both isotropic and highly anisotropic, low-order scattering.ACM Transactions on Graphics (TOG). 11/2013; 32(6).

Page 1

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-

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 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

c ? The Eurographics Association 2003.

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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

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 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.

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-

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.

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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

re-arranged 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/ds|2ds 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 non-negative, 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 Euler-Lagrange 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 z-axis 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

multiply-scattered 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

multiply-scattered 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 “quasi-MPPs” 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

Henyey-Greenstein 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 brute-force 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 non-uniform 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 Perlin-style noise function, and a

Gaussian phase function was used with µ =0.11. Again, no-

tice some slight darkening.

Figure6showsthe resultsfor anon-uniform 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 proof-of-concept

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 volume-based 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 99-78099.

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