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Beyond Renewal Approximations: A 1D Point Process Approach to Linear Transport in Stochastic Media



We present a novel approximate model for monoenergetic linear transport in stochastic media that permits correlations between successive free-path lengths of particle collisions. Our model utilizes collision times determined by a 1D point process, chosen such that perfectly forward scattering along a transect precisely matches ensemble-averaged statistics. In contrast, previous renewal-based non-classical transport formulations only guarantee the accuracy of the first collision time and assume subsequent collision times are independently and identically distributed. By accommodating non-renewal collision times, our model can account for step correlations that emerge in most variability models, including cross-section fluctuations driven by Gaussian processes , transformed Gaussian processes, and the majority of discrete mixture models. We compare multiple scattering predictions from our model to new two-dimensional benchmark simulations featuring transformed Gaussian fluctuations, demonstrating enhanced accuracy compared to renewal approximations.
M&C 2023 - The International Conference on Mathematics and Computational Methods Applied
to Nuclear Science and Engineering ·Niagara Falls, Ontario, Canada ·August 13 17, 2023
Beyond Renewal Approximations: A 1D Point Process Approach to
Linear Transport in Stochastic Media
Eugene d’Eon1
2788 San Tomas Expressway, Santa Clara, CA 95050
We present a novel approximate model for monoenergetic linear transport in stochastic media
that permits correlations between successive free-path lengths of particle collisions. Our model
utilizes collision times determined by a 1D point process, chosen such that perfectly forward
scattering along a transect precisely matches ensemble-averaged statistics. In contrast, previous
renewal-based non-classical transport formulations only guarantee the accuracy of the first col-
lision time and assume subsequent collision times are independently and identically distributed.
By accommodating non-renewal collision times, our model can account for step correlations that
emerge in most variability models, including cross-section fluctuations driven by Gaussian pro-
cesses, transformed Gaussian processes, and the majority of discrete mixture models. We compare
multiple scattering predictions from our model to new two-dimensional benchmark simulations
featuring transformed Gaussian fluctuations, demonstrating enhanced accuracy compared to re-
newal approximations.
KEYWORDS: Non-Classical, Stochastic, Point Process, Transect, Step Correlations
Particle transport in stochastic media presents significant challenges in various domains, including neutron
transport, heat transfer, remote sensing, and tissue optics. Even for homogeneous, monoenergetic, time-
independent problems, once the total macroscopic cross section Σt(x)is subject to randomness, the only
exact method for estimating ensemble-averaged observables, given the presence of scattering, is an expen-
sive double-Monte-Carlo simulation that requires averaging classical simulations across numerous system
realizations [1]. Developing non-classical equations to directly approximate transport in stochastic media
without generating ensemble realizations is, therefore, highly sought after.
One such approach is a renewal transport process, whose integral equation is that of random flights [2,3],
shown to be equivalent to a generalized linear Boltzmann equation (GLBE) [4]. This approach was inspired
by the observation that ensemble-averaged extinction in stochastic systems is often non-exponential [5]. By
replacing the Poisson process, which produces exponentially-distributed intercollision lengths in classical
theory, with a broader renewal process [6], this non-exponential behaviour can be directly exhibited in
a transport formalism. However, in a system with scattering, the renewal assumption demands that all
subsequent free-path lengths are independent, thus disallowing correlations in successive lengths between
collisions (“step correlations”). The accuracy of this renewal assumption for modeling general stochastic
media is currently not well understood.
In this paper, we introduce a novel model for monoenergetic linear transport in stochastic media that al-
lows arbitrary step correlations, offering insights into when the renewal approximation is appropriate and
providing a more general model for when it is not. Our approach utilizes exact collision-time statistics de-
rived from the case of perfectly-forward (singular) scattering, where transport is restricted to unidirectional
flow along a medium transect, yielding collision times determined precisely by a 1D point process. By
leveraging known results from time series analysis, we can efficiently simulate collisions along transects
Eugene d’Eon
Figure 1: In a realization with fluctuating Σ(x)(greyscale), scattering events (white) along a
transect (red) are typically correlated (clumpy) and given by a 1D point process. We apply these
transect statistics along histories with general scattering (blue).
and rigorously identify which classes of stochastic media are truly renewal. To approximately treat non-
forward scattering, we propose applying these transect collision times along a general history, thus creating
an efficient autoregressive transport model that directly simulates ensemble-averaged behavior by correlat-
ing future collisions with prior times but not locations along a history. We demonstrate that our model can
outperform a renewal transport process over a wide range benchmarks.
Previous discussions on step correlations in stochastic media have often attributed them to complex correla-
tions emerging when particles scatter backward into prior scenery [7]. While reversal can indeed cause step
correlations, we clearly demonstrate that nearly all forms of stochastic media, including those with Gaussian
or transformed-Gaussian density fluctuations, exhibit step correlations even in the case of perfectly forward
scattering (Figure 1), and therefore step correlations can be expected to arise in nearly all systems, including
those with highly forward scattering. To assess the impact of these correlations, we present new benchmark
simulations for monoenergetic absorption and scattering in isotropic, stationary two-dimensional (Flatland)
stochastic media with transformed-Gaussian density fluctuations. Comparisons between these benchmarks
and our new model reveal a reduction in error (relative to a renewal transport process) of up to an order of
In this section, we introduce our novel model, which directly extends the random-flight interpretation of
classical linear transport [2,8]. We review essential results from point process literature and explore the role
of point processes in determining collision times along a particle history. We then discuss the time-series
analysis of collisions along a transect and the relationship to Cox processes, which forms the foundation of
our model.
Scope We will limit the scope of the present work to consider only time-independent, monoenergetic
linear transport in an isotropic, piecewise-homogeneous medium with deterministic scattering kernel and
deterministic single-scattering albedo c. We assume that the total macroscopic cross section Σt(x)at po-
sition xis given by a stationary random field. Extension of our model to time-dependent problems is
straightforward. Multi-group, inhomogeneous systems, anisotropic media and stochastic albedo care not
presently supported by our model. Despite these restrictions, we feel that our work unveils important insight
regarding the role of step correlations in stochastic systems and the limitations of the GLBE.
2.1. Random Flights and Point Processes
We will determine our model by defining the stochastic process governing a single particle, born at time t=
0. Specifically, we will form a generalized random flight by specifying a sequence of random collision times
tialong each history. Our goal is to choose a flight whose expectation directly approximates ensemble-
averaged observables in the stochastic system. The collision times ticonstitute a 1D point process N(t):
a non-negative integer random variable at each time tthat gives the number of collisions up to that time
[6]. Once N(t)is specified, the transport is completely determined, since the scattering kernel and survival
probability care assumed to be deterministic and independent of ti. Given N(t), a Monte Carlo estimator
Beyond Renewal Approximations: A 1D Point Process Approach to Linear Transport in Stochastic Media
for our model is readily derived by sampling collision times tifrom N(t), constructing a history by sampling
the scattering kernel at each collision, utilizing implicit capture for absorption, and terminating with roulette
or upon escape/boundary interactions.
The Poisson Process of Classical Transport To gain some familiarity with the role of point processes
in classical transport, let us recall that in a deterministic homogeneous medium, the collision times along a
history are given by a Poisson point process (PPP) with a constant rate λ(t)[9]. The PPP is a memoryless
point process, where the number of points within a time interval [ta, tb]follows a Poisson distribution with
a mean of Rtb
taλ(t)dt. Assuming motion occurs at unit speed, the rate of the Poisson process is equal to the
total macroscopic cross section λ(t)=Σt(x(t)) at the particle’s current position x(t). In a homogeneous
system, we observe a constant rate λ(t) = Σtand exponential times between collisions. However, in a
system with inhomogeneous cross section, such as a realization of a stochastic system, the rate Σt(x(t)) of
the point process at time tdepends on the current position x(t)(which, in turn, is a function of both prior
times and directions of the history). Averaging this PPP over the Σt(x)ensemble becomes intractable.
The complexity of this 3D averaging casts heavy doubt on ever finding an exact random flight approach
to general stochastic media. If, however, we constrain the problem to one dimension by either considering
only absorption, or forcing scattering to be perfectly forward, exact results are possible. We consider each
of these scenarios now, in turn.
Attenuation A number of tractable results are available if we consider only absorption. In this case,
transport is restricted to a straight path up to the first collision and the problem is one-dimensional. The
attenuation law T(t), which is simply the probability of finding no collision in [0, t), is
T(t)Pr {N(t) = 0}.(1)
In a deterministic medium, this probability follows from the rate of collisions λ(t) = Σt(x(t)) governing
the PPP, giving the well-known equation for attenuation
T(x) = exp Zt
λ(t)dt= exp Zx
This equation illustrates a close relationship between the point process and particle transport literatures.
In fact, the point process community independently developed the concept of delta (Woodcock) tracking,
under the name “thinning algorithm”, for sampling events in an inhomogeneous PPP [10,11].
Similarities between these two studies are also found in the case of stochastic rate λ(t). For stochastic
systems, we denote the mean attenuation law (from an equilibrium/deterministic origin) as
Tu(x) T(x)(3)
which is the ensemble average of the deterministic result. Finding Tu(x)requires averaging Equation 2
where Σtis stochastic, which makes N(t)a doubly-stochastic Poisson process (or Cox Process) [12], which
was noted by Kostinski [5,13]. The mean attenuation law in Equation 3 is tractable for a number of fluc-
tuation models including Gaussian fluctuations, squared-Gaussian fluctuations and discrete n-ary Markov
The ’u’ label on Tu(x)refers to an unconditional ensemble average over all realizations, and distinguishes
the result from an average that is conditioned on starting from t= 0 at a scattering center [14]. For brevity,
we will consider only deterministic sources, using equilibrium (asynchronous) initialization of the point
process N(t)(see [15] for more details). However, synchronous initialization of N(t)is easily adopted for
emission from spatially correlated scattering centers, which may be appropriate in neutronics [4].
Eugene d’Eon
2.2. Transect Statistics
A Cox process provides a precise and comprehensive representation of transport in a purely absorbing
stochastic system, as the absence of scattering reduces the problem to a one-dimensional scenario, allowing
for an equivalent 1D point process. Another instance where we can take advantage of this 1D equivalence
occurs when the scattering kernel is a singular Dirac delta peak in the forward direction, thus restricting
transport to a 1D transect (see, for example, Figure 1). In this case, N(t)is also reduced to a 1D Cox
process, accounting for all collisions along the transect instead of just the first one. By eliminating all
angular dependence from N(t), we circumvent the intricacies of correlations arising from prior scenery.
Although perfectly forward scattering essentially nullifies the scattering collisions, reverting the problem
back to one of pure absorption, showcasing these transect statistics is still a crucial requirement for any non-
classical transport model to be considered accurate. For this reason, we will employ these transect statistics
to not only establish a new transport model that embodies them but also to develop new benchmarks for
assessing non-classical formalisms.
We now introduce some necessary results from point process literature. Point processes are completely
determined by their generating functions. Given a stationary point process, we will denote its equilibrium
probability generating function as
znPr {N(t) = n},(4)
which, for any time t, yields the required probabilities for N(t). The attenuation law along a transect can
then be expressed as
Tu(t) = ϕN(0; t),(5)
with higher-order probabilities recovered via
Pr {N(t) = n}=1
By giving an exact account of the full point process N(t), the generating function ϕN(z;t)is a highly
useful tool for analyzing the clustering of collisions along a transect in a medium with stochastic cross
section (Figure 1). Remarkably, it turns out that determining ϕN(z;t)is no harder than determining the
attenuation law ϕN(0; t).
For a Cox process, the optical depth along a transect is
Σt(x)dx, (7)
which is a random variable. The Cox process is then uniquely determined, given that
Pr {N(t) = k}=eτ(t)(τ(t))k/k!, k 0.(8)
The probability generating function can then be written [15, p.219]
ϕN(z;t) = exp[(z1)τ(t)],(9)
which corresponds to the moment generating function of the optical depth at time t, evaluated at z1.
Since z= 0 in Equation 9 corresponds to the attenuation law of the Cox process, and because (z1)
simply serves as a constant scaling factor for all optical depths τ(t)regardless of t, it becomes clear that
determining all collision times along the transect is, in fact, no more challenging than determining the
attenuation law itself. Said another way: ϕN(z;t)is just the attenuation law in a system where the random
field Σt(x)is scaled by a constant (1 z).
Beyond Renewal Approximations: A 1D Point Process Approach to Linear Transport in Stochastic Media
The Transect Absorption Law We can now define two new analytic benchmarks for measuring approxi-
mate non-classical models of transport. Firstly, for an absorbing and scattering system, where each collision
scatters perfectly forward with probability cand otherwise absorbs and terminates the random flight, we can
solve for the ensemble-averaged attenuation of unit flux ψ(t)along a transect using the generating func-
tion. Given that the attenuation for kcollisions occurring along the segment [0, t]is ck, we have
Pr {N(t) = k}ck=ϕN(c;t),(10)
which is just the generating function for the point process (Equation 4) evaluated at z=c. This result is also
equivalent to the attenuation law of a related purely absorbing system where Σt(x)is scaled by a constant
(1 c)in order to account for the relative probability of real (absorbing) to total (real + null/scattering)
collisions. When c= 0, the system is purely absorbing and the original attenuation law is recovered, as
expected. Likewise, for c= 1, the system is lossless and ψ(t)= 1.
Time of the mth collision Another way that we can benchmark non classical models is using the time
of the mth collision along the transect. Given the equilibrium generating function ϕN(z;t)for a stationary
point process, the probability density fm(t)of the time of the mth collision/arrival along a transect (from
equilibrium t= 0 initialization) is [16, Eq.(25)]
fm(t) =
∂zjϕN(1 z;t)|z=1.(11)
2.3. Non-Renewal Transport
Because transect statistics are a necessary and testable condition for any accurate theory of non-classical
transport, we can test the renewal assumption of the GLBE under this lens. This leads immediately to
the query: for what class of Σtfluctuations are the transect statistics renewal, which is to ask: what Cox
processes are also renewal processes? This was answered rigorously by Kingman [17]: either Σtis a two-
phase medium with one void phase and exponential chord lengths in the non-void phase, or Σtis singular
with respect to Lebesque measure. This immediately excludes renewal transport from being a generally
accurate model in all but an extremely narrow form of Markov binary mixtures or, alternatively, within
media where Σtis described by some unknown set of fractal variability models. In particular, Gaussian
and transformed-Gaussian fluctuations (e.g. Figure 1) are non-renewal and always exhibit clumping of the
collision times. It is known that some Gamma and Weibull renewal processes also have a Cox process
equivalence, but the exact fractal nature of Σtthat produces them is not known [18].
Approximation by a renewal process To illustrate the non-renewal character of Cox processes with
Gaussian fluctuations, we consider the Gauss-Poisson Cox process where Σt(x)is Gaussian with exponen-
tial autocovariance R(|st|) = r2ey|st|, which has a known generating function [6, p.183]. We consider
unit mean cross section Σt(x)and keep rsmall to make negative cross sections unlikely. In Figure 2,
we use the generating function to compare the point counts from equilibrium over various time intervals
to those predicted by an equilibrium renewal process. The renewal and Cox processes agree for n= 0
(the attenuation law), but all other collision counts differ due to the lack of step correlation in the renewal
process. We observed similar inaccuracies of renewal approximations for transformed Gaussian processes
(where the Gaussian cross sections are squared or exponentiated to avoid non-negative values), and also
when comparing the collision times of the nth collision along a transect, using Equation 11.
2.4. Scattering
Transect statistics have led to several new analytical benchmarks for testing the accuracy of a given non
classical theory of transport, but only in the contrived case of purely forward scattering. To form a practical
Eugene d’Eon
0 2 4 6 8 10 12 14
0 2 4 6 8 10 12 14
0 2 4 6 8 10 12 14
0 2 4 6 8 10 12 14
0 2 4 6 8 10 12 14
0 2 4 6 8 10 12 14
Figure 2: We compare the probabilities for finding ncollisions in [0, t]for Gaussian Σt(x)with
exponential correlation R(|st|)=0.32e0.1|st|using double-MC ground truth (blue dots), a
renewal process approximation (red dots) and analytic ground truth (continuous). While the
attenuation law n= 0 matches for the renewal approximation, all other probabilities differ due to
the lack of step correlations.
and general model of non-classical transport that exhibits transect statistics, we propose to simply apply
them along any history, regardless of the scattering kernel. Intuitively, the clustering of collision times that
arises for systems with long-range spatial correlations will be approximately achieved for a system with
highly forward scattering (Figure 1). We leave any further justification for the proposed model to numerical
benchmark comparisons that we provide in section 3.
We do not presently consider an integral transport equation for our model. This would include a cross
section Σ(t1,· · · , tk1;t)that is a Janossy density [19] for the point process N(t)up to the current time t
subject to the occurrence of k1prior collisions at times ti. Such an integral equation could be written
down in principle, but it is not immediately clear to us how useful this would be. However, as described
above, a Monte Carlo estimator for our model follows directly from the model’s definition.
Sampling Transect Collision Times Cox processes can be sampled using a variety of methods [19,20].
For simplicity, in the next section, we sample a single large square tileable auxiliary realization of the
stationary random field Σt(x)using Fourier transforms. This happens once for each piecewise homogeneous
element of the system with unique statistics. After this precomputation, traditional Monte Carlo sampling
then follows where collision times for each history are determined using delta tracking along a transect in
the auxiliary domain. So while a particle in the physical system follows a general history, a virtual particle
in the auxiliary domain begins at a random position and direction and follows a straight path in order to
determine collision times in the physical system.
Relationship to Prior Work One interesting property of our model is that it includes a number of previ-
ous transport formalisms as special cases under a common framework:
When N(t)is chosen to be a PPP, our model describes classical transport in a deterministic medium.
When N(t)is a mixed-Poisson process (where Σis random, but constant in each realization), our
model corresponds to an approximation known as the independent-column approximation in remote
Beyond Renewal Approximations: A 1D Point Process Approach to Linear Transport in Stochastic Media
sensing [21], and is an important benchmark for parametric stochastic media in the limit of infinite
correlation lengths.
When N(t)is a renewal process, N(t)depends only on the previous ti1collision time, and the
resulting integro-differential/integral transport equations are the GLBE/random-flight equations, re-
spectively [2,3,4,14,22,23].
When N(t)is a Markov-renewal process, N(t)depends only on the previous ti1collision time
and an additional integer state, and corresponds to the chord-length-sampling/Levermore-Pomraning
approximations for n-ary Markov mixtures (when we additionally extend the albedos cjto depend on
state j) [24].
By construction, the accuracy of our model is only ensured in the limited case of purely forward scatter-
ing. In this limit, scattering collisions are effectively null events and the transport is equivalent to a purely
absorbing one. To test the accuracy of our model for non-forward scattering, we rely on numerical simu-
lations. We constructed a new two-dimensional benchmark for stochastic media with homogeneous mean
density, homogeneous deterministic c, and generalized Henyey-Greenstein [25] scattering parametrized by
the mean cosine 1< g < 1. The benchmark configuration is illustrated in Figure 3 (left): a deterministic
unit monodirectional beam was applied along the normal to the boundary of a source-free disk domain (in
two-dimensional Flatland) and the leakage from the vacuum boundary at azimuth ϕ(regardless of outgoing
direction) was tallied and averaged over the sampled ensemble of disk realizations. We ran a suite of sim-
ulations for Gaussian and transformed-Gaussian Σtfluctuations with exponential correlations. In each, we
varied the radius Rof the disk, the correlation width, as well as cand g.
For each benchmark configuration, we compared the double-MC ground truth result to our new model, and
also to a renewal approximation, where the first collision is given exactly and the intercollision lengths were
determined in order to form an equilibrium renewal process [14]. For narrow correlation widths, we found
close agreement between all three models, and observed deviations as the correlation widths increased
relative to the mean free path (see Figure 5). The middle and right plots in Figure 3 illustrate selected
examples of the improved accuracy of our model over a renewal approximation for realizations of radius
R= 0.02 where Σtwas based on a transformed Ornstein-Uhlenbeck process with radial correlation er10,
where the Gaussian process was squared to create a non-negative Σtfield. These examples show substantial
improvements over the renewal approximation, both in the highly-peaked g= 0.9case, but, remarkably,
also in the more isotropic g= 0.5configuration, where it was less clear that transect statistics should apply.
We noted similar behaviour over a wide matrix of configurations, with the renewal approximation being
the worst performer overall, and in fact observed our new model consistently outperforming a renewal
approximation even in the case of backscattering with g < 0(Figure 4).
We did not perform benchmarks for Markov binary mixtures because our model reduces to the CLS algo-
rithm in such cases and comparisons to a renewal approximation have already been made in 3D [24], where
it was also noted that the renewal approximation can significantly underperform relative to a model that
includes step correlations (i.e. CLS).
We have presented a novel non-classical transport model based on transect statistics, which provides a uni-
fied framework for various existing transport formalisms. The model has been tested against a range of
benchmark configurations, demonstrating its accuracy and robustness even in cases of non-forward scatter-
ing. The improvements over the renewal approximation in both highly-peaked and more isotropic config-
urations highlight the model’s potential for practical applications. Our analysis of transect statistics also
provides a compelling argument against the use of renewal transport in stochastic systems, casting doubt
on the general applicability of the GLBE. Furthermore, the newly established analytical benchmarks for
nth collision time and attenuation pave the way for enhanced evaluation and development of future non-
classical transport methodologies in multiple fields.
Eugene d’Eon
Deterministic Source
Ground Truth
Transect Stats
-3 -2 -1 1 2 3
Ground Truth
Transect Stats
-3 -2 -1 1 2 3
Figure 3: Example benchmark results for the emergent scalar flux when a homogeneous disk
domain with stationary fluctuations of Σt(x)is subject to a monodirectional beam at the boundary.
Note how even with a low mean cosine of scattering (g= 0.5), our model still significantly
outperforms a renewal approximation.
-3-2-1 0 1 2 3
g= -0.9
-3-2-1 0 1 2 3
g= -0.7
-3-2-1 0 1 2 3
g= -0.5
-3-2-1 0 1 2 3
-3-2-1 0 1 2 3
-3-2-1 0 1 2 3
-3-2-1 0 1 2 3
-3-2-1 0 1 2 3
Figure 4: Disk benchmark values comparing ground truth (black) to our model (dots) and to a
renewal approximation (dashed) as the mean cosine of scattering gis varied. Note how our model
improves upon the renewal approximation even for g < 0, which is predominantly back scattering.
-3-2-1 0 1 2 3
-3-2-1 0 1 2 3
-3-2-1 0 1 2 3
-3-2-1 0 1 2 3
Figure 5: Disk benchmark values comparing ground truth (black) to our model (dots) and to a
renewal approximation (dashed) as the correlation parameter yis varied. Note how as yis increased
(decreasing the the correlation width of the exponential autocovariance), the variabililty of the
fluctuations averages away to a classical medium and all three predictions align.
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We introduce a non-exponential radiative framework that takes into account the local spatial correlation of scattering particles in a medium. Most previous works in graphics have ignored this, assuming uncorrelated media with a uniform, random local distribution of particles. However, positive and negative correlation lead to slower- and faster-than-exponential attenuation respectively, which cannot be predicted by the Beer-Lambert law. As our results show, this has a major effect on extinction, and thus appearance. From recent advances in neutron transport, we first introduce our Extended Generalized Boltzmann Equation, and develop a general framework for light transport in correlated media. We lift the limitations of the original formulation, including an analysis of the boundary conditions, and present a model suitable for computer graphics, based on optical properties of the media and statistical distributions of scatterers. In addition, we present an analytic expression for transmittance in the case of positive correlation, and show how to incorporate it efficiently into a Monte Carlo renderer. We show results with a wide range of both positive and negative correlation, and demonstrate the differences compared to classic light transport.
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We present a theoretical and experimental study of light transport in disordered media with strongly heterogeneous distribution of scatterers formed via nonscattering regions. Step correlations induced by quenched disorder are found to prevent diffusivity from diverging with increasing heterogeneity scale, contrary to expectations from annealed models. Spectral diffusivity is measured for a porous ceramic where nanopores act as scatterers and macropores render their distribution heterogeneous. Results agree well with Monte Carlo simulations and a proposed analytical model.
Exponential extinction of incoherent radiation intensity in a random medium (sometimes referred to as the Beer–Lambert law) arises early in the development of several branches of science and underlies much of radiative transfer theory and propagation in turbid media with applications in astronomy, atmospheric science, and oceanography. We adopt a stochastic approach to exponential extinction and connect it to the underlying Poisson statistics of extinction events. We then show that when a dilute random medium is statistically homogeneous but spatially correlated, the attenuation of incoherent radiation with depth is often slower than exponential. This occurs because spatial correlations among obstacles of the medium spread out the probability distribution of photon extinction events. Therefore the probability of transmission (no extinction) is increased.
Conference Paper
We develop a new theory of volumetric light transport for media with non-exponential free-flight distributions. Recent insights from atmospheric sciences and neutron transport demonstrate that such distributions arise in the presence of correlated scatterers, which are naturally produced by processes such as cloud condensation and fractal-pattern formation. Our theory formulates a non-exponential path integral as the result of averaging stochastic classical media, and we introduce practical models to solve the resulting averaging problem efficiently. Our theory results in a generalized path integral which allows us to handle non-exponential media using the full range of Monte Carlo rendering algorithms while enriching the range of achievable appearance. We propose parametric models for controlling the statistical correlations by leveraging work on stochastic processes, and we develop a method to combine such unresolved correlations (and the resulting non-exponential free-flight behavior) with explicitly modeled macroscopic heterogeneity. This provides a powerful authoring approach where artists can freely design the shape of the attenuation profile separately from the macroscopic heterogeneous density, while our theory provides a physically consistent interpretation in terms of a path space integral. We address important considerations for graphics including reciprocity and bidirectional rendering algorithms, all in the presence of surfaces and correlated media.
Previous proposals to permit non-exponential free-path statistics in radiative transfer have not included support for volume and boundary sources that are spatially uncorrelated from the scattering events in the medium. Birth-collision free paths are treated identically to collision-collision free paths and application of this to general, bounded scenes with inclusions leads to non-reciprocal transport. Beginning with reciprocity as a desired property, we propose a new way to integrate non-exponential transport theory into general scenes. We distinguish between the free-path-length statistics between correlated medium particles and the free-path-length statistics beginning at locations not correlated to medium particles, such as boundary surfaces, inclusions and uncorrelated sources. Reciprocity requires that the uncorrelated free-path distributions are simply the normalized transmittance of the correlated free-path distributions. The combination leads to an equilibrium imbedding of a previously derived generalized transport equation into bounded domains. We compare predictions of this approach to Monte Carlo simulation of multiple scattering from negatively-correlated suspensions of monodispersive hard spheres in bounded two-dimensional domains and demonstrate improved performance relative to previous work. We also derive new, exact, reciprocal, single-scattering solutions for plane-parallel half-spaces over a variety of non-exponential media types.
We give numerical benchmark results for particle transport in a randomly mixed binary medium, with the mixing statistics described as a homogeneous Markov process. A Monte Carlo procedure is used to generate a physical realization of the statistics, and a discrete ordinate numerical transport solution is generated for this realization. The ensemble averaged solution, as well as the variance, is obtained by averaging a large number of such calculations. Reflection and transmission results are given for several problems in both rod and planar geometry. In a separate development, two coupled transport equations are derived which formally described transport in a random binary mixture for arbitrary mixing statistics. Closing these equations by approximating their coupling terms in a low order and intuitive way leads to a model for stochastic transport previously obtained via the master equation. The present derivation, based upon approximating exact equations, allows in principle the opportunity to develop more accurate models by making higher order approximations in the coupling terms.