When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to vector flux radiance because the macroscopic cross-section @math depends on the distance @math to the previous collision or boundary. This creates a nonlocal relationship between collision rate and flux and requires revising a number of familiar deterministic and Monte Carlo methods. We generalize collision and track-length estimators to support unbiased estimation of either flux integrals or collision rates in generalized radiative transfer (GRT). To provide benchmark solutions for the Monte Carlo estimators, we derive the four Green's functions for the isotropic point source in infinite media with isotropic scattering. Additionally, new moment-preserving diffusion approximations for these Green's functions are derived and reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.
We introduce a radiative framework that takes into account the spatial correlation of scattering particles in a medium, resulting in non-exponential extinction that cannot be predicted by classic radiative transfer. 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. We formulate a new model for transport in stochastic media with long-range spatial correlations where exponential attenuation (controlling the propagation part of the transport) becomes power law. Direct transmission over optical distance τ(s), for fixed physical distance s, thus becomes (1 + τ(s) a)− a, with standard exponential decay recovered when a → ∞. Atmospheric turbulence phenomenology for fluctuating optical properties rationalizes this switch. Foundational equations for this generalized transport model are stated in integral form for d = 1, 2, 3 spatial dimensions. A deterministic numerical solution is developed in d = 1 using Markov Chain formalism, verified with Monte Carlo, and used to investigate internal radiation fields. Standard two-stream theory, where diffusion is exact, is recovered when a = ∞. Differential diffusion equations are not presently known when a < ∞, nor is the integro-differential form of the generalized transport equation. Monte Carlo simulations are performed in d = 2, a... Abstract A formalism for photon transport in leaf canopies with finite-dimensional scattering centers that cross shade mutually is developed. Starting from first principles, expressions for the interaction cross sections are derived. The problem of illumination by a monodirectional source is studied in detail using a successive collisions approach. A balance equation is formulated in R 3 and the interaction between a leaf canopy and the adjacent atmosphere is discussed. Although the details are those relating to a leaf canopy, the formalism is equally applicable to other media where the constituents cross shade mutually such as planetary surfaces, rings and ridged-ice in polar regions, i.e., media that exhibit opposition brightening.  Abstract This paper considers monochromatic radiative transfer in a diffusive three dimensional random binary mixture. The absorption coefficient, along any line-of-sight is a homogeneous Markov process, which is described by a three-dimensional Kubo-Anderson process. The transfer equation is solved numerically by Monte-Carlo simulations on a massively parallel computer (a Connection Machine) by attaching one or several photons to each processor. The implementation of the simulations on the machine is discussed in detail, in particular the association between photons and processors and the storage of the data concerning the photons and the realizations of the statistics. With a CM-2 having 8000 processors, it is possible, with an adequate strategy, to follow simultaneously millions of photons in hundreds of realizations and to reach optical thicknesses up to 100 with dispersions of order 10 -2 for the reflection and transmission coeficients. The simultations are validated, in the case of the one-dimensional rod geometry, by comparison with the exact analytical solution, constructed by averaging the solution of the non-stochastic problem (diffusion in a rod of given optical thickness) over the probability density of the optical thickness. The latter obeys a stochastic Liouville equation which is solved by a Green's function method. The influence of the dimension of the Kubo-Anderson process is studied for the case of a slab and it is shown that a slab consisting in a pile of layers (1D process) is more transparent than one which consists in a stack of lumps (3D process). A strategy for improving the efficiency of Monte-Carlo simulations, based on the distribution of the lengths of the individuals steps of the photons, is presented and discussed. This paper addresses light transport through a discrete random medium, which we define as a volume filled with macroscopic scattering geometry generated by a random process. This formulation is more general than standard radiative transport, because it can be applied to media that are made up of closely packed scatterers. A new approach to rendering these media is introduced, based on precomputed solutions to a local multiple scattering problem, including a new algorithm for generating paths through random media that moves through the interior of the medium in large strides without considering individual scattering events. A method for rendering homogeneous isotropic random media is described that generates paths using precomputed scattering solutions compressed and randomly sampled using Nonnegative Matrix Factorization. It can efficiently render discrete media, such as a large pile of glass objects, in which the individual scatterers are visible. The method is demonstrated on scenes containing tens of thousands of transparent, specular objects that are nearly impossible to render with standard global illumination techniques. Many porous media cannot be homogenized as Beerian semi-transparent media. Effective extinction, absorption and scattering coefficients can indeed have no physical meaning for small or intermediate optical thicknesses. A generalized radiative transfer equation (GRTE), directly based on the extinction cumulative distribution function, the absorption and scattering cumulative probabilities and the scattering phase function is established for this optical thickness range. It can be solved by a statistical Monte Carlo approach. For a phase of a porous medium that is optically thick at local scale, the GRTE degenerates into a classical Beerian RTE. In these conditions, a radiative conductivity tensor is directly obtained, by a perturbation method, and expressed with the radiative coefficients of this RTE and temperature. As illustrations, exhaustive radiative conductivity results are given for a set of overlapping transparent spheres within an opaque phase and for opaque rod bundles. Abstract This paper presents a derivation and initial study of a new generalized linear Boltzmann equation (GLBE), which describes particle transport for random statistically homogeneous systems in which the distribution function for chord lengths between scattering centers is non-exponential. Such problems have recently been proposed for the description of photon transport in atmospheric clouds; this paper is a first attempt to develop a Boltzmann-like equation for these and other related applications. Stochastic models of biased random walk are discussed, which describe the behavior of chemosensitive cells like bacteria or leukocytes in the gradient of a chemotactic factor. In particular the turning frequency and turn angle distributions are derived from certain biological hypotheses on the background of related experimental observations. Under suitable assumptions it is shown that solutions of the underlying differential-integral equation approximately satisfy the well-known Patlak-Keller-Segel diffusion equation, whose coefficients can be expressed in terms of the microscopic parameters. By an appropriate energy functional a precise error estimation of the diffusion approximation is given within the framework of singular perturbation theory. Abstract New radiative transfer theory is developed for stochastically inhomogeneous scattering media. The three-dimensional shapes and large scale (compared to the mean free path) structures of the media are modeled by stochastic interfaces separating regions of different scattering properties. The small scale fluctuations are characterized by a pair-correlation function. The radiative transfer equation is extended to include individual scattering and propagation probabilities of a ray for each subregion as well as the probability for a ray to cross the interface between two subregions. The propagation probability is found to depend on the entire preceding path of the ray; the present formulation accounts for the two previous scatterings. A new adding doubling algorithm is developed to solve this problem numerically. Transmission through a cloud layer and backward scattering seem to be particularly sensitive to inhomogeneities.  It is argued that, to directly target the mean fluxes through a structured medium with spatial correlations over a significant range of scales that includes the mean-free-path, one can use an effective propagation kernel that will necessarily be sub-exponential. We come to this conclusion using both standard transport theory for variable media and a point-process approach developed recently by A. Kostinski. The ramifications of this finding for multiple scattering and effective medium theory are examined. Finally, we describe a novel one-dimensional transport theory with asymptotically power-law propagation kernels and use it to shed new light onto recent observations of solar photon pathlength in the Earth’s cloudy atmosphere.