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.
In this paper Our use of GRT'' follows , but we apply it to a broader class of reciprocal transport processes with general intercollision statistics. we use GRT'' to refer specifically to any linear non-Beerian transport process that discards the notion of exponential free-path distributions (FPDs) characterized only by a mean, and instead forms semi-Markov random flights using the ensemble-averaged two-point intercollision free-path statistics exhibited by a given random medium @cite_132 @cite_44 @cite_1 @cite_49 @cite_25 @cite_133 @cite_32 @cite_52 @cite_39 @cite_64 . We intentionally exclude higher-order memory @cite_107 , stochastic absorption and phase functions @cite_61 and heterogeneous mean densities @cite_30 , all of which would significantly complicate some of the present derivations.
@cite_61 @cite_30 @cite_64 @cite_107 @cite_132 @cite_1 @cite_32 @cite_52 @cite_39 @cite_44 @cite_49 @cite_133 @cite_25
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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 derive new diffusion solutions to the monoenergetic generalized linear Boltzmann transport equation for the stationary collision density and scalar flux about an isotropic point source in an infinite d-dimensional absorbing medium with isotropic scattering. We consider both classical transport theory with exponentially distributed free paths in arbitrary dimensions as well as a number of nonclassical transport theories (nonexponential random flights) that describe a broader class of transport processes within partially correlated random media. New rigorous asymptotic diffusion approximations are derived where possible. We also generalize Grosjean’s moment-preserving approach of separating the first (or uncollided) distribution from the collided portion and approximating only the latter using diffusion. We find that for any spatial dimension and for many free-path distributions Grosjean’s approach produces compact, analytic approximations that are, overall, more accurate for high absorption and for smal... We survey research on radiation propagation or ballistic particle motion through media with randomly variable material density, and we investigate the topic with an emphasis on very high spatial frequencies. Our new results are based on a specific variability model consisting of a zero-mean Gaussian scaling noise riding on a constant value that is large enough with respect to the amplitude of the noise to yield overwhelmingly non-negative density. We first generalize known results about sub-exponential transmission from regular functions, which are almost everywhere continuous, to merely “measurable” ones, which are almost everywhere discontinuous (akin to statistically stationary noises), with positively correlated fluctuations. We then use the generalized measure-theoretic formulation to address negatively correlated stochastic media without leaving the framework of conventional (continuum-limit) transport theory. We thus resolve a controversy about recent claims that only discrete-point process approaches can accommodate negative correlations, i.e., anti-clustering of the material particles. We obtain in this case the predicted super-exponential behavior, but it is rather weak. Physically, and much like the alternative discrete-point process approach, the new model applies most naturally to scales commensurate with the inter-particle distance in the material, i.e., when the notion of particle density breaks down due to Poissonian—or maybe not-so-Poissonian—number-count fluctuations occur in the sample volume. At the same time, the noisy structure must prevail up to scales commensurate with the mean-free-path to be of practical significance. Possible applications are discussed. Although order statistics have been studied for several decades, most of the results are based on the assumption of independent and identically distributed (i.i.d.) random variables. In the literature, how to compute the mth order statistics of n correlated random variables is still a problem. This article proposes a recursive algorithm based on statistical min max operations to compute order statistics for general correlated and not necessarily identically distributed random variables. The algorithm has an O(mn) time complexity and O(m p n) space complexity. A binary tree-based data structure is further developed to allow selective update of the order statistics with O(nm2) time. As a vehicle to demonstrate the algorithm, we apply it to the path selection algorithm in at-speed testing. A novel metric multilayer process space coverage metric is proposed to quantitatively gauge the quality of path selection. We then show that such a metric is directly linked to the order statistics, and our recursive algorithm can thus be applied. By employing a branch-and-bound path selection algorithm with these techniques, this article shows that selecting an optimal set of paths for a multimillion-gate design can be performed efficiently. Compared to the state of the art, experimental results show both the efficiency of our algorithms and better quality of our path selection.