arXiv subjects

Eugene d'Eon

Publications and source records attributed to Eugene d'Eon.

15 recordsLinked to original sources

arXiv2026

Gaussian Process Implicit Surfaces as Participating Media: Realization-Free Rendering from Level-Crossing Statistics

We present a theory of light scattering that connects Gaussian Process Implicit Surfaces (GPISes) and participating media in both directions. Applying the Kac--Rice level-crossing formula under a local-conditioning approximation yields a complete anisotropic radiative transfer equation (RTE) directly from pointwise GPIS statistics. A shared projected area couples extinction and scattering, ensuring geometric consistency between the GPIS and its volumetric representation. The framework spans rough surfaces, porous and non-height-field geometries, and participating media. From the same statistical structure, we derive full-sphere Beckmann and GGX normal distribution functions supporting in-plane and out-of-plane anisotropy. These families provably recover SGGX, Beckmann, and GGX as special cases and admit exact visible-normal importance sampling. We also derive analytic masking--shadowing functions and single-scattering surface models for specular microsurfaces, with extensions to multiple scattering. In the height-field limit, we prove that the local-conditioning approximation reduces to Smith's independence assumption. Our realization-free approach improves rendering efficiency over realization-based methods and can be implemented within a standard volume renderer. In the inverse direction, we characterize families of GPISes corresponding to compatible RTE parameters and develop practical lifts for heterogeneous density fields. Existing volumetric assets thereby become renderable as GPISes, while trained radiance-field reconstructions yield surface geometry and shading normals without mesh extraction and provide a density-based representation of geometric uncertainty.

cs.GR↗

arXiv2026

Grid-Free Monte Carlo for Time-Dependent Diffusion

Many scientific applications require modeling how diffusive systems evolve over time, not merely their eventual steady states. While conventional steady-state analysis of partial differential equations (PDEs) on complex geometries is already hindered by costly volumetric meshing, transient analysis further requires sequential time stepping and careful step size selection. Grid-free Monte Carlo solvers such as walk on spheres (WoS) and walk on stars (WoSt) avoid this meshing bottleneck but remain largely limited to steady-state problems. We generalize WoS, for pure Dirichlet problems, and WoSt, for mixed Dirichlet--Neumann problems, to heat equations with initial conditions and time-dependent source and boundary data. We equip each random walk with a finite time budget and sample an exit time at every spatial step. If the exit time exceeds the remaining budget, the walk samples an interior point and evaluates the initial condition; otherwise, it continues with a reduced budget, accumulating source and boundary contributions. Our main technical contribution is a suite of kernel sampling and variance reduction techniques, including a low-bias, tabulation-free exit time sampler and efficient rejection samplers. Unlike grid-based transient solvers, our method directly estimates the solution at any requested time without volumetric meshing or sequential time marching. It also retains the parallel, progressive, and output-sensitive evaluation of WoS and WoSt while eliminating time step selection and temporal discretization bias entirely. Finally, we show how sharing walks enables efficient estimates at multiple target times.

cs.GR↗

arXiv2024

A Generalized Ray Formulation For Wave-Optics Rendering

Under ray-optical light transport, the classical ray serves as a linear and local "point query" of light's behaviour. Linearity and locality are crucial to the formulation of sophisticated path tracing and sampling techniques, that enable efficient solutions to light transport problems in complex, real-world settings and environments. However, such formulations are firmly confined to the realm of ray optics, while many applications of interest -- in computer graphics and computational optics -- demand a more precise understanding of light: as waves. We rigorously formulate the generalized ray, which enables linear and weakly-local queries of arbitrary wave-optical distributions of light. Generalized rays arise from photodetection states, and therefore allow performing backward (sensor-to-source) wave-optical light transport. Our formulations are accurate and highly general: they facilitate the application of modern path tracing techniques for wave-optical rendering, with light of any state of coherence and any spectral properties. We improve upon the state-of-the-art in terms of the generality and accuracy of the formalism, ease of application, as well as performance. As a consequence, we are able to render large, complex scenes, as in Fig. 1, and even do interactive wave-optical light transport, none of which is possible with any existing method. We numerically validate our formalism, and make connection to partially-coherent light transport.

cs.GR↗

arXiv2022

A physical model and a Monte Carlo estimate for the specific intensity spatial derivative, angular derivative and geometric sensitivity

Starting from the radiative transfer equation and its usual boundary conditions, the objective of this work is to design Monte Carlo algorithms estimating the specific intensity spatial and angular derivatives as well as its geometric sensitivity. The present document is structured in three parts, each of them dedicated to a specific derivative of the intensity. Although they are all assembled here in one document each derivative is of interest independently whether it be for radiative transfers analysis or engineering conception. Therefore, they are thought to be written as three different papers and are presented here as such. Estimating derivatives of the specific intensity when solving radiative transfers using a Monte-Carlo algorithm is challenging. Finite differences are often not sufficiently accurate and directly estimating the derivative from a specific Monte-Carlo algorithm can lead to arduous formal or numerical developments. The proposition here is to work from the radiative transfer equation and its boundary conditions to design a physical model for each derivatives. Only then Monte-Carlo algorithms are built from the derivatives differential equations using the usual equivalent path integral. Since the same methodology is applied to the specific intensity spatial derivative, angular derivative and geometric sensitivity we chose to keep the same writing structure for all three parts so that all common ideas and developments appears exactly the same. We believe this choice to be coherent to facilitate the reader's understanding. Finally, these are preliminary versions of the final papers: for each parts the theory is fully described, but, although they have been implemented, the examples and algorithms sections are not always complete. This will be mentioned in the introductions of the concerned sections.

physics.app-ph↗

arXiv2022

A Position-Free Path Integral for Homogeneous Slabs and Multiple Scattering on Smith Microfacets

We consider the problem of multiple scattering on Smith microfacets. This problem is equivalent to computing volumetric light transport in a homogeneous slab. Although the symmetry of the slab allows for significant simplification, fully analytic solutions are scarce and not general enough for most applications. Standard Monte Carlo simulation, although general, is expensive and leads to variance that must be dealt with. We present the first unbiased, truly position-free path integral for evaluating the BSDF of a homogeneous slab. We collapse the spatially-1D path integral of previous works to a position-free form using an analytical preintegration of collision distances. Evaluation of the resulting path integral, which now contains only directions, reduces to simple recursive manipulation of exponential distributions. Applying Monte Carlo to solve the reduced integration problem leads to lower variance. Our new algorithm allows us to render multiple scattering on Smith microfacets with less variance than prior work, and, in the case of conductors, to do so without any bias. Additionally, our algorithm can also be used to accelerate the rendering of BSDFs containing volumetrically scattering layers, at reduced variance compared to standard Monte Carlo integration.

cs.GR↗

arXiv2021

An analytic BRDF for materials with spherical Lambertian scatterers

We present a new analytic BRDF for porous materials comprised of spherical Lambertian scatterers. The BRDF has a single parameter: the albedo of the Lambertian particles. The resulting appearance exhibits strong back scattering and saturation effects that height-field-based models such as Oren-Nayar cannot reproduce.

cs.GR↗

arXiv2021

An unbiased ray-marching transmittance estimator

We present an in-depth analysis of the sources of variance in state-of-the-art unbiased volumetric transmittance estimators, and propose several new methods for improving their efficiency. These combine to produce a single estimator that is universally optimal relative to prior work, with up to several orders of magnitude lower variance at the same cost, and has zero variance for any ray with non-varying extinction. We first reduce the variance of truncated power-series estimators using a novel efficient application of U-statistics. We then greatly reduce the average expansion order of the power series and redistribute density evaluations to filter the optical depth estimates with an equidistant sampling comb. Combined with the use of an online control variate built from a sampled mean density estimate, the resulting estimator effectively performs ray marching most of the time while using rarely-sampled higher order terms to correct the bias.

cs.GR↗

arXiv2020

Green's functions for non-classical transport with general anisotropic scattering

In non-classical linear transport the chord length distribution between collisions is non-exponential and attenuation does not respect Beer's law. Generalized radiative transfer (GRT) extends the classical theory to account for such two-point correlation between collisions and neglects all higher order correlations. For this form of transport, we derive the exact time-independent Green's functions for the isotropic point source in infinite 3D homogeneous media with general anisotropic scattering. Green's functions for both collision rate density, which characterizes absorption and reaction rates in the system, and radiance/flux, which characterizes displacement of radiation/particles, are solved in Fourier space. We validate the derivations using gamma random flights to produce the first anisotropic scattering benchmark solutions for the generalized linear Boltzmann equation. For gamma-4 flights with linearly anisotropic scattering and gamma-6 flights with Rayleigh scattering the collision rate density is found explicitly in real space as a sum of diffusion modes.

cond-mat.stat-mech↗

arXiv2019

A Reciprocal Formulation of Nonexponential Radiative Transfer. 2: Monte Carlo Estimation and Diffusion Approximation

When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to angular flux / radiance because the macroscopic cross-section $Σ_t(s)$ depends on the distance $s$ to the previous collision or boundary. 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, which reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.

cond-mat.stat-mech↗

arXiv2019

Radiative transfer in half spaces of arbitrary dimension

We solve the classic albedo and Milne problems of plane-parallel illumination of an isotropically-scattering half-space when generalized to a Euclidean domain $\mathbb{R}^d$ for arbitrary $d \ge 1$. A continuous family of pseudo-problems and related $H$ functions arises and includes the classical 3D solutions, as well as 2D "Flatland" and rod-model solutions, as special cases. The Case-style eigenmode method is applied to the general problem and the internal scalar densities, emerging distributions, and their respective moments are expressed in closed-form. Universal properties invariant to dimension $d$ are highlighted and we find that a discrete diffusion mode is not universal for $d > 3$ in absorbing media. We also find unexpected correspondences between differing dimensions and between anisotropic 3D scattering and isotropic scattering in high dimension.

physics.class-ph↗

arXiv2019

A Reciprocal Formulation of Nonexponential Radiative Transfer. 3: Binary Mixtures

We derive the form of reciprocal generalized radiative transfer (RGRT) that includes the Levermore-Pomraning attenuation law for paths leaving a deterministic origin. The resulting model describes linear transport within multi-dimensional stochastic binary mixtures with Markovian mixing statistics and nonstochastic albedo and phase function. The derivation includes a new attenuation law and related free-path distribution between collisions, which was previously estimated using a Monte Carlo approach. We also show that previous stationary descriptions of binary mixtures with binomial mixing statistics reduce to exponential attenuations and, thus, have an exact homogenization within classical radiative transfer.

cond-mat.stat-mech↗

arXiv2018

Isotropic Scattering in a Flatland Half-Space

We solve the Milne, constant-source and albedo problems for isotropic scattering in a two-dimensional "Flatland" half-space via the Wiener-Hopf method. The Flatland $H$-function is derived and benchmark values and some identities unique to Flatland are presented. A number of the derivations are supported by Monte Carlo simulation.

physics.class-ph↗

arXiv2018

A reciprocal formulation of non-exponential radiative transfer. 1: Sketch and motivation

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.

physics.comp-ph↗

arXiv2014

Rigorous asymptotic and moment-preserving diffusion approximations for generalized linear Boltzmann transport in arbitrary dimension

We derive new diffusion solutions to the monoenergetic generalized linear Boltzmann transport equation (GLBE) 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 non-classical transport theories (non-exponential 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 small source-detector separations than either $P_1$ diffusion or rigorous asymptotic diffusion. These diffusion-based approximations are exact in the first two even spatial moments, which we derive explicitly for various non-classical transport types. We also discuss connections between the random-flight-theory derivation of the Green's function and the discrete spectrum of the transport operator and report some new observations regarding the discrete eigenvalues of the transport operator for general dimensions and free-path distributions.

cs.GR↗

arXiv2013

A Dual-Beam Method-of-Images 3D Searchlight BSSRDF

We present a novel BSSRDF for rendering translucent materials. Angular effects lacking in previous BSSRDF models are incorporated by using a dual-beam formulation. We employ a Placzek's Lemma interpretation of the method of images and discard diffusion theory. Instead, we derive a plane-parallel transformation of the BSSRDF to form the associated BRDF and optimize the image confiurations such that the BRDF is close to the known analytic solutions for the associated albedo problem. This ensures reciprocity, accurate colors, and provides an automatic level-of-detail transition for translucent objects that appear at various distances in an image. Despite optimizing the subsurface fluence in a plane-parallel setting, we find that this also leads to fairly accurate fluence distributions throughout the volume in the original 3D searchlight problem. Our method-of-images modifications can also improve the accuracy of previous BSSRDFs.

cs.GR↗