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First-return statistics in bounded radiative transport: A Motzkin polynomial framework

Claude Zeller, Robert Cordery

TL;DR

This work addresses first-return statistics in bounded radiative transport by mapping the three-dimensional anisotropic random walk under Henyey–Greenstein scattering to a tractable one-dimensional Motzkin polynomial framework. A Boundary Truncation Factor (BTF) is introduced to capture boundary-induced integral truncations, with the BTF exhibiting a Lorentzian (Cauchy-like) dependence on path length and anisotropy; the approach yields first-return probabilities in agreement with 3D Monte Carlo to within about 2% for moderate anisotropy. Theoretical development includes exact results for the two- and three-step cases, a computational 3D-to-1D mapping algorithm, and Monte Carlo validation across a wide parameter range. The method provides an efficient alternative to full 3D simulations for semi-infinite media and enables robust backscattering coefficient computation directly from phase-function integrals, with implications for tissue optics, atmospheric radiative transfer, and material appearance modeling.

Abstract

A photon entering a scattering medium executes a three-dimensional random walk determined by the Henyey-Greenstein phase function. The photon either reaches the boundary for a first passage or is absorbed. Projecting the walk onto the axial direction produces a one-dimensional alternating process whose peaks and valleys correspond to changes in the sign of the projected step. This reduction preserves first-return and first-passage events and leads to a representation in terms of Motzkin-type polynomials. The analytical formulation is complete except for boundary-constrained return terms, which appear as high-order integrals. We treat these contributions with a single truncation factor determined from Monte Carlo simulations of first-return distributions over a wide range of anisotropy g and scattering steps ms. The resulting factor follows a Cauchy distribution. Incorporating it yields first-return probabilities in agreement with full three-dimensional Monte Carlo to within 2% for g<=0.7. The approach gives backscattering coefficients from phase-function integrals and provides an efficient alternative to full three-dimensional simulations for problems of radiative transport in semi-infinite media.

First-return statistics in bounded radiative transport: A Motzkin polynomial framework

TL;DR

This work addresses first-return statistics in bounded radiative transport by mapping the three-dimensional anisotropic random walk under Henyey–Greenstein scattering to a tractable one-dimensional Motzkin polynomial framework. A Boundary Truncation Factor (BTF) is introduced to capture boundary-induced integral truncations, with the BTF exhibiting a Lorentzian (Cauchy-like) dependence on path length and anisotropy; the approach yields first-return probabilities in agreement with 3D Monte Carlo to within about 2% for moderate anisotropy. Theoretical development includes exact results for the two- and three-step cases, a computational 3D-to-1D mapping algorithm, and Monte Carlo validation across a wide parameter range. The method provides an efficient alternative to full 3D simulations for semi-infinite media and enables robust backscattering coefficient computation directly from phase-function integrals, with implications for tissue optics, atmospheric radiative transfer, and material appearance modeling.

Abstract

A photon entering a scattering medium executes a three-dimensional random walk determined by the Henyey-Greenstein phase function. The photon either reaches the boundary for a first passage or is absorbed. Projecting the walk onto the axial direction produces a one-dimensional alternating process whose peaks and valleys correspond to changes in the sign of the projected step. This reduction preserves first-return and first-passage events and leads to a representation in terms of Motzkin-type polynomials. The analytical formulation is complete except for boundary-constrained return terms, which appear as high-order integrals. We treat these contributions with a single truncation factor determined from Monte Carlo simulations of first-return distributions over a wide range of anisotropy g and scattering steps ms. The resulting factor follows a Cauchy distribution. Incorporating it yields first-return probabilities in agreement with full three-dimensional Monte Carlo to within 2% for g<=0.7. The approach gives backscattering coefficients from phase-function integrals and provides an efficient alternative to full three-dimensional simulations for problems of radiative transport in semi-infinite media.

Paper Structure

This paper contains 20 sections, 25 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and background
  2. Applications
  3. Paper organization
  4. One-dimensional first-passage theory
  5. Radiative transfer and the KM equation
  6. Catalan numbers and combinatorial structure
  7. Motzkin polynomials: extension to forward scattering
  8. Projection onto one dimension
  9. Comparison with generalized Kubelka--Munk
  10. Henyey--Greenstein phase function
  11. Boundary Truncation Factor framework
  12. Physical formulation
  13. Monte Carlo determination
  14. Mathematical representation
  15. Theoretical determination of backscattering coefficient
  16. ...and 5 more sections

Figures (1)

  • Figure 1: Comparison of Monte Carlo data with old and new theoretical fits.