In this free e-book I provide an incomplete collection of deterministic and Monte-Carlo solutions to multiple-scattering problems in linear transport theory. References to literature containing derivations and related work are provided to help the reader discover further detail. Reference Monte Carlo code, solutions and Mathematica worksheets comparing the deterministic solutions to Monte Carlo are provided on github.

This work is partially a summary of some notes on transport theory I’ve made for myself over the years but also somewhat in response to Peter Shirley’s call at EGSR2011 in Prague for someone in the rendering community to volunteer to provide benchmark scenes for the rendering community, to which I volunteered (but only for subsurface scattering-related problems). These results are probably not the types of scenes Peter had in mind, but I do hope that some of these verified benchmark results will help people test their rendering code and hopefully inspire other readers to learn more about the fascinating mathematical structure behind the transport equation, in the limited scenarios where exact solutions can be written down.

Book (pdf)


For each of the major problems we provide C++ code that produces Monte Carlo reference output. The deterministic solutions are all implemented and cross-checked against Monte Carlo in Mathematica worksheets, which you can download and use to interactively explore the various simulations. Each Mathematica worksheet is also saved as a PDF (convenient if you don’t have Mathematica).

Phase Functions [pdf]
The Rod Model

  • Infinite Rod Isotropic Scattering [pdf]
  • Half Rod Isotropic ScatteringĀ [pdf]
  • Half Rod Anisotropic Scattering [pdf]
  • Finite Rod Isotropic ScatteringĀ [pdf]
  • Finite Rod Anisotropic ScatteringĀ [pdf]

Isotropic Point Source Infinite Flatland Medium

  • Isotropic scattering [pdf]

Isotropic Point Source Infinite 3D Medium

  • Isotropic scattering [pdf]
  • Linearly-Anisotropic scattering [pdf]

Isotropic Plane Source Infinite 3D Medium

  • Isotropic scattering [pdf]

Delta Plane Source Infinite 3D Medium

  • Isotropic scattering [pdf]

The Albedo Problem

  • Vacuum Boundary, Isotropic scattering [pdf]

Caseology and the H-function

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