ACM Trans. Graph. 36, 4, Article 109 (July 2017).

A Forward Scattering Dipole Model from a Functional Integral Approximation

Roald Frederickx     Philip Dutré
Graphics group, Department of Computer Science, KU Leuven

Comparison of dipole models and ground truth images for an index matched medium with a forward peaked Gaussian phase function (g = 0.9) and varying medium densities. The first two rows show the dipole models of Jensen et al. [2001] and d'Eon [2014], respectively. The third row ('iso. ref.') gives a path traced ground truth image for an isotropic scattering function as determined from similarity theory (hence it is the ground truth image for the diffusion models above it). The fifth row ('Frisvad') shows the result of the directional dipole model of Frisvad et al. [2014], followed by the anisotropic reference solution ('reference'), with the last two rows showing the result of our forward scattering dipole model and effective BRDF, respectively. The scattering parameters are given above each of the columns and all objects are of unit size. Our subsurface scattering model accurately captures the ground truth behaviour over the entire range of parameters, and our BRDF provides a useful and computationally efficient approximation for the optically dense media on the left.


Rendering translucent materials with physically based Monte Carlo methods tends to be computationally expensive due to the long chains of volumetric scattering interactions. In the case of strongly forward scattering materials, the problem gets compounded since each scattering interaction becomes highly anisotropic and near-specular. Various well-known approaches try to avoid the resulting sampling problem through analytical approximations based on diffusion theory. Although these methods are computationally efficient, their assumption of diffusive, isotropic scattering can lead to considerable errors when rendering forward scattering materials, even in the optically dense limit. In this paper, we present an analytical subsurface scattering model, derived with the explicit assumption of strong forward scattering. Our model is not based on diffusion theory, but follows from a connection that we identified between the functional integral formulation of radiative transport and the partition function of a worm-like chain in polymer physics. Our resulting model does not need a separate Monte Carlo solution for unscattered or single-scattered contributions, nor does it require ad-hoc regularization procedures. It has a single singularity by design, corresponding to the initial unscattered propagation, which can be accounted for by the extensive analytical importance sampling scheme that we provide. Our model captures the full behaviour of forward scattering media, ranging from unscattered straight-line propagation to the fully diffusive limit. Moreover, we derive a novel forward scattering BRDF as limiting case of our subsurface scattering model, which can be used in a level of detail hierarchy. We show how our model can be integrated in existing Monte Carlo rendering algorithms, and make comparisons to previous approaches.


Preprint(*) incl. author note



Supplementary material (1/2)

Supplementary material (2/2)


Example Scenes


  • A clarifying note has been added w.r.t. Eq. (11) after publication.

  • The importance sampling routines in the code repository linked above have been improved compared to the originally published version (i.e. compared to how they are described in the paper and supplementary material). The full details and derivation of the updated importance sampling will be made available in the upcoming PhD thesis of R. Frederickx.

(*) The definitive version can be found at the ACM digital library.

Roald Frederickx and Philip Dutré