A Bayesian Inference Framework for
Procedural Material Parameter Estimation:
Supplemental Materials

In this supplemental material, we provide detailed descriptions on the procedural material models used in the paper. Additionally, we show animated results for optimization (i.e., posterior maximization) and posterior sampling (using Hamiltonian Monte Carlo). Lastly, we demonstrate another application of our posterior sampling: recovering similarity relations of material scattering parameters.

Procedural Material Models

For all models, light gives light intensity, while iSigma gives standard deviation of the vignetting falloff in centimeters. We use truncated Gaussians for all prior distributions.

Clicking the images below to control the optimization/sampling animations:
left click the images below to start/pause; right click to reset the animations.
Results
Synthetic Input

We use the neural-network-based summary function for all results in this section.

Bumpy surface

Leather

Plaster

Metallic flake

Brushed metal

Wood

Real Input

We use the neural-network-based summary function for all results in this section except for Metallic flake (Bins of radial bands) and Brushed metal (Bins of vertical bands + 1D FFT).

Bumpy surface

Leather

Plaster

Metallic flake

Brushed metal

Wood

Translucent Materials

Translucent materials allow light to penetrate their surface and scatter within the interior. Our forward model focuses on semi-infinite and homogeneous media with a Henyey-Greenstein (HG) phase function lit by a point light source depicted using the following parameters: \[ {\boldsymbol\theta} = (\sigma_s, \sigma_a, g, E), \] where \(\sigma_s\) and \(\sigma_a\) are respectively the material’s scattering and absorption coefficients, \( g \) indicates the phase function’s first Legendre moment (average cosine), and \( E \) is the light intensity. Further, we assume that the medium’s refractive index \( \eta \) is known.

To render these materials, we use Monte Carlo volumetric path tracing (VPT) with a specialized next-event estimation (NEE) technique capable of going through refractive interfaces. This makes our forward evaluation \( f \) to involve a complex rendering component \( R \) which is itself stochastic but still fully differentiable.

The radiative transfer material parameter space (i.e., \( \sigma_s \), \( \sigma_a \), and \( g \)) is known to be (approximately) over-complete, especially with the absence of sharp geometries (which applies in our flat configuration). That is, multiple combinations of these parameters can yield roughly identical appearances. This effect is mathematically captured by similarity relations. Specifically, the first-order variant of the similarity relations states that two scattering media with parameters \( (\sigma_s, \sigma_a, g) \) and \( (\sigma_s^*, \sigma_a^*, g^*) \) will have approximately identical appearances if \[ \sigma_a = \sigma_a^*, \quad \sigma_s (1 - g) = \sigma_s^* (1 - g^*). \]

The presence of similarity relations has been a challenge for solving inverse scattering problems because of the fundamental difficulty in distinguishing parameters within the same similarity class. Our technique, which provides posterior distributions rather than single estimates, is capable of automatically detecting such structures.

The figure below shows posterior distributions sampled with our method for two synthetic input images. Unlike optimization-based methods which usually converge to some arbitrary locations near the similarity curve, our technique is able to detect the full structure of these "areas of confusion". The posterior distributions provided by our method match the similarity theory predictions very well.

Demo code

We provide demo code to generate different materials.