Efficient BRDF Importance Sampling
Using a Factored Representation: Behind Figure #10

Jason Lawrence
Szymon Rusinkiewicz
Ravi Ramamoorthi

This document accompanies the SIGGRAPH 2004 paper that describes a factored BRDF representation useful for efficient importance sampling. Here, we provide the data used to generate Figure 10 in the hope that future researchers will be able to reproduce these results for improved evaluation of their own method. This document contains the followint items:
  • The images from Figure 10 rendered with a distribution path tracer that generates incoming directions for Monte Carlo estimation of scene radiance using (1) a best-fit Lafortune model and (2) our factored representation of the target BRDFs in the scene. We show rendered images for both strategies at equal sample counts of 40, 100, 300, 600 and 1200. The Lafortune strategy requires roughly 4x the number of samples to achieve the same quality as the factored BRDF representation.

  • The materials in the scene consist of 2 analytic BRDF models and 3 measured BRDF models from the Matusik dataset. We provide the factored representation of each BRDF along with their numerical Cumulative Distribution Functions (CDFs) for sampling. We also provide the parameters from a multi-lobe Lafortune model we fit to each BRDF using the non-linear Nelder-Mead simplex searching algorithm. In ease case, we used an equal number of samples of the original BRDF in both the factored representation and as the target dataset during the non-linear fit of the Lafortune model. We fit a 3-lobe (2 specular lobes + 1 diffuse lobe) Lafortune model to the Cook-Torrance BRDFs and a 2-lobe (1 specular lobe + 1 diffuse lobe) to the measured BRDF models. We used the techniques described in the Lafortune paper to sample these analytic BRDFs efficiently. We have also prepared notes on reconstruction and sampling of our factored BRDF representation.

  • The geometry in the scene along with each object's associated material index. These triangle meshes are binary PLY files.

  • The camera parameters used to image the scene.

  • The illumination in the scene.

  • A brief analysis of why sampling a best-fit Lafortune model performs so poorly for some of the BRDFS in this scene as compared to the factored representation.

Rendered Images

Lafortune (40 samples)
Lafortune Sampling (40 Samples)
Factored (40
Factored Sampling (40 Samples)
      (100 samples)
Lafortune Sampling (100 Samples)
Factored (100 samples)
Factored Sampling (100 Samples)
Lafortune (300
Lafortune Sampling (300 Samples)
Factored (300
Factored Sampling (300 Samples)
Lafortune (600 samples)
Lafortune Sampling (600 Samples)
Factored (600
Factored Sampling (600 Samples)
Lafortune (1200
Lafortune Sampling (1200 Samples)
Factored (1200
Factored Sampling (1200 Samples)

#1: Analytic Cook-Torrance BRDF---d=0.0, s=0.2, std=0.15, F_0={0.12,0.22,0.48} Lafortune Fit Factored (description) CDFs (description)
#2: Analytic Cook-Torrance BRDF---d=0.0, s=0.2, std=0.4, F_0={0.68,0.22,0.12} Lafortune Fit Factored CDFs
#3: Measured Metallic-Blue BRDF Lafortune Fit Factored CDFs
#4: Measured Nickel BRDF Lafortune Fit Factored CDFs
#5: Measured Plastic BRDF Lafortune Fit Factored CDFs

Scene Geometry
Table Cook-Torrance
Vase Cook-Torrance
Bowl Metallic-Blue
Teapot Body Nickel
Teapot Handle Plastic
Teapot Top Metallic-Blue
Teapot Wires Nickel
Teapot Bird Plastic

Camera Parameters
  • eye = (20.736, 3.9952, 0.3)
  • ref = (0.0, 1.9952, 0.3)
  • up = (0,1,0)
  • x-fov = .1963495

We use image-based illumination in this scene: the Uffizi environment captured by Paul Debevec.

One reasonable approach to sampling complex parametric or measured BRDFs is to compute a best-fit analytic BRDF that can be sampled directly. One such analytic BRDF is the Lafortune model. To explain why this strategy performs so poorly as compared to using our factored representation for certain BRDFs we use the measured nickel BRDF in this scene as an example. In this case, we fit a 2-lobe (1 diffuse lobe + 1 "specular" lobe) Lafortune model to the original BRDF sampled at the same resolution as the factored approximation. Below we show false-color image visualizations (on a logarithmic scale) of the shape of the original BRDF, the best-fit Lafortune model and our factored representation for normal incidence and near-grazing incidence (elevation angle = 89 degrees from normal). These images give evidence that the Lafortune model is unable to effectively match the shape of the original BRDF and, consequently, sample it efficiently.
In this case, its the restricted parameterization of the incoming hemisphere that the Lafortune model allows that limits its effectiveness. Simply put, the parameterization that the Lafortune model provides is roughly of the "R dot V" variety. For these BRDFs, this is not appropriate for the shape of the specular lobes. These BRDFs are better parameterized by "H dot N" which is the parameterization we selected to use for the factored representation. This illustrates why providing arbitrary parameterizations of the incoming hemisphere (e.g. both "R dot V" and "H dot N"), which the factored representation does allow, is critical for efficient representation of a wide variety of BRDFS.
In addition to parameterization, these 2D slices of the three BRDFs suggest the flexibility a numerical approximation provides over fitting a rigid, non-linear model to the original data. Large variations in the shape of the BRDF are better matched by the factored representation which directly approximates the fully tabulated BRDF over the incoming/outgoing directions considered. This flexibility results, in part, from the NMF algorithm computing a set of "basis images" (G_j in the paper) and "mixing weights" (F_j) specifically geared toward optimally approximating the set of BRDF slices contained in the original data matrix. This key difference between the factored representation and the Lafortune model would be true with any best-fit parametric model in most cases, even if it provided a more appopriate parameterization of the incoming hemisphere.
Taken together, this BRDF demonstrate the potential usefulness of a general, numerical approach to factoring BRDFs for the purpose of importance sampling. We refer to the paper for a more detailed comparison between sampling this factored representation and other best-fit analytic BRDF models.
Measured Nickel BRDF (normal incidence) Lafortune (normal incidence) Factored (normal incidence)
Measured Nickel BRDF (elevation = 89 degrees from normal) Lafortune (elevation = 89 degrees from normal) Factored (elevation = 89 degrees from normal)
These images are false-color visualizations of the logarithm of the values of the BRDF for a dense set of incoming directions at a fixed view. Each column corresponds to the original measured BRDF, the best-fit Lafortune BRDF as computed by the Nelder-Mead non-linear optimization algorithm and our factored representation (from left to right in that order). The top row of images are slices of these BRDFS for a normal view direction. The bottom row of images are slices at near grazing angle (i.e. the view is 89 degrees from normal). Some measurement noise is apparent at this grazing view (although it is largely exaggerated by the logarithmic transformation) and half of the incoming hemisphere is clipped by the horizon. The main thing to notice is how the Lafortune BRDF is unable to accurately represent the overall shape of the directional-diffuse lobe in this BRDF over its entire domain. The non-linear optimizer compensates for this by increasing the diffuse component of the BRDF in order to reduce the overall error of the fit. For sampling, these consequences are severe at grazing angles, where the actual specular lobe is undersampled in regions of high energy (producing bright white spots) and oversampled primarly in its diffuse component. For more specular materials (such as the measured plastic BRDF in our scene), this compensation is even more substantial, drastically reducing the sampling efficiency provided by a best-fit Lafortune BRDF. Even if this fit had been manually tuned to provide a more "defensive" sampling pattern it could never match the exact shape of the target BRDF nor is this manual tuning an easy process. For this reason, we argue that there are many BRDFs (such as those in Figure 10 of this paper) for which a general, numerical sampling approach is desirable.