1. Parameterization of Tabulated BRDFs Ian Mallett (me), Cem Yuksel
University of Utah, USA
Hello, my name is Ian Mallett. Our paper is entitled “Parameterization of Tabulated BRDFs”.

2. Background / Motivation
BRDFs describe light scattering
Extremely important to realism
Measured data is realistic
e.g. Matusik et al
The bidirectional reflectance distribution function—or BRDF—describes how light scatters off of a surface, thereby defining its material's appearance. The BRDF is typically posed as a three-dimensional or four-dimensional function, and has become ubiquitous in graphics as a method of describing surface properties.
Accurate material rendering is crucial to realism—second, perhaps, only to global illumination in its importance. Therefore, recently much research has focused on various analytic BRDF models, particularly those based on microfacet theory.
However, the BRDF can also be acquired directly from measurements of real-world materials, capturing minute variations in surface reflectivity for the ultimate in realism. In this paper, we'll deal only with raw measured data stored tabularly in samples, not in various analytic models derived from such data.

3. Background / Motivation
What parameterization (mapping of data to BRDF's angle space) should we use?
θout
Specifically, to use measured data, one must first define a mapping between a BRDF's intrinsic angular space into a multi-dimensional table of samples.
After picking a basis—say, the popular half-angle basis—we need to decide how corresponding axes map onto each other. This, the key component of the mapping, is the data's parameterization. By default, the “uniform” parameterization maps them linearly, but the parameterization can be, in-general, any monotonic function.
The choice of parameterization has a tremendous effect on the storage efficiency of the data. For example, it is advantageous to devote more table data to accurately storing the specular peak and retroreflection exhibited by most BRDFs, as opposed to the BRDF's diffuse component, which is low-frequency and well-represented by a smaller allotment of table cells.
An intelligent choice of parameterization can generate any such desired distortion, and yet . . .
Parameterization
φin
BRDF
θin
BRDF Data Table

4. Background / Motivation
Literature lacks general methods to compare and generate parameterizations!
. . . the literature lacks general methods to compare and generate new parameterizations!
While different parameterizations have been used previously, there is no underlying, consistent, mathematical framework to motivate them, analyze their effectiveness, and develop new ones.

5. Contributions Framework to generate and evaluate parameterizations
Provide example parameterization and analysis
Discuss related technical and theoretical issues
Therefore, in this paper, we begin by providing such a framework.
We then validate our framework by using it to develop a novel parameterization. By way of example, we present a thorough theoretical and empirical analysis of this parameterization.
Finally, we discuss sampling issues that arise in the use of measured BRDF data.

6. Mathematical Framework
Want parameterization to importance-sample BRDF:
γ is angle state vector (θin,φin,θout,φout) s is parameter vector (sθin,sφin,sθout,sφout)
We describe how to formulate this in the paper
𝑑γ 𝑑𝑠 ∝𝐵𝑅𝐷𝐹 γ
𝑑γ 𝑑𝑠 ∝𝐵𝑅𝐷𝐹 γ
We begin with a brief overview of our mathematical framework.
The framework is based on the idea of controlling sample density directly. The best choice is to allocate more sample density in areas where the BRDF value is larger. A similar idea underlies importance sampling.
Hence, we can set sample density at any point to be proportional to the BRDF value there. The left-hand-side of this equation represents sample density, and is a differential ratio of angular space and table space. Again, here we set it to be proportional to the BRDF—the right-hand-side.
Although conceptually straightforward, the above equation immediately implies useful tools to generate and evaluate novel parameterizations. We discuss these, interesting side-effects, and necessary technical details in the paper.

7. Example Parameterization
Fit a quadratic Bézier curve for each table axis P2 P1 P0
We used our framework to generate and analyze an example parameterization for glossy BRDFs. We now discuss this parameterization to demonstrate and validate our framework.
When analyzing a BRDF’s intensity, we typically observe an S-shaped distribution of values along most table axes. A simple model for representing such deviations is a quadratic Bézier curve. Using a Bézier curve for each table axis to invert the deviation creates a table that better represents certain areas of the BRDF. For example, in the diagram on the right, table density is reallocated along the specular plane by distorting the mapping of azimuthal angles.
The curve is given a free variable that can be numerically fit to the data. A range of representative curves is shown on the left. The numerical fit optimizes the approximate proportionality of the data and the BRDF, allowing the parameterization to adapt to a range of different BRDFs.
Effect on BRDF data spacing:
Higher density near specular plane (green/blue axes)

8. Example Parameterization
We then evaluated this parameterization on several analytic models and the entirety of tabular, measured data from Matusik et al.
This plot shows the predicted relative performance of different parameterizations for that entire dataset. The estimate is calculated by using the total intensity metric; a heuristic provided by our framework.
As you can see, the prediction for the Bézier curve parameterization exceeds that of the uniform parameterization for all BRDFs.

9. Example Parameterization
Moreover, we found that quantitative comparisons of renderings validated these predictions. Here, we show such a comparison on two different materials.
As you can see, the nonlinearity of the Bézier curve allows the specular peak on the top row to be more accurately captured, while the curve's adaptivity still allows it to represent more diffuse, glossy materials (bottom row).
From this, we determine that the Bézier curve makes for a highly effective parameterization, demonstrating that our framework is a powerful tool for generating and analyzing parameterizations generally.
More information on this analysis is available in the paper.
More examples in paper . . .

10. Higher-Order Reconstruction Filtering
Important for measured data
We suggest cubic B-spline
Good results
Efficient GPU implementation (Sigg and Hadwiger 2005)
Although somewhat orthogonal to the main contribution of our mathematical framework, we briefly comment on the BRDF reconstruction problem and a solution; when using any sampled data, it is important to use appropriate reconstruction filtering. This turns out to be especially true for measured data for BRDFs, and is a concern regardless of parameterization.
We tried a number of different filters. Ultimately, we found that B-spline filters work well, primarily because ringing and undershooting are especially objectionable in a BRDF. As a further advantage, efficient GPU implementations of B-spline filtering exist.

11. Higher-Order Reconstruction Filtering
Higher-Order Filter (Cubic B-spline)
Linear Filter
On this slide, we show the importance of higher-order filtering.
On the left, a simple linear filter produces objectionable banding artifacts, which are most visible on highly-specular BRDFs such as this one.
On the right, higher-order filtering eliminates the problem.

12. Conclusion
Propose new mathematical framework to generate and analyze parameterizations
Validate our approach by generating and analyzing an example parameterization
Discuss higher-order filtering for measured BRDFs
In conclusion, we introduce a new mathematical framework for generating and analyzing parameterizations of BRDFs.
We then validate our approach by using our framework to develop a new parameterization. Our framework accurately predicts this parameterization's favorable qualities.
We then discussed higher-order filtering for measured BRDFs.
BRDFs are important for realism, and by improving the efficiency of their representation using the mathematical and analytical techniques we propose in the paper, measured BRDFs can be made both more bandwidth-efficient and practical.

13. Questions
Thank you for your attention, and are there any questions?

14. External Image Sources
BRDF picture: Derived from distribution_function#/media/File:BSDF05_800.png
BRDF diagram: remade from berlin.de/rapid_prototyping_11ws/rp_brdf/img/reflectio n_t.png
University of Utah and CGI images adapted from respective websites