NVIDIA Research
Microfacet theory for non-uniform heightfields

Microfacet theory for non-uniform heightfields


Our asymmetric blending operator creates two new microfacet BSDFs (center) using the same blended NDF (GGX with roughness 0.02 and 0.8) but with different roughness applied to different elevation ranges of the microsurface (as seen in the top row profiles). The result is that while the interior of the surface remains unchanged, the silhouettes of the asymmetric BSDFs are different. At grazing angles, the spheres in the middle-right column appear rougher (note the reflection of the checkerboard onto the conductive spheres) because the rougher component of the mixture protrudes outward. This behavior cannot be achieved through a simple linear blend of two single-NDF BSDFs (as seen on the left).


We propose new methods for combining NDFs in microfacet theory, enabling a wider range of surface statistics. The new BSDFs that follow allow for independent adjustment of appearance at grazing angles, and can't be represented by linear blends of single-NDF BSDFs. We derive importance sampling for a symmetric operator that blends NDFs uniformly, and introduce a new asymmetric operator that supports NDF variation with elevation. We also extend Smith's model to support piecewise-constant NDF and material variations with elevation, and demonstrate accuracy via Monte Carlo simulations.

Assymetric heightfields

A Beckmann heightfield with symmetric displacements 𝑃(ℎ) (left) is transformed into an asymmetric heightfield (right) by applying a piecewise-linear rescaling of the displacement values. We propose a layered volumetric representation of the transformed heightfield with a height-dependent NDF. This leads to a new scattering model that accurately predicts the reflectance from asymmetric surfaces and produces novel behaviours at grazing angles.

Multiple heightfields with the same overall NDF

Five microsurfaces with the same NDF: a 50/50 linear blend of Beckmann NDFs with roughnesses 1.0 and 0.02. Two distinct two-layer asymmetric surfaces (a,b) are represented in our model by layering the two NDFs in each of the two possible orders. Two symmetric variations are made using a symmetric layering 25%, 50%, 25% shown in (c,d). A fine alternate layering of the two roughness (e) approaches a surface with a uniform blended NDF at all heights. All five microsurfaces lead to distinct BSDFs.

Damaged surfaces

Modeling inward damage of a gold microsurface using a two-layer configuration with roughnesses 0.01 and 0.8. Below each dragon render we show a render of an example microsurface that is consistent with the BSDF used on the image above. The moderately damaged renders (b-d) maintain the appearance of shiny gold while filling in the darker reflections and avoiding an adverse roughening of the silhouette.

Change in grazing angle scattering

Dual Trowbridge-Reitz NDFs (with equal split) applied to more complex geometry under smooth studio lighting. In both the conductor (a) and dielectric (b) setting, the asymmetric microsurfaces enable fine control over the reflective properties at grazing angles which cannot be achieved by a simple linear blend.


    author = {d'Eon, Eugene and Bitterli, Benedikt and Weidlich, Andrea and Zeltner, Tizian},
    title = {Microfacet theory for non-uniform heightfields},
    year = {2023},
    publisher = {Association for Computing Machinery},
    address = {New York, NY, USA},
    url = {https://doi.org/10.1145/3588432.3591486},
    doi = {10.1145/3588432.3591486},
    booktitle = {SIGGRAPH 2023 Conference Papers},
    numpages = {10},
    location = {Los Angeles, CA, USA}


Microfacet theory for non-uniform heightfields

Eugene d'Eon, Benedikt Bitterli, Andrea Weidlich, and Tizian Zeltner

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