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A demonstration how different 2D transformations (top right insets) of the microsurface affect the reflection off a rough silver plate. Top left: The original unmodified surface with shape-variant GTR distribution with tail γ = 1.7 and roughness α = 0.0121. A 10×10 grid covers the unit texture space to serve as a reference. Top right: The microsurface is stretched 5 times horizontally. This makes the surface smoother in that direction, so the highlight is compressed accordingly. Bottom left: A skew transform. Bottom right: An example of a non-linear transformation: T(u, v) = (u, v+2u5). The Jacobian matrix JT (top right inset) is a local linear approximation of T and we apply it as a local tangential transform. Notice that the prolonged and curved shape of the highlight to the right is due to more surface compression in that region.

Microsurface Transformations

Asen Atanasov Vladimir Koylazov Rossen Dimov Alexander Wilkie

Abstract

We derive a general result in microfacet theory: given an arbitrary microsurface defined via standard microfacet statistics, we show how to construct the statistics of its linearly transformed counterparts. A common use case of such transformations is to generate anisotropic versions of a given surface. Traditional anisotropic derivations based on varying the roughness of an isotropic distribution in an ellipse have a general closed-form formula only for the subclass of shape-invariant distributions. While our formulation is equivalent to these specific constructs, it is more general in two aspects: it leads to simple closed-form solutions for all distributions, including shape-variant ones, and works for all invertible 2D transform matrices. The latter is of particular importance in case of deformation of the macrosurface, since it can be approximated locally by a linear transformation in the tangent plane. We demonstrate our results using the Generalized Trowbridge-Reitz (GTR) distribution which is shape-invariant only in the special case of the popular Trowbridge-Reitz (GGX) distribution.

Teaser Video

BibTex Citation

				
					@article{https://doi.org/10.1111/cgf.14590,

author = {Atanasov, A. and Koylazov, V. and Dimov, R. and Wilkie, A.},

title = {Microsurface Transformations},

journal = {Computer Graphics Forum},

volume = {41},

number = {4},

pages = {105-116},

keywords = {CCS Concepts, • Computing methodologies → Rendering, Reflectance modeling},

doi = {https://doi.org/10.1111/cgf.14590},

url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/cgf.14590},

eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1111/cgf.14590},

abstract = {Abstract We derive a general result in microfacet theory: given an arbitrary microsurface defined via standard microfacet statistics, we show how to construct the statistics of its linearly transformed counterparts. A common use case of such transformations is to generate anisotropic versions of a given surface. Traditional anisotropic derivations based on varying the roughness of an isotropic distribution in an ellipse have a general closed-form formula only for the subclass of shape-invariant distributions. While our formulation is equivalent to these specific constructs, it is more general in two aspects: it leads to simple closed-form solutions for all distributions, including shape-variant ones, and works for all invertible 2D transform matrices. The latter is of particular importance in case of deformation of the macrosurface, since it can be approximated locally by a linear transformation in the tangent plane. We demonstrate our results using the Generalized Trowbridge-Reitz (GTR) distribution which is shape-invariant only in the special case of the popular Trowbridge-Reitz (GGX) distribution.},

year = {2022}

}