Parametric Roughness Model with an Elementary Closed Form

Determine whether a parametric model for homogeneous semi-infinite microflake media can preserve an elementary closed-form multiple-scattering response while providing a parameter that plays the role of roughness.

Background

The paper derives an energy-preserving BRDF with an elementary closed form by using a specific one-sided quadratic normal distribution function (NDF). Although this construction accounts exactly for all scattering orders and supports direct importance sampling, the quadratic NDF is parameter-free and therefore cannot represent the range of appearances typically controlled by a microfacet roughness parameter.

The unresolved issue is whether the analytic structure enabling exact marginalization and summation can be retained in a broader parametric family of NDFs that includes a meaningful roughness parameter. This is the only explicitly stated unresolved problem in the paper; the earlier question of whether any Smith microfacet BRDF can combine exact multiple scattering with an elementary expression is presented as settled by the paper’s construction.