Enforcing Kubelka–Munk constraints for opaque paints

Abstract

AbstractThe Kubelka–Munk model relates the colours of paint mixtures to the absorption and scattering coefficients (K and S) of the constituent paints, and to their concentrations (C) in the mixtures. All Ks and Ss are non‐negative, and Cs are physically constrained to be between 0 and 1. Standard estimation procedures cast the Kubelka–Munk relationships as an overdetermined linear system and apply ordinary least squares (OLS). OLS, however, sometimes produces coefficients or concentrations that are less than 0 or greater than 1. These physically impossible solutions occur because OLS projects a target vector (such as a desired reflectance spectrum) onto a vector subspace, while in fact the set of physically realisable paint combinations is a convex polytope, which is a subset of that subspace. This paper reformulates Kubelka–Munk estimation problems geometrically, as the problem of finding the point on that polytope which is closest to a target vector. The solutions to the reformulated problem are always physically realisable. If feasible, a worker could solve the reformulated problem with a ready‐made commercial solver. Otherwise, the Gilbert–Johnson–Keerthi algorithm is recommended as particularly suitable for Kubelka–Munk estimation; this algorithm has been tested on some simple cases and released as open‐source code.