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Enforcing Kubelka–Munk constraints for opaque paints
Coloration Technology ( IF 1.8 ) Pub Date : 2020-11-10 , DOI: 10.1111/cote.12497
Paul Centore 1
Affiliation  

The 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 K’s and S’s are non-negative, and C’s 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 realizable paint combinations is a convex polytope, which is a subset of that subspace. This paper reformulates KubelkaMunk estimation problems geometrically, as the problem of finding the point on that polytope that is closest to a target vector. The solutions to the reformulated problem are always physically realizable. If feasible, a worker could solve the reformulated problem with a ready-made commercial solver. Otherwise, the Gilbert-Johnson-Keerthi (GJK) algorithm is recommended as especially suitable for Kubelka-Munk estimation; this algorithm has been tested on some simple cases and released as open-source code.

中文翻译:

对不透明涂料强制执行 Kubelka-Munk 约束

Kubelka-Munk 模型将涂料混合物的颜色与组成涂料的吸收和散射系数(K 和 S)以及它们在混合物中的浓度 (C) 相关联。所有 K 和 S 都是非负的,并且 C 在物理上被限制在 0 和 1 之间。标准估计程序将 Kubelka-Munk 关系转换为超定线性系统,并应用普通最小二乘法 (OLS)。然而,OLS 有时会产生小于 0 或大于 1 的系数或浓度。 这些物理上不可能的解决方案之所以出现是因为 OLS 将目标矢量(例如所需的反射光谱)投影到矢量子空间上,而实际上物理上的集合可实现的油漆组合是一个凸多面体,它是该子空间的一个子集。本文以几何方式重新表述 KubelkaMunk 估计问题,作为在该多胞体上找到最接近目标向量的点的问题。重新制定问题的解决方案总是可以在物理上实现的。如果可行,工人可以使用现成的商业求解器解决重新制定的问题。否则,推荐使用 Gilbert-Johnson-Keerthi (GJK) 算法,因为它特别适合 Kubelka-Munk 估计;该算法已经在一些简单的情况下进行了测试,并作为开源代码发布。推荐使用 Gilbert-Johnson-Keerthi (GJK) 算法,因为它特别适用于 Kubelka-Munk 估计;该算法已经在一些简单的情况下进行了测试,并作为开源代码发布。推荐使用 Gilbert-Johnson-Keerthi (GJK) 算法,因为它特别适用于 Kubelka-Munk 估计;该算法已经在一些简单的情况下进行了测试,并作为开源代码发布。
更新日期:2020-11-10
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