Predicting Post-Foundation Skin Color With a Coverage-Dependent Optical Mixing Model Beyond the Continuous-Film Assumption

Abstract

Cosmetic foundation has traditionally been modeled as a continuous, optically homogeneous film over the skin, an assumption that underlies most color-prediction frameworks based on Kubelka-Munk theory. However, microscopic observation shows that, in practice, foundation deposits form spatially discrete particle distributions on a partially exposed substrate rather than a uniform film. Therefore, the perceived color is more naturally understood as an optical mixture of skin and foundation contributions, weighted by an effective coverage. In this work, we adopt this partial-coverage perspective and formulate a closed-form prediction of post-application CIE L*a*b* values as a smoothstep-weighted interpolation between bare-skin and foundation reference values, modulated by a small exponent that depends on the lightness difference between skin and foundation. The model was validated against 54 in vivo spectrophotometric measurements collected from the forearms of 6 subjects, covering 3 foundation lightness levels and 3 application strengths. The proposed model achieved a mean CIEDE2000 of 2.74 in-sample and 3.01 under leave-one-subject-out cross-validation, with the medium-shade regime reaching Delta E 00 approximate to 2, at or below the conventional perceptual threshold. The results further reveal a systematic dependence of the effective coverage slope on foundation lightness, with lighter foundations showing steeper saturation behavior. They also indicate that prediction accuracy is influenced by the foundation-reference measurement procedure. Although the limited subject pool does not yet permit fully generalized parameterization, the present work establishes a partial-coverage optical mixing framework and a practical closed-form model for predicting post-foundation skin color.