A modiﬁed single-constant Kubelka–Munk model for color
prediction of pre-colored ﬁber blends
Received: 20 August 2017 / Accepted: 22 January 2018 / Published online: 2 February 2018
Ó Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract The goal of this work is to propose a
modiﬁed single-constant Kubelka–Munk model for
color prediction of pre-colored ﬁber blends. The
original single-constant Kubelka–Munk model fol-
lows the Duncan’s additivity theorem, assuming that
the optical coefﬁcients of individual components in a
turbid media are linear to their respective proportions.
However, the linear assumption is invalid for the
media of ﬁber blends due to the interactions between
primary ﬁbers, causing inaccurate color prediction of
the model. Aiming at improving the accuracy, the
single-constant Kubelka–Munk model was modiﬁed
by employing a new additivity formula. The new
additivity formula was established to achieve good
linearity of the optical coefﬁcients by modeling
interactions between primary ﬁbers as conﬁgurations.
Cotton ﬁbers blending samples were prepared to
assess the color prediction accuracy. The average
color difference of the proposed model was 0.91
CIEDE2000 unit, which was signiﬁcantly better than
that of the original model (* 5.48). The results
indicate the proposed model is much more suitable for
color prediction of pre-colored ﬁber blends.
Keywords Color prediction Á Pre-colored ﬁber
blends Á Single-constant Kubelka–Munk model Á
Additivity formula Á Linearity
Blending pre-colored ﬁbers to obtain a wide variety of
colors is an important coloration method in the textile
industry. Generally, the blends of a limited number of
pre-colored ﬁbers (or primaries) can obtain any
desired color within the color gamut by formulating
recipes. In this ﬁeld, one of the most important
problems is to seek the appropriate recipe for a given
color standard. Therefore, it is fundamental to estab-
lish color prediction models to describe the color
mixing mechanism or the corresponding relationship
between the recipes and the blended colors.
There has been a long history of work to derive
color prediction models for pre-colored ﬁber blends.
The existing models can be divided into two cate-
gories, namely theoretical models and experimental
models. The former are derived from the behavior of
light propagation within the mixture, while the latter
are trained from speciﬁc experimental data.
For the purpose of color prediction, experimental
models always attempt to ﬁnd a transfer function from
speciﬁc samples so that an additive model can be
established (Park and Stearns 1944). The models can
be written as,
C. Wei Á X. Wan (&) Á J. Li
School of Printing and Packaging, Wuhan University, No.
129 Luoyu Road, Wuhan 430079, China
Cellulose (2018) 25:2091–2102