Kubelka–Munk theory

In optics, the Kubelka–Munk theory devised by Paul Kubelka[1][2] and Franz Munk, is a fundamental approach to modelling the appearance of paint films.

[4] In their article, differential equations are developed using a two-stream approximation for light diffusing through a coating whose absorption and remission (back-scattering) coefficients are known.

While Kubelka entered this field through an interest in coatings, his work has influenced workers in other areas as well.

In the original article, there is a special case of interest to many fields is "the albedo of an infinitely thick coating".

[a]) While numerous early authors had developed similar two-constant equations, the mathematics of most of these was found to be consistent with the Kubelka–Munk treatment.

[5] Others added additional constants to produce more accurate models, but these generally did not find wide acceptance.

Due to its simplicity and its acceptable prediction accuracy in many industrial applications, the Kubelka–Munk model remains very popular.

Sometimes these improvements are touted as extensions of Kubelka–Munk theory, sometimes as embracing more general mathematics of which the Kubelka–Munk equation is a special case, and sometimes as an alternate approach.

The hiding power of a coating measures its ability to obscure a background of contrasting color.

Kubelka derived many additional formulas for a variety of other cases, which were published in the post-war years.

[10] While the Kubelka–Munk coefficients are assumed to be linear and independent quantities, the relationship fails in regions of strong absorption, such as in the case of dyed paper.

By accounting for this dependency, the anomalous behavior of the Kubelka–Munk coefficients in regions of strong absorption were fully explained.

[11] The band-gap energy of semiconductors is frequently determined from a Tauc plot, where the quantity

The work of Kubelka and Munk was seen as yielding a useful systematic approach to color mixing and matching.

By resolving the Kubelka–Munk equation for the ratio of absorption to scatter, one can obtain a "remission function":[15]

[16][17] There were far more mathematics to choose from, but the name Kubelka–Munk became widely regarded as synonymous with any technique that modeled diffuse radiation moving through layers of infinitesimal size.

This led to a situation analogous to the described in the section just above for pigments, where the analyte had little effect on the scatter, which was dominated by the KBr.

However, in the field of near-infrared spectroscopy, the samples are generally measured in their natural (often particulate) state, and deviations from linearity at higher absorption levels were routinely observed.

The term "failure of the Kubelka–Munk theory" has been applied because it does not "remain valid in strongly absorbing materials".

[21] In literature related to diffuse-reflection infrared Fourier-transform (DRIFT) spectra, "particularly specular reflection" is often identified as a culprit.

[23] In 2003,[24] Donald and Kevin Dahm illustrated the degree to which the continuous theories all suffer from the fundamental limitation of trying to model a discontinuous sample as a continuum and suggested that as long as the effect of this limitation is unexplored, there is little reason to search for other reasons for "failure".

Through work with the Dahm equation, we know that the ART function is constant for all sample thicknesses of the same material.

Using a simple system (albeit rather complex mathematics), it can be shown that continuous models correctly predict the R and

in the ART function, but do not correctly predict the fractions of incident light that are transmitted directly.

While the total assembly would behave the same in either direction, in order to apply the mathematics, we will need to use an intermediate step where it does not.

Kubelka[8] has shown by theory and experiment that remittance and absorption of a non-homogeneous specimen depend on the direction of illumination, whereas transmittance does not.

from the first layer that occurs in the denominator is the remission when illuminated from the reverse (not the forward) direction, so we will need to know the value for

The K–M paint-mixing algorithm has been adapted to directly use the RGB color model by Sochorová and Jamriška in 2021.

Their "Mixbox" approach works by converting the inputs into a version of CMYK (phthalo blue, quinacridone magenta, Hansa yellow, and titanium white) plus a residue (to account for the gamut difference), performing the K–M mixing in that latent space, and then producing the output in RGB.

This RGB adaptation makes it easier for digital painting software to integrate the more realistic K–M method.