The Neugebauer equations are a set of equations used to model
color printing
Color printing or colour printing is the reproduction of an image or text in color (as opposed to simpler black and white
or monochrome printing). Any natural scene or color photograph can be optically and physiologically dissected into thre ...
systems, developed by
Hans E. J. Neugebauer.
They were intended to predict the color produced by a combination of
halftones printed in
cyan, magenta, and yellow inks.
The equations estimate the
reflectance
The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic ...
(in
CIE XYZ
The CIE 1931 color spaces are the first defined quantitative links between distributions of wavelengths in the electromagnetic visible spectrum, and physiologically perceived colors in human color vision. The mathematical relationships that defi ...
coordinates or as a function of wavelength) as a function of the reflectance of the 8 possible combinations of CMY inks (or the 16 combinations of CMYK inks), weighted by the area they take up on the paper. In wavelength form:
:
where ''R
i''(''λ'') is the reflectance of ink combination ''i'', and ''w
i'' is the relative proportions of the 16 colors in a uniformly colored patch. The weights are dependent on the halftone pattern and possibly subject to various forms of
dot gain
Dot gain, or tonal value increase, is a phenomenon in offset lithography and some other forms of printing which causes printed material to look darker than intended. It is caused by halftone dots growing in area between the original printing film ...
.
Light can interact with the paper and ink in more complex ways. The
Yule–Nielsen correction takes into account light entering through blank regions and re-emerging through ink:
:
The factor ''n'' would be 2 for a perfectly diffusing
Lambertian paper substrate, but can be adjusted based on empirical measurements. Further considerations of the optics, such as multiple internal reflections, can be added at the price of additional complexity.
In order to achieve a desired reflectance, these equations have to be inverted to produce the actual dot areas or digital values sent to the printer, a nontrivial operation that may have multiple solutions.
See also
*
CMYK color model
The CMYK color model (also known as process color, or four color) is a subtractive color model, based on the CMY color model, used in color printing, and is also used to describe the printing process itself. The abbreviation ''CMYK'' refers ...
References
Equations
Color
Printing
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