Murray-Davies formula

In printing technology, the Murray-Davies formula is a formula for calculating the tonal value on screened prints, films and printing forms.

Mathematical description

With the help of the Murray-Davies formula, the tonal value A of an area partially covered with color is determined from the reflected (previously: remitted) light quantities of the partially covered (grid area), the fully covered (full tone area) and the uncovered area (blank, paper white) . The tonal value is a physical measure, that is, it shows how dark a (e.g. rasterized) measurement spot appears to a measuring device or the human eye in relation to a fully covered measurement spot. Its value is given as a percentage of the optical effect of the full tone. Not only geometric conditions play a role (such as the fact that 50% of the total area is covered with color), but also other physical effects such as light catching . These effects mean that the tone value is higher than the value that would result from the geometric relationships.

From the reflected (previously: remitted) light rays (via blank) and (above the grid surface ), the degree of reflection (previously: degree of remission) for the measuring spot with grid is calculated. ${\displaystyle I_{\text{B}}}$ ${\displaystyle I_{\text{R}}}$ ${\displaystyle \beta _{\text{R}}}$

${\displaystyle \beta _{\text{R}}={\frac {I_{\text{R}}}{I_{\text{B}}}}}$

The same applies to the degree of reflection of the full tone. ${\displaystyle \beta _{\text{V}}}$

There is a variant with degrees of reflection for the Murray-Davies formula:

${\displaystyle A={\frac {1-\beta _{\text{R}}}{1-\beta _{\text{V}}}}\cdot 100\,\%}$

Often the optical (“densitometric”) densities are measured. The optical density of the grid spot is defined as a logarithmic measure: ${\displaystyle D_{\text{R}}}$

${\displaystyle D_{\text{R}}=\log {\frac {I_{\text{B}}}{I_{\text{R}}}}=-\log {\beta _{\text{R}}}}$

The same applies to the full tone density . ${\displaystyle D_{\text{V}}}$

The Murray-Davies equation as a variant with optical densities is:

${\displaystyle A={\frac {1-10^{-D_{\text{R}}}}{1-10^{-D_{\text{V}}}}}\cdot 100\,\%}$

This variant is found most frequently in literature and practice.

Earlier publications used the symbol for "optical effect of the grid area coverage" instead of . That tempted to see only the area coverage in it. In the Standard Offset process, the tonal value has therefore been chosen as decisive. ${\displaystyle A}$ ${\displaystyle F_{\text{D}}}$ ${\displaystyle A}$

Suitable measurement methods

In practice, measurements are carried out using densitometers or spectral densitometers . A densitometer (old design) measures luminous flux through a color filter and thereby largely selectively records the effect of cyan , magenta or yellow , or key (black). The densitometer indicates the measured values ​​as optical densities, with four-color printing usually color densities in values ​​from 0 to about 3.Modern spectral densitometers, on the other hand, measure the reflections over the entire visible spectral range and calculate the color density of the recognized process color from the spectral reflectance.

The measurable upper limit, the respective solid tone density, is included in the formula as a reference value . To determine the tone value, the measuring device must therefore first measure the solid tone density and then the grid density of the grid field of the same color. From a metrological point of view, this is the simplest method of integrally recording the luminous flux of the full tone value and the halftone value and then calculating the tone value as a percentage. ${\displaystyle D_{\text{V}}}$ ${\displaystyle D_{\text{R}}}$

A metrological alternative to integral luminous flux measurement with densitometers is planimetry . The optical influence of the light capture is not taken into account, so that the geometric tonal value is determined in this way, the practical relevance of which is limited exclusively to the determination of the tonal value transfer behavior of printing forms. In the past, the light and dark areas were measured under the microscope. Today devices are used for geometric image analysis - a digital process in which the software counts the differently bright pixels above and below a threshold value.

Yule-Nielsen formula

Because the Murray-Davies formula contains the physical phenomenon of light capture as an integral part, but does not represent it in a formula, Yule and Nielsen published a formula in 1951 with which the light capture effect can be quantified and eliminated. This formula extends or corrects the Murray-Davies formula by the light capture factor , which must be determined experimentally or estimated empirically. This makes it possible to determine the actual geometric area coverage despite the integral area coverage, i.e. the area perceived by the eye or the amount of light detected by the measuring device: ${\displaystyle n}$

${\displaystyle A={\frac {1-10^{-D_{\text{R}}/n}}{1-10^{-D_{\text{V}}/n}}}\cdot 100\,\%}$

The light capture factor (which is actually not a factor, but a divisor in the exponential correction term ) fluctuates around the value 1. The tonal value calculated using the Murray-Davies formula decreases if n> 1 and increases if n <1 is. Empirically determined examples are quoted by R. Riedl ${\displaystyle n}$ ${\displaystyle 1/n}$ ${\displaystyle A}$

n = 1.6 for an art paper, 140 g / m²,

n = 2.0 for a machine-coated paper 70 g / m²,

n = 1.70 for art paper (60L / cm),

n = 1.65 for coated paper,

n = 2.70 for uncoated paper and

n = 2.60 for DuPont Cromalin .

With the availability of image analysis devices, the determination of the light-trapping factor and thus the application of the Yule-Nielsen formula has lost its importance. The term “Yule-Nielsen” is still used as a synonym for the geometric tonal value. If no printing plate measuring device is available, the geometric area coverage on the offset printing plate can be determined with a spectral densitometer under the Yule-Nielsen function.

See also

• Dot gain

Individual evidence

1. ^ A. Murray: "Monochrome reproduction in photoengraving." - In: "Journal of the Franklin Institute" No. 221 (1936), pp. 721-744
2. ↑ Rudi Riedl, Dieter Neumann, Jürgen Teubner: "Technologie des Offsetdrucks", 1. Edition, VEB Fachbuchverlag Leipzig 1989, ISBN 3-343-00527-4 , p. 324.
3. ↑ J. Yule, W. Nielsen: “The penetration of light into paper and its effect on halftone reproduction.” - In: “TAGA Proceedings” Vol. 3 (1951), pp. 65-76
4. ↑ Rudi Riedl, Dieter Neumann, Jürgen Teubner: Technology of offset printing. 1st edition, VEB Fachbuchverlag Leipzig 1989, ISBN 3-343-00527-4 , p. 325.
5. ^ Brochure material Macbeth, a division of Kollmorgen AG, Zug, Switzerland
6. ↑ n-Value for General Conditions, Pearson, M. - TAGA Proceedings, Rochester, 1980, pp. 415-425.
7. ↑ Techkon: SpectroDensitometer SpectroDens, Software SpectroConnect. Manual 2012 ( Memento of the original from April 27, 2014 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.