The Murray-Davies Equation: An Origin Story

In 1936, Alexander Murray, Supervisor of Graphic Arts Research at Kodak Laboratories in Rochester, NY,^​2​ published a simple equation that would have a profound impact on the printing industry.^​3​ This simple equation, shown below as it was originally published, predicted the density of a halftone given the known percent area covered by halftone dots and the reflectance of the solid.

In the equation, D is the density of the tint, a is the dot area, and r is the reflectance of the solid (r may be rewritten in terms of the solid density, D_s, using the equation, r=10^-Ds ). While this equation was published by Murray, it was suggested to him in a personal correspondence by Edward Roy Davies, head of the Kodak Research Laboratory in Harrow, England.^​1​ Murray validated Davies’ equation experimentally and added an expansion to account for the physical increase in dot size that occurs when ink is transferred from plate to substrate. By a simple rearrangement of terms, the Murray-Davies equation (as it later became known) can be used to calculate the percent area of a printed halftone. In 1951, John Yule and Waldo J. Nielsen showed the Murray-Davies equation was not a great predictor of density, given a known dot area.^​4​ However, it has since become the standard equation for the calculation of dot are and dot gain for print process control.  

The exact sequence of events that lead the Murray-Davies equation to its prominent position requires additional research. It is the equation specified in ISO standard 12647-2 for the calculation of dot gain, and some have argued that the definition of “dot gain” includes the Murray-Davies equation.

Gary Field^​*​ recently suggested to me that perhaps “the equation was meant more for research purposes than process control” because “very few printers would have had the means of deriving dot area via [Murray-Davies], nor of putting it to practical use.”^​†​ Regarding the Yule-Nielsen equation, Field stated “the n-factor refined the initial model, but this probably made it even further removed from daily practice.” Access to the kinds of computational tools needed to compute the appropriate n-factor were also limited to the general printing population. Milton Pearson, of RIT, did suggest a standard range of n-values for the Yule-Nielsen equation in 1980, but the Murray-Davies equation still managed to defend its position.

Many factors affected the process of accurately reproducing a tone scale in the early 20^th century. In three-color workflows, objects were photographed through red, green, and blue filters, producing continuous tone negatives, referred to as “unstructured” negatives (using either wet or dry plates).^​5​ A “structured” positive was then produced by copying the unstructured negative through a cross-line screen onto another glass plate negative. This structured positive contained halftone dots rather than smooth gradations. The final structured negative was made using a contact exposure of the structured positive onto another negative, and was then used to expose the letterpress printing plate. After exposure, the plate was chemically etched to remove areas not exposed to light through the structured negative. Finally, a print was produced using the plate on a printing press. Tone reproduction was successful if there was a direct correspondence between the density of the original tone scale and the density of the printed reproduction.

The primary means of achieving accurate tone reproduction in the 1930s was through chemical re-etching, largely a craft process with which it was difficult to achieve repeatable results. For example, halftone highlight dots may be made larger during initial exposure, then refined with additional etching. Re-etching could be avoided by building tone reproduction into the negative used to expose the plate. The primary tool for mapping tone reproduction in photographic materials at the time was sensitometry, the science of how light-sensitive materials respond to light and chemistry.

Missing in the early 20^th century was a way to accurately relate print density to halftone dot area. This understanding was essential to calculating plate curve corrections. However, as mentioned above, it was not desirable to correct the plates themselves since that would involve re-etching, also referred to as “fine-etching.” In 1927, Herbert Ives (not to be confused with his Father, the inventor Frederic Eugene Ives), described the goal of tone reproduction as reducing the need for fine etching of letterpress plates.^​5​ Murray reiterated this idea in 1936, stating that it was desirable to produce “perfect plates without the intervention of manual methods of correction”.^​3​ Therefore, curve corrections had to be built into the negative.

Refining the tone reproduction process was essential for multi-color process printing. According to Alexander Murray, “a purely photographic process of multi-color plate-making could not be realized unless it gave a practically perfect monochrome reproduction of the density scale on each plate.” In 1936, Murray sought to determine experimentally whether it was possible to achieve perfect tone reproduction using a halftone process without re-etching.^​3​

Murray described halftones as “a structured image in which variations in density are rendered by variations in the ratio between areas of solid ink and areas of blank paper. On a paper proof in black ink the scale of densities is limited by the reflecting power of the paper itself at the highlight end, and the density of unbroken black ink at the shadow end.” Previous researchers had not stated so clearly the importance of paper white and maximum solid density on halftone tone reproduction. Murray then hypothesized that “since the scale of densities in a halftone printing plate is purely a structural differentiation between the two limits of ‘solid’ and ‘blank,’ it should be possible to arrive at density values by measuring dots under a microscope and calculating their areas.” In Murray’s experiment, he measured the dot sizes on the plate and proof using a microscope and accounted for the imperfect dot shapes by measuring along different dot dimensions and averaging across several dots.

Print density was measured using an Eastman Transmission and Reflection Densitometer, a device in which density was determined by matching the visual appearance of a sample to the appearance of a tone step from a calibrated density scale projected onto a visual field surrounding the sample.^​6​

Visual densitometers were used well into the 1960s, before handheld electronic densitometers became available. As Field recalled, “in the early 1960s, when I entered the industry, we used Kodak densitometers based upon a visual matching process to determine density.”

Having the microscopically measured dot areas of halftone patches on different proofs and the densities of those halftone patches, the next step was to model the relationship between density and tone value. It is here that Davies’ equation comes into play. The Murray-Davies equation assumes the structure of a printed dot is uniform. Both Murray and Davies knew that perfect uniformity was not a physical reality, but Murray understood that in each dot, despite the known non-uniformity, “the average density is identical with the density of a large unstructured ink area.” Murray also gave credit to Davies for pointing out that tone reproduction curves of the same plate printed with inks of different densities would be different, and then validated that claim experimentally.

Murray established baseline measurements in his experiments using a “chalked-up” engraving, the process of coating a relief letterpress printing plate with finely powdered chalk to give it the appearance of a solid print. The dots of chalked-up engravings were be perfectly shaped and of uniform density.

The plates themselves were made using the indirect method, in which a continuous tone negatives were made first (through the colored filters for multi-color printing), and then positive halftone screens were created from the continuous tone negatives. The debate between the use of the indirect and direct processes (photographing the object through a screen and filter in the same step) was discussed in a previous post.

The indirect method was the favored method of Herbert Ives due to the added flexibility afforded by that method to manufacture plates of the desired structure. Photographic plates were characterized by density as a function of log exposure, referred to as “characteristic curves,” a method originated by Hurter and Driffield in 1880.^​5​ In photographic negatives, exposure to lots of light yields a high density (high opacity) and exposure to little light yields a low density (low opacity). Unstructured and structured negatives require different characteristic curves. Unstructured negatives require a gradual change in density as a function of exposure (log exposure, in the case of H-D curves), bounded by the minimum density of the unexposed negative (the D_min, or the “fog”) and the maximum possible density a negative (the D_max). Structured negatives are very high contrast, yielding low densities for all except the highest exposures, at which high densities were achieved. The ideal characteristic curve for structured negatives, illustrated by Ives below, was necessary to produce well-defined halftone dots,^​5​ where the x-axis is log exposure and y-axis is Density.

Emulsions of the day were not capable of producing negatives with the “ideal” characteristic curve shape. Ives sough to determine how to produce such a negative with the high-contrast characteristics through experimentation. He used collodion wet plate negatives and worked with a local photo-engraver to reproduce a neutral tint wedge. Factors were varied, such as: camera aperture (e.g. f/16), screen-to-plate distance (e.g. 3/8″), exposure (e.g. determined such that the most exposed dots had a transmission of about 1%, and highest transmissions around 95%), and development. It was common practice to use multiple exposures to achieve the desired plate curve.^​‡​ Ives settled on 3-4 exposures using different apertures (affecting dot shape and amount of light), starting with exposures to form large dots and concluding with exposures to form small dots.  Large dots were not affected by the formation of small dots upon them, allowing for the layering of exposures.

The four exposure steps included:

1. An exposure with a very large aperture that yielded the largest dots and yielded negatives with a steep, high-contrast characteristic curve (all-or-nothing response).
2. An exposure with a smaller aperture that formed dots of a large size, but the dot image has a uniform appearance, preventing the formation of small dots.
3. An exposure with a very small aperture that formed small dots on the screen but did not increase in size with increased exposure.
4. An exposure to a uniform white field with a small aperture to fine-tune the base of the curve.

An example of four characteristic curves of negatives with exposures A-D is shown below, reproduced from Ives article.

Ives’ goal in proposing the multiple exposure process was to standardize the creation of plates for three-color work and reduce the need for re-etching. He stated that, for three-color work, “it is imperative that the three color plates all be correct in their tone relations.” He continued, “it is of interest in this connection to recall that there has been a long standing disagreement between ‘practical’ three color printers and the scientific exponents of color photography as to the colors of the inks to be used, and as to the adequacy of three color analysis to produce faithful results.”

In his experiments, Murray compared dot measurements on plate and proof and noticed “a gain in area for the dots in the proof at every step. This is due to spreading of the ink beyond the edges of the dot.” Ink would squeeze out of the edges of each dot when the metal plate was impressed upon the paper. Previous researchers assumed that the increase in dot size due to spreading was uniform over the tone scale. Murray showed that this was not the case.

Murray accounted for the physical spread of dots by introducing a “spreading factor” to the measured radius of each dot, where n is “the step number in a scale of 16 steps differentiated by equal density increments.” The step number is in terms of reflection density on the original. In one case, Murray gave a and b the values 0.01 and 0.001. Assuming a unit solid area, the modified dot area, A_mod, can be calculated . When the printed pattern begins to form white circles, rather than black dots, S is subtracted from the radius of the white circle, rather than added to the radius of the black circle. Thus, the Murray-Davies equation, modified for physical ink spread, may be written as,

The primary importance of Murray’s work with Davies’ equation and his modification to account for ink spread, is that it allowed for the back-calculation of the area on the plate and the creation of a tone reproduction correction curve that could be built into the negative. However, Murray noted, due to the large number of variables in the print process, “it is impractical to calculate ideal structure curves for any assumed spreading equation, for various ink densities, because the quantitative effect of spreading varies with the dot formation, which for any given per cent area, is at present unpredictable.”^​3​ The only way to achieve repeatable tone reproduction was through standardization and the specification of the “conditions, if any, of ink density and spreading, a specific plate will yield a proof giving a perfect reproduction curve.” Measuring dots under a microscope is practical for laboratory work but cannot be applied on an industrial scale. Tone reproduction can only be applied through direct measurement of density. Density measurements of the proof must consider the effects of ink spread and substrate density and agree well with microscopic measurements of the dot areas. Murray stated, “the curve of perfect reproduction for the structured image of a halftone proof is the same as that for photography with unstructured images.” The D_max­ of the print must equal or surpass the D_max of the original. Otherwise, linear tone reproduction cannot be achieved and contrast will be reduced. Dot areas on the plate must take into account ink spreading to achieve linear tone reproduction, as indicated in the below figure from Murray’s paper.

In a series of articles from the early 1940s, John Yule explored the phenomena described by Murray in more detail.^​7–9​ First, Yule examine how dots were formed on a negative through a cross-line screen.^​9​ He asked the question: “Under given conditions of camera aperture, screen distance, etc., what will be the distribution within a halftone dot of the light reaching the sensitive emulsion; or in other words, what will be the light intensity at any given point in the emulsion, and how will this be related to dot formation and tone reproduction?” Yule described methods for predicting dot shape and dot area as a function of camera aperture, screen distance, and exposure. He stated, “the most important function of the light distribution in the dot is to control the tone reproduction curve of the halftone print.” He continued, “by measuring the areas enclosed within lines of equal illumination…and converting these to density by means of [the Murray-Davies] equation…the density obtained with varying exposures can be determined and, hence, the tone reproduction curve.” The ability to accurately predict the shape and area of dots produced by an imaging system, through knowledge of both optics and the sensitometry of film, and dot spreading, enabled the estimation of print density. Yule published a modification of the Murray-Davies equation, with a rearrangement of terms and a modification that included terms for paper reflectance,

where D is the density of the tint, a is the dot area, R_b, is the reflectance of the solid, and R_w is the reflectance of the paper.

Beginning in 1943, Yule began describing the deficiencies of the Murray-Davies equation.^​7​ He observed that, for screen positives with sharp dots, calculated density agreed with measured density, but the density of paper was higher around printed dots. He hypothesized that, “possibly the vehicle spreads, or the dots cause a slight shadow within the paper,” and observed that the density of tones near the solid was higher than the density of the solid itself (likely due to errors of ink transfer). As a general principle, Yule stated “dots are reduced in size by etching and increased in size by ink spreading, and not necessarily uniformly over the entire scale.” The image below, plotting theoretical print density against the original density, illustrated the effect of this change in dot size on tone reproduction.

Yule reinforced Davies’ idea that shape of the tone reproduction curve depended on D_max, “a low ink density reduces contrast in the shadows much more than in the highlights.” An increase in dot diameter would lead to a compression in the highlights while shadows depend largely on ink density.

In 1951, Yule and Nielsen further developed Yule’s ideas on ink spreading.^​4​ It was known that light scatters when it enters paper. “Therefore, some of the light which enters a halftone pattern through a space tries to come out through a dot, and is absorbed instead of being reflected.” The internal scattering of light within a paper after entering a halftone causes the reflectance to be lower than expected, as illustrated by the image below reproduced from Yule and Nielsen’s 1951 TAGA paper. For example, a 50% dot should absorb less than 50% of the light incident upon it due to imperfect absorption, but light scattering results in the opposite effect.

The result of Yule and Nielsen’s experiments conducted to explore the effect of light scatter on the measurement of halftone density lead to a modified version of the Murray-Davies equation,

where n is an empirical factor included to account for the reduction in density due to light scatter, or what is commonly referred to as “optical dot gain.” The n-value linearizes the relationship between density calculated from physical dot area, and the measured dot area (n basically accounts for the effect of the shadow so the user can focus on the reproduction of dots). Yule and Nielsen demonstrated the effect of the n-factor on linearizing the tone reproduction curve in the following figure.

Yule and Nielsen understood that their equation was empirical and not a first-principles description of light-scattering within a paper. “These phenomena [multiple internal reflections] have very pronounced effects in color work, even in a single color. The equation applies equally well to a colored ink, and enables us to calculate more accurately the amount of red, green, and blue light absorbed.” Thus, they stressed again the importance of accurate tone reproduction models on multi-color printing.

Many have suggested standardizing n with one value or another, but selection of the optimal Yule-Nielsen n factor for a given system requires an iterative optimization process that cannot be performed easily. So, despite the improvement offered by the Yule-Nielsen modification, and the advancements of later models, the Murray-Davies equation continues to be used today in process control for the calculation of dot area and dot gain. The calculated dot area accounts for the physical enlargement of the dots, unevenness of the dot structure, and optical effects of light scatter. While the number calculated for area coverage may have arguable physical meaning, it can be a useful tool for printers to monitor the consistency of the print process, especially given its simplicity compared to equations that followed.

For further reading on the Murray-Davies Equation, the Yule-Nielsen modification, and for a general discussion on the usefulness of these equations, I recommend an excellent series of articles by John Seymour on his John the Math Guy blog.^{​10–14​} This link takes you to the first of his posts, “The color of a bunch of dots, part 1.” Additionally, you will find many insightful academic articles that explore the physics of halftone reproduction in the academic literature.

Below are brief biographies of Alexander Murray and Edward Roy Davies.

Alexander Murray

Alexander Murray was born in Scotland sometime between 1895 and 1897.^​15​ He traveled to the US with his family around 1900, later studying art and chemistry before working for a paint manufacturer, a watch-engraving company, and an ink-chemical company.^​2​ He joined Eastman Kodak, in Rochester, NY, in 1927, and in 1931 became the first director of the Graphic Arts Department of the Kodak research laboratories. In addition to his contributions to the development of the Murray-Davies Equation, Murray was influential in the development of color-correction masking techniques, publishing in a 1936 Kodak booklet called “The Modern Masking Method of Correct Color Reproduction.” Murray also filed several patents related to the photographic reproduction of halftones in the 1930s (a simple Google Patent search for Murray will reveal a list of his patents received). Most notably, Murray, along with Richard Morse, patented the first rotating drum scanner in 1936. In 1951, Murray was named outstanding person of the year in the graphic arts by the Technical Association of the Graphic Arts, and in 1952 was recognized for his 25^th year at Kodak.^​16​ He was granted more than 50 patents.  Murray passed away on June 11, 1959.^​17​

Edward Roy Davies

Edward Roy Davies was born on April 13, 1903 and graduated from King’s College, London with a degree in physics in 1925.^​1​ In 1926, he went to work for Thomas Illingworth and Co. Ltd., who manufactured paper for Ilford. Among Davies’ accomplishments at Thomas Illingworth was the development of a reflection densitometer with automatic curve tracer.^​18​ In 1931, at age 27, Davies became Director of the Kodak Ltd. Research Laboratories in Harrow. He was awarded the Order of the British Empire (O. B. E.) in 1946 in recognition of his work in aerial photography during the war. Davies stepped down from the role of director around 1964.^​19​ He passed away in 1995 at the age of 97.^​20​

Disclaimer

This article was written by Brian Gamm in his personal capacity. The views, thoughts, and opinions expressed in this article belong solely to the author, and not necessarily to the author’s employer, organization, committee or other group or individual with which the author has been, is currently, or will be affiliated.

Brian Gamm

A color scientist with a love for the history of color.