## Abstract

In a three-color print, the color of any area that is large in comparison with the size of the dots of the structured image can be regarded as an additive mixture of eight colors: the unprinted paper stock; the cyan, magenta, and yellow of the individual ink dots; the red, green, and blue that result when ink dots overlap in pairs; and the black that results when all three ink dots overlap. Since the extent of overlapping is determined by the sizes of the dots, the color of the additive mixture can be expressed by three equations. The straightforward attack on the problem of color reproduction is to solve these fundamental equations for the required dot sizes on the assumption that, in every area, the tristimulus values of the reproduction should be equal to those of the original. This has not been accomplished hitherto, because it involves solving three simultaneous equations of third degree, each containing eight terms. However, a relatively simple electronic equation-solving network has been constructed which solves these equations with ample precision in 0.001 second. By using this network in connection with a scanning machine of the type described in the preceding paper, full color correction is achieved. A corresponding set of equations can be written for the additive mixture produced in four-color printing, wherein the fourth color is black. Since the three equations now contain four unknowns, an additional condition must be imposed. From the standpoint of the printing requirements, it is desirable that at least one of the color dots be absent, or of some predetermined minimal size, in every region of the reproduction. An extension of the principles embodied in the electronic network mentioned above imposes this condition and yields a continuous solution to the three fourth-degree equations.

© 1948 Optical Society of America

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### Equations (20)

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(1)
$$\begin{array}{l}{R}^{\prime}=(1-c)(1-m)(1-y){R}_{w}\\ +\hspace{0.17em}c(1-m)(1-y){R}_{c}+m(1-c)(1-y){R}_{m}\\ +\hspace{0.17em}y(1-c)(1-m){R}_{y}+my(1-c){R}_{my}\\ +\hspace{0.17em}cy(1-m){R}_{cy}+cm(1-y){R}_{cm}+cmy{R}_{cmy},\end{array}$$
(2)
$$\begin{array}{l}{G}^{\prime}=(1-c)(1-m)(1-y){G}_{w}\\ +\hspace{0.17em}c(1-m)(1-y){G}_{c}+m(1-c)(1-y){G}_{m}\\ +\hspace{0.17em}y(1-c)(1-m){G}_{y}+my(1-c){G}_{my}\\ +\hspace{0.17em}cy(1-m){G}_{cy}+cm(1-y){G}_{cm}+cmy{G}_{cmy},\end{array}$$
(3)
$$\begin{array}{l}{B}^{\prime}=(1-c)(1-m)(1-y){B}_{w}\\ +\hspace{0.17em}c(1-m)(1-y){B}_{c}+m(1-c)(1-y){B}_{m}\\ +\hspace{0.17em}y(1-c)(1-m){B}_{y}+my(1-c){B}_{my}\\ +\hspace{0.17em}cy(1-m){B}_{cy}+cm(1-y){B}_{cm}+cmy{B}_{cmy}.\end{array}$$
(4)
$$\begin{array}{l}R={R}^{\prime}=1.000(1-c)(1-m)(1-y)\\ +\hspace{0.17em}0.146c(1-m)(1-y)\\ +\hspace{0.17em}0.442m(1-c)(1-y)\\ +\hspace{0.17em}0.857y(1-c)(1-m)\\ +\hspace{0.17em}0.398my(1-c)+0.064cy(1-m)\\ +\hspace{0.17em}0.033cm(1-y)+0.023cmy,\end{array}$$
(5)
$$\begin{array}{l}G={G}^{\prime}=1.000(1-c)(1-m)(1-y)\\ +\hspace{0.17em}0.197c(1-m)(1-y)\\ +\hspace{0.17em}0.219m(1-c)(1-y)\\ +\hspace{0.17em}0.980y(1-c)(1-m)\\ +\hspace{0.17em}0.206my(1-c)+0.196cy(1-m)\\ +\hspace{0.17em}0.014cm(1-y)+0.019cmy.\end{array}$$
(6)
$$\begin{array}{l}B={B}^{\prime}=1.000(1-c)(1-m)(1-y)\\ +\hspace{0.17em}0.613c(1-m)(1-y)\\ +\hspace{0.17em}0.185m(1-c)(1-y)\\ +\hspace{0.17em}0.148y(1-c)(1-m)\\ +\hspace{0.17em}0.012my(1-c)+0.103cy(1-m)\\ +\hspace{0.17em}0.120cm(1-y)+0.016cmy.\end{array}$$
(7)
$$\begin{array}{l}R={R}^{\prime}=1.000(1-c)(1-m)(1-y)(1-n)\\ +\hspace{0.17em}0.146c(1-m)(1-y)(1-n)\\ +\hspace{0.17em}0.442m(1-c)(1-y)(1-n)\\ +\hspace{0.17em}0.857y(1-c)(1-m)(1-n)\\ +\hspace{0.17em}0.398my(1-c)(1-n)\\ +\hspace{0.17em}0.064cy(1-m)(1-n)\\ +\hspace{0.17em}0.033cm(1-y)(1-n)\\ +0.023cmy(1-n)+0.023n,\end{array}$$
(8)
$$\begin{array}{l}G={G}^{\prime}=1.000(1-c)(1-m)(1-y)(1-n)\\ +\hspace{0.17em}0.197c(1-m)(1-y)(1-n)\\ +\hspace{0.17em}0.219m(1-c)(1-y)(1-n)\\ +\hspace{0.17em}0.980y(1-c)(1-m)(1-n)\\ +\hspace{0.17em}0.206my(1-c)(1-n)\\ +\hspace{0.17em}0.196cy(1-m)(1-n)\\ +\hspace{0.17em}0.014cm(1-y)(1-n)\\ +\hspace{0.17em}0.019cmy(1-n)+0.019n,\end{array}$$
(9)
$$\begin{array}{l}B={B}^{\prime}=1.000(1-c)(1-m)(1-y)(1-n)\\ +\hspace{0.17em}0.613c(1-m)(1-y)(1-n)\\ +\hspace{0.17em}0.185m(1-c)(1-y)(1-n)\\ +\hspace{0.17em}0.148y(1-c)(1-m)(1-n)\\ +\hspace{0.17em}0.012my(1-c)(1-n)\\ +\hspace{0.17em}0.103cy(1-m)(1-n)\\ +\hspace{0.17em}0.120cm(1-y)(1-n)+0.016cmy(1-n)+0.016n.\end{array}$$
(10)
$$\begin{array}{l}{R}^{\prime}={R}_{w}-({R}_{w}-{R}_{c})c-({R}_{w}-{R}_{m})m\\ -\hspace{0.17em}({R}_{w}-{R}_{y})y+({R}_{w}-{R}_{m}-{R}_{y}+{R}_{my})my\\ +\hspace{0.17em}({R}_{w}-{R}_{c}-{R}_{y}+{R}_{cy})cy\\ +\hspace{0.17em}({R}_{w}-{R}_{c}-{R}_{m}+{R}_{cm})cm\\ -\hspace{0.17em}({R}_{w}-{R}_{c}-{R}_{m}-{R}_{y}+{R}_{my}\\ +\hspace{0.17em}{R}_{cy}+{R}_{cm}-{R}_{cmy})cmy,\end{array}$$
(11)
$$\begin{array}{l}{G}^{\prime}={G}_{w}-({G}_{w}-{G}_{c})c-({G}_{w}-{G}_{m})m\\ -\hspace{0.17em}({G}_{w}-{G}_{y})y+({G}_{w}-{G}_{m}-{G}_{y}+{G}_{my})my\\ +\hspace{0.17em}({G}_{w}-{G}_{c}-{G}_{y}+{G}_{cy})cy\\ +\hspace{0.17em}({G}_{w}-{G}_{c}-{G}_{m}+{G}_{cm})cm\\ -\hspace{0.17em}({G}_{w}-{G}_{c}-{G}_{m}-{G}_{y}+{G}_{my}\\ +\hspace{0.17em}{G}_{cy}+{G}_{cm}-{G}_{cmy})cmy,\end{array}$$
(12)
$$\begin{array}{l}{B}^{\prime}={B}_{w}-({B}_{w}-{B}_{c})c-({B}_{w}-{B}_{m})m\\ -\hspace{0.17em}({B}_{w}-{B}_{y})y+({B}_{w}-{B}_{m}-{B}_{y}+{B}_{my})my\\ +\hspace{0.17em}({B}_{w}-{B}_{c}-{B}_{y}+{B}_{cy})cy\\ +\hspace{0.17em}({B}_{w}-{B}_{c}-{B}_{m}+{B}_{cm})cm\\ -\hspace{0.17em}({B}_{w}-{B}_{c}-{B}_{m}-{B}_{y}+{B}_{my}\\ +\hspace{0.17em}{B}_{cy}+{B}_{cm}-{B}_{cmy})cmy.\end{array}$$
(13)
$$\begin{array}{l}\mathrm{\Delta}{R}^{\prime}=-(\partial {R}^{\prime}/\partial c)\mathrm{\Delta}c,\\ \mathrm{\Delta}{G}^{\prime}=-(\partial {G}^{\prime}/\partial c)\mathrm{\Delta}c,\\ \mathrm{\Delta}{B}^{\prime}=-(\partial {B}^{\prime}/\partial c)\mathrm{\Delta}c.\end{array}$$
(14)
$$R={R}^{\prime}={R}_{w}-({R}_{w}-{R}_{c})c-({R}_{w}-{R}_{m})m-({R}_{w}-{R}_{y})y,$$
(15)
$$G={G}^{\prime}={G}_{w}-({G}_{w}-{G}_{c})c-({G}_{w}-{G}_{m})m-({G}_{w}-{G}_{y})y,$$
(16)
$$B={B}^{\prime}={B}_{w}-({B}_{w}-{B}_{c})c-({B}_{w}-{B}_{m})m-({B}_{w}-{B}_{y})y.$$
(17)
$$\begin{array}{l}-(\partial {R}^{\prime}/\partial c)=({R}_{w}-{R}_{c}),\\ -(\partial {G}^{\prime}/\partial c)=({G}_{w}-{G}_{c}),\\ -(\partial {B}^{\prime}/\partial c)=({B}_{w}-{B}_{c}).\end{array}$$
(18)
$$\begin{array}{l}-(\partial {R}^{\prime}/\partial c)=({R}_{w}-{R}_{c})-({R}_{w}-{R}_{c}-{R}_{y}+{R}_{cy})y-({R}_{w}-{R}_{c}-{R}_{m}+{R}_{cm})m\\ +({R}_{w}-{R}_{c}-{R}_{m}-{R}_{y}+{R}_{my}+{R}_{cy}+{R}_{cm}-{R}_{cmy})my,\end{array}$$
(19)
$$\begin{array}{l}-(\partial {G}^{\prime}/\partial c)=({G}_{w}-{G}_{c})-({G}_{w}-{G}_{c}-{G}_{y}+{G}_{cy})y-({G}_{w}-{G}_{c}-{G}_{m}+{G}_{cm})m\\ +\hspace{0.17em}({G}_{w}-{G}_{c}-{G}_{m}-{G}_{y}+{G}_{my}+{G}_{cy}+{G}_{cm}-{G}_{cmy})my,\end{array}$$
(20)
$$\begin{array}{l}-(\partial {B}^{\prime}/\partial c)=({B}_{w}-{B}_{c})-({B}_{w}-{B}_{c}-{B}_{y}+{B}_{cy})y-({B}_{w}-{B}_{c}-{B}_{m}+{B}_{cm})m\\ +({B}_{w}-{B}_{c}-{B}_{m}-{B}_{y}+{B}_{my}+{B}_{cy}+{B}_{cm}-{B}_{cmy})my.\end{array}$$