IntroductionModeling printing gamutsModeling color halftoningAdding trapping to the Neugebauer modelAdding dot gain to the Neugebauer modelModeling the printing primariesGamut mappingStrategy of n-tone mappingCoordinate system of n-tone mappingSteps in n-tone mappingMonotone and duotone mappingsSelecting inksInk-selection objective functionInk-selection algorithmComputing separationsSeparation objective functionSeparation algorithmConclusionReferencesFigure 1Figure 2Figure 3Figure 4Reproducing Color Images Using Custom InksEric J. Stollnitz Victor Ostromoukhov David H. SalesinUniversity of Washington Ecole Polytechnique F´ed´erale de LausanneAbstractWe investigate the general problem of reproducing color images onan offset press using custom inks in any combination and number.While this problem has been explored previously for the case of twoinks, there are a number of new mathematical and algorithmic chal-lenges that arise as the number of inks increases. These challengesinclude more complex gamut mapping strategies, more efficient inkselection strategies, and fast and numerically accurate methods forcomputing ink separations in situations that may be either over- orunder-constrained. In addition, the demands of high-quality colorprinting require an accurate physical model of the colors that resultfrom overprinting multiple inks using halftoning, including the ef-fects of trapping, dot gain, and the interreflection of light betweenink layers. In this paper, we explore these issues related to printingwith multiple custom inks, and address them with new algorithmsand physical models. Finally, we present some printed examplesdemonstrating the promise of our methods.CR Categories: I.3.4 [Computer Graphics]: Graphics Utilities; G.1.6 [Nu-merical Analysis]: OptimizationAdditional Keywords: color reproduction, color printing, gamut mapping,ink selection, Kubelka-Munk model, Neugebauer model, separations1 IntroductionIt isofinterest:::that, regardless ofthe number of impres-sions, the inks may be selected solely on the basis of theircolor gamut. Their colors need not be cyan, magenta, andyellow; nor is it required that they be transparent. The wayis therefore opened for entirely new printing processes.—Hardy and Wurzburg, 1948 [6]Fifty years ago, the promise of color printing with custom inks ap-peared imminent. The advantages of such a process are clearly nu-merous. Freed from the same fixed set of process color inks—cyan,magenta, yellow, and black—it should be possible to print more vi-brant colors for art reproductions, annual reports, and packaging.Moreover, if the inks are chosen specifically for the particular im-age being reproduced, it should be possible in many cases to achievethese vivid colors with just a small number of inks—perhaps four—and perhaps at no greater cost than using the four process colors. Inaddition, it is common today to print boxes and wrappers with fourprocess inks (for images) plus two spot colors for corporate logos orlarge areas of background. By selecting custom inks that comple-ment the required spot colors, we might achieve better quality withsix inks or comparable quality with fewer inks.In recent years, several new color printing processes have been pro-posed that use a fixed set of six or more standard printing inks [1,22, 28]. For those willing to use more inks, these new processesdo provide more vivid color reproduction. However, Hardy andWurzburg’s fifty-year-old vision of printing with arbitrary custominks remains elusive. Indeed, there are quite a few difficult prob-lems that stand in the way.For one, it is very difficult to derive a physical model that accuratelypredictshowarbitraryinks will interactwhen printedtogether, insu-perposition and in juxtaposition using halftoning. In addition to op-tical effects, the model must take into account physical effects suchas trapping and dot gain.Furthermore, the gamuts produced by multiple custom inks have ir-regular, nonconvex shapes. Creating efficient, reliable gamut map-ping algorithms for smoothly mapping image colors to the colorsthat can be achieved with a given set of inks is a nontrivial problem.Choosing the best set of custom inks to use for a given image is an-other difficult problem—in this case, acombinatorial challenge, par-ticularly as the number of inks used for printing gets large.Finally, computing ink separations becomes more difficult for mul-tiple inks. While for two inks there is always a simple analytic so-lution, for three or more inks the problem can become either over-or under-constrained. The problem becomes over-constrained whenthe color to be printed cannot be achieved with quantities of ink be-tween 0 and 100%. The problem is under-constrained when thereare two or more ways of achieving the same color, using differentink combinations. This situation arises wherever the gamut is dou-bly covered, a commonplace occurrence with four or more inks.This paper addresses these challenges in detail with new physicalmodels and algorithms, then demonstrates the potential of our ap-proach with printed examples. Although a great deal more work re-mains to be done before Hardy and Wurzburg’s vision is achievedin its entirety, this paper at least takes some steps toward that goal.Related workPoweret al. [23] showed that for duotone printing, in which just twoinks are used, choosing the optimal inks for the particular image athand can result in remarkably good reproductions. Our paper dis-cusses the many issues involved in generalizing their work to threeor more custom inks—what we refer to as n-tone printing. Theseissues can be broken down into a number of subproblems.First, for any given choice of paper and inks, we require a modelof the gamut of printable colors. Many existing models have beendeveloped for specific inks and printing processes; unfortunately,these models typically do not apply when printing with custom inks.More general models include the color halftoning model developedby Neugebauer [21], and colorant layering models such as the Beer-Bouguer, Kubelka-Munk, and Clapper-Yule equations (describedby Kang [12]). Liu describes a model for process color printingsimilar to both the Kubelka-Munk layering model and the Neuge-bauer halftoning model [16]. We model the printing gamuts of cus-tom inks using a similar approach in Section 2, where we combinethe Kubelka-Munk and Neugebauer equations while taking into ac-count the effects of dot gain and
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