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| The '''Neugebauer equations''' are a set of equations used to model [[color printing]] systems, developed by [[Hans E. J. Neugebauer]].<ref>{{cite book|title=Color technology for electronic imaging devices|first=Henry R.|last=Kang|publisher=SPIE Press|year=1997}}</ref> They were intended to predict the color produced by a combination of [[halftone]]s printed in cyan, magenta, and yellow inks.
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| The equations estimate the [[reflectance]] (in [[CIE XYZ]] 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:
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| :<math>R(\lambda) = \sum_{i=1}^{16} w_i R_i(\lambda)</math>
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| where <math>R_i(\lambda)</math> is the reflectance of ink combination ''i'', and <math>w_i</math> 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]]. <ref>Raja Balasubramanian, A spectral Neugebauer model for dot-on-dot printers, Proc. SPIE vol. 2413, (1995) [http://chester.xerox.com/~raja/papers/EI95.pdf]</ref>
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| Light can interact with the paper and ink in more complex ways. The Yule-Neilsen correction takes into account light entering through blank regions and re-emerging through ink:<ref>J. A. C. Yule and W. J. Neilsen, "The Penetration of light into Paper and its Effect on Halftone Reproduction," 1951 TAGA Proceedings, pp. 65-76</ref>
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| :<math>R(\lambda) = \left [ \sum_{i=1}^{16} w_i R_i(\lambda)^{1/n} \right ]^n</math>
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| The factor ''n'' would be 2 for a perfectly diffusing [[Lambertian reflectance|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.
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| 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.<ref>Marc F. Mahy, Insight into the solutions of the Neugebauer equations. Electronic Imaging: SPIE/IS&T International Technical Group Newsletter January 1999, p. 7,11. [http://www.inventoland.net/pdf/Reports/ei_newsletter99ocr.pdf]</ref>
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| ==See also==
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| * [[CMYK color model]]
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| ==Original paper==
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| H. E. J. Neugebauer, "Die theoretischen Grundlagen des Mehrfarbenbuchdrucks," ''Zeitschrift für wissenschaftliche Photographie Photophysik und Photochemie,'' 36:4, 1937. p 73-89.
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| ==References==
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| <references/>
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| [[Category:Equations]]
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| [[Category:Color]]
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| [[Category:Printing]]
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| {{mathapplied-stub}}
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Oscar is what my wife enjoys to contact me and I totally dig that name. Bookkeeping is my occupation. Years ago we moved to North Dakota. To gather badges is what her family and her appreciate.
My weblog :: over the counter std test; just click the next post,