In previous work on improving print standards, we recommended that a mathematical model of "ideal" tone reproduction (TR) be incorporated into future print standards. (Birkett and Spontelli, 2004) This paper describes our efforts to create such a model. Our work builds on that of Murray-Davies (Murray, 1936) and Yule-Neilsen (Yule and Neilsen, 1951). They modeled densitometric measurements of black and white halftone ink ramps. Our work models colorimetric measurements of color multi-ink ramps. The model has many practical applications, including process characterization, pressroom calibration, and color matching.
We explain the development of the model, beginning with multi-ink ramps. Colorimetric xyz measurements are a non-linear function of the ink ramp parameter. A simple transform borrowed from the xyz to Lab calculation makes the functions nearly linear for many data sets. The transformed data is then modeled using a Hermite spline. The Hermite spline has four parameters, which correspond directly to essential print characteristics, and are easy to visualize.
To validate our work, we developed a computer algorithm for fitting the model to real data. We've measured the quality of fit for an assortment of publicly available characterization data. We recommend this model for use in future industry print standards.