An early model

Tollenaar and Ernst tacitly suggested a model in their 1963 IARIGAI paper.^[7] It was

${\displaystyle \mathrm {gain} _{\mathit {TE}}=a_{\mathrm {form} }\,\left(1-a_{\mathit {vf}}\right)}$

where a_vf, the shadow critical area fraction, is the area fraction on the form at which the halftone pattern just appears solid on the print. This model, while simple, has dots with relatively small perimeter (in the shadows) exhibiting greater gain than dots with relatively larger perimeter (in the midtones).

Haller's model

Karl Haller, of FOGRA in Munich, proposed a different model, one in which dots with larger perimeters tended to exhibit greater dot gain than those with smaller perimeters.^[8] One result derivable from his work is that dot gains depend on the shape of the halftone dots.^[9]

The GRL model

Viggiano suggested an alternate model, based on the radius (or other fundamental dimension) of the dot growing in relative proportion to the perimeter of the dot, with empirical correction the duplicated areas which result when the corners of adjacent dots join.^[10] Mathematically, his model is:

${\displaystyle \mathrm {gain} _{\mathit {GRL}}={\begin{cases}a_{\mathrm {form} }-a_{\mathit {wf}},&\mathrm {for} \ a_{\mathrm {form} }\leq a_{\mathit {wf}}\\[6pt]2\,\Delta _{0,50}{\sqrt {a_{\mathrm {form} }\left(1-a_{\mathrm {form} }\right)}},&\mathrm {for} \ a_{\mathit {wf}}<a_{\mathrm {form} }<a_{\mathit {vf}}\\[6pt]a_{\mathrm {form} }-a_{\mathit {vf}},&\mathrm {for} \ a_{\mathrm {form} }\geq a_{\mathit {vf}}\end{cases}}}$

where Δ_0,50 is the dot gain when the input area fraction is 1⁄2; the highlight critical printing area, a_wf, is computed as:

${\displaystyle a_{\mathit {wf}}={\begin{cases}{\dfrac {4\Delta _{0,50}^{2}}{1+4\Delta _{0,50}^{2}}},&\mathrm {for} \ \Delta _{0,50}<0\\[6pt]0,&\mathrm {for} \ \Delta _{0,50}\geq 0\end{cases}}}$

and the shadow critical printing area, a_vf, is computed according to

${\displaystyle a_{\mathit {vf}}={\begin{cases}1,&\mathrm {for} \ \Delta _{0,50}\leq 0\\[6pt]{\dfrac {1}{1+4\Delta _{0,50}^{2}}},&\mathrm {for} \ \Delta _{0,50}>0\end{cases}}}$

Note that, unless Δ_0,50 = 0, either the highlight critical printing fraction, a_wf, will be nonzero, or the shadow critical printing fraction, a_vf will not be 1, depending on the sign of Δ_0,50. In instances in which both critical printing fractions are non-trivial, Viggiano recommended that a cascade of two (or possibly more) applications of the dot gain model be applied.

Empirical models

Sometimes the exact form of a dot gain curve is difficult to model on the basis of geometry, and empirical modeling is used instead. To a certain extent, the models described above are empirical, as their parameters cannot be accurately determined from physical aspects of image microstructure and first principles. However, polynomials, cubic splines, and interpolation are completely empirical, and do not involve any image-related parameters. Such models were used by Pearson and Pobboravsky, for example, in their program to compute dot area fractions needed to produce a particular color in lithography.^[11]