Saturation Density Spectrum

One of the spectra needed is the saturation density spectrum of the top ink. The other input spectra (solid ink density spectrum of the bottom ink, likewise for the top ink, and the density spectrum of the paper) are pretty straightforward, but this one's a but of a head-scratcher sometimes. Where do I get this spectrum from?

We've been using functions of the SID spectrum of the top ink for some time now. Or, more generally, we approximate $D\_\backslash infty\; -\; D\_p$ as a function of $D\_2\; -\; D\_p$. Not exact, but convenient.

While I've been making the argument that this violates a fundamental principle (i.e., that 0 must be a fixed point of such a function, because the density (at some wavelength) of an ink that is as dense (at that wavelength) as the paper must be invariant with respect to the layer thickness, and the masstone and undertones must be identical), Franz has had success simply adding a constant density to $D\_2$ to obtain $D\_\backslash infty$.

The saturation density spectrum in the test data was computed this way, using a constant of 2. Take a look at the screenshot of the sample session: the values in the fourth column (saturation density of ink 2) are 2 units higher than the values in the third column (density of ink 2).

You want something better? (Such as a function with a fixed point 0?) Other suggestions appear in our 2011 TAGA paper, and there are countless others. Feel free to come up with your own. Just make sure that the solid ink density for the top ink is between the density of the paper and that ink's saturation density, a necessary consequence of Tollenaar-Ernst theory. They can be ordered (smallest to largest) Dp, D2, D_\infty (normal case, with dark ink on light paper), or D_\infty, D2, Dp (wall painter's case, with light ink on dark substrate), just as long as D2 is in the middle.

The same order does not even need to apply for every wavelength. Some yellow inks may be lighter (because of scatter) than some papers in the longer wavelengths, and darker in the shorter (because of absorption).