K boost formulas

I am working away on some of the finer details of my QuadToneRIP book and there was a recent question on the QTR Yahoo Group about how the K_Boost setting works when making custom QTR Profiles. I've been meaning to dig in to see exactly what is being affected with this setting, rather than just say "it affects the tones at the shadow end of the scale". So I poured a pot of coffee and spent my morning working out what I thought was really going on here with the K_Boost setting. I don't have the formula that matches the output from the QTR curve generation program, but here is the gist of what I see happening.

Basically, The boost setting is not JUST affecting the end of the scale, from 80% to 100%, but is actually applying some added density along the entire scale based on an overall gamma adjustment or predefined curve. The lower graph shows the percentage difference between the boosted and non-boosted curves. It is based on a boost setting 20% higher than the ink limit, and that is proportioned along the whole grayscale. 

Ink Limits and Boost Settings

Your ink limit is some % of 100, and your boost is some larger % of 100 which is then multiplied by 65535 (just the 16-bit value of the total possible ink load). That is pretty obvious just from looking at the total ink limit for any ink channel or by opening the quad curve in a text editor. The next part is where it gets tricky. 

You take the difference of the black ink limit and black boost values. In the example I used the limit was 50 ( or 0.5\*65535) and the boost was 60 ( or 0.6\*65535), and the difference is 6553. 

That difference is divided into 256 steps, and each step then has some gamma adjustment applied to it (I used 2.07 in my example and to get fairly close to the QTR generated curve). The adjusted boost value for each of the 256 steps is then added to the pre-boosted K values. So it really isn't *only* adjusting the shadow end of the scale. It is adding some proportion of the boost to the entire grayscale, but since there is a gamma adjustment applied to it, it is showing more of the effect at the end of the scale.

I played with the inputs a little more and compared them to the QTR-generated curves, and I expect there is something other than a simple gamma adjustment being applied. There might just be some predefined curve built into the program that is scaled to whatever the difference between the Limit and the Boost is, but that is a black box and there is no public documentation of the exact curve or fomula. 

Here is a few graphs comparing the QTR generated boost and the one with my gamma settings. They are not exact, but I hope it illustrates some of what is happening. I only graphed a single partition K curve (essentially the Gray Curve, and without Gray shadow/gray highlight/gray gamma settings)