More Tonescale Sigmoid Ramblings
The last couple weeks I’ve been doing some more explorations on this topic. I’ll summarize some of the more interesting points and thoughtprocesses here for those rare persons who might still be following this thread.
A Tale of Two Contrasts
At the end of this meeting, I did a quick demo of the different behavior of contrast / exponent adjustments between these two functions
Daniele’s “MichaelisMenten Spring Function”
f\left(x\right)=s_{1}\left(\frac{x}{s_{0}+x}\right)^{p}
where s_{0}=e_{0}s_{1}^{\frac{1}{p}} and e_{0} is the scenelinear exposure control.
the “NakaRushton Function” I posted before
f\left(x\right)=\frac{s_{1}x^{p}}{s_{0}+x^{p}}
The difference between the two functions is essentially where the exponent is applied.
 In Daniele’s function, contrast is adjusted as a power function in displaylinear.
 In the NakaRushton function, contrast is adjusted as a power function in scenelinear.
Based on all the dumb experiments I’ve done with the above two tonescale functions, it seems necessary to have more contrast in SDR than HDR. This implies the slope through middle grey changes subtly between an SDR rendering and an HDR rendering. Logically this makes sense: Since we have more dynamic range available in HDR, we would want to have less highlight compression and less stretching of midrange values through boosted contrast. The question I’ve been exploring is how do you create a tonescale that continuously changes between SDR at 100 nits peak luminance and HDR at > 1000 nits peak luminance?
What the heck is a spring function?!
I think Daniele used this term in one of the previous meetings (or maybe I imagined it, just like I think I imagined @SeanCooper using the term “water vs balloon” to describe display gamut rendering methods). Or maybe I just have a psychological vulnerability for inventing stupid names for things.
Anyway spring just refers to a sigmoid function which can be scaled in Y without the slope through the origin changing. A simple example being f(x)=s_{1}\frac{x}{s_{1}+x}. With this function you can multiply up s_1 and the slope through the origin stays constant, while the rest of the sigmoid is scaled up vertically. This way of thinking about HDR display rendering tonescales is much more elegant and simple than the messy way I was thinking about this before.
The basic approach is to
 Set contrast with the power function
 Set “exposure” or middle grey point using the scenelinear exposure control
 Set peak white luminance using the y scale s_1.
NakaRushton Spring?
It’s easy to set up a “NakaRushton” equation in “spring” mode: f\left(x\right)=\frac{s_{1}x^{p}}{s_{0}+x^{p}}
where s_{0}\ =\frac{s_{1}}{e_{0}} and e_0 is our scenelinear scale.
As a simple example, here is a variation on the tonescale model based on the “NakaRushton Tonescale Function” I posted previously. It has a constant contrast of p=1.2, constant flare compansation of f_l=0.02, and maps middle grey to 10 nits at 100 nits peak luminance.
In this model, the output yscale is normalized so that at 100 nits peak luminance, output displaylinear = 1.0, then as peak luminance increases the output peak y value increases up to 40 at 4000 nits. To normalize into a pq range where 1.0 = 10,000 nits and 0.01 = 100 nits, you would divide by 100. This makes it simple to turn on pq normalization for HDR or turn it off for SDR.
As I hinted at before, I think we would want to reduce the contrast with increasing peak luminance. With a contrast of 1.2 at 4000 nits I think the highlights are pushed too bright. Or maybe this is a problem with the tonescale function, and the reason Daniele was asking “how does it work in HDR?”
Pivoted Contrast?
After the above description of the “NakaRushton” function, you might be thinking
Gee if that function is just applying a power function to scenelinear input data, why not turn it into a pivoted contrast function instead, so that middlegrey isn’t shifted around when adjusting contrast?!
It actually seems like a valid approach using something like a 3stage tonescale rendering:
 Scenereferred pivoted contrast adjustment (possibly with linear extension above pivot)
 Scenelinear to displaylinear rendering using pure MichaelisMenten function
 Flare compensation
Many Valid Approaches
Given the large quantity of garbage in my previous posts in this thread I thought it might be useful to assemble a list of tonescale functions into a single place.
In this notebook there are 3 categories of tonescale functions

MichaelisMenten : \frac{s_{1}x}{s_{0}+x}
Just a pure MichaelisMenten function, no exponent, no contrast. 
MichaelisMenten
DisplayPostTonemap Contrast : s_{1}\left(\frac{x}{s_{0}+x}\right)^{p}
The variation Daniele posted, with exponent applied in the displayreferred domain. 
MichaelisMenten
ScenePreTonemap Contrast : \frac{s_{1}x^{p}}{s_{0}+x^{p}}
The variation I posted above with the exponent applied in the scenereferred domain.
I have included “spring function” variations, and variations with intersection constraints where possible.
A Note on Names
Just a brief interlude to justify my decisions against @Troy_James_Sobotka 's pedantic trolling in the previous meeting.
In the original NakaRushton 1966 paper, the function they use is a classic MichaelisMenten function y=s_{1}\frac{x}{x+s_{0}}. I agree that strictly speaking using this name to refer to my above function is disingenuous.
I used this name because in this other paper the “NakaRushton equation” is referenced as f\left(x\right)=s_{1}\frac{x^{p}}{s_{0}^{p}+x^{p}}.
Also technically speaking the function Daniele posted is a MichaelisMenten function with an added contrast. MichaelisMenten refers strictly to the hyperbolic function \frac{s_{1}x}{s_{0}+x}
So yeah, maybe moving forward we call these functions by what they are: The MichaelisMenten function with contrast added in displayreferred domain or scenereferred domain.