It’s been pretty quiet here lately.

I will share here a couple of new developments in my thinking about tonescale model. (Every time I write the word tonescale I hear @Troy_James_Sobotka 's voice in my head echoing “But it doesn’t really map “tone” does it?” But unfortunately I don’t have a better commonly understood term to use so I’ll just keep using this one).

Since @daniele posted his initial model I have torn it apart and rebuilt it many times, each time understanding it a bit better. I moved all the pieces around, solved for different parts, tried to constrain output middle grey and peak. Eventually in my last post above I settled on a constraint for middle grey, so that the user effectively controls display peak luminance and display grey luminance.

But the added complexity always bugged me. In the end the grey constraint didn’t really solve anything. We still had to create a model for changing the curve over different display peak luminances. Instead of creating a model for the grey intersection point, why not just scrap the intersection constraint entirely and just make a model for the 4 core parameters of the function:

- input exposure
- output normalization
- contrast
- flare

With this in mind I started from scratch again, from the simplest form of the function.

The simple compressive hyperbola / Michaelis-Menten equation: f\left(x\right)=\frac{x}{x+1}

This function (or the Naka-Rushton / Hill-Langmuir variation of it) has been shown to describe well the response to stimulus of photoreceptor cells in the retina.

In the above form the asymptote as x approaches infinity, y approaches 1. If the hyperbola is plotted with a log-ranging x-axis, it forms a familiar sigmoid shape.

If we replace the 1 with a variable f\left(x\right)=\frac{x}{x+s_{x}}\ , we can have control over input exposure. As s_x increases, input exposure decreases.

Then we can add a power function for contrast, and another scale for output normalization: f\left(x\right)=s_{y}\left(\frac{x}{x+s_{x}}\ \right)^{p}.

This gives us 3 variables to adjust: input exposure, contrast, and output normalization. We also want some way of controlling flare or glare compensation, so we add an additional parabolic toe compression function f\left(x\right)=\frac{x^{2}}{x+t_{0}}, where t_0 is the amount of toe compression.

Here is a desmos plot with the above 4 variables exposed for adjusting: Michaelis-Menten Tonescale - The math can be abstract so I find that it’s good for us normal humans to fiddle with parameters and watch how they change.

Cool. So that leaves us with the task of coming up with some model to describe how to change these 4 parameters based on varying display peak luminance values.

I’ve done some quick models using the desmos nonlinear regression solver, based on a few data points from the previous behavior of OpenDRT: Michaelis-Menten Tonescale

However would be very curious to do some tests with the OpenDRT Params version, which I have recently modified to expose the above 4 parameters. I only have access to a 800nit peak luminance OLED tv, so I am taking a shot in the dark for HDR settings above that realm. If there’s anyone reading this who is curious and has access to something like a Sony X300…