Output Transform Tone Scale

Michaelis-Menten Scene-Contrast

Here is another idea: If we apply contrast in the scene-linear domain using a pivoted power function with linear extension, and use a “pure” michaelis-menten equation with no exponent, you can almost get reasonably consistent results from 108-4000 nits peak brightness with a constant contrast setting.

m01 above ignores SDR and treats DCI 108 nit / 7.5 nit HDR as a valid datapoint in the spectrum of peak brightnesses.

m02 is a continuous range in middle grey from 10 nits at 100 nit peak, through 16 nits at 4000 nit peak.


Please keep those coming, I’m certainly lacking of time but this is great stuff that I have been enjoying between boring builds :slight_smile:

Is there a function you prefer in your latest batches?



I will second Thomas’ comments, please do keep them coming.

Something I have been wondering about, is if only using peak output is appropriate for setting the parameters. I think we need to include the surround/adopting field luminance as one of the key parameters to allow us to understand how to bridge between dim and dark surrounds and to try rationalise the mid-grey luminance between the different outputs.

In a dark surround we assume the average picture luminance is scaled based upon the image luminance so we could make an assumption of ‘10%’ of peak for SDR when switching to Rec 1886 Output we switch our anchoring to be based on the reference surround value which by magic is also 10% of peak luminance. But when migrating to HDR this might break, so obviously we need to be a little more involved for choosing the mapping function, but this is just an off the cuff thought for consideration.


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Yes please do keep those points coming. It is an interesting discussion to be had.

I’ll add my own grain of salt with regards to mapping SDR to HDR and I found that getting the black levels in the same place is important for a consistent experience given of course that both are viewed in the same viewing conditions.

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I put together a Tonescale Model Selects colab, with the most successful models in my trash pile of experiments.

I would say they all have different pros and cons.

I like the simplicity and look of the pre-tonemap contrast with linear extension + michaelis-menten function. The Michaelis-Menten Spring Dual-Contrast model in the above colab is the one I will use moving forward I believe.

The post-tonemap contrast Michaelis-Menten Spring function is very neutral and performs very nicely in HDR, but the shadow contrast is too low in SDR. This is what I was previously fighting by modeling an exponent that started higher and decreased as peak luminance increased. I never liked this. Included in the Michaelis-Menten Spring model is an idea for a “default tonecurve lmt” which adds a bit of contrast to compensate and seems to work okay through the transition to HDR.

And I figured I would throw in a refinement of an earlier experiment with the Piecewise Hyperbolic Tonescale Model. In this one I do like that values below middle grey can be kept strictly linear if desired. It is more controllable. I also like the stronger highlight appearance, and the ease with which you can transition from SDR to HDR with a consistent contrast.

All models include a parameter for surround compensation using an unconstrained post-tonemap power function.

Tonescale_Selects_v01.nk (33.8 KB)
Here’s a nuke script with all the models as well.

I’ve also pushed OpenDRT v0.1.2 that uses the “dual contrast” tonescale model above.


While doing the OkishDRT I wanted to use the same tonescale as the ACES2 candidates use, the Michaelis-Menten Spring Dual-Contrast and needed its derivative to drive the path-to-white. It turns out there is a sudden transition in 0.18 where the tonescale changes from the linear extension to the michaelis-menten function, as seen in the following desmos plot: ACES2 MM-SDC tonecurve. Not sure if this is a problem for the tonescale but for using the derivative to do things it probably would be better if it was smooth.

I made a modified version of your Desmos plot, where C_1 is automatically calculated to make the curve continuous as you vary C_0, instead of being a slider.

You need to drop C_0 to 1.0 instead of 1.2 to make the first derivative smooth. That is probably not enough contrast for our purposes. But it is interesting to see where the kink comes from.

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My original suggestion was smooth in both derivatives.


I think that in fact when C_0 is set to 1.0, it becomes the same as your original function, does it not @daniele?

Edit: S_0 and S_1 also need to be set to 1.0, and l set to 0.05 to match the original Desmos plot you posted

Reposting this from earlier. Daniele’s tonescale in log plot to better see what the parameters do: Daniele Compression Curve

Here is my quick investigation code to fit a cubic polynomial, where by simple weighting it more closely matches at the join Google Colab.

And an updated graph ACES2 MM-SDC tonecurve

Here is a version of the tone scale formula which compensates for the reduction in exposure introduced by the (Display) Flare compensation.

As the first equation does not change the slope at zero (besides the gamma value) one could change the order without significant difference, but I find this cleaner.

t_1 as most of the parameters is there to give us leverage on many different issues at once.
Some are:

  • it compensates the display flare without moving 0.0; this is important in a relative black system
  • it compensates for the shadow wash-out, which is introduced by the first part of the equation
  • most importantly, it compensates for the kink you get from the toe of the log grading space toe, this is important for the “gradeability” of the system.

with t_1 = 0.0 we get a ACEScct to Rec.1886 mapping like this:

This makes grading in the working space very unenjoyable.

with t_1 = 0.01 we get a ACEScct to Rec.1886 mapping like this:

A bit better but still not good

with t_1 = 0.05 we get a ACEScct to Rec.1886 mapping like this:

Even better.

So the right value of t_1 is influenced by many factors, but mainly by the toe function of the default grading space and the EOTF of the display.


Seems quite easy to get very good match to the MM-SDC tonescale. I used following parameters: g = 1.1, w = 0.84, t1 = 0.075

To get exposure to match I had to adjust the w value. I haven’t checked whether HDR matches…


I’ve got a version of the ZCAM DRT (v013) that integrates @daniele’s new curve, along with @priikone’s values.

It can be found here if anyone want to have a poke:


The MM-SDC tonescale hit 1.0 quite early. With this new one ACEScct 1.0 output is 0.998 and with ACEScct 1.46799 the output is 0.9999. The toe t1 value should probably be a bit higher. But it’s all very close…

0-1 ACEScct:

Here’s Daniele’s tonescale for different peak luminances compared to MM-SDC:

SDR is a good match but HDR is different.

Nothing stops you to change w for HDR as the comment in the desmos already hints at.

Yes, and Jed figured out all this with his tonescales. So here’s a variant that now adjusts the exposure and the peak luminance also for closer match to MM-SDC. The exposure should be a match.

It adds a new parameter w1 that can be pre-computed and was derived in same fashion as in MM-SDC using Linear Regression. This parameter determines how quickly the curve hits the peak luminance.

HDR might now have a tiny bit more contrast compared to SDR. Don’t know how visible it is. Contrast too could be a parameter that changes a bit.

where do your destination values in the linear regression desmos come from.
I think it is unwise to scatter the origins of your constants in separate place.
Also it is a bit strange to have an analytical model be driven by an approximation to unknown data points.

I would try to express what those values should do and model that instead directly

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Agreed. Personally I’d like the curves to hit peak luminance at ACEScct 1.0 (or around 10 stops above mid grey) but this isn’t doing exactly that but that’s what I tried to hit with those numbers. I’m equally unsure how Jed derived his numbers.