I have been doing a bit of research and experimentation in the domain of tonescales. Specifically, simpler formula-based alternatives to the spline approach used in the ACES SSTS. Full disclosure, I’m embarked on this adventure partially because I am too dumb / don’t have the knowledge yet to implement splines myself. And I find their complexity a bit overwhelming, and have the gut feeling that there are simpler alternatives.
I will share here a few things I made along the way in case anyone finds it interesting or useful.
Hable Piecewise Power Tonemap
One of my first experiments was diving into John Hable’s Piecewise Power tonemap, which was discussed previously by @Thomas_Mansencal. I was intrigued by the (relative) simplicity of the piecewise power curve fitting, but I didn’t really understand how it worked on a mathematical / technical level. So I decided to implement it myself starting from scratch.
Above is a link to a desmos implementation. I have changed the parameterization a bit from what is proposed by Hable. The parameterization expresses
- The linear section pivot as an xy coordinate
- The slope of the linear section
- The shoulder and toe linear section length
- The shoulder asymptote position as an xy coordinate
- The toe asymptote position as an xy coordinate (Note that the Hable implementation assumes a 0,0 coordinate at the origin, and is thus a bit less flexible).
I also added an inverse transform based on the same parameterization.
This s-curve would be applied in a log domain.
There is a Nuke implementation here, and a blinkscript version here.
The big downside of this curve is that it’s not possible to control the slope or behavior of the shoulder, and the shoulder tends to compress values too much as it approaches asymptote. One can work around this by adjusting the shoulder asymptote xy position, but it’s tricky and doesn’t work super well.
PowerP Sigmoid
Frustrated by the Hable shoulder, I decided to embark on an adventure in math to make my own parameterized sigmoidal curve using the Power( p ) compression function that @JamesEggleton posted here. After a couple of weekends learning calculus I finally figured out the math to solve for the intersections, and made this:
- It has similar parameterization for the pivot, and toe / shoulder linear section length.
- Has “power” adjustments for both shoulder and toe compression curves.
- Has a solve for shoulder limit and toe limit, to specify what x value crosses y=1 and y=0.
- Has an inverse, (it’s a little buggy in the desmos implmentation but the nuke version works okay)
Note this is pretty similar to the tonescale in the NaiveDRTPivoted node, but all combined into one transform instead of being separate pieces.
The curve is very flexible, but unfortunately I wasn’t very happy with how it looked. I really struggled to get the curve for shadows to look good in images.
Here is a Nuke implementation, and a Nuke implementation applied in the log domain.
Hill-Langmuir Equation
I got distracted into implementing a few more color models in my growing collection, and read Fairchild’s HDR-IPT paper. It mentioned the Michaelis-Menten model of enzyme kinetics used in biochemistry as a way of describing the human visual system response to increasing light stimulus. This is also discussed a bit in the Kim / Weyrich / Kautz 2009 siggraph paper Modeling Human Color Perception under Extended Luminance Levels. I was very curious about this curve because it has almost the exact shape of 1D lut component of a print film emulation lut, and many show luts I have encountered over the years. I was also particularly interested in this curve because of how insanely simple it is.
I chose to use a variation of the Hill-Langmuir formulation, because it provides a bit more flexibility. I solved for the inverse, solved for the y=1 intersect.
I also did a variation with a linear extension on the toe and shoulder, but later thought I probably didn’t need this.
The Hill-Langmuir tonescale is the one I’m using in my OpenDisplayTransform project.
Nuke implementation with the linear extensions available here, and a simpler version with a log-domain wrapper available here.
Timothy Lottes
I also implemented the Timothy Lottes 2016 Tonemap Operator (and display rendering method). It is a nuke node available here.
Others
I also tested like 20 other sigmoidal type functions if anyone wants to play around with them, they are all here.
Edit - Just wanted to put a link to a desmos plot of probably the top “honorable mention” sigmoid - the 4-parameter logistic curve: