I had been thinking previously about an example how this is practically useful to Photographers:

The L curve in Lab is a somewhat contorted gamma curve. It is based on a toppled gamma 3.0 with a small linear part in the dark tones. It is meant to match practical perception of subtractive samples with a reference white brightness of 100cd/m2.

It has given us one very useful value which is the middle gray reference of 18%.

However, because it is a contorted gamma curve, it loses one of the more useful features of a true gamma curve, which is invariance to multiplication. (At least I believe that's what it's called in english). That equates to Exposure Compensation in Photographer's terms.

That property has become all the more interesting in recent times where we have true HDR imaging. That is: input values in files can exceed well beyond the usual maximum value of 1.0. In addition, technology has advanced to such extend that the 100cd/m2 brightness is really a somewhat outdated number.

So, the interesting part of the formula above therefore is the fact that, while from a completely different approach, it matches L exactly, yet it has several properties that suggest it may be applicable in extended cases. HDR editing being one of them, monitor calibration being another...

Also, we had some recent discussions about on-board RAW data histograms. Because linear gamma histograms are generally not very useful, we all tend to agree that some form of logarithmic distribution is best.

A base-2 log distribution would be useful because it equates directly to F-stops, which is obviously something that most photographers will understand. You would see F-stops on the horizontal axis, and amount of data vertically. It would immediately show you how much to open up or stop down to bring the resulting data within the desired range.

(see Guillermo's example histogram in threads about ETTR elsewhere)

However, a default log distribution requires some form of reference unit. This is doable in-camera, but once the data is in a file it becomes a bit more tricky. Now, the above formula is nothing other than the square root of the base-2 log distribution. This turns out to be a direct equivalent of the L curve in Lab. And that suddenly makes everything very interesting because the L curve in Lab was established in a totally different way.

Then there are 2 problems with the L curve:

1. it is a shifted gamma curve which likely invalidates its application beyond the maximum value. (See above).

2. it has a linear part that is supposed to represent the transition to a more sensitive B&W perception when people look at dark tones in a dark context. Problem of course is that when viewing a single black pixel in a sea of white, we do not change our perception to the more sensitive B&W viewing. Some very complex models have been designed to overcome this problem, but they are not generally useful in any real life application.

Can we simply ignore the linear part in L ? Yes, but unfortunately not for the L formula, because the shifted gamma curve in L doesn't map the origin correctly.

So, now I have a gamma curve that seems identical to L, but suggest a better applicability to larger values, and a correct rendition to zero… And given the way that the above formula was derived, and how well it matches the perceptual data that the L formula was meant to represent, it may well have more significance than this.

tl;dr…

(see, the short version really was more interesting!)