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Old May 10th, 2008, 12:45 PM
Doug Kerr Doug Kerr is offline
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Location: Alamogordo, New Mexico, USA
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Default Found: a real bicone (sort-of)

While doing research in my quest to understand the (hex)cone and bi(hex)cone metaphors often cited in connection with the HSV (aka HSB) and HSL color models, I encountered a nice Canon site discussing their Picture Style Editor software package.

In that program, one can "pick" a color (for various purposes) in terms of three color coordinate systems: "Lab" (meaning L*a*b*), RGB, and "HSL". An excellent description is given of this HSL color model. It points out that in it the full range of "S" (which they describe as "saturation") is only available when L ("luminosity") is 0.5. Above that (as we approach L=1) and below that (as we approach L=0) the available range of S declines, vanishing at L=1 and L=0.

This certainly suggests that in their "HSL" color model the gamut (plotted in its native coordinate system) is something like a bi-cone (not a bi-hexcone).

They nicely illustrate the implications of this on entering colors into the program on a figure with a cross-section of a bi-cone solid.

You can see this nice presentation here:

and a further portrayal of the bi-cone here:

Clearly this "HSL" coordinate model is not what I am so bold as to call the "standard" HSL color model (whose gamut plots, in its own coordinate system, as a cylinder).

I reverse-engineered the "Canon" HSL model (I'll call it "CHSL" here to avoid any ambiguity) by tedious examination of the relationships between RGB and (C)HSL representations in the dialog boxes of the Picture Style Editor. I finally developed a system of equations (defining CHSL in terms of RGB) that comports perfectly with those relationships.

I won't present those equations here, as this is too cumbersome in "linear text" format (all the stuff would be drowned in parentheses). I will shortly release a revised version of my article on HSV, HSL, and "the cones" that will treat CHSL in detail in an appendix.

Now, it turns out that the gamut of CHSL doesn't actually plot (in its own coordinate system) as a bi-cone (that is, two right circular cones joined at their bases). It is in fact a somewhat-similar symmetrical figure of revolution, also tapering to a point at its apex and nadir, but its surface is "concave". Here we see a cross-section of the CHSL gamut solid:

Its "axis of revolution" is the L axis.

Now, what is the significance of the CHSL coordinate "S". Well it is "something like colorfulness" (in the same spirit that L is "something like (relative) luminance" and the S of HSL proper is "something like saturation". It doesn't directly track with accepted definitions of colorfulness (just as L and the other S don't actually track with any colorimetric property).

A widely-accepted definition of colorfulness is saturation times luminance. Its available range declines to zero as we (a) go above a certain luminance (different for different chromaticities) toward a luminance of 1, and (b) as we go below such a luminance toward toward a luminance of zero.

Colorfulness has a nice attraction conceptually since the "impact" of a certain saturation declines at lower luminances.

Why Canon chose this particular mathematical definition of "sort-of colorfulness" is not known. They could have easily chosen one that would have made the color gamut solid a real bicone. But then, what value would that have had, anyway?
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