cscheid, 16 Feb 2012

I’ve been playing around with the HCL color space. HCL, if you’ve never heard of it before, is a color space that tries to combine the advantages of perceptual uniformity of Luv, and the simplicity of specification of HSV and HSL. HCL is an improvement over HSV and HSL, but it is not exactly ideal: there is a nasty discontinuity at some bits of the transformation! I have been trying to find a way around this, but I’m stumped. Let me explain, and maybe you can help me.

The transformation from RGB to HCL is somewhat complicated, and involves two intermediate color spaces, CIEXYZ and CIELUV. Going from RGB to XYZ is a simple matrix transformation: $(x,y,z) = M . (r,g,b)$. For arcane reasons, there are many possible matrices: the one most relevant nowadays is the sRGB/D65 matrix. This is a linear transformation designed to make a “brightness” coordinate, Y, while encoding the rest of the space in the other coordinates by roughly mapping them to “red” and “blue” stimuli.

To go from XYZ to CIELUV, things are a bit more complicated: this is the bit that tries to match the physiology of a typical human vision system, which is much better at telling shades of yellow and green apart than it is at telling shades of blue apart.

The full transformation behaves nonlinearly, and tries to make the euclidean distance in CIELUV correspond roughly to perceptual differences. In this space, L encodes the lightness of the color, or how bright it is, and uv encodes the chromaticity portion: the particular tint or shade of the color.

Finally, HCL is then obtained by simply transforming the UV coordinates of Luv to polar coordinates. The phase is interpreted as hue, and the length of the vector as “saturation” (specifically, it’s then called chroma).

The goal of HCL is to be perceptually uniform along its axis, and so the thing to notice is how the apparent brightness of the colors all appear roughly the same for any given slider setting; and while moving along the horizontal axis changes the hue of the color, it doesn’t change the perceived lightness or saturation. Compare this with the HSV colorspace.

So you can play with these color spaces, I’ve written a few little demos of the color spaces using Lux. The sliders control the axes which resemble brightness, and the image then shows a slice of the resulting parameter space. You will need WebGL and Chrome for these to work (sorry!). Pay attention to the boundary of the gamut.

One of the great conveniences of HSV is that no matter what you do in HSV, you will end up somewhere inside the (0,0,0)-(1,1,1) cube of valid RGB colors. That means nothing too strange happens.

On the other hand, if you play a bit with the LUV and HCL colorspaces in low luminances, you will see a discontinuity in the conversion. Although it happens outside the RGB gamut, it is still quite annoying: some paths through HCL are cut off in RGB. The issue happens when clamping the values from outside of the gamut back into (0,0,0)-(1,1,1). This is what I would like to solve: is there a simple way to create a (clamped) conversion from HCL to RGB that is continuous and reasonable?

The procedure that is used in the R package for colorspace management is the one I’m currently using in the demo above: after converting from HCL to a value, we find the closest point to the raw conversion that is inside the RGB cube.

Here’s a different approach that is continuous: instead of converting the color $c$, we instead search for the closest color in HCL space $c’$, which converts to a value inside the RGB gamut. Now the problem is: how do we actually find such a transformation efficiently? It’s easy to see that if $c$ goes outside the RGB gamut, then $c’$ will be on the boundary of the gamut. So this is “merely” a two-dimensional search problem. Except that the boundary of HCL or CIELUV in RGB space is complicated. So we’re looking for the minimum of a function constrained to a complicated 2D surface, and I don’t think there’s any simple algorithm to do this.

Or is there?

## Acknowledgements

Thanks to Hadley Wickham for teaching me about HCL, whose ggplot library uses that color space extensively. This post grew out of trying to make continuous HCL scales easier to specify.