In the CIEXYZ to CIELUV conversion, what are `u'`, `v'`, `u'ₙ`, `v'ₙ`, and `Yₙ`?

Question

I've recently been looking into color spaces and color models. I've become interested in the relationships and conversions between different spaces, and I came across CIEXYZ and CIELUV.

In the conversion algorithm, the CIELUV article mentions the variables u', v', u'ₙ, v'ₙ, and Yₙ. I understand that these are related to the white point (of which a table of common values can be found here,) but I'm not exactly sure what the relationship is.

In the article, it says:

For example, for the 2° observer and standard illuminant C, u'ₙ = 0.2009, v'ₙ = 0.4610

(The values for the 2° observer and standard illuminant C can be found in the table mentioned above, and are given as x₂ = 0.31006 and y₂ = 0.31616.)

Below this, the equations for computing u' and v' are given:

u' = (4x)/(-2x + 12y + 3)

v' = (9y)/(-2x + 12y + 3)

If we plug in the x₂ and y₂ we've been given, we get the expected result of u' = 0.2009 and v' = 0.4610. However, earlier in the article, it said that u'ₙ and v'ₙ should be those respected values. Not u' and v'.

Also, the article makes no mention of Yₙ other than stating that it's the luminance of the (u', v') chromaticity coordinates, information on which I can't find.

So, my question is, what the heck are u', v', u'ₙ, v'ₙ, and Yₙ? Assuming that u' and v' are the target chromaticity and u'ₙ and v'ₙ are the source chromaticity (or vice-versa,) then the main conversion algorithm would make no sense if the target and source are the same, as the resulting u* and v* would equal 0. If someone could shed a little light on this for me, it'd be greatly appreciated. Thanks.

Answer 1

You calculated the u', v' of white point, which are per definition u'ₙ, v'ₙ. These are constant. Computer screen (sRGB), HDTV (Rec.709), movie theater, etc. have own specific white point. You should decide it, and keep constant for the transformation.

Yₙ is the luminance of the white point. This depend on the setting of the television. More power, more luminous will be the white point. Probably you should set it as 1.0 (or 100, or the maximum digital value). You see that L*, u*, and v* are inversely proportional to Yₙ. This is just used to norm L to 100 (in the case of Wikipedia transformation). So you should just it as the maximum Y, or the luminance Y of white (if you have superwhite, like in HDTV).

CIEXYZ (you initial space, and most common "common" space to be used between transformation) have just defines X, Y, Z (or x, y, z or also more frequently x, y, Y). These values represent eye sensitivity (really a linear transformation, just to have positive values, they discovered exact coordinates later (LMS colour space), so that are independent to white space (so you have no parameter for white), but this means also that well defined chromacity (x, y) can be seen as different colour, according the actual colour adaptation.

I short (inverse order from Wikipedia).

• you have x, y, Y of a point, according CIEXYZ (linear space). x, and y are the chromacities. Y is the luminance.

• you transform to u', v' (for now just chromacities, also just in function to x, y).

• and last step is the gamma correction, so you transform from linear space into a perceived function, so you get L*, u*, v*.

Wikipedia is mixing colour model (e.g. RGB, and the generic transformation to/from RGB) with colour space (sRGB [the normal "web colour" space], which is a RGB model with well defined chromacities for Red, Green, Blue and White, and a well defined gamma function). You should treat variables with N subscript as pre-defined constant, for your specific Luv colour space.