CIELAB: From XYZ to x,y,Y and MacAdam Ellipses (Part 2)

Every color can be described with three tristimulus values X, Y, and Z. These values are always ≥ 0 and form the basis of colorimetry. Instead of these three absolute values, practice often uses the normalized color coordinates x, y, and additionally the luminance component Y. This forms the notation x,y,Y.

From XYZ to x,y,Y

The color coordinates are calculated as follows:

• x = X / (X + Y + Z)
• y = Y / (X + Y + Z)
• z = Z / (X + Y + Z)

The sum x + y + z = 1, whereby the third coordinate can always be derived (e.g., z = 1 – x – y).

In the x,y-chromaticity diagram:

• x is plotted along the horizontal axis
• y along the vertical axis
• Y functions as the third dimension (brightness), but is not displayed in the 2D diagram

Therefore, when interpreting, it should be considered that all colors in the diagram assume equal brightness (equal Y) — something that is often incorrectly ignored in many visual diagrams.

Interpretation of the x,y Diagram

Important structures within the diagram:

• The horseshoe-shaped edge contains the spectral colors (single wavelength).
• The straight line between the blue and red ends is the purple line (mixed colors that are not spectral).
• The white point (neutral light) is at x = 0.333, y = 0.333 for an ideal equal-energy light.

This diagram is widely used to visualize color relationships, but should be interpreted with caution.

Non-Uniformity: MacAdam Ellipses

In 1942, David MacAdam investigated how large a variation in color could be before a human perceives a difference (JND – just noticeable difference). For various points in the x,y diagram, he determined:

• Direction of smallest perceptible difference
• Size of this variation

The resulting areas — MacAdam ellipses — are often shown 10× enlarged on maps.

MacAdam's Conclusions

• The ellipses differ greatly in shape, size, and orientation depending on the position in the x,y space.
• This demonstrates that the x,y-chromaticity diagram is not uniform.
• An equal distance in x,y does not represent an equal perceived color change everywhere.

This lack of perceptual uniformity was an important motivation for the development of CIELAB, which specifically attempts to make color differences perceptually consistent.

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