CIE XYZ Explained

CIE XYZ is the bridge between spectra, human observer functions, chromaticity diagrams, luminance, RGB working spaces, and profile connection spaces. This workbench exposes the whole pipeline in one place: build a spectrum, choose the standard observer and reference white, integrate to XYZ, project to xyY, then test whether the result fits a real RGB display space.

CIE XYZ is a three-dimensional coordinate system in which every visible color sits at a unique point. Coordinates are non-negative real numbers, two of them roughly proportional to chromatic information (X and Z) and one proportional to luminance (Y). Two stimuli with identical XYZ triples look identical to the standard observer under the same viewing conditions, regardless of what their underlying spectra are.

That is the entire promise. Take any spectrum, integrate it against three published weighting functions called the color matching functions, and you get three numbers - X, Y, Z - that fully describe what the average human will see. Two laboratories on different continents, with different equipment, looking at different physical samples, can compute the same XYZ from a measurement and agree on the color. Before 1931, this was not possible.

In the late 1920s two British researchers ran the experiments that became the empirical foundation of XYZ. W. David Wright at Imperial College and John Guild at the National Physical Laboratory each built an apparatus with three primary lights that could be mixed in adjustable proportions. Observers looked into the apparatus at a split visual field: a test wavelength on one side, the three-primary mix on the other. They adjusted the mix until the two halves matched.

Doing this for many wavelengths across the visible spectrum, with seventeen observers in total, produced a tabulated function that said "to match wavelength λ, take this much red, this much green, this much blue." Those tabulated functions are called color matching functions. Wright used 650 / 530 / 460 nm primaries; Guild used a different set; but when both data sets were transformed to a common pair of primaries, they agreed almost exactly.

The color matching functions (CMFs) are the heart of XYZ. There are three of them: x̄(λ), ȳ(λ), z̄(λ) ("x-bar," "y-bar," "z-bar"). Each one is a curve of weight versus wavelength. Multiply a stimulus spectrum by one of them and integrate, and you get one of the tristimulus values. Three CMFs, three tristimulus values, one color description.

The Wright/Guild data, expressed in the original rgb primaries, contained negative numbers - some spectral colors required "subtracting" red, which in the real apparatus meant adding the matching primary to the test side rather than the comparison side. Calculations with negative values are awkward and lead to negative tristimulus quantities that have no physical interpretation.

The CIE solved this by choosing three new primaries - X, Y, Z - that were intentionally outside the gamut of visible colors. No physical light can produce the X, Y, or Z primaries. But because they sit outside the gamut, any real color can be expressed as a positive combination of them. The three new primaries are "imaginary" or "non-physical" in the sense that they cannot be realized with a physical light source - but they make the math clean and the tristimulus values always positive.

Now we can write the full operational definition. For a self-luminous stimulus Φ(λ) - measured in watts per nanometer - the tristimulus values are

For a reflective object, the stimulus is reflectance times illuminant, and you normalize by the illuminant's tristimulus values so that a perfect diffuse white under the same illuminant produces Y = 100.

The three tristimulus values entangle two pieces of information: how much light is there (intensity) and what color it is (chromaticity). To plot colors on a flat map - to draw the famous CIE chromaticity diagram - we project XYZ onto a plane by normalising.

The xyY representation keeps x and y as chromaticity and uses Y directly as luminance. Two stimuli with the same (x, y) but different Y are the same color at different brightnesses. xyY is the basis for almost every published color diagram you have ever seen.

The CIE published two standard observers. The original 1931 2° observer was based on Wright/Guild data, who used a 2-degree wide test field on the retina - small enough that the image fell entirely within the cone-dense fovea. The 1964 10° supplementary observer was added later using larger field experiments - representing what happens when the test area extends into the cone periphery, where rod and cone densities differ.

The two observers disagree slightly. The 10° matching functions are flatter and shift a bit because the larger field includes more peripheral cones and some rod influence. Picking the wrong observer for a given application can produce disagreements as large as a few ΔE units. Modern color management defaults to the 2° observer for most computational work but uses 10° for printed-textile and paint-chip comparisons where the sample field is genuinely large.

XYZ alone does not tell you what "white" is - it just gives coordinates. To convert between color spaces you need to specify a white point: the XYZ of whatever the system considers a neutral reference. Every standard color space picks its white. sRGB and Rec.709 use D65. Adobe RGB also D65. ICC profile-connection space defaults to D50. Cinema's DCI uses a slightly green-shifted "DCI white."

Every RGB color space is defined by a matrix transform between its linear-light RGB primaries and CIE XYZ. The standard sRGB matrix (D65, ICC v4, IEC 61966-2-1) is:

Two crucial details. First, the inputs must be linear-light RGB, not the gamma-encoded RGB that comes out of a typical image file. You have to undo the sRGB transfer function first - approximately a 2.2 power, exactly a piecewise curve. Second, the matrix is space-specific. Adobe RGB has a different matrix. Display P3, Rec.2020, ProPhoto - all different. The matrix entries encode each space's primary chromaticities and its white point in a single transform.

The Y in XYZ is deliberately the photopic luminosity function. That single design choice has enormous consequences. It means the second coordinate of every XYZ value directly encodes how bright the stimulus looks. Every contrast calculation in WCAG, every grayscale conversion, every video luminance signal - they all start from Y or a close relative.

In sRGB the Y coefficient is roughly 0.21 R + 0.72 G + 0.07 B - which is why green dominates perceived brightness and pure blue looks so dim despite reaching full digital value. Variants of these coefficients live in Rec.709, Rec.2020, the WCAG relative-luminance formula, and the L* lightness used in CIELAB. They differ slightly because each is tuned to a particular primary set, but they all descend from CIE Y.

XYZ is a reference, not a representation of how color looks. Two key shortcomings drove the development of every later color space.

Despite its limitations, XYZ is the universal pivot. Every other modern color space is defined by a transform to or from XYZ. ICC profiles store their transforms relative to a profile-connection space that is essentially XYZ at D50. CIELAB is a direct nonlinear function of XYZ. sRGB, Display P3, Rec.2020 are matrix transforms of XYZ. ΔE is computed in CIELAB, which is computed from XYZ.

Six questions on XYZ, the standard observer, and tristimulus integration. Wrong answers come with brief explanations.