When White Becomes Pale Yellow – The Fundamental Limit Of Colour Measurements

Consider two white light sources, both of them appearing white like daylight. Consider one light source to be actual daylight, known as D65, and the other a ‘fake daylight’ created by a combination of LEDs. Their emission spectra could look like this:

Though the two light sources have completely different emission spectra, they will appear indistinguishably white to the average human observer. You could swap them, and the average person wouldn’t notice.

The reason a human can’t tell the light sources apart is simple: humans don’t see spectra, they see drastically simplified forms of them. They perceive the ratios of the so-called tristimulus values and associate a colour with them. There are 3 of those tristimulus values, called X, Y and Z, and each of them is a weighted average of the light source spectra. Those weighted averages, loosely speaking, select for red (X), green (Y), and blue light (Z), and, together, they form colour vision.

The spectral weighing for each tristimulus value is described by its filter, commonly referred to as CIE X, CIE Y and CIE Z. The weighing with those filters and the computation of the tristimulus values for the above spectra looks like this:

For D65 white, the tristimulus values are

(1)  

(2)  

On the CIELUV chromaticity chart, it looks like this[1]:

In the eyes of a human observer, both spectra produce the same colour. They are called metameres.

An Imperfect Observer

Colour measurement replicates the colour response of the human eye. To do so, it commonly equips an objective lens with a wheel carrying at least 3 transmission filters. One filter changes the spectral transmittance of the objective lens such that it is identical to the X tristimulus filter; a second filter to Y, and a third filter to Z.

These filters and lenses are rarely, if ever, perfect. More often than not, the actual spectral transmittance of such a system looks something like this:

With such filters, the objective lens is, at best, an imperfect colour measurement device. It resembles human colour vision, but not quite. The spectral transmittance and the tristimulus curves are not perfect matches.

With these filters, the two white emitters are no longer perceived to emit white light. When running it through the same algorithm as before:

and computing the precise numbers:

one finds that the emitters, which are actually white, are perceived to be pale yellow and slightly darker pale yellow in colour. The imperfect filters lead to two distinct errors in the colour measurement:

Colour calibration can somewhat fix the first issue, but not the second. And this inability to fix the second issue is a hard limit.

Before moving to colour calibration, it is valuable to plot out the metameric failure caused by these imperfect filters. It gives one a feeling of just how wrong things can go with bad filters. To get this feeling, one can generate a number of white spectra, measure their percevied CIELUV colour coordinates, calculate the metameric failure

(3)  

One can now easily see that the less spectral overlap a particular white spectrum has with the D65 spectrum, the worse the metameric failure can be. Misdetecting white emitters to be green is perfectly posssible.

White Point Calibration

White point calibration, at its most fundamental level, means that one introduces calibration constants to modify the measured tristimulus values X, Y and Z such that the so-corrected tristimulus values minimise the colour error ΔE for a given reference spectrum.

By choosing D65 as this reference spectrum, the simplest possible white point calibration is expressed by the equations

(4)  

Applying such a simple white point calibration and once again plotting out the metameric failure as a function of the spectral overlap gives the following result:

One immediately notices that the colour error ΔE significantly reduces under white point calibration. The closer a spectrum to the D65 spectrum, the better the calibration works. For the D65 spectrum itself, at 100 % spectral overlap, the colour error is perfectly 0.

One also notices that colour errors smaller than 0.01 start to be hard to notice. The colours are so pale that they are hard to distinguish from true white. However, colour errors larger than 0.01 are clearly noticeable to the human eye.

Most notably, however, metameres are still not recognised as such. For small D65 spectral overlaps, the white point calibration barely works and the detected colour is still far far away from white. The only way to improve the measurement accuracy by is by having knowledge of the source spectrum. But this defeats the purpose of colour measurement entirely — if the source spectrum is known, why bother with colour measurement at all?

Some things simply cannot be calibrated away. A colour measurement system’s ability to correctly detect white is effectively limited to spectra close to the calibration spectrum. In this particular example, if one were to accept a colour error of 0.01 at most, then the spectral overlap with D65 cannot be less than 90%. The moment the source spectrum is “too unlike” the D65 spectrum, the colour measurement will be off. And no calibration can help it. Only better hardware can.

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