Color difference

Common definitions make use of the Euclidean distance in a device-independent color space.

When the result should be computationally simple as well, it is often acceptable to remove the square root and simply use

The closer approximations would be more properly (for non-linear sRGB, using a color range of 0–255):[1]

However, the non-uniformity of these spaces were later discovered, leading to the creation of more complex formulae.

[3] In 2019 a new standard for WCG and HDR was introduced, since CIEDE2000 was not adequate for it: CIEDE2000 is not reliable below 1 cd/m2 and has not been verified above 100 cd/m2; in addition, even in BT.709 blue primary CIEDE2000 is underpredicting the error.

The ΔEITP color difference metric is derived from display referenced ICTCP, but XYZ is also available in the standard.

where the components of this "ITP" is given by The Euclidean measure is known to work poorly on large color distances (i.e. more than 10 units in most systems).

[6] The International Commission on Illumination (CIE) calls their distance metric ΔE* (also inaccurately called dE*, dE, or "Delta E") where delta is a Greek letter often used to denote difference, and E stands for Empfindung; German for "sensation".

Use of this term can be traced back to Hermann von Helmholtz and Ewald Hering.

A good metric should take this into account in order for the notion of a "just noticeable difference" (JND) to have meaning.

[10] All ΔE* formulae are originally designed to have the difference of 1.0 stand for a JND.

This convention is generally followed by other perceptual distance functions such as the aforementioned ΔEITP.

[11] However, further experimentation may invalidate this design assumption, the revision of CIE76 ΔE*ab JND to 2.3 being an example.

Named after the developing committee, their metric is called CMC l:c. The quasimetric (i.e. it violates symmetry: parameter T is based on the hue of the reference

alone) has two parameters: lightness (l) and chroma (c), allowing the users to weight the difference based on the ratio of l:c that is deemed appropriate for the application.

[15] The CIE 1976 color difference definition was extended to address perceptual non-uniformities, while retaining the CIELAB color space, by the introduction of application-specific parametric weighting factors kL, kC and kH, and functions SL, SC, and SH derived from an automotive paint test's tolerance data.

and where kC and kH are usually both set to unity, and the parametric weighting factors kL, K1 and K2 depend on the application: Geometrically, the quantity

corresponds to the arithmetic mean of the chord lengths of the equal chroma circles of the two colors.

The parametric weighting factors kL, kC, and kH are usually set to unity.

The inverse tangent (tan−1) can be computed using a common library routine atan2(b, a′) which usually has a range from −π to π radians; color specifications are given in 0 to 360 degrees, so some adjustment is needed.

The inverse tangent is indeterminate if both a′ and b are zero (which also means that the corresponding C′ is zero); in that case, set the hue angle to zero.

7 and p. 23 stating most implementations on the Internet at the time had "an error in the computation of average hue".

The maximum discontinuity happens when the hues of two sample colors are about 180° apart, and is usually small relative to ΔE (less than 4%).

[25] Sharma, Wu, and Dalal has provided some additional notes on the mathematics and implementation of the formula.

[25] Tolerancing concerns the question "What is a set of colors that are imperceptibly/acceptably close to a given reference?"

This requires a perceptually uniform metric in order for the threshold to be constant throughout the gamut (range of colors).

In the CIE 1931 color space, for example, the tolerance contours are defined by the MacAdam ellipse, which holds L* (lightness) fixed.

As can be observed on the adjacent diagram, the ellipses denoting the tolerance contours vary in size.

More generally, if the lightness is allowed to vary, then we find the tolerance set to be ellipsoidal.

[26] The definition of "acceptably close" also depends on the industrial requirements and practicality.