# CIE ΔE* (Color Difference) Equations

Pictured above is **a combination handheld spectrophotometer and densitometer, the SpectroDens 4**. The intent of using deltaE (ΔE) is to describe the distance between two colors. The Just Noticeable Difference (JND) of deltaE is usually 1. In other words, if two colors have a deltaE less than 1 it is imperceptible and larger than 1 is perceivable. Unfortunately, due to the nature of human color perception (our eyes), the visual perception of colors is different. In general, our eyes are more sensitive to changes in Chroma than lightness. That means, the same deltaE between two yellows and two greens will very likely look different in our eyes. With that in mind, newer deltaE equations have been developed over the years and below is a list of 4 major formulas we might encounter in the color reproduction field.

**ΔEab (aka. ΔE76, dEab or dE76)**

The CIEL*a*b* and ΔEab was introduced by the International Commission on Illumination (CIE) in 1976. Given two colors in the CIEL*a*b* color space, (L_{1}, a_{1}, b_{1}) and (L_{2}, a_{2}, b_{2}), the ΔEab formula is defined as:

_{dEab equation from http://www.brucelindbloom.com/index.html?ColorDifferenceCalc.html}

Where,

L_{1 }– the CIE L* value of reference color

a_{1 }– the CIE a* value of reference color

b_{1 }– the CIE b* value of reference color

L_{2 }– the CIE L* value of sample color

a_{2 }– the CIE a* value of sample color

b_{2 }– the CIE b* value of sample colorThe ΔEab has been succeeded by other formulas that are discussed below, while it still bears useful information about the linear distance between two colors.

**ΔE**_{CMC}(aka. dE_{CMC}, CMC l:c)

The CIE is not the only party that defined color differencing equations. The Colour Measurement Committee of the Society of Dyers and Colourists (CMC) defined a new color difference method in 1984, named after the developing committee, CMC l:c.

_{dECMC equation from http://www.brucelindbloom.com/index.html?ColorDifferenceCalc.html}This equation takes the complexity of human color sensitivity/perception into consideration based on CIEL*C*h* – notation of colors. There are a few variations in the formula since it allows the user to assign different weights to its lightness (l) and chroma (c) factors. The CMC l:c was developed based on the visual evaluation of textile samples and human vision sensitivity levels in the lightness (l) and the chrome (c), the default ratio of l: c is 2:1, which doubles the tolerance of variation for lightness then that for chroma. The other common ratio of l:c is 1:1. Please consult with your supplier/buyer on which ratio to use if CMC l:c is selected for your production. Different ratios will result in varying sizes of tolerance ellipses, in other words, acceptability of color match.

**ΔE**>_{94 }(aka. dE94)

In 1995, the CIE revised the formula by introducing ΔE_{94}to address the color non-linearity nature under ΔEab. Like CMC l:c method, ΔE_{94 }also uses CIEL*C*h* for calculating color differences._{dE94 equation from http://www.brucelindbloom.com/index.html?ColorDifferenceCalc.html}The ΔE_{94 }formula provides two coefficients, k and S, which are mostly based on tolerance data from RIT/Dupont from automotive paint research. The k-coefficients are known as parametric factors and refer to effects including color-difference judgement. The S-coefficients account for CIEL*a*b*’s lack of visual uniformity (Billmeyer, 2000). Most of the time, those two types of coefficients are pre-selected by the software developer based on either textile or graphic arts industry users. While, due to the limitation of the ΔE_{94 }that lacking accuracy in the blue-violet region of the color space, which eventually leads to the release of ΔE_{2000 }(Habekost, 2013)_{.}

**ΔE**_{2000 }(aka. ΔE_{00}, dE2000, CIEDE2000 or dE00)

The ΔE_{2000 }was first proposed by CIE TC1-47 in CIE Publ.142 in 2001 and standardized in 2013. You might find an old ISO white paper or IDEAlliance G7 Specification that still use ΔEab as the dominant color difference formula along with ΔE_{2000 }for information purposes only. Since 2013, both ISO and IDEAlliance has adopted ΔE_{2000}as the new industry standard for calculating color differences._{dECMC equation from http://www.brucelindbloom.com/index.html?ColorDifferenceCalc.html}

It is obvious that the ΔE_{2000 }is much more complex than the ΔEab. In summary, the ΔE_{2000 }introduced weight functions: S_{L}, S_{C}, S_{H}, R_{T} and a’(Neutral), as positional corrections to the lack of uniformity of CIELAB, also parametric factors: K_{L}, K_{C} and K_{H} as corrections accounting for the influence of experimental viewing conditions (D65, 1000 lx, background gray with L*=50, etc.).

There are numerous studies indicating that ΔE_{2000 }is superior than other color differencing formulas:

“Currently, we have no candidate color space (e.g. DIN99, CAM02, OSAGP , etc.) providing statistically significant improvement upon CIEDE2000.” (CIE TC 1-55, 2016)

“From the experiments carried out in the various years it becomes clear that ΔE_{2000 }corresponds better with the way human observers perceive small color difference.” (Martin, 2013)

“The CIEDE2000 formula may not be the final word with respect to a colour difference formula for small color differences for industry. The experimental data on which the formula is based are far from perfect. However, at the present time the formula represents the best that can be achieved. In our view, CIEDE2000 is timely because there are two different formulae (CMC and CIE94) being widely used at present. This is clearly unsatisfactory. The new formula offers significant improvements over both.” (M. R. Luo, G. Cui, and B. Rigg, 2002)

**Final Verdict**

Even though the ΔE_{2000 }won’t be the final answer for the best color difference equation for the color field, it is by far the best and most widely used formula in the graphic arts industry backed by ISO and IDEAlliance. But regardless of which color differencing equation you are using, a good color match is what is approved by your customer or whoever pays the bills! That said, please allow me to end this blog using the five rules from Billmeyer (1970/1979):

- Select a single method of calculation and use it consistently.
- Always specify exactly how the calculations are made.
- Never attempt to convert between two color differences calculated by different equations through the use of averaging factors.
- Use calculated color differences only as a first approximation in setting tolerance, until they can be confirmed by visual judgments – in other words, verify all calculation visually.
- Always remember that nobody accepts or rejects color because of numbers – it’s the way that it looks that counts.

**Update from 1.25.2022:**

Within the past few years, more and more clients of ours have asked about changing the weighting functions of the deltaE equation they are using. Therefore, we have released a firmware update for our handheld SpectroDens 4 which now offers this new feature. As the picture 1 below shows, the user can modify each weighting factor value if needed. Please note, if you assign a larger value than the original factory settings (in the picture), you are “decreasing the sensitivity” of that attribute. In other words, under the same deltaE tolerance, a larger weighting function value allows more discrepancies than a smaller one.

Here are the default coefficients for each deltaE equation:

kL | kC | kH | |

ΔEab | N/A | N/A | N/A |

ΔE_{CMC} | 2 | 1 | N/A |

ΔE_{94} | 1 | 1 | 1 |

ΔE_{2000} | 1 | 1 | 1 |

Note: it is inadvisable to change from the default coefficient unless specified in the Print Quality Requirement from the print buyer

Picture 1. TECHKON SpectroDens deltaE equation settings

We have noticed an obvious tendency that more and more people are switching from ΔEab (ΔE76, dEab or dE76) to the ΔE2000 (ΔE00, dE2000, CIEDE2000 or dE00). If you are thinking about switching, or just curious about seeing the deltaE values from different equations, we have a free app to help. You can download and give it a try via the link here: **download the deltaE calculator**

Picture 2. TECHKON deltaE equation calculator

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*References*

*Bruce Lindbloom, varies screenshots of different color differencing formula, April, 2017 (http://www.brucelindbloom.com/index.html?ColorDifferenceCalc.html)*

*CIE TC 1-55 committee, Recommended Method For Evaluating The Performance of Colour-Difference Formulae, CIE217:2016 (http://www.cie.co.at/publications/recommended-method-evaluating-performance-colour-difference-formulae)*

*Martin Habekost, Which color differencing equation should be used, International Circular of Graphic Education and Research, No. 6, 2013 (https://www.hdm-stuttgart.de/international_circle/circular/issues/13_01/ICJ_06_2013_02_069.pdf)*

*Billmeyer, S. (2000): ”Principles of Color Technology”, 3rd ed. New York: Wiley & Sons.*