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The battle of the ∆E (Part III)

In the previous post of this series we analyzed ∆ECMC's strengths and weaknesses. Anyway, it would be fair to say it turned out to be a huge quality leap compared to ∆Eab, which explains why it is widely used in the industry even today. But things didn't stop there...

∆EBFD, the soldier who never entered the battle

About 3 years after ∆ECMC was published, Ming Luo and Brian Rigg analyzed several real-world data sets in order to detect weaknessess in that formula and realized that better adjustments could be done by introducing an additional term called "rotational". Let's take a closer look.

As we noted, if we move from measuring differences in the a* and b* axis to the C* and h* ones we are in fact working with "ellipsoidal" tolerances. Since one of those axis is C*, we can expect all of them to point towards L* (because C* depends of the distance to that axis). Nevertheless, empirical evidence showed that is not completely true at the blue region.

BFD
Distribution of equal-tolerance ellipses according to Luo and Rigg.

In the graph to the left, several tolerance ellipses on the a*b* plane are shown (ellipsoids are projected here onto ellipses). In general we can see that the major axis of those ellipses point to the center (L* axis) except in the blue, high saturation region (+a*, -b* cuadrant, bottom middle).

Luo and Rigg developed a new formula that "rotates" those ellipsoids when compared colors are located in that zone, hence the need of adding a rotational term to the equation (in fact, they found a better correspondence modifiying the CMC formula than developing a new one).

With this modificaction, in 1987 they presented a formula called ∆EBFD[1][2], which had the following expression:

    \[ \Delta E_{BFD} = \sqrt{\left(\frac{\Delta L_{BFD}}{l}\right)^2 + \left(\frac{\Delta C^*}{cD_C}\right)^2 + \left(\frac{\Delta H^*}{D_H}\right)^2 + R_T\left(\frac{\Delta C^*}{D_C}\right)\left(\frac{\Delta H^*}{D_H}\right)} \]

The l and c factors had the same meaning as in ∆ECMC. The last term inside the square root contains the RT factor, whose mission is to control the ellipsoid axis' rotation when compared colors are located in the "problematic" zone, and its contribution is proportional to the differences ∆C* and ∆H*.

Despite of the fact that its creators showed it fitted the data better than previous metrics, this one never really catched up. A problem it presents is that the ∆LBFD factor is not a difference in the L* values of compared colors but an entirely new parameter LBFD which in turn depends of the Y value of the underlying XYZ coordinates. This implies that a L*a*b* comparison not only requires a "forward" conversion to L*C*h*, but a "backward" conversion to XYZ as well. Nevertheless, many elements of this formula were adopted, more than a decade later, in ∆E00.

∆E94, a compromise solution

Meanwhile, CIE took ∆ECMC formula as a reference, but instead of looking for more precision, they tried to get a similar performance with a simpler algorithm. Techical comittee CIE TC 1-29 found in 1994 an optimized formula (appropiately) named ∆E94, whose general structure resembled ∆ECMC, at least at the surface:

    \[ \Delta E_{94} = \sqrt{\left(\frac{\Delta L^*}{k_LS_L}\right)^2 + \left(\frac{\Delta C^*}{k_CS_C}\right)^2 + \left(\frac{\Delta H^*}{k_HS_H}\right)^2} \]

The kL, kC y kH factors are constant and equal to 1 (one) under standard viewing conditions, with some change in specific applications (for instance, in textile industry is customary to take kL = 2). Ellipsoids semiaxis, as in ∆ECMC, are defined in this case by SL, SC y SH. Simplification here consists in assuming that for any color pair to compare is always SL = 1, while both SC and Sdepend only on the average C* of compared colors.

∆E94 vs ∆ECMC

Besides the fact of being mathematically simpler, this new formula does not have the order-dependency of the computed difference; in other words, ∆E94 is again a proper metric instead of a quasimetric[3].

How much sinpler is it? Let's see how SC and SH vary with respect to the chroma value of compared colors:

  • The SC factor has a strictly linear dependence on C*. This factor has a minor variation according to whether it is applied in graphic arts or in the textile industry. For sake of comparison, the analogous factor in CMC is also shown.
  • The SH factor, which in ∆ECMC dependes on both C* and H*, here depends just linearly on C*. There is also a minor modification in the case of the textile industry.

Performance

Given the goal of ∆E94 was to produce the same result as ∆ECMC with less computational cost, one way to assess it is to compare how much they agree when applied to large data sets. One study[4] revealed an agreement of about 90% between them can be expected, and ∆E94 seems to be better adapted to observations when compared colors have different luminance (paradoxically the no-correction in the L* axis assumed by ∆E94 had to be later revised, because this axis is not itself perceptually uniform either). On the other hand, graph suggests for color differences near the gray axis, ∆E94 tends to underestimate differences compared to ∆ECMC.

Even when ∆ECMC was quickly adopted as standard in England, the fact it was never endorsed by CIE may explain why it never reach the adoption level of ∆E94 at the rest of the world, especially in the textile industry. However, pressure from those industries made apparent the need for researchers to improve those formulas even more. Our next contender will take several ideas from all this metrics and will become The One (at least until we will be able to find a better replacement to L*a*b*...)


1 BFD(l:c) Colour-Difference Formula, Ming Luo, Brian Rigg, Schools of Studies in Computing, University of Bradford.
2 There is no official reason as to why researchers picked that name, but in the original paper it can be read the authors used several data sets to test their brand new formula, each one having an alphabetic code. One of them has BFD as its code, so it is easy to suspect that name could be transferred to the metric for some internal reason.
3 In fact, in case of ∆ECMC values of SL, SC and SH depends on the color picked as reference, so when the comparison order is inverted resulting differences are in general different.
4 Testing CIELAB-Based Color-Difference Formulas, Manuel Melgosa, Departamento de Óptica, Facultad de Ciencias, Universidad de Granada.
Published inColorimetryHistory

2 Comments

  1. James Curran James Curran

    Thank you for writing and sharing these tremendous articles - I have ready numerous papers, book chapters and other sources describing this, and have never found such clarity, combined with an entertaining and approachable tone!

    • Edgardo García Edgardo García

      Thank you James for your kind words! I hope the final post of these series will be available soon... Stay tuned!

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