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The battle of the ∆E (Part II)

In the first part of this series we discovered that our humble ∆E, born along with L*a*b* circa 1976, was not good enough as expected to measure perceptual differences between colors, and we also analyzed why, despite of being an intrinsic issue with L*a*b*, it turned out to be more practical to modify the formula than to seek for a new uniform space.

From now on we must get more "serious", so let us define what we are expecting to get when using classic ∆Eab. Given two distinct colors by means of their coordinates L*1, a*1, b*1 and L*2, a*2, b*2. we can calculate a difference in each coordinate as ∆L* = L*- L*1, ∆a* = a*- a*1 and ∆b* = b*- b*1. The design goal of L*a*b* was to let us calculate a color difference simply by using

    \[ \Delta E = \sqrt{\Delta L^*^2 + \Delta a^*^2 + \Delta b^*^2} \]

For those more visual oriented, we are just measuring the diagonal of the small cube whose opposite vertices are the colors to compare:

Definition of classic ∆E (∆Eab o ∆E76) as the simple distance between two colors in L*a*b* space, represented as the diagonal of the cube shown.

∆ECMC, the first contender

L*a*b* gained widespread adoption soon after CIE published it, chiefly in the textile industry. As experts in dye and inks manufacturing since the end of XIX century, it was one of the industries using the system the most, and also one of the first to detect some inconsistencies.

If we recall again MacAdam ellipses, it's easy to understand that, in an ideal metric, colors located at an arbitary but fixed tolerance with respect to a reference color should be located in a circle around it in a chromaticity diagram, and also in a sphere in a 3D representation. From large print jobs' measurement data, taken at different pass/fail conditions, it was found that colors located at the same tolerance didn't form a sphere, but ellipsoids inside the very L*a*b* space instead, i.e. the same flaw MacAdam originally pointed out in the CIE xy chromaticity diagram.

The gauntlet was picked up by the Society of Dyers and Colourists (SDC for short), a professional society born in 1884 and "specializing in colour in all its manifestations". They created a commitee to address this issue, the Colour Measurement Committee (CMC). After several studies, they produced in 1984 (that is to say, 10 years before the next metric appeared) a new formula for color difference calculation. This new formula took the committee's name and therefore is known as ∆ECMC, which brought a bunch of new concepts that all of its future contenders will adopt.

Opponent-process color theory, which is physiologically backed up by the existence of ganglion cells at the retina, successfully explains the color representation in the Munsell system and it has its place in the very definition of L*a*b*. Nevertheless, although that process happens as soon as the nerve impulses leave the cone cells, it is just one previous stage before final color perception takes place inside our visual cortex and its natural interpretation in terms of luminosity, hue and saturation. It seemed logical that color difference perception was closely related to that model; in fact Munsell system shares that same structure (value, hue, chroma). It was then necessary to move from a rectangular coordinate system such as L*a*b* to a polar one (in fact cylindrical) known as L*C*h*, where L* is the same in both systems, while C* (chroma) is the color's radial distance to L* axis, and h* (hue) is the angle subtended by the color and an arbitrary, fixed reference axis (+a* was chosen for that purpose).

a*, b* coordinates of a certain color (left) and their respective C*, h* coordinates (right). In this azimuthal view, L* (same in both systems) points towards the reader.

The following formulas let us convert from L*a*b* to L*C*h*:

    \[ L^* = L^* \]

    \[ C^* = \sqrt{{a^*}^2 + {b^*}^2} \]

    \[ h^* = \arctan\frac{b^*}{a^*} \]

For all the above, ∆ECMC starts converting the colors to compare to their L*C*h* equivalents and assigns a "weight" to each coordinate according to human sensitivity to a variation of that coordinate. In other words, we keep measuring distances, but each one is measured with a different scale, depending on its influence in perceived color difference.

The complete formulas can be found elsewhere[2][3], but for the purposes of this post let say ∆ECMC considers not differences between L*, a* and b*, but between L*, C* y h* instead, according to this equation:

    \[ \Delta E_{CMC} = \sqrt{\left(\frac{\Delta L^*}{lS_L}\right)^2 + \left(\frac{\Delta C^*}{cS_C}\right)^2 + \left(\frac{\Delta H^*}{S_H}\right)^2} \]

Although the general form of this equation resembles the classic one (square root of sum of squares), several features must be noted:

  • Terms ∆a* and ∆b* have been replaced by ∆C* and ∆H*, i.e. chroma and hue differences.
  • In this formula there is no ∆h* (as one might expect) but ∆H*. The reason: h* is an angle, and its difference ∆h* it's not a distance. The corresponding distance is constructed from that value and is referred to as ∆H*[4].
  • ellipsoid
    Geometric meaning of the weighting factors SL, SC and SH.

    Each difference term is divided into an associated factor SL, SC y SH which represents the tolerance to a variation in that particular dimension. These factors, which allow us to move from a spherical tolerance (as pretended by ∆Eab) to an ellipsoidal one (as empirically established), aren't fixed: they depend in turn to the colors being compared. Any factor greater than one means a greater tolerance compared to classic ∆E in the corresponding dimension, and consequently shows that the simple ∆Eab numeric distance would tend to overestimate perceived difference.

  • Last, differences ∆L* y ∆C* each have an additional parameter, l y c respectively, allowing a sort of "formula customization" for specific applications. It is customary to take l = 1 y c = 1 when looking for "just noticeable differences" (perceptibility criterion), or to take l = 2 y c = 1 when looking for "acceptable differences" (acceptability criterion). It is usual to name these cases as ∆ECMC(1:1) and ∆ECMC(2:1).

∆ECMC vs ∆Eab

Let's take a look at the behavior of these SL, SC and SH factors. At this point we should note that any single numeric analysis won't be able to tell us which metric is better; in order to achieve that, we need to compare computed values with actual, observed differences. Assuming ∆ECMC as better (so far in this post series), we want to know what those who used to work with ∆Eab should expect. Each factor has a specific behavior:

  • The SL factor corrects differences relative to the lighnness level L*. Even though ∆Eab is relatively accurate around L* = 50, we know that predicted differences do not match perceived ones above and below that value. In the following graph we show how the SL factor varies with L*. According to CMC, for values of L* greater than 43, measured distance in the L* direction tends to overestimate perceived difference with respect to the classic formula; thus, ∆ECMC will compute smaller differences to those estimated by ∆Eab. On the other hand,  for smaller values of L*, tolerance reduces up to L* = 16 and remains constant at about 0.5.
  • The SC factor does the same for chroma differences. The next graph suggests that ∆Eab overestimates differences above C* = 6, and underestimates them below that value. As a consequence, according to CMC, those using ∆Eab are computing smaller differences than perceived for colors near the gray axis.CMC SCv2
  • We finally arrive at the SH factor, the most complex of them because it depends on both h* and C*, and therefore we cannot plot its behavior with a curve; we need a surface instead, whose height varies with each point considered. The following animation aims to get an idea of the values SH takes at every point of the a*b* plane. The higher the surface, the greater the tolerance to changes in hue in that point. The first mark of the vertical axis starting from below represents 1 (one).CMC-SH-ani-v4


British in its origins, it's no surprise this formula had gained widespread adoption there, mainly at the textile industry. British Standard adopted it in 1994 and the following year became an international standard (ISO 105-J03 Textiles — Tests for colour fastness — Part J03: Calculation of colour differences). It remains very popular; those who acknowledged ∆Eab flaws but don't yet trust in ∆E00 are using it nowadays.

The numerical analysis of this formula allow us to infer the following features:

  • As expected, CMC takes into account the wider tolerance of human vision to changes in chroma in the high saturation region;;
  • Conversely, tolerance narrows near the gray axis relative to ∆Eab. If we compare colors under C* = 6 we can see that classic formula understimate differences.
  • Particularly, for darker colors (L* < 16) CMC predicts a tolerance almost half of that of ∆Eab. Nevertheless, this is not usually a problem (at least in graphic arts) because it is very unlikely to reach densities high enough as to make L* to fall in that range;
  • A big problem of CMC is that computed distances are not simmetrical (this issue cannot be seen in these graphs). In general, distance from color 1 to color 2 is different than distance from color 2 to 1, thus the order colors are compared is relevant (in math parlance, it's not a metric but a quasimetric);
  • Even though SL, SC and SH are continuous respect to their variables, SH exhibits "broken zones" where the parameter changes its slope abruptly from decrescent to crescent. It would be better if those transitions were smooth in order to not having "special zones". Several borders can be seen in the SH surface, particularly the one at h* = 275ª (close to the -b* axis) in the blue region. That zone is specially problematic and its treatment remains so even in ∆E00, as we will see later.

In the third part of this series we will talk about the arrival of ∆E94 at the battlefield.

1 In fact there was already enough empirical support for this idea at that time, and even practical applications. When in 1962 PAL color television system was patented, one of its killer features was to address an intrinsic issue of NTSC, in which errors in the transmitted signal produced changes in hue in the resulting image. PAL includes a system which, by alterning the phase of consecutive lines (and hence the name Phase Alternating Lines) those errors result in "less objectionable" saturation changes instead of hue changes. This means that by the time L*a*b* was published, color television industry already knew that changes in hue had different tolerance than changes in saturation.
4 CMC computes this "hue distance" as the geometric mean of the two chords formed by the circles of equal chroma, which is \Delta H^* = \sqrt{\Delta a^*^2 + \Delta b^*^2 - \Delta C^*^2}.
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