Explanation about the ΔC h (formerly known as ΔF)
From colour to colour difference?
Characterizing and evaluating colours differences can be done in various ways. Basically such a
colour difference is understood as a (weighted) distance between two colours [typically two
points in a coordinate system] in a reasonable colour space. While in some industries such as
the textile or automotive industry there is often a physical reference sample and a reproduction
or test sample in the graphic arts industry both colours are normally equally important and no
specimen takes precedence over the other. That is important since colour-difference equations
such as the CMC formula might not always be symmetric, i.e. ΔECMC Ref. vs. Test ΔECMC Test vs.
Ref.
The most common colour space to define colours is the 1976 CIELAB colour space, which can be
visualized by both cartesian as well as polar coordinates, see Fig. 1 for an example.
Fig. 1: Portraying two different ways to specify one colour. Left: the usage of CIEa* and
CIEb* in the so called cartesian coordinate. Right: Using the hue angle CIEhab and CIEC* (the
distance from the achromatic axis) to describe the exact colour. Note: It is only the CIEa*b*plane, so the lightness information is retained. Source: bvdm.
As colour is three-dimensional there are three coordinates (figures) needed for an unambiguous
definition. This is usually facilitated by a lightness component (e.g. CIEL*), a hue angle (e.g.
CIEhab) and a chroma component (e.g. CIEC*). Since Figure 1 is only showing two of the three
dimensions one most be cautious when interpreting colours or colour differences to not forget
the “missing dimension”. In this example it is the (neglecting of the) lightness information,
which will be important for this report. So a three dimensional visualization is strongly
recommended to get the full image. If, as it is normally the case, the CIELAB values are given
the CIEC* and CIEhab are calculated as follows:
2
*
*
*
CIE Chroma: Cab = a + b
2
CIE hue angle: hab = arctan (b*/a*).
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What is a colour difference at all?
Having now two colours, e.g. an aim CIELAB value to be printed and a measured value of a
digital print, one common way of computing a colour difference is to calculate the length
between both points (in the defined colour space). This is done by calculating the difference for
the three dimensions, typically the lightness-, hue angle- and chroma-differences or the
lightness, “redness-greenness” (CIEa*) and “yellowness-blueness” (CIEb*) differences. Based on
the well known Pythagorean theorem (“a2+b2=c2”) the resulting distance (ΔE) is calculated as
follows:
*
E ab
=
2
* 2
(L ) + (a ) + (b )
*
*
2
This is illustrated in Fig. 2 and shown by means of a concrete example in Table 1.
Fig. 2: Visualization of the colour difference components, here the lightness difference ΔL*,
the “redness-greenness” difference Δa* and the “yellowness-blueness” Δb*.
Reference
CIEL*
55
CIEa*
-37
CIEb*
-50
Print
53
-35
-48
Difference ΔL=55-53 Δa=-37-(-35) Δb=-50-(-48)
ΔL=2
Δa= -2
Δa= -2
CIEhab
= arctan(-50/-37)
= 233,5°
= 234°
= 233.5° - 234°
= Δh=0.5°
CIEC*
= SquareRoot(-37^2+-50^2)
= 62,2
= 59,4
= 62.2 – 59.5
= ΔC*=2.7
Table 1: Example calculation of two colours by means of the characteristic differences. The
reference represents the Cyan solid for FOGRA39.
That results in a total colour difference (ΔE*ab) of:
*
E ab
=
(2)
2
2
2
*
+ (2) + (2) = E ab
= 12 = 3.46 3.5
The computation of modern colour difference metrics such as CIEDE94 or CIEDE2000 is more
complex and is explained elsewhere. For information both have been calculated for this example
to CIEDE2000 2 and CIEDE94 2.1. The “E” stands for the german word “Empfindung”, which
means perception. The unit of ΔE*ab is 1 so you would normally write: “The colour difference of
my brand colour was today ΔE*ab = 2.3” and not “The colour difference of my brand colour was
2.3 ΔE’s”. Conceptually (psychophysically) ΔE=1 represents a just noticeable distance.
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Why is a total colour difference not enough?
As you can spot in Table 1 one can infer more information than only the total colour difference
ΔE. Colour difference information such as the lightness difference, the hue angle or the chroma
difference might give additional explanation in how to solve daily colour management
challenges: “Do I need more density to achieve the aim?” or “Is the hue angle achievable at
all?”. Having a closer look at the formulae defined in Table 1 one must be cautious in using the
right terminology. So the chroma difference (CIEC*1 – CIEC*2) between two colours is termed
ΔC*. This if often confused with the colour difference in the CIEa*b*-plane, which this article is
all about.
In defining a reasonable hue-angle (“colour cast”) difference, e.g. from a blue to purple blue, the
statement of Δh in degrees is not appropriate. Imagine a pale colour (CIELAB3 in Fig. 3) and a
saturated colour (CIELAB2 in Fig. 3) would have the same hue angle difference. For that reason
a more appropriate hue angle difference metric has been defined, namely the CIE hue
contribution: ΔH. It is calculated as follows:
*
H ab
=
* 2
ab
* 2
* 2
ab
(E ) (L ) (C )
.
So it can be though of the colour difference being left if the lightness and the chroma
differences have been “removed”. Technically or geometrically speaking it is the cord between
the “hue rays” of both colours on the geometrical mean. The sign indicates whether the colour
change is to the “red side” (clockwise) or to the “bluish side” (counter clockwise). If you use the
formulae above all ΔH values are positive. This reduces problems if you evaluate more ΔH values
in order to calculate an average or mean value, as it is required for the composed grey patches
of the control wedge defined in ISO 12647-7.
CIELAB3
Fig. 3: Visualization of the hue rays of two colours and the corresponding areas for a Δa*,
Δb* colour difference definition (green) and a CIELCh based definition.
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ΔCh – the most reasonable way to characterize near neutral colour differences!
So far we have seen colour differences somewhere in the colour space. Close to the grey axis, in
other words for near neutral colours (typically with a ΔC*<7), the usage of an explicit hue is not
appropriate. The closer a colour lies to the neutral axis the less meaningful a hue becomes. For
example there is no hue difference for a grey colour with CIELAB_A=50,0,0 and
CIELAB_B=50,0,-3. Anyhow one will perceive colour “A“ (compared to colour “B”) yellowish. In
order to solve this, you simply compute the distance of both colours in the CIEa*b*-plane
(neglecting the CIEL* information) as follows:
E C = F =
(CIEa1 CIEa2 )
2
2
+ (CIEb1 CIEb2 ) =
(0 0)
2
2
+ (0 (3)) = 3
This difference is called chromaticness and is defined, among others, in DIN 55981, ISO 12646
(called ΔEC) and now in ISO/DIS 12647-8.
To test your German skills some near neutral colour differences have been plotted in Fig. 4 .
All three near neutral pairs have, from a relative point of view, the same chromaticness
difference. From an absolute point of view the pair “A2-B2” appears more bluish and “A3-B3”
more greenish. This “absolute perception” will only occur if the “neutral” patch “A1” is present
in the field of view. Without this “orientation” all “B-samples” have a “yellow-reddish” cast
compared to the “A-samples”. Conversely all “A-samples” comprise a bluish-green colour cast
compared against the “B-samples” – including the neutral patch A1 compared to B1.
Fig. 4: Three near neutral grey pairs where „B“ has always the same colour shift as „A“.
Translation: Green (Grün), Yellow (Gelb), Red (Rot), Blue (Blau). Source:
http://www.farbmetrik-gall.de/images/farbsti_460.gif
In Conclusion:
For the evaluation of near neutral colour the ΔH (hue contribution) is useable but not very
appropriate. Instead the usage of the ΔCh -metric will be proposed.
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