2. INTRO
2
The determination of the difference between
the colours of two specimens is important in
many applications, and especially so in those
industries,
such as textile dyeing, in which the colour of
one specimen (the batch) is to be altered so
that it imitates / duplicate that of the other (the
standard).
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This is usually an iterative process.
3. Reliability of visual colour-
3
difference assessments
the human visual system is excellent at
assessing
Whether two specimens match.
If the supplier and customer assess its colour
difference from standard visually, they are
likely to disagree.
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4. Reliability of visual colour-
4
difference assessments
quantifying both the repeatability and
reproducibility of visual assessments of colour
differences
repeatability :is a measure of the extent to
which a single assessor reports identical
results,
Reproducibility: is the corresponding
measure for more than one assessor.
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5. Reliability of instrumental colour-
5
difference evaluation
The results from instrumental methods are
much less variable than those from visual
assessments.
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6. Development of CIELAB and
6
CIELUV colour-difference formulae
The result was the publication, in 1976,
Of two CIE recommendations,
CIELAB and CIELUV, for
approximately uniform colour spaces and
colour-difference calculations.
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7. Calculation of CIELAB and CIELUV
7
colour difference
The colour difference between a
batch (B)
and its standard (S) is defined,
in each space, as the Euclidean distance between the points (B
and S) representing their colours in the relevant space.
The formulae for the calculation
of colour difference and its components in the two spaces are
identical in all
but the nomenclature of their variables. We shall therefore detail
only those pertaining to the calculation of colour difference in
CIELAB COMPILED BY TANVEER AHMED
8. Calculation of CIELAB colour
8
difference
If L*, a* and b* are the CIELAB rectangular
coordinates of a batch,
and L*S, a*S and b*S those of its standard,
substituting ∆L* = L* – L*S,
∆a* = a*B – a*S and
∆b* = b*B – b*S
in Eqn 4.22 gives Eqn 4.23:
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10. Hue angle difference ∆h
10
However, the hue
angle difference ∆h
is in degrees,
and so is
incommensurate
with the other two
variables:
the substitution is
mathematically
invalid.
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11. Hue angle difference ∆h
11
The definition of CIELAB
colour difference
includes two methods of
overcoming the problem.
The first uses radian
measure to obtain a
close approximation to
a hue
(not hue angle)
difference ∆H* in units
commensurate with
those of the other
variables (Eqn 4.26):
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12. 12
Suppose there does exist a variable ∆H*ab
representing, in units commensurate with
the other variables of CIELAB colour
difference, the hue difference between
batch and standard,
and that it is orthogonal to both ∆L* and
∆C*ab.
Then ∆E*ab must be the
Pythagorean COMPILED BY theseAHMED component
sum of TANVEER three
differences
13. HUE Difference ∆H*ab (not hue
13
angle)
We know the value of each of the first three
variables in
Eqn 4.27------------- ∆E*ab
from the output of Eqn 4.23, ----- ∆L* as one
of the inputs to Eqn 4.23,
and ∆C*ab ---------------------from Eqn
4.24, each
without knowledge of ∆H*ab.
By rearranging Eqn 4.27 we can define ∆H*ab
(Eqn 4.28):
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14. Unfortunately, this method also has
14
its problems.
The other four components of CIELAB difference are defined
as
differences and are thus signed so that,
for example,
∆L* > 0
if L*B > L*S
but ∆L* < 0
if L*B < L*S,
while ∆H*ab is defined (by Eqn 4.28)
as a square root, the sign of which is indeterminate.
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15. Unfortunately, this method also has
15
its problems.
The CIE states that ‘∆H*ab is
to be regarded as positive
if indicating an increase in hab
and negative
if indicating a decrease’.
This may be interpreted as
implying that the sign of ∆H*ab
is that of ∆hab,
So that ∆H*ab > 0
if the batch is anticlockwise from
its standard,
and ∆H*ab < 0 if clockwise.
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16. Unfortunately, this method also has
16
its problems.
Thus, for example,
for batch B1a and
standard S1
where h,B1a = 30
and for hab,S1 = 10,
∆hab = 30 – 10 = 20
(greater than zero),
so that the sign of
∆H*ab is positive.
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17. Unfortunately, this method also has
17
its problems.
Thus, for example,
Now consider batch B1b
where h,B1b = 350
and for hab,S1 = 10,
∆hab = 350 – 10 = 340,
which is again greater than zero so
that ∆H*ab is positive.
The hue vector from S1 to B1b
must, however, clearly be
considered clockwise, so that
∆hab and ∆H*ab should be
negative.
This problem arises
whenever ∆hab > 180.
The definition therefore presents
problems, but for many years it
offered the only way of calculating
∆Hab. COMPILED BY TANVEER AHMED
18. work of Huntsman.
18
Another method was based on the work of
Huntsman.
Equating the right-hand sides of Eqns 4.23
and 4.27, followed by manipulation, yields Eqn
4.29
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19. work of Huntsman.
19
Although Eqn 4.29 provides a
simpler method of calculating
∆H*ab,
it still suffers from the
disadvantage that it outputs the
wrong sign of ∆H*ab when ∆hab >
180.
The correct sign
may, however, be determined
without knowledge of the value
of ∆hab,
by testing the relative sizes of the
two directed areas
a*B b*S and a*S b*B;
denoting the correct sign of ∆H*
by s [23] gives Eqn 4.30:
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21. CMC(l : c)
21
formula was published in 1984 under the
name of CMC(l : c),
CMC being the abbreviation commonly used
for the Colour Measurement Committee (Eqn
4.32) [24]:
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22. 22
where ∆L*, ∆C*ab and ∆H*ab
are respectively the CIELAB lightness, chroma and
hue
differences between batch and standard,
l and c are the tolerances applied respectively
to differences in lightness and chroma relative to that
to hue differences
(the numerical values used in a given situation being
substituted for the characters l and c,
for example CMC(2 : 1), whenever there be possible
ambiguity),
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23. 23
where the ki
(i = 1, 2, 3) are as defined in Eqn
4.31, and L*S, C*ab,S and hab,S are
respectively the CIELAB lightness, chroma
and hue angle (in degrees) of the standard.
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24. The mathematics of the CMC(l : c)
24
formula deserve examination
Because they well illustrate the general principles of
optimised formulae, currently so important in industrial
applications.
In CIELAB space, Eqns 4.27 and 4.28 define the shell containing
all shades equally acceptable as matches to (or perceived as
equally different from) a standard at a given colour centre.
Arising from the non-uniformity of CIELAB space, the magnitudes
of each of ∆L*, ∆C*ab and ∆H*ab are not usually equal,
and ∆E*ab is therefore a variable which is assumed in CMC(l : c)
and most other optimised formulae to define an ellipsoidal shell
with its three axes orientated in the directions of the component
differences.
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25. The non-uniformity of CIELAB
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space further dictates
The non-uniformity of CIELAB space further dictates that an
equally acceptable (or perceptible) ∆E*ab, at another colour
centre, is unlikely to define a similar shell.
For a formula to allow SNSP, however, we require the overall
colour difference to be a constant, so that its locus describes
a spherical shell of equal radius at all colour centres.
The ellipsoid in CIELAB space may be converted into a
sphere by dividing each of its attribute differences (∆L*,
∆C*ab and ∆H*ab), in turn, by the length of the semi-axis of
the ellipsoid in the direction of the relevant attribute difference
(SL for lightness, SC for chroma and SH for hue).
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26. 26
The inclusion in Eqns 4.27 and 4.28 of the
relative tolerances (l and c) yields the first line
of the
CMC(l : c) formula (Eqn 4.32).
This line therefore converts a usually
ellipsoidal tolerance volume in CIELAB space
into a spherical one in a CMC(l : c)
microspace.
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27. 27
The principal difficulty in the
design of optimised colour-
difference formulae,
however, is to derive
mathematics allowing the
generation of the systematic
variation
in the relative magnitudes of
attribute differences judged
equally acceptable (or
equally
perceptible) at different
centres. These mathematics
occupy the whole of the
remainder of the formula.
Their effect is demonstrated
in Figure 4.9. COMPILED BY TANVEER AHMED
