There are two uses of distance measures. The first is to compare similarity of one or more entities to a single reference; e.g. how similar is this color (or these colors) to Red?". The second use is to compare a collection of entities with each other; e.g., all the colors in set to each other. For this a metric is needed. Without one: cluster diagrams, dendrograms, and in general graphs with less than n-1 leaves will be misleading at best because only paths of length 1 with a common vertex will be comparable.​For colors, the Mathematica built-in CMC and CIE2000 are very good for the 1st use. Also I have found that CIE76 (the Mathematica default) is very good for comparisons to Black. For the 2nd use with colors, CIE76 is the only built-in measure that qualifies as a metric. Counter-examples of the others are given in the table below.​ColorDistance Metric TestsDataset: ColorData["HTML","ColorRules"] without Aqua and Fuchsia (synonyms of Cyan and Magenta)Error tolerance ϵ = 0.00001 used in comparison tests.Decimal numbers displayed below with 3 digits of precision.
Measure name
CIE76
CIE94
CIE2000
CMC
DeltaH
Metric?
Yes
No
No
No
No
Dataset min, max
0.0130, 2.50
0.0102, 1.49
0.00655, 1.16
0.00748, 2.47
0, 2.42
Colors tested / total
138 / 138
138 / 138
138 / 138
138 / 138
138 / 138
identity failures
0 / 138
0 / 138
0 / 138
0 / 138
0 / 138
positive definite failures
0 / 18906
0 / 18906
0 / 18906
0 / 18906
290 / 18906
commonest error source
n/a
n/a
n/a
n/a
Black
violations min, max
n/a
n/a
n/a
n/a
0., 0.
symmetric failures
0 / 18906
18816 / 18906
0 / 18906
18906 / 18906
0 / 18906
triangle inequality failures
0 / 2571216
46755 / 2571216
74664 / 2571216
114888 / 2571216
543938 / 2571216
commonest error source
n/a
White
Black
DarkBlue
Black
violations histogram
n/a
0.00
0.15
0
2000
4000
6000
8000
10000
12000
0.0
0.3
0
5000
10000
15000
20000
0.0
0.4
0.8
1.2
0
20000
40000
60000
80000
​​