A relational approach to color

© Copyright 2016 HP, Inc.
A RELATIONAL APPROACH TO COLOR
Peter Morovič
HP Inc.
24th Color & Imaging Conference
San Diego (CA), November 2016
[Redacted]

OUTLINE
• Metamer Sets
• HANS
• Beyond print control

METAMER SETS

COLOR FORMATION (RETINAL)
• Color is the result of the
relationship between light,
(a surface) and an observer 
[necessary conditions for
color perception]
• Arbitrary* changes in any one
of the components results in
a change of the resulting XYZ
• However, possible to ﬁnd S/
E/Xi so that XYZ doesn’t
change: metamerism
E(λ)
S(λ)
Xi(λ)
∫E(λ) S(λ) X1(λ) dλ = X1
∫E(λ) S(λ) X2(λ) dλ = X2
∫E(λ) S(λ) X3(λ) dλ = X3

METAMERISM
5
Human Observer 
under daylight
All reﬂectances that match:
Diﬀerent observer,
camera or light
[...and vice versa]
XYZ

MACBETH COLORCHECKER

MACBETH COLORCHECKER
30 38 43 35 43 52
22 38 22 25 36 24
32 37 14 22 23 43
8 39 47 40 32 23
Worst case DE76 (D65 to CIE A)

CONVEX COMBINATIONS
Given all possible reﬂectances, color a unit area with a subset in varying
proportions (adding up to 1) to match a given color: naturalness constraint
The set of all metameric reﬂectances (for a given illuminant
and observer) form a convex set

THE PURPOSE OF A COLOR PIPELINE
Primaries
(e.g. inks)
Mechanism
& Constraints
Drops/ink
Pixels/color
Substrate (paper)
Ink-limits
Spatial resolution
System Error
…
Desired Result
+ grain, banding, smoothness, 
detail, speed, ink-use (eﬃciency), …

HANS
Side view
Subtractive
Additive
Halftone Area Neugebauer Separation
Side view
Colorants
Substrate
Appearance
W
C M Y
Neugebauer primaries
Materials
Example halftone
C M Y CMY
+ + =
NP areas
W=1/9
C=1/9
M=2/9
CM=2/9
CY=1/9
MY=1/9
CMY=1/9Colorant vector
[C,M,Y]=[5/9, 6/9,3/9]

NEUGEBAUER PRIMARY AREA COVERAGES
A pattern C, formed by combining some of a printing system’s NPs,
can be characterized by its NP area coverage (NPac) vector – NPacC
1/3! 2/9! 2/9! 2/9!
k – number of colorant levels / colorant / pixel
n – number of colorants,
(i.e., the weights are convex)
NPi – the i-th NP
T() – color (e.g., Yule-Nielsen-corrected XYZ)

ASSOCIATIVITY
• Convex combination of relative area coverage weighted NP
colors can also be seen as the convex combination of two
constituent patterns – CA and CB, where wCi = wCBi + wCAi
=! *1/3+! *2/3!
1/3! 2/9! 2/9! 2/9! 1/3! 2/3! 1/3! 1/3! 1/3!
1/9! 2/9! 2/9! 2/9!2/9!
*1/3! *2/3!

CONSEQUENCES OF ASSOCIATIVITY
• As a consequence of
associativity, we can perform
convex combinations not only
of at-pixel states (i.e., the
Neugebauer Primaries), but also
of a pattern’s sub-patterns
• New NPacs can be constructed
by convexly combining other
NPacs, and this process can, in
principle, go on ad inﬁnitum.
0!
10!
20!
30!
40!
50!
0! 20! 40! 60! 80!
lightness!
chroma!
ink!
area coverage!
Yn
0!
25!
50!
areacoverage(%)!
C!
W!
CM!
K!
M!
Xn

HERZOG’S CUBE-SHAPED GAMUT

THE 8-VERTEX COLOR SEPARATION
BLACK
YELLOW
GREEN
BLUE
CYAN
RED
MAGENTA
WHITE
BLACK
RED
MAGENTA
WHITE
CYAN
GREEN
BLUE
YELLOW
Device RGB Interface
Yule-Nielsen XYZ measurments of HANS NPac characterisation chart
Print & measure
NPs and NPacs
Select 8 vertices in
correspondence
with RGB cube
Apply tetrahedral
RGB tessellation to
8 vertices in XYZn
Do
tetrahedra
in XYZn
overlap?
RGB-NPac-
XYZ LUT
YES
NO

A HANS PIPELINE (SIMPLIFIED VIEW)
Print & measure
Neugebauer primary
(NP) CIE XYZs
Compute convex
hull &
tetrahedralize  
hull NPs
Find printable
color’s
enclosing
tetrahedron
Printable color
20% W
30% C
25% M
0% Y
25% CM
0% CY
Barycentric
coordinates are
vertex NP
areas
Select one NP
per pixel &
diffuse NPac-NP
error
Due to linearity in
XYZ/XYZN
W
C
CM
M

NICE THEORY… DOES IT ACTUALLY WORK?
HANS 
+31% Gamut
Colorant 
pipeline
Current HANS

TESSELLATIONS
Points (NP colorimetries) Convex Hull
Example Tessellations
[triangular/rectangular/...]
42 possible polygons if we allow overlapping
• Tessellating NPs can be done in
diﬀerent ways - not a unique solution
• A given XYZ (within the convex hull) is
contained in many tessella
• Each tessella gives rise to a new NP area
NP1
NP2
NP3
NP4
NP5
NP6
NP7
?

COMBINATORIAL SOLUTION
• Given a set N NPs and their measured
XYZs there are 
∑[ p=4 to N](N over p) polyhedra
• In the case of CMY, there are 70
tetrahedra, 56 pentahedra, 28
hexahedra, etc... = 163 polyhedra
Tetrahedra All polyhedra
CMYK@1dpp 1,820 64,839
CMYK@2dpp 1,663,740 ~1024
CMYKcm@2dpp A lot!
Even in CMY (8 NPs) there are many polyhedra, 
leading to metamerism.

• How many NPacs are there?
• Given a set of N NPs and an encoding precision d 
(e.g. d=255 for 8bits), how many vectors w are there? 
w = [w0, w1, w2, … wN-1] 
such that SUM(wi) = d and all wi are 
in the range of [0 … d]
• Solution: (N + (d-1)) choose N
• How many ink-vectors are there?
• For M inks there are d
M
ink vectors
NPAC SPACE CARDINALITY
Ink-vectors NPacs
CMY 1dpp 8bit 1.7 x 107 4.9 x 1014
CMYK 2dpp 8bit 4.2 x 109 ~1.5 x 1079
CMYK 2dpp 12bit 2.8 x 1014 ~1.5 x 10172

COMPUTATIONAL OPTIMISATION
• For a given XYZ (sampling printable
gamut)
• check all polyhedra that contain it
• compute the resulting NPac
• evaluate each NPac for optimality 
(e.g. ink-use)
• Conceptually: we compute the metamer
set and choose the best candidate from
within
• Challenging: large number of tetrahedra;
large metamer sets (still only sampling)

TRICHROMACY
• Trichromatic color reproduction is the use of one
colorant/light-source per cone type
• additive: RGB light sources, varying in intensity of
output, control intensity of response from LMS
cones
• subtractive: CMY ﬁlters, varying in level of
absorption, control intensity of response from LMS
cones
• For each color that can be matched, there is one and
only one RGB / CMY combination that matches it
• Alternatives (metamers) are only available when
more than three colorants/light-sources are
available (e.g., adding K to CMY, adding W to RGB)

CMY METAMERS
115 NPacs (formulations) that produce a mid-gray
12% W
35% C
0% M
29% Y
24% CM
0% CY
0% MY
0% CMY
12% W
51% C
3% M
32% Y
2% CM
2% CY
10% MY
0% CMY
Each column represents an NPac that matches a mid-gray, the color of the
segments corresponds to NPs and their length to the relative area coverages

TWO EXAMPLES (HALFTONES)
Two out of 115 metamers: left patch uses 11 base NPacs (out of
14) – right patch uses 5 base NPacs [shown in pseudo-color]
Target LAB

INK-USE OPTIMISATION
• Print and measure 544 LAB-
uniform samples spanning the
whole CMY color gamut
• Perform computational
optimisation (tetrahedral search
for each sample over extended
base NPac set - 244 samples)
• 14,4 x106
tetrahedra evaluated
• Min vs Max ink-use over all 544
samples = 12.66% ink use range
−60 −40 −20 0 20 40 60
−40
−20
0
20
40
60
80
a*
b*
Printed and Measured LABs

6-INK INDUSTRIAL PRINTING SYSTEM
CMYKcm ink-set,
up to 3 drops per
pixel = 4096 NPs

NOT JUST PRINT-CONTROL
Simulation
Print photo
Mimír: a display pixel preview of a halftone corresponds to a local distribution of NPs in the halftone,
which is an NPac. Zoom-dependent, dynamic re-computation of halftone NPac to display pixels + a
color model allow for a WYSIWYG halftone preview (ICC proﬁle level accuracy)

NOT JUST PRINTING
Display design: decompose backlight into atomic states (monochromatic
signals) and convexly combine them to generate full variety via optically
additive mechanism.

THANK YOU ☺
Ján Morovič, Graham Finlayson, Jordi Arnabat, Juan
Manuel Garcia Reyero, Xavier Fariña, Hector Gomez,
Pere Gasparin, David Gaston, Annarosa Multari, Africa
Real, Rafael Gimenez, Albert Serra, Ramon Pastor

A relational approach to color

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