K-M analysis applied to droplet-color variation

What is the color science impact
of droplet size variation?
See the Appendix for a description of the color science model
Robert Cornell
Print System Science
March 2011

Some Halftone Simulation Examples

The Procedure
Use the halftone model to simulate
the spectral response over a range
of drop mass values.
(nominal +/-3pL in 0.02pL steps)
Select digital CMY values
Determine the droplet mass
range such that DE is less
than, or equal to 1.0, 2.0, 3.0….
Repeat across
color space
Compute L*a*b* for
Each spectral response curve
Droplet mass range such
that DE < 1 for this
particular CMY value

The Mid-Tone Colors Are Most Affected By Droplet Mass Variation
Mid-Tone Color Nearly Saturated Color
Saturated color
Additional ink does not
substantially affect DE.

To prevent visible color shifts in
the mid-tones, the droplet size
needs to be closely controlled DE = 3
DE = 2
DE = 1
Halftone Simulation Results
pL1.0-/
unitsE8.6

D
Slope
+/-0.12pL
+/-0.27pL
These values are nearly identical
to the empirical direction
given by Rich ReelWhite
Saturated
Color
+/-0.87pL
MonteCarloResult
6.2 DE

To prevent visible color shifts in the
[CMYRGBK] mid-tones, the droplet
mass needs to be held to +/-0.1ng
As the colors
approach saturation,
drop mass variations
have less effect
As the color approaches
paper-white, drop mass
variations have less effect
This is from a similar study (2005), when the 1200 dpi drop size target was 2.5 pL.

Color Shifts as a Function of Chip
Temperature Variation
• It is well-known that the ejected droplet mass is a
function of chip temperature.
• Similarly, it is well-known that the color produced on the
media is a function of the ejected droplet mass.
• Knowing the mass-temperature relationship and the spot
size-mass relationship, it is possible to simulate the
effect that loose temperature control has on the resulting
visible shifts across the entire color space.
• The following simulations are for Newman. The intent is
to provide guidance and justification for precise chip
temperature control.
• The Mariner analysis is a work in progress. Until that is
complete, it is reasonable to use these results as a
proxy.

Some Typical Simulations: C,M,Y InputSpectral ResponseCIE(XYZ)L*a*b* Output
As the temperature shifts
so does the spectral response
Some colors shift greatly with temperature.
A 10C DT line-line for this CMY would be
visibly banded.
Some CMY combinations are insensitive to
temperature shifts (usually saturated colors).
In general: The mid-tones are very sensitive to temperature shifts. The more saturated colors are insensitive.

5000 Random (C,M,Y) Simulations Used to Represent 2563 Color Space
The results were fitted to various probability density functions.
Weibull PDF provided an excellent fit to the results listed in the table.
Table-1: Color Space Variation Fitted to Probability Distribution Parameters
Temperature Control Error (C) Weibull Shape Factor (a) Weibull Scale Factor (l)
-10 2.2527 2.375
-9 2.2371 2.1295
-8 2.2398 1.8895
-7 2.2276 1.6478
-6 2.2176 1.4079
-5 2.2049 1.169
-4 2.1953 0.9319
-3 2.1853 0.6964
-2 2.1761 0.4626
-1 2.165 0.2305
1 2.1568 0.2296
2 2.1456 0.4579
3 2.1347 0.6846
4 2.1245 0.9098
5 2.1136 1.1335
6 2.1028 1.356
7 2.0927 1.5771
8 2.0832 1.7968
9 2.0734 2.0151
10 2.0633 2.2325
  
)(
)(
exp1
T
T
x
xTEP
a
l 





D
P = probability that the color error DE is
less than, or equal to, value (x) when the
control temperature error equals (T)
Color Space Simulated

Visible color shiftDesired
Range
DT = 10C
DT = 8C
DT = 5C
DT = 3C
DT = 4C
DT = 6C
DT = 7C
DT = 9C
If we allow a 10C temperature control error, we can
only print 17% of the color space without visible color shifts
P(DE<1) = 21%
P(DE<1) = 17%
P(DE<1) = 26%
32%
41%
54%
71%
89%
For each C-M-Y selection, the color and its sensitivity to temperature change were simulated.
The results nicely followed the Weibull probability distribution.
Temperature variation effects over the color space are shown in the above plot.
Table-1 Represented Graphically Looks Like This
DE < 1

Appendix
Color simulation technique

Kubelka-Munk Mixing Theory (1)
 












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
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


R
R
S
K
S
K
S
K
S
K
R
2
1
21
2
2
Reflected Light Incident Light
Media
Colorant K = absorption coefficient
S = scattering coefficient
R = reflectance = J/I
The single constant K-M equation is often used for optically thick colorants. It has also been used
successfully in color mixing experiments on paper with foam brushes.(2)
The following discussion will illustrate
how to extend K-M mixing theory into
digital halftone printing applications.
(1) Judd & Wyszecki, Color In Business, Science and Industry, (1975).
(2) Kang, H.R., Journal of Imaging Technology, Vol. 17, No.2, (1991).
For over half a century, K-M has been used to predict color in cases involving papers, dyes,
plastics, paints and textiles.
SjdxidxKSdi
jdxKSSidxdj


)(
)(J I
dx

Kubelka-Munk Mixing Theory Applied To Halftone Printing
n
n
MIX S
K
c
S
K
c
S
K
c
S
K
























......
2
2
1
1
When several colorants are mixed together the (K/S) of the mixture is the
sum of the component parts:
cn = chemical concentration of colorant n
(K/S)n = absorption/scattering coefficient of colorant n
In digital printing, we do not mix the colorants and concentrations externally.
The colorants are mixed, on demand, at the media surface.
For digitally printed halftone colors, let’s pose the following equation*
fn = fractional fill of colorant n = f[(spot diameter/pixel); digital color 1-256]
Cn = concentration function for colorant n = C(fractional fill)
Unit(K/S)n is derived from the measured spectral response of primary colorants
N = number of primary colors, e.g. CMY = 3; CMYcmk = 6
Eq. [1]
























 N
n n
nn
PaperHalftone S
K
UnitCf
S
K
S
K
1
*Ref: US Patent 6,211,970: Filed 11/24/1998 by R.W. Cornell

Typical Ink-Paper Spread Factors
 
  417.0
441.0
picoliters17.57m)(SizeSpot
picoliters95.18m)(SizeSpot





Computing The Area Fill Fraction By Analyzing Pixel
Overlap Types In An Error Diffusion Pattern
Type A
Type B
Type C
Type D
Type E
Type F
Type G
Type H
Blank neighbor from
previous scan
Filled neighbor from
previous scan
Current pixel
being printed
Error diffusion
neighbor
x
y
Scan directions
Pel
Size
Spot
Diameter sizePel
diameterSpot
F
Type J
Type K
Analytical geometry functions are derived to
determine the dot fill area for each overlap type
(F < 2.0)

Fractional Area Fill (f) as a Function of Digital Color
Number (L) and Spot/Pel Ratio (F)
Pel
Size
Spot
Diameter sizePel
diameterSpot
F
(L)

Measured Spectral Response Of Primary Colors
Cyan Magenta
Yellow
Using a spectra-photometer,
the tint ladders can be measured
for each primary color.
paper-white
saturated
paper-white
saturated
paper-white
saturated

Modeling The Ink: (K/S) Transformations
For each primary color and the blank paper, compute the following:
  
 
T
T
Paper
T
S
K
S
K
S
K
Unit
S
K
S
K
S
K
R
R
S
K
hwavelengtabsorptionpeakAt
FillSolid
2
Area FillFractional
2
1




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



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































l
l l
l
R(l) = measured spectral reflectance (400 < l < 700)nm
Eq. [2]
Eq. [3]
Eq. [4]

Unit (K/S) Curves For CMY
Cyan
Magenta
Yellow
Using Eq.[3,4] these curves are
derived from the (K/S) response
of the primary colors.

(K/S)T At The Peak Absorption Wavelength Determines The
Concentration Functions
Cyan Magenta
Yellow
Concentration Functions:
 
758.1
236.1
164.1
)FractionFill(616.7
)FractionFill(823.7
FractionFill24.7
YellowYellow
MagentaMagenta
CyanCyan
C
C
C


 Eq.[5a]
Eq.[5b]
Eq.[5c]

K-M Model For Digital Halftone Simulations
5.02
21)(
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
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
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

HalftoneHalftoneHalftone S
K
S
K
S
K
R l
Now we know:
•Fractional area fill (f) as a function of F(spot/pixel) and digital color L(1-256)
•Unit(K/S) for each primary colorant
•Concentration function (C) for each primary colorant
So we can compute (K/S)Halftone:
From this we can compute the halftone spectral response R(l):
Knowing the spectral reflectance of the halftone [R(400nm<l<700nm)],
it is possible to compute the CIE (X,Y,Z) tristimulus values and
the L*a*b* color coordinates.
Eq.[1]
Eq.[6]

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

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 N
n n
nn
PaperHalftone S
K
UnitCf
S
K
S
K
1

Yosemite Color Data (From Colin Maher)
Delta E vs Dot Area Coverage; Yosemite GPP Cyan EVT Ilford Glossy Paper
y = 0.04390x + 4.25906
R
2
= 0.98688
0
10
20
30
40
50
60
70
80
90
0 200 400 600 800 1000 1200 1400 1600
Dot Area (µm²) per 1/600 in²
DeltaE
Series1
Linear (Series1)
(The reference color in this DE calculation was blank paper)

Does The K-M Halftone Model Still Apply?
• Characterizing the spectral response, K/S and concentration
functions for an ink-media set takes several weeks, or more.
• Ink formations and media have changed many times over since this
K-M halftone method was developed in 1997.
– Is it reasonable to use 1997 spectral response data of CMY-dyes and
their K/S and concentration functions in an analysis of modern day
CMY-pigment inks?
• Droplets and pixels are now much smaller too.
– While the K-M halftone model worked well for 18ng~80m spots, is it
appropriate for today’s smaller spots?
• It will be shown on the following pages that the present K-M
halftone model with the spectral response, K/S and concentration
functions already presented will compare well with data taken on
Yosemite-color (4-5ng) with GPP ink on Ilford glossy paper.
• Thus it is quite reasonable to use the K-M halftone model as a
means of predicting the allowable droplet mass variation on a
Newman ejector.

Comparing Yosemite DE Data To
Kubelka-Munk Halftone Simulations
259.467.78
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1792
fractionFill
m1792pixel600dpiaofareatheSince
data]empiricalsColin'[from259.40439.0
2




fy
f
x
f
xy

Before plunging into the simulations to show the expected DE across the entire
color space due to droplet mass variations on Vulcan, let’s first do a sanity test
of the K-M halftone model against recent Yosemite data.
To put the empirical-numerical results on the same scale, let’s convert the
regression equation’s independent variable into fractional area fill (f).
Instead of using blank paper, let’s use 1400m2 (f = 0.78) as the reference color.
The empirical equation is transformed to:
fE 67.7846.61 D
It is this equation, derived from Colin’s Yosemite empirical data, that we will
compare the K-M halftone simulations to.

K-M Halftone Simulation Results
Digital Level
(1-256)
Fractional Area
Fill (f)
L* a* b* DE
256 0.785
(1400m2/pel)
62.8 -26.9 -43.9 0
230 0.710
(1272m2/pel)
65.0 -27.3 -42.0 3.0
200 0.612
(1097m2/pel)
68.2 -27.2 -39.0 7.4
170 0.520
(932m2/pel)
71.6 -26.4 -35.5 12.2
140 0.433
(776m2/pel)
75.2 -24.6 -31.5 17.7
110 0.333
(597m2/pel)
79.9 -21.0 -25.9 25.6
80 0.249
(446m2/pel)
84.2 -16.4 -20.3 33.6
60 0.187
(335m2/pel)
87.6 -12.1 -15.7 40.4
Reference Color

Comparison Between Yosemite GPP-Cyan Data And The
K-M Halftone Simulations
The most important aspect of this analysis is the DE(Dmass) effect.
The f(mass) effect is known from the historical ink spread equations
shown earlier. So as long as the empirical d(DE)/df slope and the
slope resulting from the simulation is similar, we may use the existing
K-M halftone model for the Vulcan DE(Dmass) analysis.
(f)
DE
d(DE)/df = 78.7
d(DE)/df = 73.3

Using The K-M Halftone Model To Quantify
DE(Dmass) For Newman
• Since d(DE)/df of the model is within 7% of the
empirical result, we now have a relatively good
numerical means of simulating halftone color shifts,
and we know it is applicable to modern inks, media
and droplet sizes.

K-M analysis applied to droplet-color variation

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