INK-PAPER INTERACTION A study in ink-jet

INK-PAPER INTERACTION A study in ink-jet
Linköping Studies in Science and Technology
Dissertations No. 806
INK-PAPER INTERACTION
A study in ink-jet color
reproduction
Li Yang
Department of Science and Technology
Linköping University, SE-601 74 Norrköping, Sweden
Norrköping, April 2003
INK-PAPER INTERACTION
A study in ink-jet color
reproduction
c Li Yang
Department of Science and Technology
Linköping University
SE-601 74 Norrköping
Sweden
ISBN 91-7373-613-9
ISSN 0345-7524
Printed in Sweden by UniTryck, Linköping, 2003
iii
iv
Abstract
An ink jet printing system consists of three fundamental parts: inks, printing
engine, and substrates. Inks are materials creating color by selectively absorbing and scattering the visible illumination light. The printer acts as an ink
distributor that governs the ink application. Finally, the substrate acts as a
receiver of the inks and forms the images. Ink setting on the substrate is a
complex process that depends on physical and chemical properties of the inks
and the substrates, and their bilateral interactions. For a system consisting of
dye based liquid inks and plain paper, the ink moves together with the liquid
carrier before the pores absorb the liquid. This process contributes to serious
ink spreading on the surface along the paper fibers. At the same time the
ink spreads down into the pore structure. This causes severe dot deformation,
physical dot gain and ink penetration. Understanding the consequences of
these phenomena and above all being able to characterize their impact on color
reproduction is of great importance. Moreover this knowledge is fundamental
for finding solutions to ink-penetration related problems. This thesis presents
studies of some important issues concerning image reproduction quality for dye
based ink-jet printing on ordinary plain paper (office copy paper), such as ink
penetration, optical dot gain, and even physical dot gain. The thesis begins
with theoretical developments to the Kubelka-Munk theory, which allows one
to study even non-uniform ink penetration into the substrate. With the knowledge of scattering and absorption coefficients and ink thickness, reflectance can
be computed by solving differential equations. Three forms of ink penetration, uniform, linear, and exponential have been studied. A method is then
presented for obtaining fundamental properties of the inks from spectral reflectance measurements, like the scattering- and absorption-power of inks, ink
layer thickness, and ink mixing scheme for the generation of secondary colors.
The method is further developed for modelling the ink penetration in printing
systems consisting of dye based liquid inks and plain paper. By combining
the spectral reflectance measurements with theoretical simulations, quantities
like the depth of ink penetration is determined. These quantities, in turn, are
used to predict the spectral reflectance of prints. Simulated spectral reflectance
values have been in fairly good agreement with experimental results. Models
dealing with light scattering inside the substrate resulting in optical dot gain
for halftone printing, in the case of existing ink penetration, have been developed for both mono- and multi-color printing. It is shown that the optical dot
gain leads to higher color saturation than predications from Murray-Davis approximation. Additionally, tentative studies for physical dot gain were made.
Finally, an evaluation of the chromatic effects of the ink penetration for printing on office copy paper has been carried out based on both experimental data
and simulations. It is found that ink penetration has a dramatic impact on
chroma and hue of the color, and the color saturation is significantly reduced
v
by the ink penetration. Consequently, the capacity for color representation, or
the color gamut, is dramatically reduced by the ink penetration.
Acknowledgements
During the years spent on this thesis work I got a lot of help from many people
and in many ways.
First of all, I would like to express my sincere gratitude to my supervisor
Professor Björn Kruse for giving me the opportunity to pursue the study in
his group, sharing his broad and deep knowledge and experiences in Graphic
Arts. His suggestions, comments, and inspiration have sparked initiatives of
the researches. His continuous efforts for establishing contacts with research
institutes and industries have been very helpful for promoting and improving
our work. His encouragement, appreciation, and sharp view of the subjects and
works have been particularly important. Words like “I trust you” have meant
a lot.
Associate professor Reiner Lenz, has acted as co-supervisor in the last couple
of years. His questions, comments, criticisms, and discussions have been very
important inputs to the researches and the formulation of the dissertation. His
enthusiasm and research style have been strong influence.
Senior researcher, Nils Pauler, in M-Real Research (Sweden), has been a
particularly important person outside of the university. He, together with
Professor Kruse, initiated collaboration in the studies of ink penetration. His
help in spectral reflectance measurement has been very important for having
a good start. His kindness and hospitality when I visited Örnsköldsvik have
made the research visits not only rewarding but also enjoyable. He and his team
member, Jerker Wågberg, have been wonderful people to collaborate with.
I would like to thank all group members, for creating an amicable and active research atmosphere, providing courses and holding interesting seminars.
Thanks Arash, Daniel, Linh, Sasan, and Thanh for pleasant coffee breaks and
free talks, and interesting discussions of various topics, from football to universe.
Special thanks to Professor Hans Ågren at Royal Technology Institute
(KTH) for inviting me to Sweden, and for very fruitful collaborations during
the time when we were at Physics Department (IFM) of Linköping University.
Many thanks to our research engineers, especially Sven Franzén, for maintaining the office- and Lab-systems. Thanks to our secretaries, especially Sophie Lindesvik, for being very helpful in arranging conferences, travel affairs,
and taking care of administrative tasks.
Thanks associated professor Stan Miklavcic who made a careful linguistic
reading and valuable technical comments during the time he had to meet a few
deadlines of himself. Thanks Dr. Sasan Gooran for a helpful proof reading.
I also wish to thank all my Chinese friends in Linköping, Norrköping, and
other places, for their friendship and constant help, especially Fang Hong and
Lin Dan, Luo Yi and KeZhao, and QinZhong and ZhuangWei.
Thanks to Swedish Foundation for Strategic Research for financial support
vii
through the Surface Science Printing Program (S2P2).
At last I wish to express my deepest gratitude to my wife Yan and our son
YiChen (Mikael), for their understanding and support, and the joys of our life.
List of publications
1. L. Yang and B. Kruse, Scattering and absorption of light in turbid
media, in Advance in Printing and Science and Technology 26 (2000)
199-218;
2. L. Yang and B. Kruse, Ink penetration and its effects on printing,
in Proc. IS&T SPIE Conf., 3963, 365-375, Jan. 2000, San Jose, CA;
3. L. Yang and B. Kruse, Yule-Nielsen effect and ink-penetration in
multi-chromatic tone reproduction, in Proc. IS&T NIP16 Conf.,
363-366, Oct. 2000, Vancouver, Canada;
4. L. Yang, R. Lenz, and B. Kruse, Light scattering and ink penetration effects on tone reproduction, J. Opt. Soc. Am. A, 18 (2001)
360-366;
5. L. Yang, S. Gooran and B. Kruse, Simulation of optical dot gain in
multi-chromatic tone production, J. Imaging. Sci. Tech., 45 (2001)
198-204;
6. L. Yang and B. Kruse, Chromatic variation and color gamut reduction due to ink penetration, in Proc. TAGA Conf., 399-407, May
6-9, 2001, San Diego, CA;
7. L. yang, B. Kruse, and N. Pauler, Modelling ink penetration in inkjet printing, in Proc. IS&T NIP17 Conf., 731-734, Oct. 2001, Fort
Lauderdale, FL;
8. L. Yang, R. Lenz, and B. Kruse, Light scattering and ink penetration effects on tone reproduction, in Proc. IS&T PICS Conf.,
pp.225-230, Mar. 26-29, 2001, Oregon, PL;
9. L. Yang, Characterization of the inks and the printer in ink-jet
printing, in Proc. TAGA Conf., 255-265, Apr. 2002, Asheville, NC;
10. L. Yang, Modelling ink-jet printing: Does Kubelka-Munk theory
apply ?, in Proc. IS&T NIP18 Conf., 482-485, Sep. 2002, San Diego,
CA;
11. L. Yang, Color reproduction of inkjet printing: model and simulation, J. Opt. Soc. Am. A, 2003 (accepted for publication);
12. L. Yang, Determination for depth of ink penetration in ink-jet
printing, to be presented in TAGA Conference, May 2003, Montreal,
Canada.
Contents
Abstract
iv
Acknowledgements
vi
List of publications
viii
Table of Contents
ix
1 Introduction
1.1 Goal of the study . . . . . . . . . . . . . . . . .
1.2 Status of studies and our contribution . . . . .
1.2.1 Extension to Kubelka-Munk theory . . .
1.2.2 Evaluation of effects of ink penetration .
1.2.3 Optical dot gain . . . . . . . . . . . . .
1.3 Structure of the dissertation . . . . . . . . . . .
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2 Paper
2.1 Structures and properties of paper . . . . . . . . . . .
2.1.1 Fibres, fillers and coating . . . . . . . . . . . .
2.1.2 Density and porosity . . . . . . . . . . . . . . .
2.2 Optical properties and measurements . . . . . . . . . .
2.2.1 Brightness, opacity and gloss . . . . . . . . . .
2.2.2 Optical measurements . . . . . . . . . . . . . .
2.3 Paper permeability and mechanism of ink penetration
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3 Ink-jet printers and inks
3.1 Ink-jet technologies . . . . . . . .
3.1.1 The continuous ink-jet . .
3.1.2 Drop-on-demand ink-jet .
3.2 Characteristics of ink-jet printers
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Contents
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5 Extended Kubelka-Munk theory and applications
5.1 Assumptions in Kubelka-Munk theory . . . . . . . .
5.2 Differential equations . . . . . . . . . . . . . . . . . .
5.2.1 Boundary conditions . . . . . . . . . . . . . .
5.2.2 Boundary reflection . . . . . . . . . . . . . .
5.3 Models of ink penetration . . . . . . . . . . . . . . .
5.3.1 Uniform distribution . . . . . . . . . . . . . .
5.3.2 Linear distribution . . . . . . . . . . . . . . .
5.3.3 Exponential distribution . . . . . . . . . . . .
5.4 Solutions of the differential equations . . . . . . . . .
5.4.1 Uniform ink distribution . . . . . . . . . . . .
5.4.2 Linear ink distribution . . . . . . . . . . . . .
5.4.3 Exponential distribution . . . . . . . . . . . .
5.5 Simulations for uniform- and linear-ink distribution .
5.5.1 Convergency of the series expansion. . . . . .
5.5.2 Optical effects of ink penetration . . . . . . .
5.5.3 Correction for boundary reflection . . . . . .
5.5.4 Effect on color gamut . . . . . . . . . . . . .
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . .
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3.3
3.2.1 General . . . . . . . . . . . .
3.2.2 HP970Cxi ink-jet printer . .
Ink-jet ink technologies . . . . . . .
3.3.1 General . . . . . . . . . . . .
3.3.2 Dye-based and pigment-based
4 Optical modelling: an overview
4.1 Radiative Transfer Theory . . .
4.2 Phase function . . . . . . . . .
4.3 Multi-flux theory . . . . . . . .
4.4 Kubelka-Munk method . . . . .
4.5 Monte-Carlo simulation . . . .
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6 Characterization of inks and ink application
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Experiment, data analysis and simulation . . . . . . . . . . . .
6.2.1 Samples and measurements . . . . . . . . . . . . . . . .
6.2.2 Data analysis and simulation . . . . . . . . . . . . . . .
6.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Spectral characteristics of the primary inks . . . . . . .
6.3.2 Spectral reflectance values and relative ink thicknesses of
the primary inks . . . . . . . . . . . . . . . . . . . . . .
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Contents
xi
6.3.3
6.4
6.5
Spectral reflectance values and relative ink thickness of
secondary colors . . . . . . . . . . . . . . . . . . . . . .
Remarks for application of Kubelka-Munk theory . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Characterization of ink penetration
7.1 Optical properties of plain paper . . . .
7.2 Assumptions and notations . . . . . . .
7.2.1 Assumptions . . . . . . . . . . .
7.2.2 Notations . . . . . . . . . . . . .
7.3 Simulation of print on office copy-paper
7.3.1 Primary colors . . . . . . . . . .
7.3.2 Secondary colors . . . . . . . . .
7.4 Optical effect of ink penetration . . . . .
7.5 Summary . . . . . . . . . . . . . . . . .
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8 Dot gain in black and white
8.1 Introduction . . . . . . . . . . . . . . . . .
8.1.1 Murray-Davis equation . . . . . . .
8.1.2 Yule-Nielsen equation . . . . . . .
8.1.3 Status of the studies . . . . . . . .
8.2 Model and methodology . . . . . . . . . .
8.2.1 Point spread function approach . .
8.2.2 Probability approach . . . . . . . .
8.2.3 Impacts of the optical dot gain . .
8.3 Overall dot gain of monochromatic colors
8.3.1 A model for overall dot gain . . . .
8.3.2 Simulation of the overall dot gain .
8.4 Summary . . . . . . . . . . . . . . . . . .
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gain in color
Reflectance of a multi-color image . . . . . . . . . . . .
Optical dot gain in multi-color tone reproduction . . . .
Simulation for multi-layer color image . . . . . . . . . .
9.3.1 Two inks of round dots: dot on dot . . . . . . . .
9.3.2 Two inks of square dots: dot on dot . . . . . . .
9.3.3 Two inks of round dots: random dot distribution
The effects of optical dot gain on color reproduction . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 Dot
9.1
9.2
9.3
9.4
9.5
xii
Contents
10 Chromatic effects of ink penetration
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10.1 Basics in colorimetry . . . . . . . . . . . . . . . . . . . . . . . . 121
10.1.1 CIEXY Z color space . . . . . . . . . . . . . . . . . . . 121
10.1.2 Chromaticity diagram . . . . . . . . . . . . . . . . . . . 122
10.1.3 CIELAB color space . . . . . . . . . . . . . . . . . . . 122
10.2 Evaluation of chromatic effects from experimental data . . . . . 123
10.2.1 Parallel comparison of prints on two types of substrates 123
10.2.2 Two-dimensional representations of chromatic effects . . 125
10.3 Evaluation in 3D color space: simulations . . . . . . . . . . . . 129
10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
11 Summary and future work
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11.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
11.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
12 Appendix
137
A
Mathematical derivation for Equation (6.4) . . . . . . . . . . . 137
B
Probability model for optical gain . . . . . . . . . . . . . . . . . 138
Bibliography
140
Chapter 1
Introduction
Ink-jet printing a is commercially young but rapidly developing printing technology. Success in making printers of high print resolution, color capacity, yet
very affordable price has made ink-jet printers available not only for big companies, but also for private users and small business units. According to Cap
Ventures [Ash01], of all the printing applications, 60% was printed on uncoated
stocks, like plain paper or office copy paper during year 2000. This figure will
reach 90% by year the 2005. Studies of ink-jet printing on uncoated substrates
is therefore of great importance not only from an academic perspective but also
from application perspective.
1.1
Goal of the study
To build up a model that can be used for prediction of ink jet color reproduction,
methods that help characterize the printing materials, printing systems, and
final printout are necessary. This thesis thus consists of various study phases
and goals.
One goal is to establish a general method that deals with various types
of ink penetration, uniform or non-uniform, whatsoever. Because of existing
widely diversified ink-paper combinations, mechanisms that are responsible for
ink-paper interaction differ from one combination to another.
A second goal is to characterize printer and optical properties of inks. It includes information about scattering and absorption characteristics of the inks,
the volume of the inks being printed, and color mixing schemes for the generation of secondary colors. In the case of having ink-penetration, this information
serves as an input for further studies.
The underlining phenomenon of ink penetration is formation of a layer of
an ink-paper mixture. Based on knowledge from the first two phases, the study
2
Introduction
moves naturally to the third phase. The third goal is to understand the fundamentals of how the ink-paper interaction affects color reproduction and to
characterize optical effects of ink penetration. Basic quantities that characterize ink-penetration, such as penetration depth, is indirectly determined from
experimental spectral reflectance values. A model that takes into account ink
and paper mixing is developed allowing prediction of spectral reflectance of real
prints.
That ink penetration into an uncoated substrate impact strongly on color
representation of the printed images is an experimentally known fact. One of
the goals is therefore to evaluate the impact of ink penetration from experimental data and simulations. The focus of the evaluation is on color, i.e., chroma,
hue, and color gamut etc.
In order to correctly predict color of halftone images, the dot gain probably has to be considered. Thus, model development for dot gain description,
including optical dot gain and physical dot gain, is an important feature of our
study.
Modelling and simulation are important tools contributing to our understanding, deeper insights into the problems. Moreover, they serve also to guide
us on the way toward finding solutions.
1.2
Status of studies and our contribution
This section briefly outlines the status of studies on the topics related to the
work presented in this thesis. It also provides a brief description of our contribution as well.
1.2.1
Extension to Kubelka-Munk theory
The original theory of Kubelka-Munk (K-M) was developed for light propagation in parallel colorant layers of infinite xy-extension [KM31, Kub48]. The
fundamental assumptions of the K-M theory are that the layer is uniform and
that light distribution inside the layer is completely diffused. From these assumptions, the light propagation in the layer was simplified into two diffuse light
fluxes through the layers, one proceeding upward and another simultaneously
downward. After its introduction in the 1930’s, K-M theory was subjected to
extensions by removal of some of the assumptions. Among others, the boundary reflection at the interface bewteen two adjacent media was introduced by
Saunderson [Sau42], i.e, the well-known Saunderson correction. Kubelka himself also made an attempt to extend the applicability of the theory to optically
inhomogeneous samples [Kub54]. However, this extension was only applied to
a special case of inhomogeneous media, in which the ratio of the absorption to
the scattering is constant.
1.2 Status of studies and our contribution
Recently, Emmel and Hersch introduced an elegant mathematical formulation of the Kubelka-Munk theory, based on matrices [Emm98, EH99]. They
proposed also a mathematical framework unifying the K-M model with the
Neugebauer model [EH00]. This allowed them to apply the K-M theory to a
halftone image. Therefore optical dot gain was studied. Very recently, Mourad
extended the 1-dimensional K-M theory representation (2 flux) to 3-dimensions
(6 flux). Such an extension made it possible to account for the light scattering
in the substrate and therefore the optical dot gain.
The object of our studies is mainly on optical performance of a layer consisting of non-uniform media concentration. This is a topic that has not been
explored theoretically. One of the applications of the study is ink penetration,
where the ink distribution inside the substrate may be nonuniform, depending
on the mechanism of ink penetration. Phenomenon of non-uniform ink penetration has been observed experimentally (cross section image) by means of
microtomy, in ink jet printing [GKOF02]. In Chapter 5, we work out a framework that is applicable to both uniform and non-uniform ink penetration cases.
Expressions for reflectance and transmittance of three types of ink penetration,
uniform, linear, and exponential (ink penetration) are derived. Moreover, applications of the K-M theory to ink jet printing has substantiated the applicability
of the theory. Explanations and discussions around these issues are given in
Chapter 6.
1.2.2
Evaluation of effects of ink penetration
Absorption of ink constituents by the substrate, or ink penetration is significant
over a range of timescales, from the first stages of ink-transfer and ink-drying
by absorption, through to long-term stability [Voe52, Oit76, Str88, MK00].
Studies of ink penetration related issues have long been important topics in
offset printing and cover a wide range of topics, such as print gloss and print
density, print defects (unevenness and mottle), print through, separation of ink
constituents [SGS00, Rou02], etc.
Studies have so far mainly concentrated on understanding the mechanisms
of ink penetration and developing materials for paper coating. Reported studies of the optical and chromatic effects of ink penetration are few, even though
such studies have recently intensified [McD02, PL02, NA02]. Among others,
Bristow and Pauler proposed a method by which the depth of ink penetration could be determined indirectly from spectral reflectance measurement
data [Bri87, Pau87], in offset printing. The method was based on K-M theory, the additivity assumption and a uniform ink distribution. In recent years,
Pauler and his research group have been very active in modelling and simulating
ink penetration in ink jet printing [PWE02a, PWE02b, PWE02c].
Experimental measurements and theoretical simulations for the optical and
chromatic effects of ink penetration have not been conducted without difficulty.
3
4
Introduction
From the experimental side, the measurements may be made by comparing
prints having ink penetration with those that dot not. Unfortunately, one
can not obtain prints with and without ink penetration (into the cellulose
porous structure of the substrate) by using the same ink-substrate combination.
Different types of substrate, like plain paper and high grade photo paper have
to be used when comparisons are made. For halftone images, differences in
color and optical dot gain characteristics for different substrates, will contribute
to the color difference between the prints. Moreover, differences in surface
characteristics between the substrates used for the prints may lead to significant
difference in physical dot gain.
Theoretical simulations have the advantages that one can use exactly the
same ink-substrate combinations when the effects of ink penetration is evaluated. One has the possibility to manipulate the ink penetration by switching it
on or off in the simulation. The underlining difficulty is to establish a theoretical model that can properly describe a complex problem like ink penetration.
So far K-M theory has been the only model that has been applied to the ink
penetration problems, even though more sophisticated theories like Multi-flux
Radiative Transfer Theory (or Discrete Ordinate Radiative Transfer (DORT)
theory [Eds02]) may be possible candidates in the future.
Studies carried out in this dissertation represent a combination of the KM model with spectral reflectance measurements. We present a systematic
framework stretching from the characterization of inks and ink application,
to the modelling and the simulation of ink penetration. It begins with the
determination of the scattering and absorption characteristics of pure ink layers
of primary colors, as well as thickness of the ink layers. The scattering and
absorption characteristics of the primary inks are then applied to determine the
color mixing scheme for the generation of secondary colors. The applications
serve not only as tests to the quality of data, but also to the applicability of
the model (see Chapter 6, for details).
The characteristics of the inks (scattering and absorption, and ink thickness)
and that of the paper, are in turn used to simulate ink penetration (ink-paper
mixture). The additivity assumption is modified by considering the correlation
between the light scattering (from the paper) and the absorption (from the
inks). With the help of our model, the depth of ink penetration is indirectly
determined from the measured reflectance values. Spectral reflectance of ink
jet prints on office copy paper (in both primary- and secondary-colors) have
fairly well been reproduced by the simulation (see Chapter 7, for details).
Evaluation of the impact of ink penetration is obtained from both experiment and simulation perspectives. Color difference between prints with and
without ink penetration are represented in chromaticity diagrams (2D) and
CIELAB color space (3D). It shows that ink penetration has a dramatic impact on the chroma of the print and even on the hue. This leads to a dramatic
reduction of color gamut which is absolutely negative for color reproduction
1.2 Status of studies and our contribution
(see Chapter 10, for details).
1.2.3
Optical dot gain
The physical phenomenon behind the so-called optical dot gain, or Yule-Nielsen
effect, is basically light scattering within the substrate paper. Light scattering
inside the substrate of a printed image is a very complex process. Studies of
the effects on color rendition of a print has attracted constant research interest,
since it was first studied by Yule and Nielsen in 1950’s [YN51]. Yule and Nielsen
found that the optical dot gain can be well approximated by introducing an
empirical factor, n, into the Murray-Davis equation (see Eq. (8.2) and (8.5)),
although the physical meaning of the n factor was not clear. Shortly after,
Clapper and Yule extended the work of Yule and Nielsen by including a contribution from multiple internal reflections between upper and lower boundaries
of the substrate [CY53]. The assumptions made in the Clapper and Yule model
are that the ink layer is uniform and that the light is completely scattered by
bulk scattering. Complete light scattering is a physical approximation when
the average lateral light scattering distance is much greater than the size of the
halftone element.
In 1978, Ruchdeschel and Hauser [RH78] provided a physical explanation for
the n factor and showed that 1 ≤ n ≤ 2, if only optical dot gain involved. n = 1
and 2 represent two extremes for the light scattering, where n = 1 corresponds
to no light scattering while n = 2 corresponds to complete scattering. The
original intention of the Yule-Nielsen model was to involve the optical dot
gain. Nevertheless, it has often been applied to cases where there was also
physical dot gain involved. Unsurprisingly, they got n factor bigger than 2 and
sometimes much bigger than 2 [BT96]. Consequently, the physical meaning of
such n factor is difficult to explain.
In the past 10 years, Kruse and Wedin [KW95], among many others, proposed an approach which was thoroughly studied and fully implemented by
Gustavson [Gus97a, Gus97b]. The approach simulated the light scattering
process from a fundamental level. It was based on direct numerical simulation
of scattering events which depend on the optical properties of the materials,
the halftone frequency and the halftone geometry. This approach is similar in
nature to the Monte-Carlo method that is briefly explained in Sec. 4.5. Statistics are recorded over a large number of light scattering events. From these the
probability of an event can be established. Arney [Arn97] and Hübler [Hüb97]
independently proposed similar models based on probability descriptions of the
light scattering. In their models, the light scattering inside the paper was described by the probabilities that a photon emerges from the inked and non-inked
areas. These probabilities depend on the positions where the photon enters the
paper and emerges from the paper, as one will see in Sec. 8.2.2. A point spread
function (PSF) is a different representation of light scattering. Using the PSF
5
6
Introduction
approach, Rogers [Rog97, Rog98a, Rog98b, Rog98c] presented a method dealing with the light scattering process. He proposed a matrix approach where
the tristimulus values of a halftone image could be calculated as the trace of
a product of two matrices. So far, the studies have not provided any explicit
expression for the reflectance or tristimulus as a function of dot percentage
(as it was given by the Neugebauer equation). Moreover, the studies have so
far been limited to the mono-color or black-and-white case and no studies for
multi-color have been reported. Furthermore, the subject of ink penetration
(into the substrate) has barely been touched.
Our work makes contributions in these three areas, i.e., explicit expressions
for reflectance and optical dot gain have been worked out, studies of the optical
dot gain have been extended to multi-color cases, and ink penetration has been
included in the model. Detailed descriptions can be found in Chapters 8 and
9.
1.3
Structure of the dissertation
This thesis consists of 6 parts and 11 chapters. The contents of the presentation are organized in such a way that they follow a logical path for reasoning.
Apart from the introduction (Chapters 1-4), the thesis begins with a theoretical extension of Kubelka-Munk theory (Chapter 5). This provides us with a
general description of ink penetration. It is then applied to determine ink characteristics, ink application, and ink penetration in full tone prints (Chapters
6 and 7). Furthermore, the model and application are extended to halftone
cases, where dot gain plays an important role (Chapters 8 and 9). Finally, the
chromatic effects from the ink penetration and dot gain are evaluated with help
of experimental data, as well as simulations.
Each chapter is intended to be self consistent and sufficiently independent
of all others, that a reader only interested in a given topic needs only view
that particular chapter. However, the chapters are also arranged according to
a common theme.
A brief description about the contents of the thesis is given as following.
Part I consists of the first four chapters. Chapter 2 describes the basics of
paper structures, properties, and relevant measurement technologies. Chapter
3 briefly describes ink jet technologies, inks, and printers used in the thesis
study. Chapter 4 provides an overview of optical modelling and simulations.
Part II consists of a single chapter (Chapter 5). A theoretical framework
is proposed as an extension of the Kubelka-Munk theory. The extension allows
one to deal with both uniform and nonuniform ink distribution in cases of ink
penetration. Expressions for reflectance and transmittance of three types of
ink penetration, uniform, linear, and exponential, have been worked out.
Part III has the goal of characterizing ink penetration for solid tone printed
1.3 Structure of the dissertation
patches and consists of two chapters, Chapters 6 and 7. In studying ink penetration, one actually deals with an ink-paper mixture. It is therefore essential
to know what kind of ink, in terms of its optical properties (scattering and
absorption), has been printed and how much ink has penetrated into the substrate. Chapter 6 aims at characterizing inks and ink application controlled
by ink-jet printer, such as scattering- and absorption-power of the inks, the
amount of ink printed onto a substrate, and color mixing scheme in generation
of secondary colors. Results of Chapter 6 serve as input for Chapter 7 where
ink penetration into plain paper is studied. Quantity alike depth of the ink
penetration has been determined from simulations.
Part IV presents models and simulations for halftone-image-related issues, such as optical dot gain in both monochromatic (Chapter 8) and multichromatic (Chapter 9) printing cases. Even overall dot gain (physical- plus
optical-dot gain) has been tentatively studied (Chapter 8). The model has
been applied to simulating optical dot gain for different dot shapes and locations. It is found that the optical dot gain results in more saturated color
sensation. Applications of the model to overall dot gain is tested by applying
it to a digital image.
Part V consists of two chapters, Chapters 10 and 11. In Chapter 10, the
impact of the ink penetration on capacity for color representation was evaluated
from both experimental and simulation perspectives. The impact is represented
in (2D) chromaticity diagram and 3D color gamut (in CIELAB color space).
It is found that ink penetration has a dramatic impact on chroma and even
on hue, and leads to a dramatic reduction in color gamut. Chapter 11 gives a
summary of the thesis.
Part VI consists of an Appendix and the Bibliography.
7
Chapter 2
Paper
2.1
Structures and properties of paper
In the Graphic Arts industries, paper is the most commonly used substrate
that receives the inks and colorants to form images. The properties of the
paper substrate are important partly because the substrate is visible between
the printed areas, and partly because the substrate defines the background reflectance for the ink layer. Moreover, optical properties, mechanical properties,
permeability to liquids, and so on, directly affect the quality of the images and
the production practices. Paper-making is a multi-disciplinary subject involving mechanics, physics, optics, chemistry, etc. Interested readers can find more
detailed descriptions elsewhere [NKP98, Ran82, Bri86, Bak97]. In this chapter
we give a brief overview of the structures and properties of paper which are
either directly or indirectly related to the current work.
2.1.1
Fibres, fillers and coating
Paper is a stochastic network of fibers as seen in Fig. 2.1. Since the fibers
are much longer than the thickness of the paper sheet, the network is more or
less flattened out [Nor91] and therefore almost two-dimensional (in xy-plane).
For paper-making fibers, the basis weight is about 5 − 10g/m2 , which means
that ordinary printing and writing papers consist of normally 5-20 “layers”
of fibers [NKP98]. The two-dimensional (2D) structure governs many paper
properties, such as in-plane mechanical properties. The fibre network in paper
is normally built up of mechanical pulp fibres and/or chemical pulp fibres.
They form the backbone of the paper.
Graphic Arts industries have continuously placed stringent demands on
the paper-making industries, such as high productivity, wide diversity, and
10
Paper
Figure 2.1:
Micrograph of paper surface area of ca. 1 mm2 (from K. Niska-
nen [NKP98]).
improved product quality etc. Paper that only consists of fibres can hardly meet
these demands. As a result chemicals and post paper making processes are now
commonly applied. For example, by adding fillers such as kaolin clay or calcium
carbonate to the furnish one can increase the specific scattering strength and
in turn improve the opacity. Additionally, a post paper-making process which
is usually called coating is employed in order to improve both mechanical and
optical properties. The main constituents of the coating layer are pigments
(∼ 50 v − %), polymers (∼ 20 − 30 v − %) and air (∼ 25 − 35 v − %) [Rou02].
The coating can be composed of one structure or of multi-layer structures,
containing one or several pigments and a binder. Detailed studies and reviews
about the relation between the addition of fillers, coating constituents and the
properties of the paper can be found in references [FBP90, Bro85, Lep89]
2.1.2
Density and porosity
Closer examination using microphotography reveals that paper is actually three
dimensional [Fay02]. The z-directional structure and material distribution affect paper properties such as bulk, bending stiffness, optical properties, and
surface roughness. The distribution of fines and fillers are particularly important.
Density and thickness are basic macroscopic characteristics of paper structure. Density, ρ (g/m3 ), is defined as the ratio of the basis weight, b (g/m2 ),
2.1 Structures and properties of paper
11
and thickness, d (m).
ρ = b/d.
(2.1)
Its inverse value, termed bulk (m3 /g), is more convenient to use in the papermaking industry.
Figure 2.2: Micrograph of pore structure of a paper sheet (from A. Fayyazi [Fay02]).
In a 3D perspective (see Fig. 2.2), the paper is not merely a network of
fibers but more strictly an aerogel. The network of fibers embraces and creates
a network of pores, and paper is thus a two-phase system in which the pores or
voids between the fibers are an important part of the paper structure [Bri86].
The 3D pore structure controls the density and optical properties directly, while
the mechanical properties and dimensional stability are indirectly controlled
through the relative bonded area [NKP98]. It is therefore useful to introduce
a term called porosity, φ, that is defined as the ratio of pore volume to total
volume,
ρ
V − Vf
=1−
(2.2)
φ=
V
ρf
where V is the volume of the entire sheet, Vf is the volume occupied by the
fibres, ρ and ρf are the paper and the fibre wall densities.
Paper porosities range from 0.1 for glassine to 0.87 for filter paper. The
variation of φ is controlled by the paper-making furnish and its beating level.
Mechanical pulps with stiff and bulky fibers usually give higher paper porosity
than chemical pulps. The beating of chemical pulp decreases porosity since
12
Paper
fiber flexibility and collapse increase during beating. Therefore, a sandwich
type of layer structure using chemical pulps in the upper- and lower-surface
layers and a mechanical pulp in the middle layer, will make paper with good
surface properties (smoothness, printability, and bending stiffness) and high
porosity at the same time. In practice, for example, newsprint often primarily
consists of mechanical pulp while copy paper mostly of chemical pulp. Mixture
of mechanical and chemical pulps finds use especially in printing papers and
multiply boards [RNN98].
The pore size distribution is also influenced by operation such as calendering, the mean pore size becoming smaller as a result of such compressive
treatments. Additionally, measurements made, for example by immersing a
sheet in a low viscosity oil show that virtually all the pore volume is accessible
to the liquid and it can be assumed that there are no isolated inaccessible voids
within the structure [Bri86].
Surface roughness is another important characteristic of paper. It influences
the optical properties such as gloss. High roughness reduces the contact area
between the ink film and paper and gives low ink transfer in Offset printing.
On the other hand a rough surface contains leaks and holes that lead to ink
penetrating into the paper sheet. Ink penetration determines how much of the
transferred ink remains on the surface of paper. Small penetration gives high
print density. It has been observed that unevenness in offset printing comes
from the spatial variation in surface roughness and ink penetration [Kaj89].
Detailed studies on the ink penetration for ink jet printing will be presented in
Chapter 7.
2.2
Optical properties and measurements
Optical properties such as opacity, brightness, and gloss are important to users
of most paper and board grades. In order to successfully manufacture paper
with desired optical properties, it is important to understand the physical principles of sheet structure and composition that determine these characteristics.
Measurements of these optical properties can in turn provide information for
characterizing the sheet structure.
2.2.1
Brightness, opacity and gloss
When light of intensity, I0 , hits a paper surface, a fraction of intensity, Isurf ,
reflects back (surface reflection), and the remainder enters the sheet. Inside the
sheet, light spreads and scatters in all directions. Some light (Ibulk ) eventually
reflects back from the sheet, another part transmits through the sheet (Itran ),
and the rest is absorbed. The reflectance, R, is defined as the ratio of the
2.2 Optical properties and measurements
13
reflected intensity to the incident intensity, i.e.
R=
Isurf + Ibulk
.
I0
(2.3)
Similarly transmittance, T , is defined as the ratio of the transmitted intensity
to the incident intensity,
Itran
T =
.
(2.4)
I0
The sum of R and T is less than unity when there is absorption.
Brightness, R∞ (λ = 457nm), is the reflectance of an infinitely thick pile
paper sheet, which is measured by adding sheets to the pile until there is no
change in the intensity of reflected light at wavelength of 457 nm (blue light).
The use of blue light arises because paper-making fibers have a yellowish color
and because the human eye perceives blue colors as brightness.
Opacity characterizes the ability of a single sheet to hide text or pictures
on the back side of the sheet. Quantitatively, opacity is defined as a ratio of
reflectance of a single sheet, R1 , to that of an infinite number of sheets, R∞ ,
at wavelength λ = 557 nm [Les98a]
Opacity =
R1 (λ = 557nm)
R∞ (λ = 557nm)
(2.5)
One should be aware that opacity is defined at a different wavelength (λ =
557nm) from that of the brightness (λ = 457nm).
Gloss is an optical phenomenon caused by light reflection from a smooth
top surface. A glossy material is characterized by a high reflectance in the
direction of regular reflection or close to that direction. If the illumination
is white, the glossy reflection is normally colorless despite the color pigments
under the surface of the print [Gra01]. Every day experience tells us that the
gloss of printed paper depends considerably on the illumination and detection
angles.
Gloss paper has a high specular reflectance that is closely related to the
surface smoothness or in other words, surface roughness of the paper. The
surface roughness of paper can be sorted into three categories according to the
in-plane resolution: Optical roughness at length scale < 1 µm; micro roughness
at 1−100 µm; and macro roughness at 0.1−1 mm. Gloss is a combination of the
effects of micro roughness and optical roughness of the paper surface. Micro
roughness affects gloss because titled surface facets reflect light in different
directions as shown in Fig. 2.3
The scale of optical surface roughness has the same magnitude as the wavelength of the light. Optical roughness therefore causes light diffraction. The
total reflection, Rt , from a surface of normally distributed height is [Mic84]
Rt = R0 exp[−
25 π 4 σ 4
(4πσ)2
]
+
R
( ) (∆Θ)2
0
λ2
m2 λ
(2.6)
14
Paper
where λ is the wavelength of light, σ the root mean square (RMS) surface
roughness, m the mean gradient of the surface, R0 the reflectivity of fibers or
coating color, and ∆Θ the solid angle of measurement. The first term is the
specular component responsible for the gloss of paper. Clearly, as roughness,
σ, increases the exponential function decays to zero and gloss vanishes. In
contrast, the diffuse reflectance increases with roughness as σ 4 .
Figure 2.3: Schematic effects of micro roughness on paper gloss. The randomly titled
surface facets reflect light to different directions and reduce gloss.
High print gloss on paper also requires that the paper itself has high gloss.
To achieve high gloss or good surface smoothness, coating is usually needed,
and higher coating weight generally gives better gloss. However, proper choice
of pigments and binders, together with the average size and distribution of the
coating materials controls the behavior of the coating layer and the gloss [Lep89,
Les98a].
2.2.2
Optical measurements
Optical measurements is an important issue in paper-making and Graphic Arts
industries. In general, the measurements need to be relevant to what the human
sees. For a reflective sample, an image for example, the light that stimulates
human vision depends not only on the optical properties of the image but
also on the illumination and observation geometries. International standards
specify the procedures for the reflectance measurements including the spectral
2.2 Optical properties and measurements
15
characteristics of the incident light and standards for the calibration and light
collection has been established by CIE and ISO.
b)
a)
broadband
light source
mono
chromatic
broadband
light source
detector
detector
mono
chromatic
o
45o
45
(a)
(b)
d)
c)
light
source
mono−
chromator
light trap
detector
detector
gloss trap
light
source
monochromator
detector
sample
(c)
mono−
chromator
sample
(d)
Figure 2.4: Schematic diagrams for different illumination and detection geometries,
(a) (45o /0o )-geometry, (b) (0o /45o )-geometry, (c) (D/0o )-geometry, and (d) (0o /D)geometry.
The instruments that measure the spectral reflectance values share the name
of Spectrophotometer despite their differences in illumination (monochromatic
or white) and illumination-detection geometries. On the market, there are different types of instruments available depending on the needs of the application.
Figure 2.4 shows some examples of schematically different implementations for
illumination and detection geometries (taken from Ref [Ryd97]). They are also
recommended by CIE for reflectance measurements and are usually noted in
abbreviation as (0o /45o ), (45o /0o ), (0o /D) and (D/0o ), respectively, according to the positions or types of the light sources and detectors. For example,
(0o /45o ) specifies the angles of the detector (45o ) and light source (0o ) to the
normal of the sample, and (D/0o ) stands for the sample being illuminated by
diffuse light (D) and being measured along the sample’s normal. In the following, we will take a closer look at both instrument based on the (D/0o ) and
16
Paper
(45o /0o ) geometries as these are the instruments we have used in our studies. More information about other geometric implementations may be found
in Ref. [Ryd97], and references therein.
Elrepho 2000 belongs to the (D/0o ) category and is one of the most widely
used instruments in paper making industry. A typical implementation of the
instrument is shown in Fig. 2.5. The instrument consists of an integrating
sphere with holes for illumination, sample, and detector. The inside surface of
the sphere is covered with white pigments that ideally scatter light isotropically.
Inside the sphere there are baffles whose surfaces are also covered by the white
pigments. Using such an arrangement, light from the light sources is scattered
onto the inside surface of the sphere and no light from the light sources can
shine directly onto the sample.
detector
gloss trap
light
light
baffle
sample
Figure 2.5: Instrument of D/0o geometry for measurement of reflectance.
The accuracy of which the illumination mimics the ideal diffuse light depends on the optical properties of the white pigments and the sizes of the holes
on the sphere. Generally speaking, the smaller the holes the closer the illumination to ideal diffuse light. Nevertheless, too small holes may cause larger
errors in the measurement. For example, too small holes for the sample may
result in lower signal/noise ratio. Additionally, it may also lead to less statistically meaningful measurements when the characteristics of a relatively large
2.2 Optical properties and measurements
17
area, say reflectance of a piece of paper, is of interest (this is exactly the case
for paper-making and Graphic Arts industries). Therefore, the implementation
of diffuse illumination into a real instrument is a compromise between many
factors. More detailed descriptions of the (D/0o ) geometry instrument may be
found in the international standard ISO 2469 [24694]. It should be pointed out
that measurements using Elrepho2000 are very time consuming because each
patch has to be manually positioned.
In Graphic Arts, (45o /0o ) or (0o /45o ) -observing geometry (Fig. 2.6) is recommended by CIE and well applied in practice. The illumination unit consists
generally of a cone shape glass illuminated by up to three different unpolarized light sources. This setup leads to an approximately circular illumination
condition which helps to diminish the measurement dependencies on the sample orientation. The spectrophotometer Gretag MachbethT M Spectrolino used
in our measurements belongs to this category. In a reflectance measurement,
the sample gets circularly illuminated from the top with the collimated flux
I0λ . By using this instrument in combination with the software called SpectroChart [GM01], one can measure many patches automatically in sequence.
This makes the measurements used for system calibration, such as color gamut
mapping, much easier. Comparative measurements made by applying both
Elrepho 2000 and Spectrolino have shown excellent agreement in spectral reflectance values for plain paper [Pau01].
monochromator
& detector
broadband
light source
broadband
light source
45o/0o
o
o
0 /45
Figure 2.6: Instrument of 45o /0o and 0o /45o geometry for reflectance measurement.
An angle-resolved spectrophotometer operating with a collimated illumination (laser) [Gra01] is one of the high grade instruments. With it one can
measure the monochromatic reflectance of a sample at any illumination- and
detection-geometries, which is particularly useful when there is specular re-
18
Paper
flection involved. Nevertheless, it is still too early to generally implement such
measurements in industrial practice because of the limitation of the instrument
itself (monochromatic illumination and extremely time consuming for the space
sampling). A more serious obstacle, though, is the lack of a simple theoretical
model that copes with the measurement.
2.3
Paper permeability and mechanism of ink
penetration
Most end uses of paper involve transport phenomena in some form that requires
a specific level of permeability. Printing paper has medium permeability, filter
papers, facial tissues, and sanitary papers must have high permeability, and in
many packaging papers and so-called barrier papers, low permeability [Les98b].
The permeability of paper is closely related to its porosity, φ. One usually
assumes that the structure consists of ellipsoidal cavities connected through
narrow channels. Comparing the ideal model system with measurements of
fluid penetration or fluid flow through paper determines the apparent pore size
distribution [Les98b].
For graphic arts industry, liquid (water, oil, etc.) penetration into paper has
tremendous impact on the quality of the print. Absorption of ink constituents
by paper is driven by thermodynamic interaction between the ink and the paper, by capillary forces and by chemical diffusion gradients. Capillary pressure
is acknowledged to be the main driving force in the offset ink oil transport in a
typical paper coating porous structure. With increased latex content diffusiondriven transport of ink chemicals into the latex counterpart of the coating layer
becomes important [Rou02]. The penetration can also be driven by the printing nip during the printing and converting processes. Therefore, theoretical
handling of the penetration process is complicated. Moreover, experimental
verifications of theory are challenged by the small dimension and by the relevance of both short and long time-scale. In the following, we will only mention
some theoretical background for capillary driven penetration process that may
be relevant to ink penetration for ink jet printing on un-coated paper.
The capillary force (pressure) is described by Young’s equation,
p=
2γcosθ
r
(2.7)
where γ the surface tension, r the capillary radius, and θ the contact angle
between the liquid and the capillary wall as shown in Fig. 2.7. According
to the Young equation, penetration occurs when the contact angle is smaller
than 90o (f > 0). However, experiments showed that even if the contact
angle θ > 90 and the external pressure zero, surface tension can drive a liquid
2.3 Paper permeability and mechanism of ink penetration
19
into paper, if there are converging capillaries. With irregularly shaped pores,
capillary penetration accelerates in converging parts of the pore and decelerates
in diverging parts [Lyn93].
Air
θ
Liquid (ink)
Solid (paper)
Figure 2.7: Contact angle, θ, of a wetting liquid on a solid surface.
In the printing processes, the capillary force together with the external
pressure govern the penetration. The time dependence of the penetration is
described by the Lucas-Washburn equation,
h2 =
r2 t 2γcosθ
(
+ pE )
4η
r
(2.8)
where h is the thickness of the ink penetration, η the fluid viscosity, and pE
the external pressure difference. The first term is the penetration driven by the
capillary force. It says that the depth of penetration is proportional to square
root of the capillary radius, but inversely proportional to the square root of the
viscosity of the ink solvent.
In practice, the mechanism of the ink penetration depends significantly on
the printing process. For offset printing, it is the nip pressure that dominates
in the printing and converting processes. This is followed up by the capillary
driven penetration afterwards. Similar process occurs in Toner Fusing Printing
process [Hwa99]. For an ink jet, the external pressure comes from the kinetic
impact (a pulse) when the ink droplet hits the paper substrate. Comparatively,
capillary driven penetration may play an important role.
Chapter 3
Ink-jet printers and inks
3.1
Ink-jet technologies
Ink-jet is a non-impact, dot-matrix printing technology. Ink droplets are emitted from nozzles of the printer directly to a specified position on a substrate
to create an image. The operation of the ink-jet printer is easy to visualize: a
printhead scans the page in horizontal strips, using a motor assembly to move
it from left to right and back, as another motor assembly rolls the paper in
vertical steps (see Fig. 3.1). A strip of the image is printed, then the paper
moves on, ready for the next strip. To speed things up, the printhead does not
print just a single row of pixels in each pass, but a vertical row of pixels at a
time.
Figure 3.1: A four-color ink-jet printer. The printhead moves along the drum perpendicular to the rotation of the drum facilitating the deposition of ink droplets of
all colors in each pixel (from L. Palm [Pal99]).
Ink-jet printing is a relatively young commercial industry. It began about
22
Ink-jet printers and inks
20 years or so ago, even though the mechanism for breaking up a liquid stream
into droplets was described more than 120 years ago by Lord Rayleigh [Ray78].
Efforts to make an ink-jet printer started about 50 years ago [Elm51]. After
that, continuous efforts have been made to improve the reliability of drop formation, to reduce the size of the ink droplets, while at the same time to increase
the jetting speed etc. This continuous development has led to continuous improvement of printed images [Le98]. The state of the art ink-jet technology can
generate ink droplet as small as 2 pico-liter (pL) in volume. Today, the ink-jet
printers can produce images of photo-quality with reasonable printing speed.
Ink Jet Printing
Continuous Ink Jet
Drop on Demand
Raster
Binary
Airbrush
Electrostatic
Hertz
DuPont
Iris Graphics
Stork
Domino
Imaje
Linx
Videojet
Scitex
Videojet
LAC
Sign Tech
Vutex
Piezoelectric
Epson
iTi
Aprion
Brother
Dataproducts
Epson
MIT/Xaar
Spectra
Tektronics
Triden
Thermal
Canon
HP
Lexmark
Olivetti
Xerox
Figure 3.2: Ink-jet technology map. Vendors that employ the technologies have been
listed in the figure.
Ink-jet printing has been implemented in many different designs with a
wide range of potential applications. Fundamentally, the technologies for ink
application is divided into two groups, continuous and drop-on-demand (DOD)
as shown in Fig 3.2.
3.1.1
The continuous ink-jet
In a continuous ink-jet, the creation of ink droplets is controlled by periodic
signals (not the printing signal) which lead to a constant ink droplet genera-
3.1 Ink-jet technologies
tion. The generated droplets are selectively charged, a feature controlled by
the printing signals. The charged droplets correspond to no print and are deflected into a gutter for recirculation when they pass through the electric field,
while the uncharged droplets fly directly to the media and form an image (see
Fig. 3.3). The advantage of the continuous ink-jet technology lies at its higher
printing speed compared to the drop-on-demand technology.
Figure 3.3: Schematic drawing of the principle for controlling droplet flight in a
continuous ink-jet printer through charging and deflection (a binary deflection system)
of individual droplets. (From L. Palm [Pal99]).
3.1.2
Drop-on-demand ink-jet
In contrast to the continuous ink-jet technology, impulse ink-jet technology
generates ink droplets when they are needed for printing. In the literature,
this technology is more commonly called drop-on-demand (DOD).
In DOD design, technologies of ink drop formation and ejection can be
categorized into four major methods: thermal, piezoelectric, electrostatic, and
acoustic. The first two are the dominant technologies for products on market
today, while the other two are still in the developmental stage.
In a printhead of a thermal ink-jet, there is an electric heater which is
attached to the ink chamber. Heat is transferred from the surface of the heater
to the ink. The heater is controlled by an electric current pulse. When the
current is on, the ink is superheated to the critical temperature for bubble
nucleation (about 300 o C for water based ink). When nucleation occurs, a
water vapor bubble instantaneously expands to force the ink out of the nozzle.
Once the current is off and all the energy stored in the ink is used, the bubble
begins to collapse on the surface of the heater. Concurrently, with the bubble
collapse, the ink drop breaks off and accelerates toward the paper as shown
in Fig. 3.4. Once the ink droplet is ejected, ink is refilled into the chamber
and the process is ready to begin again. Because the ink droplet is essentially
generated by the growth and the collapse of the vapor bubble, the thermal
23
24
Ink-jet printers and inks
Printhead Body
Ink Supply
Heater Chip
Nozzle Plate
Substrate
Figure 3.4: A schematic diagram of thermal jet technology.
jet is also called a bubble jet. The simple design of a bubble jet printhead
along with its semiconductor fabrication process allow printheads to be built
at low coat and with high packing density. These made the thermal ink-jet
technology the most successful one on the market today [Le98]. Moreover, the
bubble ink-jet with a high printing resolution and color capacity is available
at a very affordable price. On the market, there are many vendors who have
adapted this technology in their ink-jet products, such as, Hewlett-Packard,
Lexmark, Olivetti, Canon, and Xerox.
In the printhead of a piezoelectric ink-jet, there are piezo-ceramic plates
bonded to the diaphragm (electrodes) as shown in Fig. 3.5. Similar to the
bubble jet, the piezoelectric material is also controlled by a current pulse. In
response to the electric pulse, the piezo-ceramic plates deform in shape which
causes the ink volume change in the chamber to generate a pressure wave that
propagates toward the nozzle. Consequently, an ink droplet is ejected out.
Depending on the mode of shape deformation of the piezoelectric plates, there
are different printhead designs, such as, push-mode, bent-mode, shear-mode,
etc, and Fig. 3.5 is an example of the shear-mode design. Piezoelectric ink-jet
is also very popular among the ink-jet manufacturers, such as, Epson, Xaar,
Tektronix, etc.
Unlike the continuous ink-jet, drop-on-demand ink-jet technology means
3.2 Characteristics of ink-jet printers
Figure 3.5: A shear mode piezoelectric ink-jet design [Le98].
that ink droplets are generated and ejected when they are used in imaging. This
technology eliminates the complexity of drop charging and deflection hardware
as well as the inherit unreliability of the ink recirculation systems required for
the continuous ink-jet technology. The drop-on-demand ink-jet technology is
the most common technology on the market today.
The trend in the development of ink-jet technology is toward jetting smaller
droplets for image quality, fast drop frequency and a higher number of nozzles
on the printhead for print speed.
3.2
3.2.1
Characteristics of ink-jet printers
General
The performance of an ink-jet printer may be characterized by its printing
speed and resolution. The speed depends above all on the jetting frequency or
the time interval between two consecutive ink-jets. On an ordinary ink-jet, the
printhead takes about half a second to print a strip across a page. Since A4
paper is about 21 cm wide and the ink-jet operates at a minimum of 300 dpi,
this means there are at least 2,475 dots across the page. The printhead has,
therefore, about 1/5000th of a second to respond to whether or not a dot
needs printing. Nevertheless, higher printing speed may also be achieved by
25
26
Ink-jet printers and inks
adapting bigger printheads with more nozzles which enables it to deliver native
resolutions of up to 1200 dpi and print speeds approaching those of current
color laser printers: 3 to 4 pages per minute (ppm) in color, and 12 to 14 ppm
in monochrome.
Table 3.1: Descriptions for some desk ink-jet printers
Jet type
Thermal
drop size
(picoliter)
5
Piezo
4
Thermal
N/A
Thermal
N/A
Canon S300
Number of nozzles
Resolution (dpi)1
black
color
black
color
320
128×3
600×600 2400×1200
Epson C60
144
48×3
up to 2880×720
HP 920c
N/A
N/A
up to 2400×1200
Lexmark Z53
N/A
N/A
up to 2400×1200
Speed (ppm)2
black color
11
7.5
12
8
9
7.5
16
8
1 - it consists generally of 3 categories, draft, normal and best. Only the
highest possible resolution of the printer has been listed here.
2 - the speed of printing, pages per minute (ppm), decreases when higher resolution is chosen in the printing. Only the higher possible speed (draft) is listed
here.
Resolution of the print depends on the volume of the ink droplet. The
smaller the ink droplet the higher the possible printing resolution. The volume
of the ink droplet is determined by the diameter of the nozzle as well as the
width of the current pulse (in time). Therefore, to be able to manufacture
the printhead with a very fine nozzles is of great importance for achieving
high printing quality. For example, for 10 pL droplet, HP DeskJet 890C color
printhead has nozzle diameter of around 20 µm. Apart from the printhead,
the substrate onto which the ink drops can also have great impact on the
printing resolution, due to the ink-substrate interaction, such as ink spreading
and penetration. Ink bleeding can actually have significant impact on the
image resolution. Finally, it should be borne in mind that print speeds may
vary, depending on the document, software program, and computer settings.
Table 3.1 is a collection of technical descriptions for some ink-jet printers that
have been available on the market over recent years.
3.3 Ink-jet ink technologies
3.2.2
HP970Cxi ink-jet printer
In this dissertation, the test charts were printed by employing a HP970Cxi
ink-jet printer, which is of the drop-on-demand ink-jet design. It uses dyebased liquid inks (water resolution). The technological characteristics of this
printer is summarized in Tab. 3.2.
Table 3.2: Description of HP970Cxi ink-jet.
Document type
Draft
Normal
Best
∗
Resolution (black-white, dpi )
300 x600
600 x 600
600 x 600
Resolution (color, dpi)
300 x 600 Color layering 2400 x 1200
Print speed (black text, ppm)
12.0
6.5
4.7
Print speed (Mixed text/graphics)
10.0
5.0
3.1
Print speed (Full page color)
2.9
0.6
0.3
Type of ink
Water-based organic dyes
Printer command language
HP PCL level 3 enhanced
∗
dpi - dots per inch.
3.3
3.3.1
Ink-jet ink technologies
General
Inks are probably the most critical components in the ink-jet printing. The
development of ink-jet inks has been an important part of ink-jet technology.
This is because the ink properties not only dictate the quality of the printed
images, but they also determine the characteristics of the drop ejection and the
reliability of the printing system.
The ink-jet inks consist normally of colorants, an ink vehicle, and additive
materials. The ink vehicle like water, oil, solvent, resin, etc., governs the dynamic properties of ink distribution. The ink vehicle is the major component
of the ink, whose amount in percentage varies from 40 − 90% depending on
the ink type. The colorants are the materials that create color of the printed
image, whose amount lies between 1 − 10%. The rest of the ink is generally
referred to as additives. These improve the chemical and physical properties of
the ink, such as ink viscosity, adhesion strength, heat stability, cure rate (for
light inks), surface tension (for liquid ink), etc.
Ink-jet inks can be sorted into different groups based on the different perspectives, as shown in Tab. 3.3. Ink vehicle is one perspective: ink-jet inks
are usually divided into 4 groups. They are, aqueous-based, nonaqueousbased, phase-change, and reactive.
27
28
Ink-jet printers and inks
Aqueous and nonaqueous inks use water or other solvents as ink vehicles
whose drying mechanism depends on penetration and absorption of the receiving media (substrate). When ordinary copy or plain paper is used, the ink
together with the ink-vehicle are absorbed by the porous material. The mixing
of the ink with the pores, lowers the color density and spot resolution.
Phase-change ink is also called solid ink which is solid at room temperature. The ink is jetted out from the printhead as a molten liquid. When the
molten ink drop hits the substrate surface, it solidifies immediately. The quick
solidification prevents the ink from spreading or penetrating the substrate, and
ensures good image quality for a wide variety of substrates.
Detail descriptions of the groups of inks may be found in references [Le98,
HF97, III99].
Table 3.3: An ink-jet ink technologies map.
Category according to ink bases
Aqeous-Based
Solution, Dispersed, Microemulsion
Nonaqeous-Based
Oil, Solvent
Phase-change
Liquid to Solid, Liquid to Gel
Reative
UV Cured, Two parts
Category according to colorants
Organic Dyes
Direct, Acid, Reactive, Disperse, Solvent
Polymeric Dyes
Aqueous, Nonaqueous, Polymer Blend
Pigments
Carbon Black, Organic
3.3.2
Dye-based and pigment-based inks
Inks can be divided into 3 groups based on colorants: organic dye, polymeric
dyes, and pigments. The organic dyes consist of organic dye molecules, while
the polymeric dyes consist of dye polymers. The pigmented inks are mainly
inorganic powders even though there are few organic pigments.
In the case of color the use of a dye or a pigment is one of the most widely
debated topics in the industry. Dyes and pigments are different in many ways,
which contributes to their different color performances for the printed images.
In this section, we briefly compare these in terms of their color performance.
Most dyes are soluble synthetic organic materials, as opposed to pigments
which are generally insoluble. Chemically, dyes exist in the ink as individual
molecules, while the pigments exist as clusters that consist of thousands of
colorant molecules.
Generally speaking, the dye-based inks have superior color representation
capability or greater color gamut than the pigment-based inks. By printing on
3.3 Ink-jet ink technologies
high grade substrates, dye-based ink-jet printing can deliver images of compatible quality as those produced by silver halide. The disadvantages of the
dye-based inks are their relatively poor (long term) image performance which
includes light fastness (light fading stability), dark storage stability, humidity
fastness, and water fastness.
One explanation to poor image stability for images produced by dye-based
inks, is that the dye consists of individual molecules which are chemically less
stable in terms of light exposure, oxidation, and humidity. Being a cluster
of many molecules, the pigment inks have greater resistance to the impact
of the environment and therefore possess better light fastness and humidity
fastness. In terms of dye design, chromophore chosen has a dominant impact
on the spectral characteristics and color stability achieved. Additionally, their
physical and chemical properties also have great impact on color stability.
Trying to achieve a high degree of light fading stability and large color gamut
at the same time can pose a challenge for ink development due to the rarity of
pigments and dyes that have both of these desirable properties [ADT+ ]. Over
the years, it has been a hot topic for debate in ink chemistry studies and ink-jet
technologies. Hopefully, we are now seeing a rapid closing of the gap in color
and image performance, between pigments and dyes [IB01].
29
Chapter 4
Optical modelling: an
overview
Reflectance, such as, brightness, opacity, etc, characterize the paper sheet quantities, but not the general material properties of the paper. Optically, the
fundamental events that govern reflection are light scattering and absorption.
Therefore, quantities that parameterize the fundamental processes, such as
coefficients of scattering (σs ) and absorption (σa ), are fundamental material
properties. To link measured reflectance values with the material properties,
one needs an optical model.
Basically, there exist two groups of models for optical modelling. One group
of models is based on Radiative Transfer Theory [Cha60]. Another group uses
Monte-Carlo methods [Jam80, Rub81] to simulate light scattering and absorption. In this chapter we highlight the fundamental concepts of these methods
as well as their application to Paper Optics and Graphic Arts.
4.1
Radiative Transfer Theory
Radiative Transfer Theory (RTT) based approaches start with solving integrodifferential equations which describe light propagation in media. According to
RTT, the radiance L(.r, û) (W · m−2 · sr−1 ) of light at position .r travelling in
a direction of unit vector û is decreased by absorption and scattering but is
increased by light that is scattered from û into the direction û. The radiative
transfer equation which describes this light interaction is [Cha60]
û · ∇L(.r, û) = −(σa + σs )L(.r, û) +
σs
4π
4π
q(û, û )L(.r, û )dω (4.1)
32
Optical modelling: an overview
where σa (m−1 ) is the absorption coefficient, σs (m−1 ) is the scattering coefficient, dω is the differential solid angle, and q(û, û ) is a phase function.
The total extinction coefficient, σt , is a sum of the absorption and scattering
coefficients,
(4.2)
σt = σa + σs
The phase function, q(û, û ), describes angular distribution of a single scattering event. If the integral of the phase function is normalized to equal to
unity, i.e.
1
q(û, û )dω = 1
(4.3)
4π 4π
then it is the probability density function of scattering from direction, û to
direction û. Assume the directions of the incident and the scattered light
are û(θ, φ) and û (θ , φ ). It is reasonable to assume that the phase function
depends only on the scattering angle Θ (cosΘ = û· û ) rather than the incoming
or the outgoing angles, i.e.
where
q(û, û ) = q(cosΘ)
(4.4)
cosΘ = cosθcosθ + sinθsinθ cos(φ − φ)
(4.5)
A complete description of light transfer requires knowledge of σa , σs , and
q(û, û ). These quantities depend not only on the properties of the raw materials, but also on their distribution (or the structure of the system). To obtain
these quantities for various papers is not trivial and remains an open problem.
4.2
Phase function
The key in solving the integro-differential equation (Eq. 4.1) depends largely on
the form of the phase function, q(cosΘ). Different phase functions have been
proposed to physically describe different types of scattering. Among the wellknown are the Rayleigh phase function [Ray71], Mie phase functions [Mie08],
and Henyey-Greenstein phase function [HG41]. These phase functions were
originally proposed for studying radiative transfer in atmospheric gaseous systems and in the galaxy. Later, they found applications in other fields.
Mie theory describes light scattered by isolated spherical particles of arbitrary size and refractive index. Taking particle size (radius r) and refractive
index as input parameters, Mie theory calculates efficiency parameters, i.e.,
scattering efficiency Qsca and absorption efficiency Qabs . The angular distribution of the scattered light, or the phase function, is calculated by
q(cosΘ) =
i(Θ)
2πα2 Qsca
(4.6)
4.3 Multi-flux theory
33
where α = 2πr/λ is the particle size parameter relative to the wavelength of
the light, i(θ) is called the Mie theory intensity function. Mie theory found its
original application in gaseous systems where the particles are well isolated.
However, it has also been applied to systems that are closely packed such as,
paint film [JVS00].
Henyey-Greenstein phase function is a one parameter analytical approximation to a real phase function. It is expressed as [HG41]
q(cosΘ) =
(1 +
g2
1 − g2
− 2gcosΘ)3/2
(4.7)
where g is called an asymmetry factor which controls the scattering pattern.
g = 0 corresponds to isotropic scattering, which is the case in the KubelkaMunk approach. Clearly, Θ = 0, π are two singular points for g = ±1, respectively. It is easy to see that when Θ → 0 and g → 1
lim q(cosΘ → 1)
g→1
=
→
(1 + g)
g→1 (1 − g) (1 + g 2 − 2gcosΘ)
δ(Θ)
lim
(4.8)
Consulting the normalization condition given by Eq. (4.3), one can conclude
that g = 1 corresponds to complete forward scattering. Similarly, g = −1
corresponds to complete backward scattering. One of the greatest advantage
of the Henyey-Greenstein phase function is its simple form under the Legendre
expansion,
q(cosΘ) = 1 + 3gcosΘ + 5g 2 P2 (cosΘ) + 7g 3 P3 (cosΘ) + ...
(4.9)
This makes it a popular choice in applications, for example, in studying radiative transfer in biological tissues (dermal and aortic tissues) [Pra88, Yoo88,
CPW90]. Very recently, it has been even considered for application in simulations for light transfer in paper [Eds02].
4.3
Multi-flux theory
Mathematically, the radiative transfer equation (Eq. 4.1) has no analytic solutions except in a few special cases, such as q(cosΘ) = const. To solve the
problem with the help of computers, one must to divide the direction in space
into channels as shown in Fig. 4.1. All light travelling in a direction within a polar angle θ1 of the positive direction of the axis perpendicular to the boundary
plane is said to be in channel 1. All light travelling at a polar angle between θ1
and θ2 is allocated to channel 2, etc. Such a discrete ordinate method is called
Discrete-Ordinate-Method Radiative Transfer [Cha60, STWJ88] or Multi-flux
Radiative Transfer Method [MR71].
34
Optical modelling: an overview
Figure 4.1: The division of the directions in space into channels (provided by
H. Granberg).
The number of channel divisions depends on the nature of the application.
Many papers involving radiative transfer calculations have been written by
authors using 2, 3, or 4 channels [Sch05, MR71]. A general mathematical
treatment using this coordinate discretion, was first developed by Wick [Wic43].
It was then thoroughly exploited by Chandrasekhar [Cha44] and applied to the
problem of radiative transfer.
Mudgett and Richards [MR71, MR72] reformulated this method in a more
comprehensive way and applied it to parallel layered media, such as paint
film [Ric70]. Their work was well followed up by other authors in modelling
and predicting the optical characteristics of paint films [JVS00, All73].
In principle, by applying the Multi-flux approach with a sufficient number
of channels, one can accurately solve the radiative transfer problem. Nevertheless, the solution depends directly on the knowledge of the phase function.
Therefore, finding the proper phase functions for different types of papers is
essential for optical modelling and simulations.
4.4
Kubelka-Munk method
The Kubelka-Munk (K-M) approach is actually a two-flux version of the multiflux method for solving the radiative transfer problem. Here the ordinate is
only divided up into an up- and a low-hemisphere by the bounding plane (paper
4.5 Monte-Carlo simulation
sheet). In the K-M approach, the light propagation depends on the K-M coefficients of light absorption (k) and scattering (s). The fundamental assumption
of this approach is that light distribution in the media is ideally diffuse. It requires the media to be of little absorption, while at the same time of involving
strong and angle-independent scattering (q(cosΘ) = 1). Indeed, a comparison between the more accurate multi-flux calculation with the Kubelka-Munk
approximation reveals that the two-flux approximation gives excellent results
provided the absorption is small compared to scattering, and the optical thickness is greater than 5 [MR71]. A key factor for a successful application of
this method lies at finding the so-called K-M absorption coefficients, k, which
depends not only on the physical properties of a medium but also on light distribution in the medium. Detailed discussions about this issue is presented in
Secs. 5.2 and 7.2.
The K-M approach has probably been the most widely used approach in
paper-making and color-using industries since it was introduced more than
70 years ago [KM31, Kub48]. The continued popularity of this approach is
attributable to the simple analytical solutions. Nevertheless, the solutions provide insights to the processes of the light transfer and can be used to predict the
reflectance of the specimen with reasonable accuracy. In addition, the simple
principles involved in the theory are easily understood by the non-specialist. An
excellent review about the advantages and disadvantages of the K-M method
can be found in [Nob85].
In the ordinary K-M method, the surface reflection is usually neglected,
even though it may be an important contribution sometimes. In addition, the
analytical solution that has widely been used, is only applicable to layered
media that has a uniform concentration along the ordinate z-axis within the
layer, and the layer has infinite horizontal extension in the xy-plane. When the
distribution is non-uniform one has to work directly with the integro-differential
equation. In the next chapter, we extend the K-M method to cope with the
non-uniform ink distribution and surface reflection.
4.5
Monte-Carlo simulation
The Monte-Carlo approach belongs to another category of radiative transfer
simulations. It was first used by Fermi, Von Neumann and Ulam who developed
it for the solution of problems related to neutron scattering during the development of the atomic bomb. The name Monte-Carlo is used since the method
is based on the selection of random numbers. In this sense it is related to the
gambling casinos at the city of Monte Carlo in Monaco. As shown in Fig. 4.2,
the light scattering process can be considered as a “random walk” which consisting of straight paths between points of interaction with the media, such
as fibres or fillers in paper and colorant particles in print. The Monte Carlo
35
36
Optical modelling: an overview
method can be considered as a very general mathematical method to solve a
great variety of problems. In the following, we will only highlight the basic
concepts of this method for the simulation of light propagation. More detailed
description of this method may be found in references [Jam80, Rub81, Gus97b].
1
3
2
1’
A
l1
B’
A’
D
l
3
B
l
2
θ
C
3’
Figure 4.2: Random walk of photons in a turbid medium (2D diagram).
For simplicity of description, we shall trace the random walks of photons in
a 2D scene (Fig. 4.2). The length, li , of an undisturbed straight path of one
photon is a stochastic variable. Its statistical expectance value, lp , is called the
mean free path and is inversely proportional to the extinction coefficient, σt ,
i.e.
N
1
i=1 li
=
lp = lim
(4.10)
N →∞
N
σt
When the photon hits the media, it will either be absorbed (like photon 2 at site
B ), or be scattered randomly. On average, the probability of a photon being
absorbed is related to the relative strength of the absorption coefficient to the
total extinction coefficient, i.e. σa /σt , and similarly σs /σt , for the scattering.
From a physics point of view, scattering means that a photon is absorbed
and then re-emitted with the same (elastic scattering) or different (inelastic
scattering or Raman Scattering) energy in a different direction. The latter
case is out of the scope of this work. The direction of the re-emission is at
random. In a 3D scenario, the direction is specified by a pair of angles, polar
and azimuth angles, (θ, φ).
4.5 Monte-Carlo simulation
37
By applying a large number of photons or equivalently by repeating the one
photon process a large number of times, one can obtain statistically meaningful
quantities (probability) that characterize the properties of the studied system.
Assume that the total number of incident photons is Ntot . If there are Nref
photons that have returned to the same hemisphere (1 ) as the incident photons, and Ntran photons that have reached the opposite hemisphere (3 ), then
reflectance (R) and transmittance (T ) can be computed as
R
=
T
=
Nref
Ntot
Ntran
Ntot
(4.11)
(4.12)
A useful concept that describes light propagation in the media is the point
spread function (PSF). If a photon hits the surface of a sheet of paper at .r, the
probability that the photon exits the paper at .r is described by the PSF and
denoted as p(.r, .r ). If the PSF of the paper is known, quantities like reflectance
and transmittance can easily be computed. Therefore, to obtain the PSF is
more essential in the Monte-Carlo simulation. Recently, Gustavson [Gus97b]
developed methods for computing the PSF of paper and simulated the optical
dot gain in halftone prints.
Chapter 5
Extended Kubelka-Munk
theory and applications
5.1
Assumptions in Kubelka-Munk theory
The original theory of Kubelka-Munk (K-M) was developed for light propagation in parallel colorant layers of infinite xy-extension [KM31, Kub48]. The
fundamental assumptions of the K-M theory are that the layer is uniform and
that light distribution inside the layer is completely diffused. From these assumptions, light propagation in the layer was simplified into two diffuse light
fluxes through the layers, one proceeding upward and the other simultaneously
downward. After its introduction in the 1930’s, K-M theory was extended by
removing some of the original assumptions. Among others, a correction to the
boundary reflection at the interface of two adjacent media was introduced by
Saunderson [Sau42], i.e., the well-known Saunderson correction. Kubelka himself also made attempt to extend the applicability of the theory to optically
inhomogeneous samples [Kub54]. However, this extension applied only to a
special case of the inhomogeneous media, in which the ratio of the absorption
to the scattering is constant.
Considering the original K-M theory together with the following extensions,
the assumptions remaining in the K-M theory may be summarized as follows:
1. The sample consists of a turbid medium and forms a plane layer (perpendicular to the z-axis) whose size in the xy-extension is much larger than
its thickness. Edge effects are therefore negligible.
2. The sample is optically homogeneous. In the case of inhomogeneous
sample, the ratio between its scattering and absorption must be constant.
40
Extended Kubelka-Munk theory and applications
3. The scattering in the sample is isotropic, i.e., it is independent of the
angle between the incident and scattering directions.
4. The light flux in both forward and reverse directions is uniformly diffuse.
The first 2 assumptions are about the material distribution in the sample and
the last 2 are about light propagation and light-medium interaction.
This chapter presents an extension of the K-M theory by completely removing the restriction of assumption No. 2. Such an extension allows one to
study cases where the scattering and absorption coefficients are any functions
of z-coordinates.
Figure 5.1: Principle of the Kubelka-Munk theory.
5.2
Differential equations
When people view an image on a piece of paper, they actually receive the light
that is reflected from the image. Therefore, it is natural to separate the light
that is reflected from the image, from that goes to the image. An equivalent
expression is to divide the light that travels toward the upper hemisphere (reflected from the image) as one group, and the light that travels toward the
lower hemisphere as another. According to the assumption No. 4, the light in
either group is uniformly diffuse. Consequently, one only need trace the light
flux at (any) one direction in each hemisphere to obtain complete information
about the light propagation. It is therefore convenient to choose the direction
along the z-axis and the inverse direction as representatives for light propagating toward the upper and the lower hemispheres, respectively (see Fig. 5.1).
Mathematically, one can then simplify the actual three-dimensional light propagation by considering one-dimensional streams of light intensities, Iup and Idn ,
5.2 Differential equations
41
representing light travelling upwards and downwards, respectively (they correspond to ir and it in Fig. 5.1). As shown in Fig. 5.1, in any differential layer, dz,
inside the material, the streams, Iup and Idn , are attenuated by absorption and
backscattering. On the other hand, Iup is enhanced by the backscattering from
the stream Idn and vice versa. From these arguments, Kubelka-Munk [KM31]
derived the following expressions for the gradients of the light intensities,
dIdn
dz
dIup
−
dz
= −(s + k)Idn + sIup
(5.1)
= −(s + k)Iup + sIdn
(5.2)
Applying the K-M theory to a semi-infinite medium layer, one obtains the
reflectance of the medium layer, when the surface reflection is omitted [Nob85],
k
k
k
(5.3)
R∞ = 1 + − ( )2 + 2
s
s
s
where k and s are called K-M coefficients. They are phenomenal descriptions to
the light absorption and scattering in the medium. Therefore, they are closely
related to the fundamental optical properties of the medium, the absorption
(σa ) and scattering (σs ) coefficients per unit path length of the material. However unlike (σa ) and (σs ) that depend solely on the physical properties of the
medium, the K-M coefficients, k and s, depend also on the light distribution in
the medium and even illumination geometries [Ste55]. Physics behinds this is
that k and s do not describe the absorption and the scattering in any specific
direction but their averages over the upper or lower hemisphere. Theoretical
analysis [Kub48, WH66, Kor69] has shown that, for the completely diffused
light distribution, the averaged path length of the photon is 2dz. Therefore,
k = 2σa
(5.4)
Awareness of the dependence of the K-M coefficients, k and s, on the light
distribution in the medium is particularly important when K-M theory is applied to a mixture of different media. Dye-based ink, for example, has little
scattering, in which the light propagates essentially along a straight path. However when the ink mixes with paper materials, like ink penetration, the light
propagates instead in a zigzag fashion in the ink-paper mixture, because of
very strong scattering of the paper. Consequently, the light appears to have
greater probability to be absorbed if it passes through the same vertical depth
of the mixture as that of the ink layer. This explains the experimental observations [Foo39, BS76] that the k appears differently when the dye mixes with
different pulps (because of different scattering power of the pulps).
Mudgett and Richards [MR71] investigated the relation between k and s
and σa and σs for an optically thick, weakly absorbing layer. By means of
42
Extended Kubelka-Munk theory and applications
the multi-flux method, they calculated accurate values of the reflectance of a
semi-infinite film (R∞ ) and of a finite film of the same material over an ideal
black backing. The values of k and s were calculated from these results. The
results confirmed Eq. (5.4) and they found that to a good approximation:
(5.5)
s = 0.75σs
z
I
I R
0
0
r
0
D
r1
I
Iup
dn
medium layer
0
Rg
backing
Figure 5.2: A schematic diagram of light propagation in a triple-layer system.
5.2.1
Boundary conditions
Figure 5.2 shows the layered structure of the medium. The layer has an interface with air at z = D on top, while at z = 0 it is in optical contact with
a backing of reflectance, Rg . If surface reflectance at the interfaces is r0 and
r1 , respectively, as shown in the figure, one may then obtain the following
boundary conditions at the z = D interface,
b
(D)
Idn
I0 R
b
= I0 (1 − r0 ) + Iup
(D)r1
= I0 r0 +
b
Iup
(D)(1
− r1 ).
(5.6)
(5.7)
At z = 0 there is
a
a
(0) = Idn
(0)Rg .
Iup
(5.8)
In Eqs. (5.6-5.8), the superscripts, a and b, denote the corresponding values
above (a) and beneath (b) the interface.
5.2 Differential equations
43
The background of the boundary conditions are the continuation of the
light streams across the interface. For example, Eq. (5.6) reveals that the light
propagating downwards (the term on the left of the equation) consists of two
light streams, i.e., the illumination that passes through the interface (first term
on the right) and the internal reflected light at the interface (second term on
the right). Following such principles one can establish boundary conditions for
systems consisting of any number of media layers [YK00].
1
0.9
n /n =1.5
2
1
0.8
Reflectance, r
0.7
0.6
0.5
0.4
0.3
r
||
0.2
r
0.1
0
0
r⊥
10
20
30
40
50
60
Angle of incidence, α
70
80
90
Figure 5.3: Fresnel reflectance as a function of angle, α, of incidence on a boundary
for which the refractive-index ratio is 1.5.
5.2.2
Boundary reflection
An ordinary printed sample consists of parallel-sided layers (ink, backing, etc.)
as shown in Fig. 5.2. Additionally, the sample itself interfaces with air. Because
of the discontinuity of refractive index at the boundaries between the adjacent
media (air, ink, and paper, etc.), a portion of the incident light falling on the
interface will be reflected on the interface. For a given ray, the amount reflected
can be computed according to Fresnel’s equations [Cha60],
2
cosα − (n2 /n1 )2 − sin2 α
(5.9)
r =
cosα + (n2 /n1 )2 − sin2 α
2
(n2 /n1 )2 cosα − (n2 /n1 )2 − sin2 α
r⊥ =
(5.10)
(n2 /n1 )2 cosα + (n2 /n1 )2 − sin2 α
44
Extended Kubelka-Munk theory and applications
Clearly, the boundary reflection depends on the refractive indices of the media,
the light polarization and the angle of incidence with respect to the interface
normal (see Fig. 5.3). For unpolarized incident light, the reflectance, r, is a
simple average of r and r⊥ :
r = (r + r⊥ )/2.
(5.11)
For clarity of discussion, we denote the reflection from within the side of
having greater refractive index, the internal reflection, as r1 . On the other side
we have the external reflection denoted by r0 .
0.9
r
0.8
r||
Boundary reflectance
0.7
0.6
r⊥
Internal
0.5
0.4
0.3
r||
0.2
External
0.1
0
1
r
r
⊥
1.1
1.2 1.3 1.4 1.5 1.6 1.7 1.8
Relative index of refraction, n=n2/n1
1.9
2
Figure 5.4: Internal and external reflectance for completely diffuse incident flux as
a function of the relative refractive-index, n = n2 /n1 .
On the assumption that the flux falling on a surface of the sample from
air is completely diffuse, the Fresnel equations must be integrated over all
(incident) angles in order to compute the total amount reflected by the surface
(external reflection, r0 ). For a relative index n = n2 /n1 = 1.5, typical of glass
or polymeric materials bounded by air, the external boundary reflection is r0 =
0.0919. One may compare this with the value of reflectance at perpendicular
incidence, 0.040. For diffuse light incident on the boundary from within, the
sample is partially reflected (internal reflection, r1 ). A fraction (1 − 1/n2 ) will
be incident on the boundary at angles greater than the critical angle and will be
totally reflected. The total internal boundary reflectance for diffusely incident
light is given by [Nob85],
1
r0
r1 = 1 − 2 + 2 .
(5.12)
n
n
5.3 Models of ink penetration
45
For n = 1.5, according to Judd [Jud42], r1 = 0.5960.
The dependence of the internal and external reflections on the relative index
of refraction, n, is shown in Fig. 5.4. It is seen that the internal reflection increases dramatically with n. For the external reflection, however, the increases
with the refractive index, n, are rather moderate.
5.3
Models of ink penetration
For ink jet printing, absorption of ink constituents by the paper is driven by
thermodynamic interaction between the ink and the paper, by capillary forces
and by chemical diffusion gradients. Capillary pressure is acknowledged to be
the main driving force in the offset ink oil transport in a typical paper coating
porous structure. With increased latex content, diffusion-driven transport of
ink chemicals into the latex counterpart of the coating layer become important [Rou02]. Different mechanisms of ink penetration result in different forms
of ink distribution in the paper.
Suppose that the ink distribution (density) varies only in the z direction.
Denote its value at position z as ρ and its value at z+dz as ρ+dρ, respectively.
The increment in the ink density may be written as
dρ = f (z)dz
(5.13)
Here, the function f (z) describes the variation in ink penetration which is
closely related to the properties of the substrate paper, on the one hand, and
those of the ink on the other hand. In the following we will investigate three
forms of f (z).
5.3.1
Uniform distribution
The simplest case is of course, f (z) = 0, the ink penetrates uniformly into the
paper (between z = 0 and z = D), with density
ρ(z) = ρ0
(f or 0 < z ≤ D)
(5.14)
If the coefficients of absorption and scattering are kp , sp and ki , si , for paper and ink, respectively, the corresponding coefficients for the ink penetrated
paper may be expressed as,
k = kp + µki
s = sp + si
(5.15)
(5.16)
where µ is a constant depending on the strength of the scattering of the paper
and illumination geometries as discussed in Sec. 5.2. Detail discussions will be
46
Extended Kubelka-Munk theory and applications
given later and in Chapter 7 as well. Because µ is only a multiplier of ki , we
can continue the study for the case of µ = 1 without losing generality.
One shortcoming of this model seems to be the discontinuity in the ink
distribution at the lower interface (z = 0) of the penetrating ink layer. Nevertheless, it may still be a reasonable approximation to a system consisting
of liquid ink and plain paper whose pore size between cellulose fibres is much
bigger than the size of the ink molecules or micro-ink cells (100Å). In such an
ink-paper system the dominant mechanism of ink penetration is the capillary
force. Additionally, the model reflects the fundamental fact that there exists
a layer of an ink-paper mixture. Furthermore, it is the only case in which one
can solve the differential equations for the light propagation analytically.
5.3.2
Linear distribution
The simplest model of non-uniform ink distribution is possibly a linear model,
in which the ink concentration in the paper decreases linearly with depth of
the ink penetration, i.e.,
f (z) = ρ0 /D
(5.17)
or
ρ(z) = ρ0
z
D
(0 ≤ z ≤ D)
(5.18)
The coefficients of the absorption and scattering in the ink-penetrated paper
may then be written as,
z
D
z
s = sp + si
D
k
= kp + ki
(5.19)
(5.20)
Compared to the uniform model, the discontinuity at the lower interface (z =
0) is removed in this model. On the other hand, the differential equations
become relatively more complicated, and no analytic solution in known any
longer. Fortunately, as discussed later one may find a good approximation in
polynomial form, and in some cases even be able to handle this model in an
analytical manner.
5.3.3
Exponential distribution
In the case of ink penetration driven by a concentration gradient, the variation
of the ink distribution is proportional to the ink distribution itself, i.e.,
dρ ∝ρdz
(5.21)
5.4 Solutions of the differential equations
47
Considering the fact that at the interface between the ink and the ink-penetrated
paper, ρ(D) = ρ0 , we may write the distribution as
(0 < z ≤ D)
ρ = ρ0 eα(z/D−1)
(5.22)
Thus, the absorption and scattering coefficients of the ink-penetrated paper
read,
k
= kp + ki eα(z/D−1)
(5.23)
α(z/D−1)
(5.24)
s = sp + si e
Compared to the uniform- and linear-ink penetration models, the exponential
model is most mathematically complex. In turn, it causes complexities in
solving the differential equations that govern light propagation in the ink-paper
mixture.
5.4
Solutions of the differential equations
Equations (5.1) and (5.2) provide the general descriptions to light propagation
in either direction (upward or downward). Substituting the expressions for
k and s (Eqs. (5.15, 5.16), Eqs. (5.19, 5.20), and Eqs. (5.23, 5.24)) into the
differential equations and solving these, one obtains the intensity of the light
reflected from the ink layer and then the reflectance of the prints with ink
penetration in uniform, linear, and exponential ink distributions.
5.4.1
Uniform ink distribution
In this sub-section we derive the reflectance for a medium layer (ink penetrated
paper layer) with and without backing. In practice, the layer with backing (see
Fig. 5.2) serves as a model for the print where the substrate paper is only partially penetrated by the ink and the remaining portion (bottom layer) acts as
a backing to the top layer. The case of no backing may be considered as paper
being fully penetrated by the ink. It is a freely suspended layer. Because transmittance measurements for prints often provide useful information, expressions
for transmittance of the freely suspended media layer has also been derived.
Reflectance of the ink-penetrated layer with backing
In the case of the uniform ink distribution, the general solutions of the differential equations (Eqs. (5.1) and (5.2)) have the form,
Iup
= a1 eBz + a2 e−Bz
(5.25)
Idn
= b1 eBz + b2 e−Bz
(5.26)
48
Extended Kubelka-Munk theory and applications
where B is a function of s and k,
B(s, k) =
s
A(s, k)
1
(
−
)
2 A(s, k)
s
(5.27)
and
A(s, k)
= sR∞
= s+k−
k 2 + 2ks
(5.28)
For simplicity, A(s, k) is hereafter denoted as A, if not otherwise stated. In
K-M theory, the reflectance (R) is usually expressed as a function of s and
R∞ [KM31, Pau87]. One advantage of replacing R∞ by A is that one avoids
possible numerical problems when the system undergoes little scattering.
Inserting the solutions (Eqs. (5.25) and (5.26)) into Eqs. (5.1) and (5.2),
one can get the following correlation relations,
A
a1
(5.29)
s
s
a2
b2 =
(5.30)
A
Therefore, there exist only two undetermined coefficients, say a1 and a2 , which
can be determined by applying the boundary conditions given in Eqs. (5.6-5.8).
Replacing Iup and Idn in the boundary conditions with Eqs. (5.25) and (5.26),
one obtains
s2 −A2
s2 −A2
Aa1 s2 −A2 D sa2 − s2 −A2 D
e 2A
e 2A
+
]
a1 e 2A D + a2 e− 2A D = I0 (1 − r0 ) + r1 [
s
A
(5.31)
Aa1 s2 −A2 D sa2 − s2 −A2 D
e 2A
e 2A
+
]
IR = I0 r0 + (1 − r1 )[
s
A
(5.32)
A
s
a1 + a2
(5.33)
Rg (a1 + a2 ) =
s
A
By solving these equations, one obtains the expression for the reflectance
of the medium with backing (Rg ) as
=
b1
R = r0 +
(1 − r0 )(1 − r1 )[s(A − sRg )e−
2
2
− s −A
A
(A − sr1 )(A − sRg )e
D
s2 −A2
A
D
− A(s − ARg )]
− (s − Ar1 )(s − ARg )
(5.34)
If the medium is thick enough, i.e, D → ∞ then R → R∞ , and
R∞ = r0 +
(1 − r0 )(1 − r1 )R∞
.
(1 − r1 R∞ )
(5.35)
Clearly, R∞ is a special case of R∞ when the interface reflection, r0 = r1 = 0,
or, are negligible.
5.4 Solutions of the differential equations
z
I0
49
I0R
r
0
D
r
1
I
Iup
dn
medium layer
r
1
0
r
0
I T
0
Figure 5.5: A schematic diagram of light propagation in a freely suspended medium
layer.
Reflectance and transmittance of the freely suspended layer
If the layer is freely suspended in space (without backing) as shown in Fig. 5.5.
The reflection at z = 0 interface is purely due to the surface reflectance, r1 .
Consequently, the boundary condition given by Eq. (5.8) should be replaced
by
Aa1
sa2
+
= r1 (a1 + a2 ).
(5.36)
s
A
The reflectance of the medium layer, R, is therefore
R = r0 +
(1 − r0 )(1 − r1 )[s(A − sr1 )e−
2
2
− s −A
A
(A − sr1 )2 e
D
s2 −A2
A
D
− A(s − Ar1 )]
− (s − Ar1 )2
.
(5.37)
For the freely suspended medium layer, one may obtain valuable information
about the layer by measuring its transmittance. From the continuity of the light
stream (propagating downward), one may obtain an extra boundary condition
beneath the (z = 0) interface,
I0 T = (a1 + a2 )(1 − r1 ).
(5.38)
Combining Eqs. (5.31), (5.36), and (5.38), one may derive the expression
for the transmittance, T, for the freely suspended medium layer,
T =
(1 − r0 )(1 − r1 )(s2 − A2 )e−
s2 −A2
2A
2
2
− s −A
A
(s − r1 A)2 − (A − sr1 )2 e
D
D
(5.39)
50
Extended Kubelka-Munk theory and applications
A special case of our discussions is that the medium has no light scattering
(s = 0), Eqs. (5.34), (5.37), and (5.39) are simplified into
R = r0 + (1 − r0 )(1 − r1 )
Rg e−2kD
1 − r1 Rg e−2kD
(5.40)
r1 e−2kD
1 − r12 e−2kD
(5.41)
for the layer with backing (Rg ), and
R
= r0 + (1 − r0 )(1 − r1 )
T
=
e−kD
1 − r12 e−2kD
(1 − r0 )(1 − r1 )
(5.42)
for the layer without backing.
5.4.2
Linear ink distribution
Unlike the case of uniform ink distribution, the differential equations for the linear ink penetration are more complicated. Generally speaking, these equations
have no simple and analytical solutions, and one has to solve them numerically.
However, there exist a possibility that one can expand the solution in series in
the vicinity of z = D.
z
I0
I0R
r
0
D
I
dn
r1
I
medium layer
up
0
Rg
backing
Figure 5.6: A schematic diagram of a linear ink distribution (with backing).
5.4 Solutions of the differential equations
51
Series solutions to the differential equations
For simplicity, we adopt a new coordinate variable,
Z=
z
D
(5.43)
Then the absorption and scattering coefficients become
k
= kp + ki Z
(5.44)
s = sp + si Z
(5.45)
where Z varies in a range between 0 and 1.
Let
Iup
Idn
=
=
∞
an Z n
(5.46)
bm Z m
(5.47)
n=0
∞
m=0
and insert the series solutions into the differential equations. One gets the
following algebraic equations,
∞
1 mbm Z m−1
D m=1
= −[k + s]
∞
1 nan Z n−1
D n=1
= −[k + s]
−
∞
bm Z
m
+s
∞
an Z n
(5.48)
bm Z m
(5.49)
m=0
∞
n=0
∞
n=0
m=0
an Z n + s
Reflectance of the linear ink-penetration (with backing)
There is a standard way to mathematically solve Eqs. (5.48) and (5.49). Comparing the terms of the same order (Z n , n = 0, 1, 2, ...) on both sides of the
equations, one finds,
a1
= D[(kp + sp )a0 − sp b0 ]
(5.50)
−b1
= D[(kp + sp )b0 − sp a0 ]
(5.51)
and for n ≥ 2
an
=
−bn
=
D
[(kp + sp )an−1 + (ki + si )an−2 − sp bn−1 − si bn−2 ] (5.52)
n
D
[(kp + sp )bn−1 + (ki + si )bn−2 − sp an−1 − si an−2 ] (5.53)
n
52
Extended Kubelka-Munk theory and applications
These relations reveal two facts. First, there are only two undetermined
coefficients, say a0 and b0 . All the other coefficients an or bn (n ≥ 1) are functions of them. Second, both coefficients, an and bn , are inversely proportional
to the order of n (actually, an , bn ∝ 1/n!). Therefore, they decrease monotonically with increase n. Thus, it provides us with the possibility of truncating
this series expansion up to a certain order of the expansion. Simulations to
various mixtures of media have confirmed these observations as one can see
in Fig. 5.8 and 5.9 in Section 5.5. More examples can be found in a previous
publication [YK00]. Because the variable, 0 ≤ Z ≤ 1, is independent of the
range of ink penetration these conclusions are generally meaningful.
Imposing the boundary condition at Z = 0 (Eq. (5.8)), we have,
b0 = Rg a0
(5.54)
Thus, there is only one undetermined coefficient remaining (say a0 ) which can
be determined by applying the boundary condition at Z = 1 interface. Inserting
Eqs. (5.46) and (5.47) into the boundary at Z = 1 (Eqs. (5.6) and (5.7)), one
has
∞
an
= I0 (1 − r0 ) + r1
n=1
I0 R
= I0 r0 + (1 − r1 )
∞
n=1
∞
bn
(5.55)
bn
(5.56)
n=1
The reflectance of the linear ink penetration system is then obtained,
R = r0 + (1 − r0 )(1 − r1 ) ∞
∞
m=0 bm
m=0 (am
− bm r1 )
(5.57)
If the series expansion is truncated in a order of M, the reflectance of penetrating ink layer can be expressed as,
M
R = r0 + (1 − r0 )(1 − r1 ) M
m=0 bm
m=0 (am
− bm r1 )
(5.58)
Simulations show that the computed reflectance, R, generally has good convergency (see Sec. 5.5 and Ref. [YK00]).
Without difficulty one can see, from Eqs. (5.50-5.54), that both an and
bn are proportional to a0 . It makes the fraction in Eq. (5.58) be actually
independent of a0 . Thus, one can compute the reflectance simply by setting a0
an arbitrary value (say, a0 =1).
5.4 Solutions of the differential equations
z
I0
53
I0R
r
0
D
r1
Idn
I
medium layer
up
r
2
0
r0
I0T
Figure 5.7: A schematic diagram of a linear ink distribution (freely suspended).
Transmittance of the linear ink-penetration (freely suspended)
When the linearly ink-penetrated paper is freely suspended as shown in Fig. 5.7.
The boundary condition at Z = 0, Eq. (5.54), should be replaced by
b0 = r2 a0
(5.59)
Even though the remaining expansion coefficients (an , bn , n ≥ 1) change upon
a0 and b0 , the expression of the reflectance remains unchanged as given in
Eq. (5.58). It is worth noticing that the internal boundary reflection at the
upper interface (r1 ) is probably different from that at the lower interface (r2 )
because of the non-uniform ink distribution.
The transmittance of the freely suspended layer, T , can be worked out by
applying the boundary condition at Z = 0,
I0 T = a0 (1 − r2 )
It results in
(5.60)
a0 (1 − r0 )(1 − r2 )
T = M
(5.61)
m=0 (am − bm r1 )
Although a0 exists explicitly in the expression, the transmittance, T , interestingly enough, is actually independent of a0 because am and bm in the denominator are proportional to a0 as we have previously noticed.
54
Extended Kubelka-Munk theory and applications
5.4.3
Exponential distribution
Substituting k and s in Eqs. (5.1) and (5.2) by their exponential expressions,
Eqs. (5.23) and (5.24), one obtains the differential equations for exponential
penetrating ink distribution. Strictly speaking, no analytical solution to these
equations is known.
Series expansions for s and k
Here we propose an approximate approach by expanding the exponential functional,
z
ρ = ρ0 eα( D −1) = ρ0 e−α eαZ
(5.62)
into a Taylor series to a maximum order, Nc . Accordingly, the absorption and
the scattering coefficients can be expressed as,
α2 2
αNc Nc
Z + ... +
Z )
2!
Nc !
α2 2
αNc Nc
Z + ... +
Z )
s = sp + si e−α (1 + αZ +
2!
Nc !
= kp + ki e−α (1 + αZ +
k
(5.63)
(5.64)
where 0 ≤ Z ≤ 1. Evidently, the accuracy of the expansion depends on the
quantity αNc /Nc !. As long as α Nc , the difference between the exponential
functional and its Taylor expansion becomes negligible.
Reflectance and transmittance of the exponentially ink-penetrated
paper
As in the case of a linear ink distribution, the light intensities, Iup and Idn ,
are here expanded into series as shown in Eqs. (5.46) and (5.47). Substituting
k and s, and Iup and Idn , in Eqs. (5.1) and (5.2) with their series expansions,
one can obtain the following equations for the expansion coefficients,
a1
= D(Y1 a0 − Y2 b0 )
(5.65)
b1
= D(Y2 a0 − Y1 b0 )
(5.66)
and (for n ≥ 1)
an+1
=
Ns
Ns
αm
αm
D
[Y1 an − Y2 bn + Y3 (
an−m ) − Y4 (
bn−m )]
n+1
m!
m!
m=1
m=1
(5.67)
bn+1
=
Ns
Ns
αm
αm
D
[Y2 an − Y1 bn + Y4 (
an−m ) − Y3 (
bn−m )]
n+1
m!
m!
m=1
m=1
(5.68)
5.5 Simulations for uniform- and linear-ink distribution
55
where
Y1
Y2
= kp + ki e−α + sp + si e−α
= sp + si e−α
(5.69)
(5.70)
Y3
Y4
= ki e−α + si e−α
= si e−α
(5.71)
(5.72)
and
Ns =
n,
Nc ,
if n ≤ Nc
if n ≥ Nc
(5.73)
From Eqs. (5.65-5.68), we observe two facts similar to the case of a linear ink
distribution. First, there exist only two undetermined coefficients (a0 and b0 ),
which depend on the boundary conditions at z = D (Z = 1) and z = 0 (Z = 0).
Second, both sets of coefficients, an and bn , decrease monotonically with n.
Therefore, one may expect quick convergence of the polynomial solutions as in
the case of the linear model.
In addition to these common observations about the expansion coefficients,
an and bn , the expressions for the reflectance and transmittance for the exponential ink distribution actually share the same forms as those of the linear ink
distribution (Eqs. (5.57) and (5.61)). It should be noted, however, that the
progressive relations between an and bn (Eqs. (5.50-5.53) and Eqs (5.65-5.68))
are different for the different types of ink distribution.
5.5
Simulations for uniform- and linear-ink distribution
Rather than focusing the discussions on any specific paper or ink, we will
study the effect of ink penetration (on the reflectance) in a general sense. The
discussions concern not only applications, but also the different approaches. In
addition, it is worth noticing that the quantities, k and s, and consequently R
and T , are wavelength dependent. For simplicity, but without losing generality,
we limit the simulations to a monochromatic band. k and s are therefore treated
as constants.
5.5.1
Convergency of the series expansion.
To test the convergence of the series solution for the case of linear ink penetration, calculations for various combinations of (kp ,sp ) and (ki ,si ) have been
carried out. As expected from the theoretical analysis in Sec. 5.4.2, the calculated reflectance values demonstrate rapid convergence with respect to the
order of the expansion (see Fig. 5.8). Moreover, the coefficients {an } and {bn }
56
Extended Kubelka-Munk theory and applications
0.8
Kp:Sp:Ki:Si=1:10:50:1
K :S :K :S =1:100:50:1
p p i i
K :S :K :S =1:100:50:40
0.7
p
p
i
i
Reflectance, R
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
18
20
Order of expansion, n
Figure 5.8: Variation of computed reflectance with respect to the order (n) of polynomial expansion (linear ink distribution, absorption power of the ink ki D = 0.5).
Variation of the expansion coefficients, a and b
n
6
n
K_p:S_p:K_i:S_i=1:10:50:1
K_p:S_p:K_i:S_i=1:100:50:1
K_p:S_p:K_i:S_i=1:100:50:40
an
4
2
0
0
5
10
15
20
25
10
15
20
25
4
bn
2
0
−2
0
5
Order of expansion, n
Figure 5.9: Variation of coefficients, an and bn , with respect to the order (n) of
polynomial expansion (ink distribution, absorption power of the ink ki D = 0.5).
5.5 Simulations for uniform- and linear-ink distribution
also show good convergency (to zero, see Fig. 5.9). For example, when the depth
of the ink penetration (D) corresponding to an absorption power ki D = 0.5, R
converges up to 4 digit accuracy if the series expansion is truncated at order
n = 10. For a smaller depth of ink penetration, D, the series truncation can be
sufficient even at lower order which makes it possible to express the solution
in the handy form of a polynomial. In other words, an analytic expression for
the reflection is possible for a thin ink layer. For deep ink penetration, (D is
large), however, the order of expansion increases rapidly and a computer has
to be used. Furthermore, Fig 5.9 shows another fact that the coefficients, an
and bn , converge slower when the scattering power of the ink-paper mixture
(sp + si ) becomes stronger.
5.5.2
Optical effects of ink penetration
To clarify understanding and to emphasize the importance of accounting for
the effect of the ink penetration, we have made comparative calculations in
two extreme cases, i.e., in one extreme, there exists no ink penetration at all,
in the second, the ink completely penetrates into the paper (no pure ink layer
being left over the substrate surface). In the calculations, the density of the ink
distribution of the ink layer on the paper is treated as uniform (or constant),
while the ink penetration is either uniform or linearly decreases with depth,
along the direction of ink penetration, as noted in the figures (Figs. 5.2 and
5.6).
In the calculations, the clean substrate paper (or part of the paper without
ink penetration) was considered as a background reflector with reflectance,
Rg = 0.85. Such reflectance requires the paper having the scattering power
about 100 (for the former) times stronger than its absorption while kp : sp =
1 : 40 corresponds to a backing reflectance, Rg = 0.8.
Results of simulations for ink penetration into these two types of substrates
are shown in Fig. 5.10, in the case of uniform ink penetration. The thickness of
the ink layer (in the case of no ink penetration) or the depth of ink penetration
(in case of having ink penetration) is expressed by a dimensionless quantity, absorption power of the ink, ki D. The following systematic behavior of computed
reflectance and transmittance are observed.
• the reflectance of the print with ink penetration is generally larger than
that without, while the transmittance is generally smaller;
• the deeper the ink penetration, the bigger the differences between the
calculated values;
• the difference gradually approaches a constant when the ink thickness
further increases.
57
58
Extended Kubelka-Munk theory and applications
• the transmittance, T, corresponding to the prints with and without ink
penetration are identical, at D = 0 and ∞, but different otherwise.
Reflectance
Transmittance
1
0.8
1
a) k :s :k :s =1:100:25:1
p
p
i
i
0.8
0.6
0.6
complete ink pene.
0.2
0.4
no ink pene.
0.5
1
1.5
2
2.5
1
Transmittance, T
Reflectance, R
0.4
0
0
0
0
p
0.6
0.6
0.4
0.4
0.2
0.2
1.5
0.5
1
1.5
2
2.5
2
2.5
d) k :s :k :s =1:40:25:1
0.8
1
no ink pene.
1
0.8
0.5
complete ink pene.
0.2
b) kp:sp:ki:si=1:40:25:1
0
0
c) kp:sp:ki:si=1:100:25:1
2
2.5
0
0
p
0.5
i
i
1
1.5
Absorption power of ink, k D
i
Figure 5.10: Comparisons of computed reflectance and transmittance values with
(dash lines) and without (solid lines) ink penetration (uniform ink penetration model)
for paper-ink mixtures of different optical parameters, kp , sp , ki , si , as noted in the
figure. The scattering strengths of the substrate relative to its absorption are different
between the upper and lower panels.
Closer inspection of Fig. 5.10 further reveals clear correlation between the
optical parameters kp , sp , ki and si and its optical performance. Comparing
panel a with b in Fig. 5.10, one finds that the scattering power of the paper
plays a crucial role for light reflection, i.e., the stronger the scattering power of
the paper, the more profound the optical effect of the ink penetration, which
is in line with experimental observations (Arney and Alber,1998). Similar
observation also holds for the transmittance as shown in panels c and d.
Results shown in Fig. 5.11 reveal that the strong absorption power from the
ink reduces the effect of ink penetration on the reflectance and the transmittance.
It is natural to make a comparison between different models of ink penetration and to investigate to what extent the optical characteristics of the layer
depend on the form of the ink distribution inside the paper (ink penetration).
However, such a comparison between the uniform model and the linear model
is not a trivial task, because they are actually not directly comparable. To
5.5 Simulations for uniform- and linear-ink distribution
Effect of absorption power of ink
1
1
0.8
a) k :s :k :s =1:100:50:1
p p i i
0.8
0.6
Reflectance, R
0.4
complete ink pene.
0.2
0
0
no ink pene.
0.5
1
1.5
2
1
0.8
0.2
0
0
1.5
2
1.5
2
0.8
0.4
0.4
0.2
0.2
1
1
1
0.6
0.5
0.5
d) k :s :k :s =1:100:5:1
p p i i
b) k :s :k :s =1:100:5:1
p p i i
0.6
0
0
c) kp:sp:ki:si=1:100:50:1
0.6
Transmittance, T
0.4
59
1.5
2
0
0
0.5
1
Absorption power of ink, kiD
Figure 5.11: Comparisons for reflectance and transmittance values computed with
complete ink penetration (uniform ink penetration model) and that without. The
absorption strengths of the ink are different between the upper and lower panels.
a certain depth of ink penetration, for example, in the case of complete ink
penetration (no ink layer being left on the surface of the paper), the linear
model contains only half the amount of ink as that of the uniform model. In
other words, the same amount of ink in both models requires the ink in the
linear model to reach twice the depth of the uniform model. Comparisons by
considering the same penetrating depth (Fig. 5.12a, b) and the same amount
of the penetrating ink (Fig 5.12c, d) have been carried out. These are shown
in Fig. 5.12.
Summarizing the discussions, a dilemma exists in trying to limit the effects of ink penetration. A natural choice for getting rid of ink penetration
is to improve the surface properties of the substrate paper and the rheologic
properties of the ink as well. In the paper making industry, post processes
called surface modification can greatly reduce ink penetration. However, such
processes have significantly increased the price of the paper which has largely
limited its popularity in producing ordinary print, such as newspaper. Although applying the paper with weaker scattering power may reduce the effect
of the ink penetration, it reduces the opacity of the paper at the same time and
is therefore unacceptable. Nevertheless, the use of ink having a higher absorption can reduce the effect of ink penetration (Fig. 5.11). It requires invention
of new types of colorants (dye or pigment). Hopefully, a better understanding
60
Extended Kubelka-Munk theory and applications
Reflectance, R
a)
0.8
Reflectance, R
k :s :k :s =1:100:25:1
p p i i
1
c)
0.8
1
0.6
Linear ink pene.
0.4
Uniform ink pene.
0.2
0
0
0.5
1
1.5
0.6
0.4
0.2
0
2
1
0
1.5
2
d)
0.8
Transmittance, T
Transmittance, T
1
1
b)
0.6
0.4
0.2
0
0.5
0
0.5
1
1.5
Depth of ink penetration,D
2
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
Absorption power of ink, kiD
Figure 5.12: Dependence of the reflectance on the distribution of ink penetration
(uniform model vs. linear model). a) and b) both models reach the same depth of
ink penetration; c) and d) both models have the same amount of ink.
of ink penetration will be helpful in finding the solution.
5.5.3
Correction for boundary reflection
Reflectance and transmittance computed with and without including the contribution of boundary reflection, are shown in Fig. 5.13. In the computation,
the indexes of refraction are assumed as n0 = 1 and n1 = 1.5 for the air and
paper-ink mixture, respectively. This means that the external and internal
boundary reflection is, r0 = 0.0919 and r1 = 0.5960, respectively, for diffuse
illumination.
The computed reflectance with consideration of the boundary reflection is
smaller than that without, and the difference is generally independent of the
thickness of the ink layer. This agrees with intuition, the internal reflection prevents a significant amount of light from passing through the ink/air interface.
On the other hand, the transmittance with the boundary reflection is smaller
than that without. Nevertheless, their difference gradually vanishes when the
thickness increases because the layer becomes less transparent in both cases.
Boundary reflection may have little influence on dye based ink jet printing,
because the ink layer consists of loosely piled up dye molecules. The refractive
index of the ink layer is close to unity. However, in the case of ink penetration,
5.5 Simulations for uniform- and linear-ink distribution
61
the refractive index of the ink-paper mixture may be remarkably bigger that
that of air. Therefore, boundary reflection must be taken into consideration.
Effect of surface reflection correction
Reflectance, R
1
0.8
with surface ref.
without surface ref.
a) k :s :k :s =1:100:25:1
p p i i
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
Transmittance, T
1
with surface ref.
without surface ref.
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
Absorption power of ink, kiD
2
2.5
Figure 5.13: Comparisons for reflectance and transmittance values computed with
and without boundary reflection.
5.5.4
Effect on color gamut
Before concluding the discussion, we will demonstrate how ink penetration
affects the color rendition of the printed image. According to definition [WS86],
the X-stimulus can be expressed as
X=
R(λ)S(λ)x(λ)dλ
(5.74)
where R(λ) is the spectral reflectance of the illuminated and viewed object,
S(λ) the spectral concentration of the radiant power of the source illuminating
the object, and x(λ) the tristimulus function. If we denote the reflectance for
the bare paper, as R0 (λ), and Rnp (λ), for the print with no ink penetration at
all, and Rcp (λ) for the case with complete ink penetration, their X-stimulus
62
Extended Kubelka-Munk theory and applications
values, are, respectively.
X0
=
Xnp
=
R0 (λ)S(λ)x(λ)dλ
(5.75)
Rnp (λ)S(λ)x(λ)dλ
(5.76)
Rcp (λ)S(λ)x(λ)dλ
(5.77)
Xcp
=
From the above discussion (Fig. 5.11), one may obtain the estimation:
Xnp ≤ Xcp , or (X0 − Xnp ) ≥ (X0 − Xcp ). In the case of printing, the differences from paper white to full tone, (X0 − Xnp ) and (X0 − Xcp ), represent
ranges of color reproduction for prints without and with ink penetration, respectively. Therefore, the latter inequality means that the ink penetration
reduces the color gamut of color reproduction. This theoretical conclusion is
in line with experimental observation.
5.6
Summary
In this chapter we present models and solutions for three types of ink penetration, uniform, linear and exponential, that may correspond to different
ink-paper combinations. In addition to the uniform model whose differential
equations can be solved analytically, series solutions corresponding to the linear
and exponential models have been worked out. Generally, rapid convergence
of the series expansions have been found from simulations. Theoretically, for
a certain amount of printed ink commanded by a printer, the printed image
with ink penetration has greater reflectance than that without. In other words,
the image becomes less saturated in color because of ink penetration. Consequently, the capacity for color reproduction (color gamut) is reduced due to
ink penetration.
The intention of the studies was to highlight the method for finding solutions rather than providing universal answers to complicated ink penetration
as a whole. Additionally, one model may be appropriate to some ink-substrate
combinations but not others. For example, for dye based liquid inks, uniform
ink penetration may be a proper approximation when the pore size of the substrate is much bigger than that of the micelles of the dye [MK00] (about 100Å).
It is therefore important analyzing the characteristics of the ink-substrate combination before applying the models.
Chapter 6
Characterization of inks
and ink application
6.1
Introduction
Ink-jet is a non-impact dot-matrix printing technique in which droplets of ink
are ejected from a small aperture directly to a specified position on a substrate
to create an image [Le98]. An ink jet printing system may be divided into three
critical components, the printer, the inks, and the substrate [vBKZT01]. Briefly
speaking, the printer acts as a distributer for the ink droplets and the substrate
acts as a receiver. It is the ink droplet that results in the observed color by
selectively reflecting (or more precisely, selectively absorbing and scattering)
the illuminating light. These three components together control the qualities
of the printed image. It is important to bear in mind that these components
are strongly correlated with each other.
Studies on substrate related issues, such as optical dot gain (or Yule-Nielsen
effect) and ink penetration, have recently been reported [Hüb97, Rog98a, AA98,
YLK01, YK01, YKP01, EH00]. The printer- and ink-related issues include,
among others, location and volume of the ink droplets, the scheme of inkmixing for generating the secondary colors, and of course, the optical properties
of the inks (scattering and absorption coefficients). Some of the printer- and
the ink-related issues can strongly correlate with the substrate related issues.
For example, the depth of ink penetration is largely dependent on the ink volume. In addition, the color rendition also depends significantly on the optical
properties of the inks when there exists ink penetration. Therefore, characterizations of the output print in terms of ink distribution and volume, the scheme
of ink-mixing, light absorption and light scattering are of essential importance
in controlling and understanding the quality of the color reproduction.
64
Characterization of inks and ink application
In this chapter, we present a method to quantitatively characterize the inkand printer-related issues. The goal of the study is to determine the following
quantities: the absorption and scattering properties, the thickness of printed
ink-layers when different ink-level specification in printing process is used or to
quantify the ink-level specification of the printer; and, the color mixing scheme
for generating secondary colors. This knowledge is important for future studies
when there is ink penetration (print on ordinary paper) because one needs prior
knowledge about the ink (in terms of its absorption and scattering properties)
and how much ink has been printed. For example, different amounts of printing
ink is possibly related to the different depths of ink penetration. Additionally,
this information can probably help to fingerprint the printer which may find
applications in identifying ink-jet counterfeiting. This is an new topic of research that is becoming more and more important in light of the continuous
improvement in image quality of ink jet printing [Har02, Wol02, JC02].
6.2
Experiment, data analysis and simulation
The strategy of the study is first to determine the scattering and absorption
powers (sz, kz, i.e. products of the scattering and absorption coefficients, s
and k, with the ink thickness, z) of a primary ink-layer of a certain thickness.
These values are then used to predict the spectral reflectance of an ink-layer
of any thickness. By varying ink-volume specification in the printer driving
software (commonly available from printer manufacturers), one can print the
ink-layer in different thicknesses. Finally, these coefficients are used to predict
spectral reflectance of secondary colors. In this way one can test the quality
(or reliability) of the values and the applicability of the model, as well.
6.2.1
Samples and measurements
The full tone samples of primary (cyan, magenta, and yellow) and secondary
colors (red, green, and blue) were printed, with the secondary colors being
mixtures of the primary colors. In order to prevent the inks from penetrating
into the substrate, as in ordinary office copy-paper which significantly modifies
the color, ink-jet transparency (3M CG3460) were used as substrates. Therefore, the sample consists of a macroscopically uniform ink-layer and a plastic
substrate. By varying ink-level specification in the printer driving software,
one can adjust the parameter relating to the ink volume (ink-level) and print
samples with up to 5 ink-levels (the ink-volume increases from ink-level 1 to
5).
Measurements were carried out by applying a spectrometer of the 45o /0o
geometry, with the collimated illumination from the top of the sample, as
described in Sec. 2.2.2. The measured spectra covers a spectral range of 380
6.2 Experiment, data analysis and simulation
65
to 730 nm at a step of 10 nm. To achieve high reflection from the samples,
a white and opaque background (a stack of white paper(s)) was placed under
the samples.
6.2.2
Data analysis and simulation
According to the radiative transfer theory [Cha60] and Kubelka-Munk approximation, the spectral reflectance value (Rjq ) of an ink-layer is a function of
its scattering and absorption powers, sq (λ)zjq and kq (λ)zjq , being products of
the scattering (sq ) and absorption (kq ) coefficients (m−1 ) and thickness of the
ink-layer, zjq (m), i.e.
Rjq (λ, zjq ) = f (sq (λ)zjq , kq (λ)zjq )
(6.1)
Here the subscript q = c, m, y represent cyan, magenta, and yellow, and j =
1 − 5, denotes the ink-level. According to the expression for reflectance of
uniform media (Eq. (5.34)), the function in Eq. (6.1) may be expressed as
f (sz, kz)
=
s(A − sRg )e−
s2 −A2
A
2
2
− s −A
A
A(A − sRg )e
−
=
z
sz(Az − szRg )e
− A(s − ARg )
z
− s(s − ARg )
(sz)2 −(Az)2
Az
(sz)2 −(Az)2
−
Az
Az(Az − szRg )e
− Az(sz − AzRg )
− sz(sz − AzRg )
where Rg (λ) is the spectral reflectance of the bare substrate and
A = s + k − k 2 + 2ks
(6.2)
(6.3)
Therefore, by fitting to two sets of measured spectral reflectance values, one can
obtain the scattering power of the ink-layer (sz) and Az. From Eq. (6.3) one
can in turn obtain absorption-power of the ink-layer (kz). In the present study,
one set of data (RI ) was obtained from samples that were printed with primary
inks at ink-level 3 (defined by the printer driving program), and another set
of data (RII ) was from twice printed samples with the same ink-level. For the
latter case the second round of printing was carried out 10 minutes later. The
sample was measured 24 hours after printing. Assuming that the ink thickness
of the latter is twice that of the former, we then find (see Appendix A for
detailed derivation)
(sz − AzRI )2 (Az − szRg )
(sz − AzRII )
=
(Az − szRII )
(sz − AzRg )(Az − szRI )2
(6.4)
Combining Eq. (6.4) with the following equation
RI =
sz(Az − szRg )e−
Az(Az − szRg
(sz)2 −(Az)2
Az
)e−
(sz)2 −(Az)2
Az
− Az(sz − AzRg )
− sz(sz − AzRg )
(6.5)
66
Characterization of inks and ink application
one can determine the scattering and absorption powers of the ink-layer, sz and
kz (through Eq. (6.3)). Equations (6.4) and (6.5) may have more than one pair
of solutions. Therefore, an extra constraint has been added to the solutions to
minimize the difference between the simulated- and experimental-values,
{[RI − f (sz, kz)]2 + [RII − f (2kz, 2sz)]2 }
(6.6)
∆=
λ
When the scattering and absorption-power of the primary ink-layer of inklevel 3, sq (λ)z3q and kq (λ)z3q , are known, these values can then be utilized
to predict reflectance values of an ink-layer of any given ink thickness, αjq =
zjq /z3q , by applying
Rjq (λ, αjq z3q ) = f (αjq sq (λ)z3q , αjq kq (λ)z3q )
(6.7)
where αjq (q = c, m, y, and j = 1 − 5) is the relative ink thickness of the inklayer. On the other hand, Eq. (6.7) can be applied inversely, i.e. for known
spectral reflectance values, Rjq , ink thickness of the ink-layer, αjq , can be
estimated by fitting Eq. (6.7) to the Rjq . In this way, we obtained the relative
ink thicknesses of the samples printed with ink-levels, 1, 2, 4, and 5. Because
the spectral reflectance values of each sample consist of reflectance values at
31 wavelengths (400 − 700nm), the agreement between the computed spectral
reflectance values and the measured ones can serve as a quantitative test of the
quality of the scattering and absorption values obtained from ink-level 3. The
agreement may also serve as a test of the applicability of the present method.
Finally, the thickness of the ink-layer is proportional to the printed ink volume.
Therefore, one can actually characterize the ink application controlled by the
printing engine.
After obtaining the scattering and the absorption powers of the primary
inks, one can compute the spectral reflectance values of the secondary colors by
applying the additivity assumption [Pau87]. For example, color red is composed
of ink magenta and ink yellow. Its scattering and absorption powers, according
to additivity, can be expressed as
sr zjr
kr zjr
r
r
= βjm
sm z3m + βjy
sy z3y
r
r
= βjm km z3m + βjy ky z3y
(6.8)
(6.9)
where sq z3q and kq z3q (q = m, y, and j = 1 − 5) are the scattering and absorption powers that were obtained from the primary inks (ink-level 3). Therefore,
by substituting sr zjr and kr zjr into Eqs. (6.2), one can compute the spectral
reflectance values of red, Rjr . Inversely, the contribution from the primary inks
t
(t = r, g, b denote the secondary color and q = c, m, y the primary colors)
βjq
can be determined by fitting to the measured spectral reflectance values of the
r
r
and βjy
, are defined in
secondary color (red). Observe that the quantities, βjm
6.3 Results and discussions
67
exactly the same way as for αjm and αjy in Eq. (6.7). Therefore, they are the
primary ink amounts (relative to z3m and z3y , respectively) that are needed for
generating the secondary color, red.
6.3
Results and discussions
In this section we present experimental measurements together with our theoretical analysis and simulations, for a dye-based ink-jet printing system. As
mentioned above, the spectral absorption and scattering powers (kz and sz) of
the primary inks were obtained by fitting to experimental spectral reflectance
values of samples printed with ink-level 3, according to the specification of the
printer driving software.
Absorption power, kz
Ink layer printed in ink level 3
3
2
1
0
400
Scattering power, sz (10−3)
cyan
magenta
yellow
a)
450
500
450
500
550
600
650
700
550
600
650
700
b)
6.2
6
5.8
400
Wavelength, λ(nm)
Figure 6.1: Scattering and absorption powers of primary inks obtained by fitting to
the measured spectral reflectance values of samples printed in ink-level 3 (specified
by printer driving program).
6.3.1
Spectral characteristics of the primary inks
The scattering and absorption powers of the primary colors (ink-level 3) are
shown in Fig. 6.1. As expected, the inks cyan, magenta and yellow show strong
absorption in the long-, middle-, and short-wavelength regions, respectively.
68
Characterization of inks and ink application
In other words, cyan has a transparent window in the short to middle wavelengths, yellow has its transparency window in the middle to long wavelengths,
and magenta has two windows of short and long wavelengths, respectively.
Nevertheless, the scattering power of the inks is rather weak and has values
of about 0.006. This means that for dye-based ink-jets, the scattering power
of the printed ink-layer is practically negligible, which is favorable for creating
color of high saturation.
1
Simulation
0.9
Measurement
yellow
0.8
magenta
0.7
Reflectance,R
cyan
Ink level 1
0.6
0.5
0.4
Ink level 5
0.3
0.2
0.1
0
400
450
500
550
600
650
700
Wavelengths, λ(nm)
Figure 6.2: Simulated (solid lines) and measured (dots) spectral reflectance values
of samples printed with different ink-levels specified by the printer driving program.
6.3.2
Spectral reflectance values and relative ink thicknesses of the primary inks
The scattering and absorption powers (sq z3q and kq z3q ) were obtained by fitting
the computed spectral reflectance values of samples printed with ink-level 3
(correspondingly, ink thickness z3q , q = c, m, y), to corresponding experimental
values. It is worthwhile to quantitatively test their reliability. The test was
made by employing the scattering and absorption powers to compute spectral
reflectance values of samples printed with the other four ink-levels (ink-level 1,
2, 4, and 5) and to compare their measured values. The different printing ink
volumes (ink-levels) result in ink-layers of different thicknesses on the substrate.
It may be proper to express the scattering and absorption powers of the samples
6.3 Results and discussions
69
as αjq sq (λ)z3q and αjq kq (λ)z3q , where the quantity, αjq , represents the relative
ink thickness of the sample.
Table 6.1: Quality evaluation for the simulated spectra in terms of color difference
(∆E) compared to experimental values.
ink-level
1
2
3
4
5
Color
Cyan
0.3093
0.3647
0.2753
0.8851
1.5858
Difference, ∆E
Magenta Yellow
2.1339
1.6610
1.4876
2.0168
0.6321
1.5422
3.1225
1.2522
3.5155
0.5329
4.5
Relative ink thickness, α
4
cyan
magenta
yellow
3.5
3
2.5
2
1.5
1
1
2
3
4
5
Ink level printer , j
Figure 6.3: Actual ink volumes αjq = zjq /z1q -vs-printer driving program specified
ink volumes (q = c, m, y, and j = 1 − 5). The actual ink volume for the program
specified ink-level 1 (z1q ) has been set to unity for each color.
The simulated spectral reflectance values (solid lines) together with the
measured ones (dots), of all the 5 (primary ink) levels, have been shown in
Fig. 6.2. For a simpler comparison between the simulation and the measurements, both the simulated and experimental spectra have been converted to
their color coordinates in CIELAB color space. Their differences in terms of
color appearance (∆E) are listed in Tab. 6.1. From Fig. 6.2, as well as Tab. 6.1,
70
Characterization of inks and ink application
one may conclude that the simulation is in fairly good agreement with experiment over the whole range of visible light (31 wavelength-sampling points over
this range in the measurements). This may be considered as a confirmation of
the validity of the method and the reliability of the sq (λ), kq (λ) values. It is
worth noting that the 5-ink-level specification in the printer driving program
does not always mean 5 different printing ink-levels. For ink cyan there are
indeed 5 different ink-levels, but in practice there are only 3 and 4 different
ink-levels for yellow and magenta, respectively. The correlation between the
actual ink volumes (αjq ) and the (printer driving program) specified ink-levels
(j = 1 − 5) is shown in Fig. 6.3. As shown, the practical ink volumes vary nonlinearly with respect to the ink-level specification (for simplicity, the actual
volume of ink-level 1, of each color, has been set to unity).
Composition of the 2nd colors from the primary inks
4
cyan
magenta
Relative ink amount, β
3.5
yellow
Red
3
2.5
Blue
Green
2
1.5
1
0.5
1
2
3
Printing ink levels, j
4
5
Figure 6.4: Ink composition of secondary colors of different printing ink-levels (j=15). The arrows indicate the primary components (see the legend) of the secondary
colors. The actual ink volume of the primary components has been normalized to z1q
(q = c, m, y), as defined for the primary colors (Fig. 6.3).
6.3.3
Spectral reflectance values and relative ink thickness of secondary colors
Because samples of secondary colors, red, green and blue, are obtained by
mixing two of the three primary colors in the printing process, they may serve as
excellent examples for testing the additivity of the scattering and/or absorption
powers as stated in Eqs. (6.8) and (6.9).
6.3 Results and discussions
71
Absorption power, kz
3.0
2.0
1.0
0
400
Scattering power, sz (10−3)
Red
Green
Blue
a)
450
500
450
500
550
600
650
700
550
600
650
700
8.7
b)
8.6
8.5
8.4
400
Wavelength, λ(nm)
Figure 6.5: Scattering and absorption powers of the secondary colors (ink-level 3)
obtained by applying the additivity assumption.
The relative amounts of the primary inks used to obtain the secondary
t
(j = 1 − 5, t = r, g, b and q = c, m, y), have been determined and
colors, βjq
shown in Fig. 6.4. It shows that the relative amounts of the primary inks
strongly depend on the colors. The color red, for example, is formed by nearly
equal mixing of magenta with yellow. The color blue, however, is predominated
by cyan relative to magenta, and the color green is something in between. In
addition, the relative amounts of the primary inks vary modestly from one
ink-level to another, which results in similar hue but different color saturation.
Furthermore, the amounts of the primary inks used to generate the secondary
colors are not a simple superposition of the amounts of the primary samples
printed. Color blue of ink-level 4, for example, consists of cyan (3.8) and
magenta (0.8). A simple superposition, would involve the mixture, cyan (2.8)+
magenta (3.2) (see Fig. 6.3). This is an example of the flexibility in the inkjet printing technique compared to traditional offset printing. This flexibility
provides the printer manufacturer the possibility to achieve an optimum color
tone, while at the same time avoiding too much ink being printed, which in
turn avoids a serious print through (on plain paper) and a prolonged drying
process, etc. Finally, similar to the observation for primary colors, the ink
amounts vary nonlinearly with respect to the printer specified ink-levels.
The scattering and absorption powers of the secondary colors (ink-level
3) are shown in Fig. 6.5. Plots in Fig. 6.5 clearly show the existence of absorption/window structures that match well with intuition. Referring to the
absorption/scattering characteristics of the primary colors (Fig. 6.1), one can
72
Characterization of inks and ink application
easily see correlations between these absorption/window structures of the secondary colors and those of their primary components, for example color red,
the strong absorbing band (400 − 600 nm) consists of two sub-absorbing bands
from magenta (480 − 600 nm) and yellow (400 − 480 nm), respectively. Naturally, the scattering power of each ink-layer is very weak and varies little with
respect to wavelength.
1
0.9
Simulation
Red
Measurement
0.8
Reflectance, R
0.7
0.6
Green
Blue
0.5
0.4
0.3
0.2
0.1
0
400
450
500
550
600
650
700
Wavelength, λ(nm)
Figure 6.6: Simulated and measured spectral reflectance values of samples in secondary colors. The samples were printed with program specified ink-level 3.
The simulated spectral reflectance values together with the corresponding
experiment values of ink-level 3 are shown in Fig. 6.6. As can be seen, the
simulations agree fairly well with experimental data. This may imply that the
additivity assumption for the absorption and scattering power holds quit well.
6.4
Remarks for application of Kubelka-Munk
theory
Kubelka-Munk (K-M) theory has been the most widely applied theory for the
colorist in research and in industry since its introduction in the 1930’s. Over
the years, the K-M theory has been subjected to very close scrutiny. The result
is an appreciation of the limitations and strengths of the theory [Nob85]. Of
all the original assumptions made by Kubelka and Munk, that of uniformly
diffuse forward and reverse flux through the sample is the most possible source
of imprecision, when it is applied to such a system as an ink-layer with strong
6.5 Summary
absorption. The reason behind this is that rays that propagate in different
directions will be attenuated differently. For example, oblique rays will be
attenuated more than vertical rays, which invalidates the assumption that we
have diffuse light everywhere in the medium. Despite such a limitation of the
theory, it remains the most widely applied theory in paper-making and printing
industries, because of its simplicity and above all because it works well in the
majority of cases. It is therefore worthwhile to closer examine the consequences
when the conditions of the theory are not well satisfied.
As shown in Fig. 6.1, the light-ink interaction (or the ink spectrum) shows
clear transparent- and absorption-band structures. For light whose wavelength
lies well away from the absorption band, the absorption is less important. Correspondingly, the light distribution for this band will probably remain diffuse if
the original illumination is diffuse. Therefore, this portion of light can be properly described by the K-M theory applied to human color vision. On the other
hand, the portion of the illumination whose wavelength lies in the absorption
band will mostly be filtered out by ink absorption after passing through an
optically thick ink-layer, even though the light may not completely satisfy the
conditions of K-M theory. Therefore, as far as color reproduction is concerned,
K-M theory can be a simple and reasonably accurate approach.
6.5
Summary
We have developed a method of characterizing the printed ink volume and the
properties of the inks by means of spectral reflectance measurements. The
measured data were analyzed with the help of theoretical simulations. The
printed ink volume (equivalently, thickness of the printed ink-layer) of the
primary colors and their absorption and scattering characteristics were determined. Spectrally, the inks show clearly absorption- and transparency-band
structures with respect to the wavelengths of the illumination. The scheme of
color composition for the generation of secondary colors (from the primary inks)
was determined. In addition, the additivity assumption for obtaining the scattering and the absorption power of secondary colors from their primary color
components has been tested and seems to hold well (with regard to computed
spectral reflectance values of the secondary color). Simulations of the spectral
reflectance values have been carried out for both primary and secondary colors.
The simulations are in fairly good agreement with the measurements.
73
Chapter 7
Characterization of ink
penetration
When ink is printed on ordinary (plain) office copy-paper, it penetrates into
the substrate, while at the same time it spreads on the paper surface. The
ink penetrated paper is an ink-paper mixture which appears significantly different from either constituent, in particular the pure ink (with paper backing).
This arises because of presence of the paper materials (fillers, fibres, etc) that
generally have strong scattering power. Ink penetration modifies the ink performance significantly and results in serious undesirable consequences for the
printed color.
In Chapter 6, we determined the optical properties of the inks (sq zjq and
t
, for primary
kq zjq ) and the ink volume controlled by the printer (αjq and βjq
and secondary colors). The sub and super scripts, q = c, m, y and t = r, g, b
denote the primary and secondary colors, respectively, and j = 1−5 refers to the
ink levels or equivalently the ink volumes specified by the printer driver. This
knowledge forms one fundamental as past study in ink penetration . Another
aspect concerns from the optical properties of the bare paper.
7.1
Optical properties of plain paper
The office copy-paper that is widely used in offices normally contains brightening materials which absorbs illuminated UV light and re-emits it as fluorescence
whose wavelength lies in the blue light region. To avoid complexities in the determination of scattering and absorption properties of paper, a UV filter was
used in the measurements in order to eliminate the UV light from the illumination.
To determine the scattering and absorption powers of the paper, sp D and
76
Characterization of ink penetration
A sheet of plain paper with white and black backing
1
Reflectance, R
0.9
0.8
0.7
0.6
0.5
white backing
black backing
0.4
400
450
500
550
600
650
700
Wavelengths, λ(nm)
Figure 7.1: Spectral reflectance of a sheet of plain paper with white (a stack of white
paper) and black backings. UV filter was used in the measurement.
kp D, two sets of spectral reflectance values are needed. Here sp and kp are
the scattering and absorption coefficients of the paper, and D, the thickness
of a single paper sheet. These were obtained by measuring a single sheet of
the plain paper for two types of backings, black and white. We denote the
spectral reflectance values of the white and black backings as, Rgw and Rgk ,
respectively. Correspondingly, the reflectance values of a sheet of paper having
white and black backings are, Rw and Rk , respectively (see Fig. 7.1). According
to Eq. (5.34), one has
−
Rw
= r0 +
(1 − r0 )(1 − r1 )[sp (Ap − sp Rgw )e
−
(Ap − sp r1 )(Ap − sp Rgw )e
2
s2
p −Ap
D
Ap
−
Rk
= r0 +
(1 − r0 )(1 − r1 )[sp (Ap − sp Rgk )e
s2 −A2
− pAp p
(Ap − sp r1 )(Ap − sp Rgk )e
where
Ap = sp + kp −
D
2
s2
p −Ap
Ap
D
− Ap (sp − Ap Rgw )]
− (sp − Ap r1 )(sp − Ap Rgw )
(7.1)
2
s2
p −Ap
Ap
D
− Ap (sp − Ap Rgk )]
− (sp − Ap r1 )(sp − Ap Rgk )
(7.2)
kp2 + 2kp sp
(7.3)
7.1 Optical properties of plain paper
77
Scattering power, spD
p
Absorption power, k D
Solving these equations numerically, one gets sp D and Ap D if the boundary
reflection values, r0 and r1 are known. From Eq. (7.3), one can then obtain
the scattering and absorption powers, sp D and kp D. It has been commonly
assumed that the refractive index of plain paper, n, is not much larger than
unity [Mou02a]. However, simulations of clean paper and ink penetrated paper
reveal the existence of boundary reflectance at the air/paper interface. Additionally, the simulations show that, n = 1.2, fits the data best. This should
not be so surprising since the paper contains fibres and fillers, which may contribute to a greater dielectric constant and consequently, a greater refractive
index compared to the air.
0.4
a)
0.3
0.2
0.1
0
400
450
500
550
600
650
700
450
500
550
600
650
700
10
b)
8
6
4
2
400
Wavelengths, λ (nm)
Figure 7.2: Scattering and absorption powers of a sheet of office copy-paper.
The scattering and absorption powers of a single sheet of plain paper is
shown in Fig. 7.2. The paper has little absorption in green and red parts of the
spectrum, but absorbs blue light somewhat. Nevertheless, the absorption is still
much weaker than that of inks as found in Chapter 6. On the other hand, the
paper has much a stronger scattering power than do the inks. The scattering
power of the paper is nearly constant in the green to red light regions. It gradually increases from the green to the blue light regions, reaching its maximum
at about λ = 430nm, and then decreases sharply for shorter wavelength. This
observation generally coincides with the fluorescence distribution observed by
bi-spectral measurements [Mou02b] and may indicate the existence of residual
fluorescence which survives from the UV filtering. A combination of increasing
absorption with decreasing scattering at the short blue region makes far weaker
reflectivity in this wavelength region, if no brightening materials are added.
78
Characterization of ink penetration
The possible failure in completely removing the fluorescence from the measured spectral reflectance will cause errors in the determination of the scattering
and absorption powers of the paper. Consequently, these errors will result in
errors in the simulated spectral reflectance of the ink-paper mixture. Indeed,
there exists a remarkable discrepancy between the simulations and the measurements of cyan and blue, as will be seen later on.
7.2
Assumptions and notations
Ink penetration into the substrate paper forms an ink-paper mixture. For
simplicity and clarity of the description, assumptions and notations used in
this chapter have been summarized or defined as fellow.
7.2.1
Assumptions
1. Paper making materials, fibres, fillers, etc., are uniformly distributed in
the paper;
2. The ink concentration in the paper is uniform and independent of depth
of ink-penetration;
3. No pure ink is left on the paper surface and consequently the depth of
ink penetration is proportional to the amount of the printed ink;
4. Light becomes completely diffused once it enters the paper or ink-paper
mixture;
5. The scattering and absorption powers of the ink-paper mixture fulfills
the additivity assumption (expressed by Eqs. (5.15, 5.16) and Eqs. (7.4,
7.5)).
Introduction of the first two assumptions is for simplicity of the study in
a first attempt to simulate ink penetration. With the help of mathematical
treatments for non-uniform ink penetration, developed in Chapter 5, these
model can readily be extended to non-uniform cases, if needed. Additionally,
the second assumption may be a reasonable approximation for a combination
consisting of dye-based liquid inks and plain paper. Because the size of the
cellulose pores in the paper is generally much larger than that of the dye microcell (100Å), ink can easily penetrate into the substrate, which results in little
gradient in ink concentration. On the other hand, the inks are completely
absorbed by the substrate which forms the grounds for the third assumption.
Physical considerations behind assumption No. 4 are that the paper materials have a very strong scattering power. The strong scattering power of the
paper materials lead to the ink-paper mixture having a very strong scattering
7.2 Assumptions and notations
79
power, at least in the transparent bands of the ink. Therefore, the light becomes diffuse in the ink-paper mixture. Such a consideration is of fundamental
importance in the current study, when the additivity assumption (assumption
No. 5) is applied.
As explained in Sec. 5.2, quantity k is a phenomenological description of
the light absorption in the medium, which not only depends on the physical
properties of the medium but also on the light distribution in the medium.
As shown in Sec 6.2.1, the instrument used for the measurement of the inklayer was of 45o /0o geometry with collimated illumination from the top of the
sample (see Sec. 2.2.2 for detailed description). Considering that the pure ink
has little scattering power (see Chapter 6), light propagates essentially through
a straight path in the ink-layer. Correspondingly, the absorption power, kq zq ,
is responsible for the light extinction along that path. However, in the inkpaper mixture, the light (in both transparent and absorption bands) becomes
scattered and propagates in a zigzag fashion. Consequently, light has greater
possibility of being absorbed, if it passes the same vertical depth in the inkpaper mixture as that in the ink-layer. Therefore, the averaged absorption
power of the ink-paper mixture becomes 2kq zq , if the light becomes completely
diffused in the layer. It was experimentally observed that the absorption (k)
of colored paper is approximately twice that of the cellophane compared at a
given amount of dye [BS76].
The additivity assumption stated in assumption No. 5 has been confirmed
valid in color mixing (see Sec.6.3.3). As one will see late, it holds even for the
ink-paper mixture, if a factor of 2 is introduced into the absorption coefficient
of the ink, because of the diffuse light distribution in the ink-paper mixture
(see Eq. (7.5)).
7.2.2
Notations
• zjq – the ink thickness (or equally ink volume) of ink level specified by the
printer deriving program, j = 1 − 5, and color q = c, m, y.
• αjq = zjq /z1q – the relative ink thickness to ink level 1.
t
• βjq
– the amount of the primary inks (q = c, m, y) used to form the secondary colors (t = r, g, b) of ink level j. For example, the ink thickness
(volume) of color red (r) of ink level j reads
r
r
z1m + βjy
z1y
zjr = βjm
(j = 1 − 5)
• D– the thickness of a single sheet of paper.
• djq –the thickness of ink penetration into the paper (in percentage of a
single sheet paper thickness).
80
Characterization of ink penetration
• sq –the scattering coefficient of the (pure) primary inks.
• kq –the absorption coefficient of the (pure) primary inks.
• sp –the scattering coefficient of the paper.
• kp –the absorption coefficient of the paper.
• sqp –the scattering coefficient of the ink-paper mixture (q = c, m, y). The
extra subscript, p, denotes paper.
• kqp –the absorption power of ink-paper mixture.
t
(j = 1−5, q = c, m, y, and
Among these quantities, sq , kq , zjq (or αjq ) and βjq
t = r, g, b), were already known from the studies of the inks on foil (Chapter 6).
The rest, kqp , sqp , and djp , will be determined in this chapter.
7.3
Simulation of print on office copy-paper
The full tone samples (solid patches) were printed on office copy-paper with
both the primary and the secondary colors, the secondary colors being mixtures
of two of the three primary colors. By varying ink-level specification in the
printer driving software, one can obtain samples printed with up to 5 ink levels
(the ink-volume increases from the ink-level 1 to 5). The measurements were
carried out by applying a spectrometer, which covers a spectral range of 380 to
730 nm at a interval of 10 nm. A UV filter was employed in order to minimize
the impact of fluorescence.
7.3.1
Primary colors
Following the notations and assumptions listed in Sec. 7.2, the scattering and
absorption powers of the ink-paper mixture (color q and ink level j) may be
written as
sqp Ddjq
= sq zjq + sp Ddjq
(7.4)
kqp Ddjq
=
(7.5)
2kq zjq + kp Ddjq
where q = c, m, y represents for the primary colors, and djq the depths of ink
penetration .
If the depth of ink penetration of one ink level is known, say ink level 1,
as d1q , the depth of ink penetration of another ink level j, according to the
assumption No. 3, may be written as,
djq = αjq d1q
(7.6)
7.3 Simulation of print on office copy-paper
81
where αjq is the relative ink thickness obtained in Chapter 6. The task of the
simulation is, therefore, to determine the depth of ink penetration of a single
ink level (say d1q ) for each primary color. The depths of ink penetration of
other ink levels can be computed from Eq. (7.6).
In practice it is d3q , the depth of ink penetration of ink level 3, that is
first determined instead of d1q . The determination of the d3q is very similar to
t
(see Section 6.2.2). When the inks
what was done in the determination of βjq
were printed on foil they formed pure ink-layers on the foil surface. Because
the refractive index difference between the air and the ink was negligible, no
correction for boundary reflection was needed. Nevertheless, when the inks are
printed on the office copy-paper the inks actually go down into the substrate
paper. Due to the refractive index discontinuity at the air/paper interface, the
boundary reflection between the air and the paper (or ink penetrated paper)
has to be considered. Therefore, the following formula that takes care of the
boundary reflection (Eq. (5.34)) is used, i.e.
−
R = r0 +
(1 − r0 )(1 − r1 )[sqp (Aqp − sqp Rg )e
s2 −A2
− qpAqp qp
(Aqp − sqp r1 )(Aqp − sqp Rg )e
2
s2
qp −Aqp
Aqp
Ddjq
− Aqp (sqp − Aqp Rg )]
Ddjq
− (sqp − Aqp r1 )(sqp − Aqp Rg )
(7.7)
where r0 and r1 are the external and internal boundary reflection, Rg , the
reflectance of the plain paper, and
2 + 2k s .
Aqp = sqp + kqp − kqp
(7.8)
qp qp
Our simulations show that n = 1.2 provided the best fit to the measured
spectral reflectance for both printed and non-printed paper. The external and
internal boundary reflection in the case of diffuse light distribution are, r0 =
0.0443 and r1 = 0.3363, respectively.
The scattering and absorption powers of the ink-paper mixture of print of
ink level 3 are depicted in Fig. 7.3. Evidently, the scattering power of the
ink penetrated paper bears a general similarity to that of the bare paper. On
the other hand, the absorption power of the ink-paper mixtures are of similar
shape to that of the inks. These observations reflect the facts that the scattering
characteristics of the ink-paper mixture are predominated by the paper and the
absorption characteristics by the inks.
A comparison between the simulated and the measured spectral reflectance
values of the primary inks of ink level 3 is shown in Fig. 7.4. Generally speaking,
agreement between the experiments and the simulations is fairly good for all
the colors, and over the entire visible spectrum. Color differences between the
measurements and the simulations are just about visible and ∆E = 3.28, 4.14,
and 5.15, for cyan, magenta, and yellow, respectively. A remarkable discrepancy occurs for ink cyan at around λ = 460 nm. This may be a consequence of
82
Characterization of ink penetration
Ink level=3
Absorption power, kz
6
4
2
0
400
Scattering power, sz
cyan
magenta
yellow
a)
0.8
450
500
550
600
650
700
450
500
550
600
650
700
b)
0.6
0.4
0.2
0
400
Wavelengths, λ (nm)
Figure 7.3: Scattering and absorption powers of the ink-paper mixture (ink level 3).
Ink level=3
1
Simulation
0.9
Measurement
0.8
yellow
Reflectance, R
0.7
magenta
cyan
0.6
0.5
0.4
0.3
0.2
0.1
0
400
450
500
550
600
650
700
Wavelengths, λ (nm)
Figure 7.4: Spectral reflectance values of the primary colors printed on the office
copy-paper (ink level 3).
7.3 Simulation of print on office copy-paper
83
the residual fluorescence from the brightening substances in the paper as being
observed from the bare paper. Bispectral fluorescence measurements made by
Mourad [Mou02b] revealed that, for brightened paper, the fluorescence has its
peak at about λ = 450 nm and the fluorescence had even remarkable contribution to the spectra of the printed cyan and blue in λ = 400 − 480nm region.
Therefore, there is a need for independent studies on the efficiency of the UV
filter employed in the present measurements. Such studies may provide clues
leading to a better understanding of the discrepancies.
Closer observation of the spectra of all the primary colors reveals that the
minima of the reflectance values are generally independent of the color and far
from zero. This is in remarkable contrast to those observed from the ink-layers
(Fig. 6.2) where the minima were essentially zero. According to the analysis in
Sec. 5.5.3, the possible explanation for the difference can lie with the boundary
reflection occurring at the boundary between the air and the ink penetrated
paper. Another possible origin lies with the strong scattering power of the
ink-paper mixture that contributes to the light reflection.
Ink level=3
Absorption power, kz
5
4
a)
red
3
2
1
green
0
400
blue
450
500
550
600
650
700
450
500
550
600
650
700
Scattering power, sz
1.2
b)
1
0.8
0.6
0.4
0.2
400
Wavelengths, λ (nm)
Figure 7.5: Scattering and absorption powers of the ink-paper mixture of secondary
colors (ink level 3).
7.3.2
Secondary colors
The reliability of the quantities, sqp djq , kqp djq , obtained from the primary colors (printed on the paper) is directly tested when they are applied to predict
84
Characterization of ink penetration
the spectral reflectance of the secondary colors. For secondary colors, according
to the assumption No. 5, the scattering and absorption powers of the ink-paper
mixture may be expressed as, (color red (r), for example)
srp Ddjr
r
r
= βjm
smp Ddjm + βjy
syp Ddjy
(7.9)
krp Ddjr
r
r
= βjm
αjm Dsmp d1m + βjy
αjy syp Dd1y
r
r
= βjm kmp Ddjm + βjy kyp Ddjy
r
r
= βjm
αjm kmp Dd1m + βjy
αjy kyp Dd1y
(7.10)
The scattering and absorption powers of the ink-paper mixture for the secondary colors (ink level 3) are shown in Fig. 7.5. Similarly to the primary
colors, scattering is predominated by the paper, and absorption by the inks.
r
In Eqs. (7.9) and (7.10) all the parameters are known, namely βjq
and
αjq from the ink on foil (Chapter 6), and sqp d1q and kqp djq from Sec. 7.3.1.
Therefore, by applying srp djr and krp djr to Eq. (7.7) one can predict the
spectral reflectance of the secondary colors printed on the paper. As there is
no data-fitting involved in predicting the spectra for the secondary colors, a
comparison between the predicted and the measured spectral values provides a
test on the validity of the model and the reliability of the parameters obtained.
The predicted spectral reflectance values of the secondary colors (ink level 3)
are depicted in Fig. 7.6 together with the corresponding experimental ones. As
seen, the prediction is in fairly good agreement with the experiment for all the
secondary colors and over the entire visible spectrum.
It is convenient to depict the depth of ink penetration in terms of a percentage of a single sheet paper thickness. The depths of ink penetration for both
primary and secondary colors and all 5 ink levels are demonstrated in Fig. 7.7.
Note that only the depths of the primary colors of ink level 3 were obtained
by fitting to the measured spectral reflectance data, all the rest were actually
computed according to Eqs (7.6), (7.9), and (7.10), respectively.
7.4
Optical effect of ink penetration
In order to evaluate the consequence of ink penetration, it is desirable to directly compare prints with and without ink penetration by using the same type
of substrate. Unfortunately, there exists no substrate which allows for ink penetration to be switch on or off at will. In practice, to avoid ink penetration,
the paper surface is modified by coatings. In other words, a high grade paper
other than copy-paper is used. Such a modification to the paper surface can
indeed reduce or even eliminate ink penetration into the cellulose structure.
Nevertheless, this changes the (optical, fluid dynamic, etc.) properties of the
paper which essentially implies another type of substrate.
7.4 Optical effect of ink penetration
85
Ink level=3
1
0.9
Simulation
Measurement
0.8
Reflectance, R
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
400
450
500
550
600
Wavelengths, λ (nm)
650
700
Figure 7.6: The spectral reflectance of the secondary colors printed on office copypaper (ink level 3).
Ink penetration into the plain paper
15 a)
10
cyan
magenta
yellow
Ink penetration, d (%)
5
0
1
2
3
4
5
20
b)
15
red
green
blue
10
5
1
2
3
Ink level, j
4
5
Figure 7.7: Depthes of ink penetration of the primary and secondary colors.
86
Characterization of ink penetration
Although there is no direct experimentally possible comparison, it can be
achieved with help of simulations. The comparison was made for print of ink
level 3 (the default ink level of the printer). Because the scattering and absorption powers of the inks (sq z3q , kq z3q ), and the ink-paper mixture (kqp djq , kqp djq )
are known from our studies, the spectral reflectance values of prints on the office copy-paper with and without ink penetration can be simulated according
to Eq. (7.7). In the case of no ink penetration, boundary reflection between the
air/ink interfaces is ignored (r0 = r1 = 0), while r0 = 0.0443 and r1 = 0.3363,
in the case of having ink penetration.
Simulation results for primary and secondary colors are depicted in Fig. 7.8.
From the figure one can clearly see the significant effect of ink penetration.
Interestingly enough, the effect shows a strong wavelength dependence. For
convenience of discussion, we refer to the band containing the local maximum
as the transparent band and that containing the local minimum as the absorption band. In the transparent band, the print for the case of no ink penetration
has greater reflection compared to that with ink penetration. In contrast, in
the case of the absorption band, the print with ink penetration shows stronger
reflection than that without. These observations reflect the collective contribution to light reflection from the substrate, the ink-layer, and ink-paper mixture.
In the case of no ink penetration, the print consists of an ink-layer and a substrate backing (plain paper), while it consists of an ink-paper mixture layer and
the substrate backing in the case of having ink penetration. As is known the
ink-layer (no ink penetration) has little scattering power, the reflection in the
transparent band is essentially due to reflection from the substrate. Comparatively, the layer of the ink-paper mixture has a much stronger scattering power
which blocks the light from reaching the substrate backing to some extent. On
the other hand, the absorption power of the ink-paper mixture (there exists
absorption even in the transparent band of the ink) is twice as great as that of
the pure ink-layer. These factors work together and result in weaker reflection
for the case of the print with ink penetration. In the absorption band, on the
other hand, the light is dramatically attenuated by absorption when it passes
through the ink-layer (in order to be reflected by the substrate). Nevertheless,
the light may return to the air before it passes through the ink-paper mixture
due to scattering of the paper materials which makes the ink-paper mixture
more reflective.
Ink penetration also has significant impact on the color appearance of the
print. Because the saturation of the color depends on the contrast between the
peaks of the transparent bands to the bottoms of the absorption bands, the
comparison suggests that ink penetration significantly reduces the saturation of
the printed color. At the same time, it causes even color shift (hue variation)
because of nonlinear modifications to the spectral reflectance values by ink
penetration. A quantitative measurement of the color difference induced by ink
penetration is given in Tab. 7.1. Clearly, ink penetration has significant effect
7.4 Optical effect of ink penetration
1
0.9
87
Ink level=3
Ink penetration
No ink penetration
0.8
Reflectance, R
yellow
cyan
0.7
magenta
0.6
0.5
0.4
0.3
0.2
0.1
0
400
450
500
550
600
Wavelengths, λ (nm)
650
700
(a) Primary colors
Ink level=3
1
0.9
Ink penetration
No ink penetration
0.8
red
0.7
Reflectance, R
green
0.6
blue
0.5
0.4
0.3
0.2
0.1
0
400
450
500
550
600
Wavelengths, λ (nm)
650
700
(b) Secondary colors
Figure 7.8: Comparison of colors printed on office copy-paper with and without ink
penetration.
88
Characterization of ink penetration
on color reproduction. Detailed discussions about the impact of ink penetration
on color rendition and correspondingly, on their graphical representations will
be given in Chapter 10.
Table 7.1: Color differences (∆E 1) ) induced by ink penetration.
color
∆E
1) ∆E =
cyan
13.13
magenta
31.98
yellow
30.61
red
29.20
green
36.97
blue
25.40
(∆L∗ )2 + (∆a∗ )2 + (∆b∗ )2 , detailed description about the CIELAB
color system may be found in Chapter 10.
7.5
Summary
In this chapter a model accounting for optical effects of ink penetration in a
combination of dye-based liquid ink and office copy-paper is presented. The
model uses optical properties (scattering and absorption) of the ink and the
paper as inputs to simulate the ink-paper mixture (ink penetration). A method
used for determining fundamental quantities like depth of ink penetration by
combining spectral reflectance measurements with simulation has been proposed and tested. Parallel comparison between prints with and without ink
penetration has been made and it shows that ink penetration causes significant
color saturation reduction and hue shifting.
Chapter 8
Dot gain in black and white
8.1
8.1.1
Introduction
Murray-Davis equation
For a small area of a halftone image as shown in Fig. 8.1, the light detected by
a sensor (a human or an instrument), is a mixture of the light reflected from the
halftone dots and from the paper between the dots. Assume that the intensity
of illumination is I0 . Under uniform illumination, it is natural to assume that
the illumination onto the dots and the substrate is proportional to the friction
of area, σ and 1 − σ, and can be written as I0 σ and I0 (1 − σ), respectively.
Subsequently, one can express the total light reflected from an area, Ir , as a
sum of these two parts, i.e.
Ir = I0 Rg (1 − σ) + I0 R10 σ
(8.1)
where Rg and R10 are the reflectance of the substrate paper and that of solid
print, respectively. The superscript 0 in R10 , denotes hereafter that the reflectance of the solid print differs from that of halftoned ink dots (R1 ). According to the definition, the average reflectance of image area, R, is therefore
written as
R = Rg (1 − σ) + R10 σ
(8.2)
Equation (8.2) is usually called Murray-Davis (M-D) equation [Mur36]. Despite
being simple in derivation, the Murray-Davis equation has historically been a
milestone in Graphic Arts, because it initiated an era of modelling the print
mathematically.
From its definition the CIE tristimulus values (X,Y,Z) can be computed
90
Dot gain in black and white
from the spectral reflectance, say,
X = R(λ)S(λ)x(λ)dλ
(8.3)
and thus from Eq. (8.2) one can obtain
X = X0 (1 − σ) + X1 σ
(8.4)
where S(λ) is the spectral distribution of the illumination and x(λ) is the stimulus function. X0 , X1 , and X are the tristimulus values of the substrate, the
ink solid, and the halftone image, respectively. Equation (8.4) is a representation of Eq. (8.2) in chromatic perspective and is known as the Neugebauer
equation.
a)
sensor
1’
2
1
2’
Σ0
Σ
1
b)
2
1
sensor
1’
2’
Paper
Figure 8.1: Murray-Davis description of light reflection from half tone images. Ink
dots are grouped into Σ1 and paper between the dots into Σ0 .
8.1.2
Yule-Nielsen equation
Although the M-D equation was conceptually correct, it was soon clear that
it gave only a gross approximation to the experimental results. In 1951 Yule
and Nielsen [YN51] reported their study on discrepancies between the measurements and theoretical predictions (by the M-D equation). They found that
light scattering within the paper substrate (see Fig. 8.2) was responsible for
8.1 Introduction
91
the discrepancies, which is known as the Yule-Nielsen effect. They proposed a
modification to the M-D equation, which is known as the Yule-Nielsen Equation,
1
1
(8.5)
R = [(R10 ) n σ + (Rg ) n (1 − σ)]n
The exponent n is usually called the Yule-Nielsen factor and can be obtained
by fitting to the experimental data (such as optical density). Unfortunately,
the Yule-Nielsen equation provides only a better numerical approximation. It
provides no physical insight into the real process. In 1978 Ruchdeschel and
Hauser [RH78] obtained an estimation of the exponent n from their point spread
function (PSF) analysis. Their study showed that 1 ≤ n ≤ 2, when only light
scattering involved. In practice, there exists almost always physical dot gain.
In this case n can be significantly larger than 2, if the Yule-Nielsen equation is
used for data fitting. Due to light scattering within the substrate the printed
dots appear bigger than they geometrically are. Consequently, the Yule-Nielsen
effect is also referred to as optical dot gain and mere recently as tone value
increase.
A similar modification to the Murray-Davis equation was also introduced
to the Neugebauer equation by Pobboravsky and Pearson [PP72] in the 1970’s.
a)
sensor
2
1
1’
2’
Σ0
Σ1
scensor
2
b)
1
1’
2’
Paper
Figure 8.2: The Yule-Nielsen effect resulting from light scattering within the substrate. In the figure, light rays enter the substrate from one area (Σ0 or Σ1 ) and exit
from another (Σ1 or Σ0 ).
Optical dot gain readings take into account the optical illusion that comes
92
Dot gain in black and white
naturally with the printing/viewing process. Besides the optical dot gain, there
almost always co-exists another type of dot gain, physical dot gain, that takes
into account the dot extension due to mechanical and/or physical origins, such
as ink spreading, film exposure to plate, etc. Dot gain measurements take into
account both the physical and the optical dot gain. In other words an overall
dot gain is measured. Moreover, as we shall see later on, the optical dot gain
and the physical dot gain correlate with one another. Finally, the optical dot
gain and physical dot gain depend not only on the print processes but also on
the materials used.
8.1.3
Status of the studies
Light scattering in substrate is a very complex process and has attracted constant research interest over a long period of time. Kruse and Wedin [KW95],
among many others, proposed an approach which was thoroughly studied and
fully implemented by Gustavson [Gus97a, Gus97b]. The approach simulated
the light scattering process from a fundamental level. It was based on direct numerical simulation of scattering events which depend on the optical properties
of the materials, the halftone frequency and the halftone geometry. This approach is similar in nature to the Monte-Carlo method that is briefly explained
in Sec. 4.5. Statistics are recorded over a large number of light scattering events.
From these the probability of an event can be established. Arney [Arn97] and
Hübler [Hüb97] independently proposed similar models based on probability
descriptions of the light scattering. In their models, the light scattering inside
the paper was described by the probabilities that a photon emerges from the
inked and non-inked areas. These probabilities depend on the positions where
the photon enters the paper and emerges from the paper, as one will see in
Sec. 8.2.2. A point spread function (PSF) is a different representation of light
scattering. Using the PSF approach, Rogers [Rog97, Rog98a, Rog98b, Rog98c]
presented a method dealing with the light scattering process. He proposed
a matrix approach where the tristimulus values of a halftone image could be
calculated as the trace of a product of two matrices.
So far the studies have not provided any explicit expression for the reflectance or tristimulus as a function of dot percentage (as was given by the
Neugebauer equation). Moreover, the studies were limited to mono-color, or
black and white case. Finally, the effect of ink penetration into the substrate
was barely been touched upon. This chapter describes our studies in two of
these three perspectives, a description of multi-color print is presented in Chapter 9.
In Chapter 5, we established a theoretical approach to account for effects
of ink penetration in which it was assumed that the substrate paper was uniformly covered by ink layers. In this chapter we generalize this by investigating halftone images where light scattering is important. In Section 8.2, we
8.2 Model and methodology
93
describe a model that includes both light scattering (Yule-Nielsen effect) and
ink-penetration. In Section 8.3, we further extend the model to simulate the
overall dot gain (optical plus physical dot gain) and derive expressions for the
reflectance and overall dot gain (and the corresponding tristimulus values) as
functions of the dot coverage. Finally, the approach is further illustrated with
application to a digital (color) image.
8.2
Model and methodology
The basic geometry used here is shown in Fig. 8.2 where the surface of the
substrate paper has been divided into two sets: Σ1 , the paper under the dots
(or ink penetrated paper) and, Σ0 , the bare paper (or paper between dots).
For clarity, only the case of a single layer of dots is analyzed. Extension to a
multi-layer case will be given in Chapter 9. Also for simplicity, we assume that
the ink layer has uniform thickness irrespective of wether or not there is ink
penetration or not.
8.2.1
Point spread function approach
Let’s first examine the process of light reflection ifrom a microscopic viewpoint
→
(see Fig. 8.2). We consider several separate steps. First, we assume that −
r1 is
→
r0
an arbitrary position on the surface of the paper under the dots (Σ1 ) and −
an arbitrary position on the surface of the paper between the dots (Σ0 ). Now
→
r1 . The flux of the
we consider an element light, I0 dσ1 , that strikes the dot at −
→
→
→
−
r1 to −
r0 , may
light detected at r0 , due to scattering of the incident light from −
be written as
→
→
r0 , −
r1 )T I0 dσ0 dσ1
(8.6)
d2 J10 = p(−
→
−
−
→
where p( r , r ) is the point spread function and, T , the overall transmittance
0
1
of the print. If the transmittance of the ink layer on the substrate is T1 , then
for no ink penetration, T = T1 , and T I0 dσ1 is thus the amount of light entering
the substrate under the dot. When the ink penetrates partially (or even fully)
into the substrate [YGK00, YLK01], then
T = T1 γ
where γ describes the optical effect of the ink penetration, defined as
Rg
γ=
Rg
(8.7)
(8.8)
where Rg is the spectral reflectance of the clean substrate, modified into Rg
by the ink penetration. γ may be considered as an induced transmittance by
94
Dot gain in black and white
the ink penetration. Figure 8.3 sketches out this approximation. By using the
overall transmittance expressed in Eq. (8.7), there is no need to distinguish
wether there is ink penetration or not. One thing that should be kept in mind
is that γ depends not only on the ink but also on the substrate.
The point spread function is the probability of photons entering the paper
→
under the dots at position −
r1 and exiting from the paper between the dots at
→
−
the position r0 . It is worth noting to be noticed that ink penetration destroys
the uniformity of the substrate. The assumption
→
→
→
→
p(−
r0 , −
r1 ) = p(| −
r0 − −
r1 |)
(8.9)
which is generally applied in non-ink penetration analyses [RH78, Rog97], becomes invalid.
T
1
T1
γ
R’ =γ2R
R’0
0
0
(a)
R
0
(b)
Figure 8.3: An approximate treatment of the effect of ink penetration. (a) remaining
ink layer (transmittance T1 ) and ink penetrated substrate (reflectance, Rg ); (b) the
ink penetrated substrate is approximated by introducing an extra ink layer (transmittance γ) and the clean substrate (reflectance Rg ), where Rg = γ 2 Rg .
Next, suppose an extended light source covering the area of the dots and
such that the incident intensity, I0 , is unchanged, while the paper between
→
the dots is not illuminated. Then, the flux of the light detected at −
r0 , due to
scattering of the incident light from the dot is the integration of Eq. (8.6) over
the dots area (Σ1 ),
→
→
p(−
r0 , −
r1 )dσ0 dσ1
(8.10)
dJ10 = I0 T
Σ1
Therefore,
Σ1
→
→
p(−
r0 , −
r1 )dσ1 ,
(8.11)
is the probability of the incident light entering the substrate under the dots and
→
being scattered into position −
r0 . Finally, by performing the integration over
8.2 Model and methodology
95
the whole area of the paper between dots (Σ0 ) one obtains the total amount
of light detected at the paper between dots that is scattered from the incident
light on the dots,
→
→
J10 = I0 T
p(−
r0 , −
r1 )dσ1 dσ0
(8.12)
Σ1
The double integral
Σ1
Σ0
Σ0
→
→
p(−
r0 , −
r1 )dσ1 dσ0
(8.13)
is the overall probability of photons being scattered from Σ1 into Σ0 , which is
therefore a measure of the Yule-Nielsen effect. In the case of no ink penetration,
a general expression of the integral has been carried out by Rogers [Rog98c].
Now we exchange the position of the light source with that of the detector.
→
r0 and the
For example, in the first step we put the light source, I0 dσ1 , at −
→
−
detector at r1 , keeping other conditions unchanged. We then have
→
→
r1 , −
r0 )T I0 dσ1 dσ0
d2 J01 = p(−
(8.14)
From the optical reciprocity one obtains the following relation,
→
→
→
→
r0 ) = p(−
r0 , −
r1 )
p(−
r1 , −
(8.15)
J10 = J01
(8.16)
Then we have
This means that under uniform illumination of the halftone sample, the amount
of the light being scattered from Σ1 (halftone dots) into Σ0 (bare paper) is
equal to the light being scattered from Σ0 to Σ1 . Calculation of the flux, J10 ,
requires knowledge of the point spread function which is usually not available,
especially, in the case of existing ink penetration. However, if the mean value
p of the integrated point spread function, defined as,
1
→
→
p=
p(−
r1 , −
r0 )dσ1 dσ0
(8.17)
σ(1 − σ) Σ1 Σ0
is available, then the scattered light J10 can be calculated from
J01 = J10 = I0 T pσ(1 − σ)
(8.18)
Evidently, p depends not only on the physical properties of the substrate paper
and the ink, but also on the geometric and spatial distribution of the ink dots.
As shown later, p is closely related to the optical dot gain, and is therefore
experimentally measurable.
96
Dot gain in black and white
8.2.2
Probability approach
Here, we study the process of light reflection from a macroscopic point of view.
Suppose a photon enters the substrate within Σ1 (i.e. paper under the dots).
The conditional probabilities that it re-emerges from Σ1 , and Σ0 , are denoted
as P11 and P10 . Similarly, for a photon entering the substrate in Σ0 , P01 and
P00 are defined as the probabilities that the photon leaves the surface of the
substrate from Σ1 and Σ0 , respectively. These probabilities fulfill the following
constraint conditions [YLK01],
P11 + P10
P01 + P00
= Rg
= Rg
(8.19)
(8.20)
where Rg is the reflectance of the clean paper. The physics behind Eq. (8.19)
is that the sum of P11 and P10 is the total probability of photons striking the
surface of the substrate at Σ1 and then returning to the air. This is exactly
the definition of the reflectance, Rg , from a probability perspective. A similar
argument holds also for Eq. (8.20). It is worth noting that the summation of
the conditional probabilities is normally smaller than unity because light may
be absorbed in the substrate or be transmitted through to another side of the
media.
Mathematically, it can be proven (see the Appendix B) that
P01
= pσ
(8.21)
P10
= p(1 − σ)
(8.22)
Furthermore, it can be proven (see Appendix B) that for a halftone image, that
the reflectance measured from the paper between the dots, R0 , and that from
the dots, R1 , respectively, are
R0
R1
= Rg − pσ(1 − T )
2
= T Rg + p(1 − σ)T (1 − T )]
(8.23)
(8.24)
Clearly, the first terms on the right in Eqs. (8.23) and (8.24) correspond
to the reflectance of the bare paper and the solid print, respectively, and the
second terms to the light scattering. Consequently, the reflectance measured
from the paper between the dots is no longer a constant, as shown in Fig. 8.4a;
the greater the dot percentage, the smaller the reflectance. Similarly, the reflectance measured from the dots is not a constant either and is generally greater
than that of the solid ink. Finally, the average reflectance of the halftone image,
R, can be computed as
R
= R0 (1 − σ) + R1 σ
= RM D − ∆Ropt
(8.25)
8.2 Model and methodology
where
97
RM D = Rg (1 − σ) + Rg T 2 σ
(8.26)
is the computed reflectance of the halftone sample under the Murray-Davis
approximation (excluding the light scattering), and
∆Ropt = (1 − T )2 pσ(1 − σ)
(8.27)
results from the light scattering inside the substrate paper. Recalling the fact
that the overall transmittance, T , is the product of transmittance of the ink
layer (T1 ) on the substrate and the ink penetration induced transmittance (γ),
i.e. T = T1 γ, Eq. (8.27) reveals that ∆Ropt depends on
• T1 : optical properties of the ink layer on the substrate;
• γ: the optical effect of the ink penetration;
• p: the light scattering in the substrate, and
• σ: the ink coverage.
Reflectance, R
a)
Ink penetration
Rp
0.6
R
0.4
Ri
0.2
0
Optical dot gain, ∆Ropt
No ink penetration
0.8
0.08
20
40
60
80
100
40
60
80
100
b)
0.06
0.04
0.02
0
0
20
Dot percentage, σ (%)
Figure 8.4: Simulations of the spectrally averaged reflectance and optical dot gain
of ink magenta printed on office copy-paper, in the case of complete light scattering
in the substrate. Solid lines represent the values without considering ink penetration
and the dashed lines are the values with ink penetration.
Since the overall transmittance, T , is wavelength dependence, ∆Ropt will
be wavelength dependent. Spectroscopically, ∆Ropt reaches its maximum in
98
Dot gain in black and white
the absorption band of the ink and its minimum in the transparent band.
Experimentally, it may be convenient to take a spectral average of the measured
quantities (∆Ropt or T ) when white light illumination like D65 is used. The
spectral average of any quantity, A(λ), is defined as
A(λ)dλ
< A >= dλ
In the following, the same symbol A will be used for both a spectrally dependent
function and its spectrally average whenever there is no confusion.
Because ∆Ropt > 0, the true reflectance, R, is smaller than its Murray-Davis
value, RM D , and the halftone image appears to be darker (more saturated in
color). Accordingly, it appears to have a larger dot coverage than predicted
when light scattering is ignored. It is for this reason that this effect is known as
optical dot gain. If scattering is not modelled, then the measured reflectance R
seems to originate from a dot size, σ + ∆σopt , instead of the true dot size, σ.
From R(σ) = RM D (σ+∆σopt ), one can then obtain the optical dot gain, ∆σopt ,
as the function of the optical properties of the materials and ink penetration:
∆σopt
=
=
∆Ropt
Rg (1 − T 2 )
(1 − T )p
σ(1 − σ)
(1 + T )Rg
(8.28)
The quantity, ∆σopt , provides a phenomenological description of optical dot
gain. However, due to its spectral dependence, it differs fundamentally from
any physical extension such as physical dot gain. Moreover, optical dot gain
differs for different colors (inks).
Because ∆σopt and ∆Ropt are the geometric and optical representations of
the same origin, we will not distinguish between them, and hereafter refer to
them both as optical dot gain. A similar convention applies to the physical
dot gain. Besides the spectral dependence, the optical dot gain, ∆σopt , is
proportional to p. From the measured optical dot gain profile, one can possibly
estimate p and obtain valuable information about the point spread function.
From Eq. (8.27), the maximum of the optical gain can be obtained from
p σ(1 − σ) + p(1 − 2σ) = 0
(8.29)
where
p =
dp
dσ
If p = 0 or p is independent of the dot percentage, we have p(σ) = p(σ = 0).
In the case of no ink penetration, for the white substrate (σ = 0), the average
8.2 Model and methodology
99
probability, p, is essentially its reflectance, Rg . Then, the optical dot gain has
its maximum at σ = 50% (see Fig. 8.4b) and can be computed from
(∆Ropt )max = (1 − T )2 Rg /4
(8.30)
p = constant is usually called complete light scattering [Rog97], and the
optical dot gain can be computed from
∆σopt =
(1 − T )
σ(1 − σ)
(1 + T )
(8.31)
which has a single maximum at σ = 50% and a symmetric form around the
maximum. It is easy to prove that here the complete light scattering corresponds to the Yule-Nielsen model with the Yule-Nielsen factor of n = 2.
Spectral dependence of the optical dot gain
1
T
c
T
m
T
Transmittance, T
a)
0.8
y
0.6
0.4
0.2
Optical dot gain, ∆σ
0
400
450
500
550
600
650
700
∆σc
∆σm
∆σy
<∆σc>
<∆σ >
m
<∆σ >
b)
0.2
0.1
0
400
y
450
500
550
600
Wavelengths, λ (nm)
650
700
Figure 8.5: Spectral dependence of optical dot gain (curves) for primary inks, in the
case of σ = 0.4 and the complete light scattering. a) spectral transmittance of the
inks; b) computed optical dot gain according to Eq. (8.31). The spectral average of
the dot gain values (lines), < ∆σopt >, have also been included.
To demonstrate the spectral dependence, the optical dot gain computed
according to Eq. (8.31) is depicted in Fig. 8.5, in the case of σ = 0.4 and complete light scattering. The spectral transmittance values of ink cyan, magenta,
and yellow obtained from Chapter 7 are shown in Fig. 8.5a. Clearly, ∆σopt
shows a disctinct correlation with its spectral transmittance. In the figure, the
spectrally averaged dot gain values, < ∆σopt >, of different colors have also
100
Dot gain in black and white
been included. Comparisons between the colors show that the light scattering
in the substrate results in the smallest spectrally averaged optical dot gain for
ink yellow, and the greatest spectrally averaged optical dot gain for ink cyan.
X0−X
X −X
MD
0
X −X
0
0
10
20
30
40
50
60
70
80
90
100
Dot percentage, σ (%)
Figure 8.6: Schematic diagram of X0 −X vs. σ variation of a mono-chromatic image.
X0 − X (solid line) is computed by present model (complete light scattering), and
X0 − XM D by Murray-Davis model (dash-dot line)
8.2.3
Impacts of the optical dot gain
Effects of optical dot gain on the color appearance of the printed images
can readily be seen from its color coordinates. According to the definition
(Eq. (8.3)), one can compute the tristimulus values X, Y, Z of the halftone
image from its reflectance (Eq. (8.25)). For example,
X = XM D − ∆Xopt
where
(8.32)
XM D =
RM D (λ)S(λ)x(λ)dλ
(8.33)
is the contribution from light following the Murray-Davis’ assumption, and
∆Xopt = ∆Ropt (λ)S(λ)x(λ)dλ
(8.34)
8.3 Overall dot gain of monochromatic colors
101
corresponds to the Yule-Nielsen effect. From Eq. (8.32) and the non-negativity
of ∆Xopt we find that for any tristimulus value X0 ,
X0 − X ≥ X0 − XM D
(8.35)
This inequality is particularly interesting when X0 is the tristimulus value of
bare paper, because X0 − X stands for the range of variation of the tristimulus
value upon printing. Since the presence of the ink layer reduces the reflectance
(compared to the bare paper), both sides of Eq. (8.35) are positive. We therefore have the situation shown in Fig. 8.6, which demonstrates that, for any dot
percentage (except for σ = 0, 100%), the range of variation computed with
including light scattering effect is bigger than that excluding this effect. Because X0 , Y0 , Z0 stand for the white point of the substrate, in monochromatic
case the quantity, X0 − X together with Y0 − Y and Z0 − Z are direct measures to the color saturation. Therefore, Eq. (8.35) implies that for any dot
percentage the printed image is viewed more saturated due to the effect of the
light scattering in the substrate. It is worth noting that Eqs. (8.25) and (8.32)
are the mathematical expressions of optical dot gain in optical and chromatic
perspectives, respectively, and Eq. (8.35) is a direct derivation of them. In
multi-chromatic tone reproduction, experimental observation [And97] and numerical simulation [Gus97a, Gus97b] have shown that light scattering actually
leads to a bigger color gamut.
8.3
Overall dot gain of monochromatic colors
Besides optical dot gain, a printed image is subject to various of distortions resulting from printing processes. Distortions originating from distortions of the
printed dots (shape and size) are generally termed physical dot gain. The physical dot gain is closely related to the printing processes, physical and chemical
properties of the printing materials (colorants and substrate etc), and printing
environment. For ink-jet printing the physical dot gain comes mainly from
ink-substrate interaction. The surface properties of the substrate play a dominant role for ink setting. In any printed image there almost always exists
some physical dot gain. In addition, the reflective optical measurements to the
halftone images always contain both physical and optical dot gains. Therefore,
the physical dot gain has to be considered before a meaningful comparison
between the simulations and measurements can be made.
8.3.1
A model for overall dot gain
Assume the commanded dot percentage is σ which distorts to σ + ∆σphy after
printing, due to the physical dot gain. According to Eq. (8.25) its effect on the
102
Dot gain in black and white
reflectance can be expressed as,
R(σ + ∆σphy )
= R(σ) − ∆Rphy (σ)
= RM D (σ) − ∆Rtot (σ)
(8.36)
where ∆Rtot (σ) is the overall dot gain for dot percentage σ and
∆Rtot (σ) = ∆Rphy (σ) + ∆Ropt (σ)
(8.37)
is a summation of the geometrical and the optical dot gain. The term of
the optical dot gain, ∆Ropt , has been derived in Eq. (8.27), and the term
corresponding to the physical dot gain can be expressed as
∆Rphy (σ) = ([Rg (1 − T 2 ) + (1 − T )2 p(1 − 2σ)]∆σphy
(8.38)
where the first term on the right comes from the Murray-Davis approximation.
The second term is the response of the optical dot gain to the physical dot
gain, and it shows a correlation between the optical dot gain and the physical
one.
The physical dot gain, ∆σphy , is a function of the commanded dot percentage (σ) and is subject to the constraints, ∆σphy = 0 at σ = 0 and 1,
respectively. The constraints are automatically fulfilled if we express ∆σphy as
∆σphy = ð(σ)σ(1 − σ)
(8.39)
where ð(σ) is a function that describes the characteristics of the physical dot
gain.
Recalling the expressions given by Eqs. (8.27), (8.38), and (8.39), the overall
dot gain can be rewritten as
dRtot (σ)
= {Rg (1 − T 2 )ð(σ) + p(1 − T )2 [ð(σ)(1 − 2σ) + 1]}σ(1 − σ)
= Q(σ)σ(1 − σ)
(8.40)
The function Q(σ) describes the overall dot gain characteristics of the print.
Additionally, due to its dependence on the transmittance of the inks, Q(σ)
differs for different colors. Furthermore, Q(σ) may be approximated by a polynomial expansion, such as,
Q(σ) = c0 + c1 σ + c2 σ 2
(8.41)
where the expansion coefficients, ci , can be obtained by fitting to the experimental dot gain curves of the calibration patches.
8.3 Overall dot gain of monochromatic colors
8.3.2
103
Simulation of the overall dot gain
The calibration charts consist of 21 patches for each primary color. The commanded dot percentage of the patches ranges from 0 to 100% at an interval
of 5%. The patches were created with an HP970Cxi ink-jet. Ink-jet films and
office copy-papers were used as substrates. For comparative purpose, the same
settings in the printer driver interface have been adopted for the prints on both
substrates. These are print quality: best, substrate type: plain paper, and
ink volume: 3. The patches were measured with a spectrophotometer.
Settings: Plain paper, best quality, ink level 3
8
6
cyan
magenta
yellow
a) plain paper
4
Overall dot gain, ∆R
tot
2
0
−2
0
20
40
60
80
100
80
100
4
b) OH film
2
0
−2
−4
0
20
40
60
Commanded dot percentage, σ (%)
Figure 8.7: Overall dot gain values (measurements) for prints with the following
settings, Substrate: plain paper, Print quality: best, and Ink volume: 3. a) Print on
plain paper; b) Print on ink jet (OH) film.
Figure 8.7 depicts the measured values of ∆Rtot of prints on plain paper and
on ink-jet film, respectively. As shown, the ∆Rtot varies irregularly with respect
to the commanded dot percentages, for different colors. Moreover it is negative
when the commanded dot percentage is greater than 70% (Fig. 8.7a) for cyan
and yellow. This observation is in contradiction to the common experiences
about the dot gain. One explanation might be that the printer has invoked
the built-in software designed for reducing physical dot gain. Because of its
smoother surface and less ink spreading, the correction due to the built-in
software for plain paper setting, leads to an over correction to the print on
the film. Hence the controversial characteristics becomes more remarkable in
Fig. 8.7b.
In order to study the effect of the substrate settings, the test patches were
also printed by employing other OH film as substrate setting. The prints seems
104
Dot gain in black and white
Settings: other OH film, best, ink level 3
25
a) plain paper
cyan
Simulation
Measurement
20
magenta
15
10
Overall dot gain, ∆R
tot
yellow
5
0
0
20
40
60
80
100
80
100
15
b) OH film
magenta
cyan
10
yellow
5
0
0
20
40
60
Commanded dot percentage, σ (%)
Figure 8.8: Overall dot gain values (measurements:dashed lines, simulation: solid
lines) for prints with the following settings, Substrate: other OH film, Print quality:
best, and Ink volume: 3. a) Print on plain paper; b) Print on ink jet film.
to be little influenced by the built-in software. A possible explanation may
be that no built-in program is invoked if the substrate is not well defined.
The experimental values of the overall dot gain, ∆Rtot , are demonstrated in
Fig. 8.8. Clearly, they are significantly greater than that with the plain paper
setting (Fig. 8.7). The characteristics of the dot gain curves agree well with
intuitive understanding of dot gain. The magnitudes of the overall dot gain
demonstrate a clear substrate dependence. Print on plain paper shows stronger
dot gain than that of the ink jet film, mainly because of greater ink spreading on
the surface of the plain paper and, in turn, stronger physical dot gain. On the
other hand, the dot gain characteristics suggest a remarkable ink dependence.
Ink cyan and ink magenta have competitive dot gain while ink yellow has
only about a half of their magnitudes. Finally, the dot gain curves are well
approximated by the polynomial expansion given by Eq. (8.41) for prints on
both substrates.
The successful parameterization of the overall dot gain characteristics for
the printer-substrate combination provides us with possibilities for controlling
and improving qualities of printed images. Furthermore, the parameterization
can greatly reduce the computing time involved in dot gain correction (sort of
color management) compared to that of using a look-up table (LUT). As an
example of this application, an image of 4.9 M pixels taken by a digital camera
(see Fig. 8.9) was processed. The computing time of the dot gain correction
8.3 Overall dot gain of monochromatic colors
(a) Print of the original image
(b) Print of the corrected image
Figure 8.9: Paradise bird flower printed from original and corrected images.
105
106
Dot gain in black and white
was 58 minutes when the LUT was used. This was reduced to 25 seconds with
the polynomial parameterization. Additionally, compared to the print of the
original image, the print with dot gain correction reveals many more details.
More discussions and examples about the model may be found in Ref. [YN02].
8.4
Summary
A model for analyzing properties of tone reproduction was derived. Reflectance
properties of a print were described by four types of parameters, γ for ink penetration, p for light scattering, Ti for transmittance of the remaining ink layer on
the paper, and Rg for reflectance of the bare substrate. Due to light scattering
inside the substrate, the reflectance becomes a nonlinear function of the commanded dot area. Moreover, the model predicts that the ink penetration leads
to a decrease in optical dot gain and that light scattering in a paper results in
a printed image appearing more saturated in color.
Relation between the optical dot gain, ∆σopt , and the scattering function,
p, can be used in both ways. From known scattering (p) the relationship can be
used to predict the optical dot gain. From measured optical dot gain profiles,
on the other hand, the measured data can be used to obtain estimates of the
scattering properties as described by p.
Analysis of the overall dot gain was made. It was shown that the overall
dot gain can be parameterized by polynomials. Application to a digital image
helped to reveal many more details of the image in the dark tone regions.
Chapter 9
Dot gain in color
Optical dot gain in multi-chromatic reproduction has rarely been studied because of its complexity and possibly also because of the lack of theoretical
guidance. From a theoretical point of view, the model that developed for the
mono-chromatic case in Chapter 8, can be extended in a straight forward fashion to the multi-chromatic tone reproduction case. By such an extension one
may make some sense of optical dot gain in multi-color printing.
9.1
Reflectance of a multi-color image
Figure 9.1 shows an image having a two ink-layer structure. The transmittances of the ink layers are, TI and TII . The image can be divided up to 4
chromatically distinct regions, denoted Σ0 through Σ3 , corresponding to white
(Σ0 ), primary (Σ1 ,Σ3 ) and secondary (Σ2 ) color, respectively. If the dot percentages of the two inks are a and b, the area of the ith chromatic region, σi ,
depends on the model of color mixing adopted by the printer. For example,
when the ink dots are placed at random, σi can be computed from
(1 − a)(1 − b)
σ0
=
σ1
σ2
= a(1 − b)
= ab
σ3
=
(1 − a)b
(9.1)
As in the monolayer case described in Chapter 8, we define Pij (i, j =
0, ..., 3) to be the probability that a photon exits the substrate from Σj given
that it enters the substrate at Σi . If the image is uniformly illuminated by light
of intensity, I0 , the outgoing flux of light from Σj due to scattering of incident
108
Dot gain in color
Figure 9.1: Halftone image consists of two ink layers, a) a side view; b) an overview.
The dots are marked by solid lines and the dashed lines denote regions of the optical
dot gain. Four chromatically different regions, are denoted as Σ0 through Σ3 .
light at Σi may be written as,
Jij = I0 Ti Tj Pij σi
(i, j = 0, .., 3)
(9.2)
where the Ti are the combined transmittance values describing the transmittance of the ink layer(s) and ink penetration of the region Σi , as defined in
Eq. (8.7). For example,
T0
=
1
(9.3)
T1
T2
= TI
= TI TII
(9.4)
(9.5)
T3
= TII
(9.6)
Similar to a monolayer system (see Sec. 8.2.2), it is easy to show that the
probabilities Pij (j = 0 − 3) are constrained by the reflectance of the substrate,
Rg , thus,
3
Pij = Rg (i = 0, ..., 3)
(9.7)
j=0
In addition, the probability, Pij , and its counterpart, Pji , fulfill the following
reciprocity relation,
Pij σi = Pji σj
(i, j = 0, ..., 3)
(9.8)
9.1 Reflectance of a multi-color image
109
Due to light scattering, photons entering the substrate at Σi can exit from
Σj . Thus, the total flux of the light outgoing from Σj may be expressed as
Jj
3
=
Jij
i=0
3
=
I0 Ti Tj Pij σi
(j = 0, ..., 3)
(9.9)
i=0
Applying the constraint conditions and the correlation relation (Eqs. (9.7) and
(9.8)), one can further write the flux as
3
Jj = I0 Tj2 Rg σj − I0
Tj (Tj − Ti )Pji σj
(j = 0, ..., 3)
(9.10)
i=0,i=j
Accordingly, the reflectance of the Σj region is calculated from
Rj = Tj2 Rg −
3
Tj (Tj − Ti )Pji
(j = 0, ..., 3)
(9.11)
i=0,i=j
Thus, the regional reflectance Rj depends directly on the transmittance (Tj )
of the ink layer. It depends also on differences of the transmittance between
the incident and exit regions, (Tj − Ti ), and the probability of light transfer
between the two regions, Pji .
Knowing the reflectance, Rj , one can calculate the average reflectance of
the image area,
R
=
3
Rj σj
j=0
=
3
Tj2 Rg σj −
3
j=0
Tj (Tj − Ti )Pji σj
j=0 i=j
j=0
=
3 3
Tj2 Rg σj −
3 3
(Ti − Tj )2 Pji σj
(9.12)
j=0 i<j
where the first term on the right is the reflectance of the halftone image calculated according to the Murray-Davis assumption, and the second is due to the
light scattering that causes darker tone, namely the optical dot gain.
When ink penetration occurs, the transmittance, Ti , in Eq. (9.12) should
be replaced by Ti γi where
R0
(9.13)
γi2 = i
Rg
110
Dot gain in color
and Ri0 is the reflectance of the substrate under the ink dot. Therefore, the
quantity, γi , describes the modification to the substrate due to ink penetration.
In the case of no ink penetration, γi = 1. Detailed description of this approach
has been given in Ref [YGK00].
9.2
Optical dot gain in multi-color tone reproduction
Equation. (9.12) can be generalized to images consisting of any number of ink
layers or colors, that is,
R = RM D − ∆Ropt
(9.14)
where
RM D =
N
−1
Tj2 Rg σj
(9.15)
j=0
is the reflectance of the halftone image under the Murray-Davis assumption,
and
N
−1 N
−1
∆Ropt =
(Ti − Tj )2 Pji σj
(9.16)
j=0 i<j
is the term corresponding to the Yule-Nielsen effect, or the optical dot gain. In
Eqs. (9.15) and (9.16), N , represents the number of distinct color regions in the
image. For images consisting of 3 or 4 ink layers, N ≤ 23 , or, N ≤ 24 , respectively. From Eqs. (9.14)-(9.16) one can draw the conclusion that the optical
dot gain is a general phenomenon in multi-chromatic tone reproduction. Since
∆Ropt is a non-negative quantity, Eq. (9.14) means that the real reflectance
of the halftone image, R, is smaller than that predicted by the Murray-Davis
equation. In other words, the tone of the print becomes darker due to light
scattering. This explains why this effect is also named tone value increase.
It is worth noting that it is not the ink dots themselves but the distinct
chromatic regions that are directly related to the color appearance of an image. Therefore, the term optical dot gain loses the intuitiveness it held in the
monochromatic case, because the distinct regions are not necessarily bigger
optically than geometrically. For example, due to light scattering from Σ1
into Σ0 , Σ1 appears to be extended towards Σ0 along the Σ0 /Σ1 border (see
Fig. 9.1b). However, Σ1 appears to be compressed due to light scattering from
Σ2 into Σ1 (i.e. the region of Σ1 close to Σ2 appears as if it is of secondary
color). The total effect of the light scattering to the regional reflectance of Σ1 ,
R1 , is a combination of these opposing contributions. This is where the term
(Tj − Ti ) in Eq. (9.11) comes. The term optical dot gain should refer to the
image area as a whole rather than any individual chromatic region.
9.3 Simulation for multi-layer color image
9.3
111
Simulation for multi-layer color image
The model presented above shows that all the regional reflectance values, Rj ,
the overall reflectance value of the image area, R, and the corresponding optical
dot gain, ∆Ropt , depend on a set of probabilities Pij which are N (N − 1)/2 in
number, where N is the number of distinct color regions. For example, in the
two inks case there are 3 pairs of probabilities, P01 and P10 , P02 and P20 , and
P12 and P21 .
Figure 9.2: A mask contains 3 × 3 halftone cells. Contributions from the neighboring
dots to the center one are included in convolution (see Eq. (9.18)).
As defined in Eq. (9.2), Jij represents the flux of light that enters the
substrate in the region, Σi , and then exits from Σj . By use of the point spread
function, Jij can also be written as,
Jij = I0 Ti Tj
p(xi − xj , yi − yj )dσi dσj
(9.17)
Σi
Σj
Comparing Eq. (9.2) with Eq. (9.17) one gets
1
p(xi − xj , yi − yj )dσi dσj
Pij =
σ i Σi Σj
(9.18)
Because the PSF is closely related to the optical properties of the substrate,
the quantity Pij depends on these properties as well. For example, if the PSF
is Gaussian,
2
2
2
p(xi − xj , yi − yj ) = κe−[(xi −xj ) +(yi −yj ) ]/δ
(9.19)
112
Dot gain in color
the optical properties of the substrate are characterized by the Gaussian parameter δ (κ is a normalization factor). This kind of PSF has been shown to
fit the experimental data of Yule, et al., fairly well [RH78, YHA67]. As the
variables of the PSF, (xi −xj ) and (yi −yj ), are related to the relative positions
of regions Σi and Σj , Pij depends on the spatial distribution of the printed ink
dots. Furthermore, the integrated value, Pij , depends on the size and shape
of the integration areas (Σi and Σj ). Finally, the magnitude of the optical dot
gain depends on the combined transmittance values of the related regions and
their differences, (Ti − Tj )2 , as can clearly be seen from Eq. (9.16). To examine
to what extent these factors affect the computed reflectance values, simulations
have been carried out by applying a Gaussian type of PSF to images printed
with two inks.
1
R
MD
0.8
0.6
0.4
0.2
0
0
20
40
60
a (%)
80
100 0
20
40
60
80
100
b (%)
Figure 9.3: Computed RM D , two inks (cyan and magenta), dot on dot. In the figure
a and b are the dot percentages of the inks.
9.3.1
Two inks of round dots: dot on dot
For simplicity, we first assume that the ink dots are concentric circular disks
(dot on dot). The simulations are carried out by choosing a mask that contains
3 × 3 halftone dot cells (see Fig. 9.2). Thus, influences to the convolution
(Eq. (9.18)) from the nearest neighboring dots have been included. Figures 9.3
and 9.4 demonstrate computed reflectance values (RM D ) under the MurrayDavis assumption, and the optical dot gain (∆Ropt ). The printed image consists
of four ink spots, white, two primary colors (substrate covered by either ink
9.3 Simulation for multi-layer color image
113
1 or 2), and one secondary color. The transmittance values corresponding
to these regions are shown in Eq. (9.3)-(9.6). Because the reflectance value
computed according to the Murray-Davis model, RM D , is a bilinear function
of the dot percentages, a and b, it has a roof like structure with maxima along
the diagonal a = b (see Fig. 9.3).
δ=0.12Lc
δ=0.07Lc
0.06
∆R
∆R
opt
opt
0.02
0.04
0.04
0.02
0
100
0.02
100
80
60
40
20
0 0
b (%)
20
60
40
a (%)
80
80
100
60
40
20
0 0
b (%)
(a)
80
100
a (%)
δ=∞
0.15
opt
0.1
0.05
∆R
opt
∆R
60
40
(b)
δ=0.20Lc
0.1
20
0
100
80
60
40
20
b (%)
0 0
(c)
20
40
a (%)
60
80
100
0.05
0
100
80
60
40
b (%)
20
0 0
20
40
60
80
100
a (%)
(d)
Figure 9.4: Computed ∆Ropt with different Gaussian parameters, δ, two inks (cyan
and magenta), dot on dot (round dots), Lc is the length (width) of a halftone cell. a
and b are the dot percentages of the inks.
Figure 9.4 presents computed optical dot gains, ∆Ropt , for different values
of the Gaussian parameter, δ. For generality, δ is measured by the length (or
width) of a halftone cell, Lc . Fig. 9.4a-c correspond to δ = 0.07Lc , 0.12Lc ,
and 0.20Lc , respectively. An extreme case, corresponding to complete light
scattering, δ = ∞, is also given in Fig. 9.4d. The following facts are observed.
1. ∆Ropt has local maxima when the printed dots have identical sizes, a = b
(they completely overlap with each other).
2. The local maxima become wider, and therefore less prominent, when
the Gaussian parameter, δ, gets larger, or equivalently the PSF becomes
broader and more flat.
3. The magnitude of ∆Ropt becomes greater when δ is larger (Note the
114
Dot gain in color
different scales used in the sub-figures).
The appearance of local maxima is a hybrid consequence of the difference
in transmittance values between adjacent regions, and the effective extension
of the PSF in space. For simplicity, suppose that the area covered by ink 1
→
(Σ1 ) has a fixed radius, | −
r1 | (Fig. 9.5). Now, consider what happens as the
→
→
r2 |, increases from | −
r2 |= 0 to
radius of the area covered by ink 2 (Σ2 ), | −
→
−
→
−
| r2 |→| r1 |. In Fig. 9.4, we are actually looking at a cross-section of the
∆Ropt surface and a plane, say a = constant. According to Eq. (9.16), ∆Ropt
is a sum of three terms,
∆Ropt
=
+
2(1 − T1 )2 P01 σ0
2(T1 − T2 )2 P12 σ1
+
2(1 − T2 )2 P02 σ0
(9.20)
Light
r2
r1
Σ2
Σ
1
Σ0
Figure 9.5: A schematic diagram of the point spread function and dot geometries.
The effective regions of light scattering from one region into another are marked by
dotted lines.
Clearly, the first term comes from light scattering between regions Σ0 and
Σ1 , the second from that between Σ1 and Σ2 , and the third from that between
→
r1 | is fixed in the current consideration,
Σ0 and Σ2 . As we have assumed that | −
the first term remains constant. Considering a photon that enters the substrate
at (x, y) in Σ2 (region producing secondary color), the PSF that describes the
probability
of finding the photon at a point (x , y ) becomes very small when
2
(x − x )2 + (y − y )2 ≥ 2δ. Therefore, only when the photon strikes the
substrate at a point in Σ2 close enough to the Σ2 /Σ1 border (inside the region
marked by dotted line circles, in Fig 9.5), there is a significant probability of
9.3 Simulation for multi-layer color image
115
finding it in the adjacent region Σ1 . In other words, the main contribution to
the second term comes from photons that hit the region between the dotted
circle lines. At the same time, there is little chance of the photon exiting the
substrate from the non-inked region (Σ0 ), i.e., the third term is negligible, if
→
→
→
→
→
r1 | and δ | −
r1 − −
r2 |. However, the third term grows when | −
r2 |
|−
r2 || −
→
−
approaches | r1 | (or a → b). Considering the fact that ∆Ropt is proportional
to the quantity (Ti − Tj )2 (see Eq. (9.20)) which has the biggest value in the
third term (i.e. (1 − T1 )2 ≥ (T1 − T2 )2 ), ∆Ropt grows at a quicker rate when
→
→
→
→
→
r1 | (| −
r2 |→| −
r1 |). However, if | −
r2 | continue to
| −
r2 | gets close to | −
→
−
→
−
→
−
increases beyond | r1 |, ie., | r2 |≥| r1 |, ∆Ropt falls again. Thus, ∆Ropt
reaches its maximum when a = b. This explanation is consistent with observed
fact No. 2 mentioned above. When δ gets larger, the PSF becomes broader
and more flat. Correspondingly, the area marked by the dotted circle lines
(in Fig. 9.5) becomes wider and therefore the local maxima of ∆Ropt become
broader and (relatively) less prominent, even though its absolute magnitude
increases. Because of the greater probability of photon entering the substrate
in one region and exiting from the other (or even others), in the case of having
a larger Gaussian parameter, δ, ∆Ropt becomes greater. An extreme case is
when δ → ∞. In this case, the PSF becomes constant (and therefore Pij as
well) over the whole paper. It means that the photon has equal probability of
being found anywhere on the paper, no matter where the photon enters the
paper. Therefore, the photon is said to be “completely scattered” [Rog97].
Consequently, the local maxima disappear and only a global maximum is built
up (see Fig. 9.4d). As shown in Fig. 9.4, the location of the global maximum
moves towards a = b = 50%, when the Gaussian parameter (δ) increases.
9.3.2
Two inks of square dots: dot on dot
In reality, printers may generate dots of different geometries, other than round
dots in halftoning [ZVL+ 03]. It is therefore, necessary to study shape dependence of the optical dot gain. For this purpose, simulations of images printed
with square dots have been carried out. Unlike a print with round dots, the
color appearance of a print with square dots depends not only on the areas
of ink dots but also on relative orientation (described by screening angles, α,)
between the primary colors. Figure 9.6a is a prototype of a geometric formation of the square dots, where each square within the solid lines represents a
square dot with area, a or b. Figure 9.6b corresponds to the case where there
is a screen angle between the dots. Naturally, to clarify the dependence of the
Yule-Nielsen effect (∆Ropt ) on the dot shapes, comparisons with the images
printed with round dots will be made. For easier comparison, identical Gaussian parameter, δ = 0.12Lc , has been chosen in the simulations, where Lc is
the length of a halftone cell, as defined before. Furthermore, dependence of the
Yule-Nielsen effect on the angle between the screen lines will also be explored
116
Dot gain in color
by choosing different α values.
in coming
light
in coming
light
Σ1
Σ
2
out going
light
out going
light
Σ
1
Σ0
Σ
2
α
(a)
(b)
Figure 9.6: Two inks print, solid line squares represent two square dots (area a and
b, respectively). The screen angle between the dots is α in the figure to the right.
Figure 9.7a-d show the computed ∆Ropt for α = 0o , 15o , 30o , 45o , respectively. Compared to the print with round dots under the same light scattering
parameter (δ = 0.12Lc , Fig. 9.4b), little difference has been observed when
α = 0. However, ∆Ropt appears remarkably different when α = 15o and the
differences become even more significant when the screening angle, α, further
increases. The differences can be summarized into the following,
1. The local maxima along the diagonal a = b, which is prominent in the
round dots case, becomes broader and less well defined.
2. The global maximum, which is a sharp peak in the case of round dots
becomes a broad plateau in the square dots case (where α = 0).
Since the point spread function has a limited effective extension (characterized by the Gaussian parameter δ) as shown in the Fig. 9.6a, there is no
significant probability that a photon is scattered from one region (say Σ2 ) into
another (say Σ1 ), unless the photon hits the substrate at a point close enough
to the border of the incident region (marked by dotted square lines). In the
case of α = 0o , there is even less probability for a photon to transfer from Σ2
into Σ0 or vice versa, if a ≈ b. In other words, it is most likely to happen
only when the two ink dots have similar areas (b ≈ a). Consequently, it forms
narrow local maxima along the diagonal a = b, as shown in Fig. 9.7a. However,
9.3 Simulation for multi-layer color image
117
when α = 0, if a photon strikes the substrate near a corner of the inner square
(Note: the rotated square has the same size as that in Fig. 9.6a), the photon has a better chance to be scattered from Σ2 into Σ0 , and vice versa, even
though the areas of the ink dots are not similar (see Fig. 9.6b). For this reason,
the local maxima become broader and more flat and therefore less prominent.
This argument holds also for the broader appearance of the global maximum.
The global maximum actually appears to be a flat plateau. The simulations
also show that the quantities of computed ∆Ropt decrease as the screen angle,
α, increases. Therefore, increasing the screen angle will possibly be helpful in
reducing the optical dot gain.
α=0o
0.1
α=15o
opt
0.05
∆R
∆R
opt
0.06
0.04
0.02
0
100
0
100
80
60
40
20
0 0
b (%)
20
40
60
80
80
100
60
40
20
0 0
b (%)
a (%)
(a)
40
60
80
100
a (%)
(b)
o
α=45
o
α=30
0.04
opt
∆ Ropt
0.04
∆R
20
0.02
0
100
0.02
0
100
80
60
40
20
b (%)
0 0
(c)
20
40
a (%)
60
80
100
80
60
40
20
b (%)
0 0
20
40
60
80
100
a (%)
(d)
Figure 9.7: Computed ∆Ropt of prints with square dots (dot on dot) of different
screening angles, α = 0o , 15o , 30o , 45o , and δ = 0.12Lc . a and b in the figure are the
ink percentages.
9.3.3
Two inks of round dots: random dot distribution
In conventional printing processes, the dot on dot can hardly be achieved due
to difficulties in registration. Additionally, to avoid artifacts, such as Moiré,
different screen angles are used when the different primary inks are printed.
This kind of dot arrangement leads approximately to a random dot overlap
distribution that can well be described by the DeMichel equations [De’24]. For
118
Dot gain in color
a 2-color print, the ink coverages of different chromatic areas, are given in
Eq. (9.1).
To study the optical dot gain in response to the random dot distribution,
simulations of two colors of round dots have been carried out. For ease of
comparison, the same Gaussian parameters (δ = 0.12Lc ), as in the case of dot
on dot, have been chosen in the simulations. Figure 9.8a depicts the computed
reflectance, under the Murray-Davis approximation. Similar to the case of dot
on dot, the reflectance decreases bi-linearly with respect to the primary ink
percentages, a, b. Nevertheless, the difference is also remarkable, i.e. there
exists no evident local maximum along the diagonal (a = b) in the random dot
distribution.
1
δ=0.12Lc
0.8
0.02
∆R
opt
RMD
0.6
0.4
0.01
0.2
0
0
0
100
20
40
60
80
b (%)
100
80
100
40
60
20
80
0
60
40
20
b (%)
a (%)
0
(a)
40
60
80
100
a (%)
(b)
δ=0.2Lc
0.04
0
20
δ=∞
0.08
0.03
opt
opt
0.06
∆R
∆R
0.02
0.01
0.04
0.02
0
100
0
100
80
60
b (%)
40
20
0
0
(c)
20
40
60
a (%)
80
100
80
60
40
b (%)
20
0
0
20
40
60
80
100
a (%)
(d)
Figure 9.8: Computed reflectance values, two inks (cyan and magenta) randomly
overlap with each other. (a) Reflectance under the Murray-Davis approximation
(RM D ); (b)-(d) Computed optical dot gain (∆Ropt ) with different Gaussian parameters δ, Lc is the length of a halftone cell.
The reflectance resulting from light scattering in the substrate or the optical
dot gain, ∆Ropt , is computed and depicted in Fig. 9.8b-d. In contrast to the
case of a dot on dot (see Fig. 9.4) where the ∆Ropt has its narrow (local and
global) maximum along the diagonal, here the ∆Ropt has rather a flat plateau
due to the diversity of the overlap between the dots. In addition, there exists
a substructure on the plateau. Finally, the magnitude is relatively smaller
9.4 The effects of optical dot gain on color reproduction
119
than that of a dot on dot, in correspondence. Considering the fact that dot
on dot has the greatest possibility of dot overlapping between the colors, this
observation implies that the optical dot gain can be reduced by reducing dot
overlap between the primary colors. This is in line with our observation from
the square dot case, i.e., the optical dot gain decreases as the difference between
the screening angles increases (for square dots).
9.4
The effects of optical dot gain on color reproduction
The discussion about the effects of the optical dot gain on the color appearance
of printed images can readily be extended to multi-color images. According to
the definition, one can compute the tristimulus values X, Y, Z of the halftone
image, from its reflectance, R, (see Eq. (9.14)),
X = XM D − ∆Xopt
(9.21)
where
XM D =
RM D (λ)S(λ)x(λ)dλ
(9.22)
is the contribution from light following the Murray-Davis assumption, and
∆Xopt = ∆Ropt (λ)S(λ)x(λ)dλ
(9.23)
corresponds to the optical dot gain. Because of the non-negativity of ∆Xopt ,
Eq. (9.21) implies that the image appears to be more saturated in color.
9.5
Summary
This chapter presents a model for simulating optical dot gain in multi-chromatic
tone reproduction, which allows us to analyze properties of images printed with
any number of inks, and in any halftone scheme. By applying a Gaussian type of
point spread function (PSF) the optical dot gain has been simulated for images
printed with 2 inks of different dot geometries (round and square dots), different
dot locations (dot-on-dot, random dot distribution), and different screen angles.
The optical dot gain shows a strong dependence on the optical properties of
the substrate and the inks, and on the geometric distribution of the printed
dots (shape, size, locations, and relative orientation of the dots). The present
model is independent of the halftone scheme, and it is therefore applicable to
images produced with any kind of halftone algorithm.
Chapter 10
Chromatic effects of ink
penetration
In this chapter, we evaluate the chromatic effects of ink penetration. The evaluation will be carried out in two ways, from experiment and from simulations.
The first one (Sec. 10.2) is done by comparing two sets of experimental data
of prints on plain paper and on photo-quality paper, respectively, in terms of
their color representations. The second comparison (Sec. 10.3) is based on simulations of prints with and without ink penetration. Each approach has its
advantages and disadvantages.
10.1
Basics in colorimetry
To make the following easier for readers, fundamentals in Colorimetry is briefly
presented.
10.1.1
CIEXY Z color space
The color of a print can be quantitatively represented in different color spaces.
Among others, CIEXY Z and CIELAB are the most popular ones. According
to definition, the tristimulus values of the color are computed as
X = k S(λ)R(λ)x(λ)dλ
Y = k S(λ)R(λ)y(λ)dλ
(10.1)
Z = k S(λ)R(λ)z(λ)dλ
122
Chromatic effects of ink penetration
where R(λ), S(λ), and x(λ) (y(λ), z(λ)) are spectral reflectance of the print,
the energy distribution of the illumination, and the averaged tristimulus functions, respectively. In this thesis, measurements were carried out under D65
illumination and at 10o viewing geometry. k is a normalization factor and is
defined as
100
(10.2)
k=
S(λ)y(λ)dλ
Eq. (10.1) provides the color a representation in a 3-dimensional color space.
10.1.2
Chromaticity diagram
To provide a convenient 2-dimensional representation of the color, a chromaticity diagram was developed. The transformation from the tristimulus values to
the chromaticity coordinates is accomplished through a normalization that removes the luminance information, i.e.
x =
y
=
z
=
X
X +Y +Z
Y
X +Y +Z
Z
X +Y +Z
(10.3)
Clearly, there are only two dimensions of information in chromaticity coordinates, the third chromaticity coordinate, can always be obtained from the other
two (for example, z = 1 − (x + y)). Therefore, the color is represented in a
2-dimensional chromaticity diagram, as shown in Fig. 10.1
10.1.3
CIELAB color space
Coordinates in CIELAB color space are defined by Eq. (10.4) for tristimulus values normalized to the white point of the illumination (specified by
X0 , Y0 , Z0 ) are greater than 0.008856 [Fai98].
L∗
=
a∗
=
b∗
=
Y 1/3
) − 16
Y0
X 1/3
Y
500[(
) − ( )1/3 ]
X0
Y0
Y
Z
200[( )1/3 − ( )1/3 ]
Y0
Z0
116(
(10.4)
where L∗ represents lightness, a∗ approximates redness-greenness, and b∗ ap∗
proximates yellowness-blueness. Furthermore, chroma (Cab
) and hue (hab ) can
10.2 Evaluation of chromatic effects from experimental data
be computed from the CIELAB coordinates.
∗
Cab
=
a∗ 2 + b∗ 2
hab
10.2
−1
= tan
∗
∗
(b /a )
123
(10.5)
(10.6)
Evaluation of chromatic effects from experimental data
To experimentally evaluate the effect of ink penetration is not a trivial task.
Ideally, one would directly compare two printing samples on the same type
of substrates, one has ink penetration, another not. To accomplish such a
comparison one needs to be able to switch on and off ink penetration at will,
which is experimentally impossible. One alternative that is usually applied is to
print on different types of substrates, such as plain (office copy) paper for print
having ink penetration and photo quality paper for not having ink penetration.
10.2.1
Parallel comparison of prints on two types of substrates
When different substrates are used for printing, it seems to be reasonable to
compare the color of the test patches in parallel, i.e., to compare patches of the
same (commanded) ink percentage but on different substrates. Nevertheless,
two things at least should be bore in mind when the evaluation ink penetration
is made. First, different substrates have different optical properties (spectral
reflectance, point spread function etc.) which will affect the color of the prints,
and optical dot gain characteristics. Secondly, substrates have different surface
properties that will influence the ink distribution on (even in) the substrates.
This will, for example, result in different physical dot gain, when halftone images are printed. To some extent, color differences resulting purely from the different substrate colors may be minimized by choosing the substrates of having
similar color under certain illumination condition (metamerism). Despite these
potential effects, the chromatic effect of ink penetration can still be evaluated,
at least qualitatively, by such a parallel comparison, because ink penetration
has by far much stronger impact on the printed color when dye-based inks are
printed on ordinary office copy paper.
In the present study, plain paper (StoraEnso, 80g/m2 ) and photo gloss paper (Hewlett Packard, 175g/m2 ) were chosen as substrates. Their tristimulus
values were, (84.03, 88.78, 94.75) and (86.49, 91.16, 96.10), respectively. Their
color difference equals ∆E = 1.32, which is therefore hardly noticeable, even
124
Chromatic effects of ink penetration
though they are evidently different in gloss. Test patches of primary and secondary colors were created by printing on these substrates. The commanded
ink percentages for each color range from 0 to 100% at a step of 5%. The
tristimulus values in these charts were measured by employing a spectrophotometer.
Table 10.1: Color differences (∆E 1) ) between the photo gloss paper and the office
copy-paper before and after printing (solid patches only).
color
∆E
1) ∆E =
paper white
1.32
cyan
12.37
magenta
19.30
yellow black red
23.67 13.44 24.92
green blue
22.59 20.75
(∆L∗ )2 + (∆a∗ )2 + (∆b∗ )2
Dependence of color gamut on substrates (PL)
0.6
G
photo paper
0.55
plain paper
0.5
Y
0.45
y
0.4
0.35
R
0.3
0.25
0.2
M
C
0.15
0.1
0.1
B
0.2
0.3
0.4
x
0.5
0.6
0.7
Figure 10.1: Chromaticity diagrams of prints on office copy-paper (dashed line) and
on photo gloss paper (solid line), drawn from the experimental data. The pairwise
points corresponding to the same (commanded) dot percentage but different substrates are connected by − . C, M, Y, R, G, and B mean cyan, magenta, etc.
Color differences between the substrates, and the solid prints on the substrates, have been collected in Tab. 10.1. As shown, the color difference between
the photo gloss paper and the plain paper before printing is hardly noticeable
(∆E = 1.32). However after printing with full tone colors, their color difference
dramatically increases to ∆E = 12.3 − 25, depending on the printed colors. As
10.2 Evaluation of chromatic effects from experimental data
125
only solid printed patches were compared, the color differences arise mainly
from ink penetration.
Chromatic effect of ink penetration (plain paper setting)
plain paper
100
photo paper
Y
80
60
G
*
40
b
R
20
0
M
−20
−40
−60
−60
C
−40
B
−20
0
20
40
60
80
*
a
Figure 10.2: Color gamut in a∗ b∗ coordinate system. The images were printed
on different substrates, plain and photo glossy papers, but with the same substrate
setting (plain paper) in the printer driving program.
10.2.2
Two-dimensional representations of chromatic effects
Chromaticity diagrams for halftone patches on different substrates have been
plotted in Fig. 10.1. To explicitly illustrate the color difference between prints
on photo gloss paper and prints on the plain paper (office copy paper), colors
(noted with triangles) corresponding to patches of the same commanded ink
percentage but on different substrates have been connected by solid lines in pair
in the figure. Moreover, areas possibly covered by prints on the photo gloss
paper as well as by those on the copy paper are marked with solid and dashed
lines, respectively. As these areas represent the possible color that can be
produced by printing on the substrates, the photo gloss paper has significantly
greater capacity to represent color than the plain paper.
Chromatic effects of ink penetration can be further examined in terms of
chroma and hue of the colors. Figure 10.2 is a 2-dimensional representation
of the CIELAB color space. The colors corresponding to the same ink and
substrate but different ink percentages have been joined up with solid lines and
126
Chromatic effects of ink penetration
dashed lines, respectively, for prints on the photo gloss paper and on the copy
paper. Observations from the figure may be summarized as:
• chroma increases with respect to increasing ink coverage;
• even hue changes somewhat with respect to the ink coverage;
• prints on different substrates appear similar in the light tone but differ
significantly in both chroma and hue, in mid to dark tone colors.
• prints on the photo gloss paper produce colors of significantly greater
chroma.
These observations may be explained as the following. First as the ink
coverage increases, the white light reflected from the non-printed substrate
decreases and the color becomes more saturated. Second the hue variation
is partly due to the non-linear transformation from CIEXY Z to CIELAB
color system (Eq. 10.4) and partly due to dot gain. Under the Murray-Davis
assumption, the tristimulus values vary linearly with the ink percentage. However, existence of the optical dot gain leads to nonlinearity between XY Z and
the dot percentage, as known from Chapter 8. This, in turn, causes nonlinearity in the hue of the color. Third, because the two substrates have almost the
same color, curves corresponding to these substrates overlap each other when
the ink coverage is small. However, the curves gradually separate when the ink
coverage increases, because ink penetration increases. Finally, ink penetration
forms an ink-porous mixture which has strong scattering-power and leads to
the reflected light being less saturated in color (as been discussed in Chapter 7).
Color gamut of a printing system, is a measure of the color capacity that
can be delivered by the device. Color gamut of the system is usually obtained
by creating massive test patches with various combinations of ink percentages.
The test patches are measured and their color coordinates represented in a color
space (say CIELAB). The topology of the volume spanned by the measured
colors is called the color gamut. This process is usually called gamut mapping.
It is worth noticing that the ink jet printer is more flexible, which allows users
to adjust parameters that control the printing, compared to conventional offset
printing. These flexibilities provide redundancy in mapping the color gamut for
the ink jet printing system. Figure 10.3 shows colors printed on the same type
of substrate (photo gloss paper) but using different substrate settings (plain
paper and photo gloss paper, respectively) in the printer driving program.
One actually obtains slightly different color gamuts as shown in Fig. 10.3.
Another example of the redundancy is related to the ink level specification.
When a different ink level is specified in the printer driving program, the color
gamut mapped from the test patches can be significantly different as shown in
Fig. 10.4.
10.2 Evaluation of chromatic effects from experimental data
127
125
Photo paper setting
Plain paper setting
100
75
50
b
*
25
0
−25
−50
−75
−50
−25
0
25
a*
50
75
Figure 10.3: Dependence of the gamut mapping on substrate settings in the printer
driving program. Two sets of test patches were printed on the same type of photo
gloss paper but with different substrate settings, plain paper setting and photo gloss
paper setting, respectively.
yellow
100
75
red
green
4
5
50
b*
3
25
2
1
0
−25
magenta
cyan
−50
blue
−75
−75
−50
−25
a*
0
25
50
75
Figure 10.4: Dependence of the gamut mapping on the ink level specification in the
printer driving program. Color coordinates corresponding to ink level 1, 3, and 5 are
connected by solid, dash, and dotted lines, respectively.
128
Chromatic effects of ink penetration
1
0.9
0.8
Reflectance, R
0.7
Ink level 1
0.6
0.5
0.4
Ink level 5
0.3
0.2
0.1
0
400
450
500
550
600
650
700
Wavelengths, λ(nm)
Figure 10.5: Spectral reflectance of ink cyan, printed with ink level specification
from 1 to 5, in the printer driving program. Spectral reflectance of the white backing
is plotted in dotted line
Examination of Fig. 10.4 reveals that the chroma in the primary color increases from low to mid ink levels (1-3) and tends to be saturated when the
ink level (or ink volume) is further increased. Such a variation with respect to
the ink level builds up a hock type of structure for the primary colors (most
evidently seen from ink magenta, see Fig. 10.4). This phenomenon is a consequence of light absorption from the printed ink layer. As the ink has little
scattering power, light reflected from the print is essentially due to light reflection from the substrate (white) paper. As known, the spectral reflectance
of the ink cyan consists of absorption and transparent bands (see Fig. 10.5).
The reflected light from the absorption band is rapidly attenuated as the ink
level increases (from ink level 1 to 3), while the reflected light through the
transparent band decreases little. In other words, increasing absorption in the
absorption band is the dominant factor responsible for the increasing chroma.
After being almost completely attenuated in the absorption band at ink level 4,
the chroma of the print depends mainly on the amount of light reflected through
the transparent band. Because of existing absorption even in the transparent
band, the chroma decreases if the ink level is further increased. Consequently,
this leads to decrease in the chroma. This argument holds for other primary
colors, as well for the secondary colors. The variations in color composition of
different ink levels (see Fig. 6.4), however, contribute to the irregularity in the
secondary colors.
10.3 Evaluation in 3D color space: simulations
10.3
129
Evaluation in 3D color space: simulations
The chromatic effects of ink penetration are further investigated by simulations.
Since one can switch on or off ink penetration in the simulation, there is no
need to use different types of substrates as is the case in the experimental
evaluation. Therefore, the differences in optical and physical dot gain resulting
from using different types of substrates are avoided. In other words, the sole
effect of ink penetration can be studied.
To accurately map the color gamut of an ink-jet printing system, information like PSF (point spread function) of the substrate, ink spreading characteristics on the substrate, ink penetration of halftone dots, etc., are needed.
To obtain such information requires carefully designed measurements, which is
beyond the scope of this thesis. Nevertheless, we made attempts to model ink
penetration in halftone images [YK01, YKP01]. This will be further developed
in the future.
In the simulation, we took a simplified approach. The printer consists of 4
inks, cyan (c), magenta (m), yellow (y), and black (k). The inks are distributed
randomly and the distinct colored areas described by the DeMichel equation,
aw
ac
= (1 − c)(1 − m)(1 − y)(1 − k)
= c(1 − m)(1 − y)(1 − k)
am
ay
=
=
ab
ag
= cm(1 − y)(1 − k)
= c(1 − m)y(1 − k)
(10.11)
(10.12)
ar
ak1
ak2
= (1 − c)my(1 − k)
= cmy(1 − k)
= (1 − c)(1 − m)(1 − y)k
(10.13)
(10.14)
(10.15)
ak3
ak4
= c(1 − m)(1 − y)k
= (1 − c)m(1 − y)k
(10.16)
(10.17)
ak5
ak5
= (1 − c)(1 − m)yk
= cm(1 − y)k
(10.18)
(10.19)
ak7
ak8
= c(1 − m)yk
= (1 − c)myk
(10.20)
(10.21)
ak9
= cmyk
(10.22)
(1 − c)m(1 − y)(1 − k)
(1 − c)(1 − m)y(1 − k)
(10.7)
(10.8)
(10.9)
(10.10)
The reflectance is then computed by applying superposition
R=
ax Rx
(10.23)
130
Chromatic effects of ink penetration
where the subscript, x, denotes the colors in Eqs. (10.7)-(10.22), and Rx reflectance of the colors. For the primary or secondary colors, Rx is approximated
by reflectance of its solid print, while the remainders (x = k1, ..., k9) are approximated by reflectance of a solid black.
In the case of no ink penetration, Rx is computed by
Rx = Tx2 Rg
(10.24)
where Tx is the transmittance of the ink layer obtained in Chapter 6, and Rg
the spectral reflectance of the copy paper. In the case of ink penetration, Rx is
simply the spectral reflectance value of the printed colors (solid print on plain
paper).
In the simulation, the commanded ink coverage of the primary inks ranges
from 0 to 100 % at a step 4 %. To account for dot gain, the overall dot gain
characteristics obtained (for plain paper) in Chapter 8 were applied in the
simulation.
Figure 10.6 depicts the simulated color gamut of printing with (inner volume) and without (outer volume) consideration of ink penetration, in CIELAB
color space. Figure 10.6b is obtained from Fig. 10.6a by rotating it 180o around
the L∗ axis. As seen from these figures, the two color volumes have no points
in common expect for the paper white. In other words, prints of the same ink
coverage (but different ink penetration status) have different color coordinates
(lightness, chroma, and hue). Differences, that appear in the vicinity of solid
black is a consequence of increasing amount of pigmented black in ink composition for the dark tone printing. It is observed that from light to mid tone, the
gray is created by mixing the primary inks. However, for the dark tone gray
(from about 65%), the pigmented black ink is gradually added, in increasing
amounts as the tone values increase. Since pigment particles are much larger
in size than the dye molecules, they do not penetrate much into the substrate
(the dye compositions still do). Therefore, the simulations predict similar color
coordinates for the solid black in both cases. Consequently, a nail-type structure is formed in the vicinity of the black point. Furthermore, the evident
difference between the inner and the outer volumes demonstrate the dramatic
impact of ink penetration on the capacity of the color reproduction or the color
gamut. It was shown experimentally that the color gamut of printing on high
quality special paper can be up to 50% larger compared to prints on plain
paper [NA02].
10.4
Summary
This chapter presents an evaluation of the chromatic effects of ink penetration.
The evaluation is based on both experimental data analysis as well as simulation. On the experimental side, experimental color coordinates of patches
10.4 Summary
131
(a) Color gamut viewed from one angle
(b) 180o rotation around the L∗ axis from (a)
Figure 10.6: Simulated Color gamuts for prints, on office copy-paper, with (inner)
and without (outer) considering ink penetration. b) is a 180o rotation of a) around
the L∗ axis.
132
Chromatic effects of ink penetration
printed on two types of substrates, one with ink penetration and another without, have been compared. It is observed that ink penetration has significant
effect on the chroma and hue of the printed colors. Issues related to different optical and surface characteristics of the different substrates have been
discussed.
With help of simulation, one can study the pure effect of ink penetration.
Contributions from different substrates (color, physical and optical dot gain)
that exist in the experimental evaluation can be avoided. The simulations show
that the color gamut of the printing system is dramatically reduced because of
ink penetration.
Chapter 11
Summary and future work
11.1
Summary
An ink-jet printing system consists of three fundamental parts, inks, printing
engine, and substrates. Creation of images is determined by underlying physical
processes, i.e. ink application and ink setting. These processes govern ink
distribution (dot size and shape), ink spreading (physical dot gain), and ink
penetration.
Visual appearance of printed images is determined by the underlying physical phenomena, i.e., spectral light absorption and scattering by the substrate,
the inks, and ink-substrate interaction. Being able to understand and characterize the printing processes and the consequences (physical phenomena) for
image color appearance are essential in order to improve image quality and to
realize faithful image reproduction.
This thesis presents a few systematic methods for studying those issues
that mostly affect image creation and color rendition such as, ink penetration,
physical and optical dot gain.
The thesis rirst presents a theoretical method that deals with different forms
of ink penetration (uniform and non-uniform ink distribution in the direction perpendicular to the paper sheet). Expressions for spectral reflectance
of printed images as a function of scattering and absorption coefficients have
been given.
Second, the thesis describes a method to characterize properties (scattering and absorption powers) of the inks and printed ink volume, by combining
spectral reflectance measurements with simulations. The model has been successfully applied to simulate the measured spectral reflectance. It was shown
to be capable of predicting spectral reflectance of primary and secondary color
ink layers of any ink thickness.
134
Summary and future work
Third, the characteristics of the inks (scattering and absorption powers, ink
volume, etc.) are used for modelling ink penetration. Spectral reflectance of
solid prints on plain paper has been well simulated, and a method to determine
the depth of ink penetration is presented. Moreover, effects of ink penetration
are studied and discussed in detail.
Fourth, a model describing light scattering in substrates subject to ink
penetration resulting in optical dot gain has been developed. The model is
applicable to both mono- and multi-chromatic images. It has been found that
optical dot gain is a general phenomenon that exists in both mono- and multichromatic printed images. Theoretical analysis shows that optical dot gain
makes the color appear more saturated. Additionally, preliminary studies of
the overall dot gain (physical plus optical dot gain) has been carried out. The
studies show that the overall dot gain characteristics can be parameterized in
terms of polynomials. Application to a digital image has demonstrated that
such a parameterization can dramatically reduce the processing time in dot
gain correction.
Finally, the chromatic effects of ink penetration has been evaluated using
experimental data analysis and simulations. It has been found that ink penetration has significant impact on the chroma and hue values of printed images.
Because of strong scattering of the ink-paper mixture, the color of a print becomes much less saturated than that without ink penetration. Consequently,
the color capacity or the color gamut of a printing system is dramatically reduced by ink penetration.
11.2
Future work
Ink penetration or ink-paper interaction, in general, has a dramatic impact on
color reproduction. It deserves considerably more study. On the other hand,
the study of ink-paper interactions is a very complicated research topic involving experimental measurements and theoretical treatments. Both demand
knowledge of physics, chemistry, fluid dynamics. etc.
In terms of follow-up work, the following topics are of interest. First, applications of the models to different ink-paper combinations. Since ink-paper
interactions depend strongly on the properties of the ink and the paper, studies of different ink-paper combinations will increase our knowledge and understanding of the principal factors governing the quality of printed images. A
second extension of the models and methodologies is to halftone images. This
requires careful characterization of non-uniform ink spreading and ink penetration mechanisms. For example, the depth of ink penetration under the
center of an ink dot is possibly bigger than that under the edge of the dot. A
third followup can possibly be model testing and improvement. The models
developed in this dissertation should be subjected to more tests, experimental
11.2 Future work
135
and numerical examinations, in order to evaluate their power and limitations.
Conversely, such evaluations will be very helpful for improving the models.
Chapter 12
Appendix
A
Mathematical derivation for Equation (6.4)
According to Eq. (6.2), the reflectance values (RI and RII ) corresponding to
ink thicknesses z and 2z, respectively, can be expressed as
RI
=
s[A − sRg ]e−
s2 −(A)2
A
z
s2 −(A)2
−
A
A[A − sRg ]e
− A[s − ARg ]
z
− s[s − ARg ]
(A-1)
RII
=
s2 −(A)2
−
A
s[A − sRg ]e
2z
s2 −(A)2
−
A
A[A − sRg ]e
− A[s − ARg ]
2z
− s[s − ARg ]
(A-2)
where
A=s+k−
k 2 + 2ks
(A-3)
From Eq. (A-1) and (A-2) one can derive the following relations
e−
e−
s2 −A2
A
s2 −A2
A
z
2z
=
=
(s − ARg )(A − sRI )
(s − ARI )(A − sRg )
(s − ARg )(A − sRII )
(s − ARII )(A − sRg )
(A-4)
(A-5)
Observe that the term on left side of Eq. (A-5) is the square of that of Eq. (A-4).
Making use of this relation, one can therefore obtain the following relation,
(A − sRII )
(s − ARg )(A − sRI )2
=
(s − ARII )
(s − ARI )2 (A − sRg )
(A-6)
138
Appendix
Replacing s and k with sz and kz, one can readily obtain the relation given by
Eq. (6.4), i.e.
(Az − szRII )
(sz − AzRg )(Az − szRI )2
=
(sz − AzRII )
(sz − AzRI )2 (Az − szRg )
B
(A-7)
Probability model for optical gain
Assume that the percentage of the paper under the dots is σ (Σ1 ) and the paper
between the dots (Σ0 ) is (1 − σ), respectively. If the intensity of irradiance onto
the whole system is I0 , the flux of photons striking the dots (Σ1 ) and the paper
(Σ0 ) areas are I0 σ and I0 (1 − σ), respectively. Then, the flux Jmn of photons
scattered from Σm into Σn (m, n = 0, 1) is given by
J11
= I0 T 2 σP11
J10
J01
= I0 T σP10
= I0 T (1 − σ)P01
(A-9)
(A-10)
J00
= I0 (1 − σ)P00
(A-11)
(A-8)
Due to the optical reciprocity (see Sec. 8.2.1), there is
J10
= J01
= I0 T pσ(1 − σ)
(A-12)
Comparing Eq. (A-12) with Eqs. (A-9) and (A-10), one obtains
= pσ
= p(1 − σ)
(A-13)
(A-14)
P10 σ = P01 (1 − σ)
(A-15)
P01
P10
and
The total flux of the photons emerging from the paper between the dots
(Σ0 ) , J0 , is a summation of J10 and J00 , i.e.
J0
= J10 + J00
= I0 [T P10 σ + P00 (1 − σ)]
(A-16)
Consequently, the reflectance measured from the paper between the dots can
be calculated from
R0
J0
I0 (1 − σ)
= P00 + T pσ
=
(A-17)
B Probability model for optical gain
139
Applying the constraint conditions (Eqs. (8.19) and (8.20)),
P11 + P10
= Rg
(A-18)
P01 + P00
= Rg
(A-19)
one obtains
P00
= Rg − P01
= Rg − pσ
(A-20)
Equation (A-17) can then be simplified into
R0 = Rg − pσ(1 − T )
(A-21)
Similarly, the total flux of the photons emerging from the dots (Σ1 ), J1 , is
therefore a summation of J01 and J11 ,
J1
= J11 + J01
= I0 [T 2 P11 σ + T P01 (1 − σ)]
(A-22)
and the corresponding reflectance reads
R1
J1
I0 σ
= T 2 P11 + T p(1 − σ)
= T 2 Rg + pT (1 − T )(1 − σ)
=
(A-23)
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