INK-PAPERINTERACTION Astudyinink-jetcolor …...INK-PAPERINTERACTION Astudyinink-jetcolor reproduction c Li Yang Department of Science and Technology Link¨oping University SE-601

Linkoping Studies in Science and TechnologyDissertations No. 806

INK-PAPER INTERACTION

A study in ink-jet colorreproduction

Li Yang

Department of Science and TechnologyLinkoping University, SE-601 74 Norrkoping, Sweden

Norrkoping, April 2003

INK-PAPER INTERACTION

A study in ink-jet colorreproduction

c© Li Yang

Department of Science and TechnologyLinkoping UniversitySE-601 74 Norrkoping

Sweden

ISBN 91-7373-613-9 ISSN 0345-7524

Printed in Sweden by UniTryck, Linkoping, 2003

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Abstract

An ink jet printing system consists of three fundamental parts: inks, printingengine, and substrates. Inks are materials creating color by selectively absorb-ing and scattering the visible illumination light. The printer acts as an inkdistributor that governs the ink application. Finally, the substrate acts as areceiver of the inks and forms the images. Ink setting on the substrate is acomplex process that depends on physical and chemical properties of the inksand the substrates, and their bilateral interactions. For a system consisting ofdye based liquid inks and plain paper, the ink moves together with the liquidcarrier before the pores absorb the liquid. This process contributes to seriousink spreading on the surface along the paper fibers. At the same time theink spreads down into the pore structure. This causes severe dot deformation,physical dot gain and ink penetration. Understanding the consequences ofthese phenomena and above all being able to characterize their impact on colorreproduction is of great importance. Moreover this knowledge is fundamentalfor finding solutions to ink-penetration related problems. This thesis presentsstudies of some important issues concerning image reproduction quality for dyebased ink-jet printing on ordinary plain paper (office copy paper), such as inkpenetration, optical dot gain, and even physical dot gain. The thesis beginswith theoretical developments to the Kubelka-Munk theory, which allows oneto study even non-uniform ink penetration into the substrate. With the knowl-edge of scattering and absorption coefficients and ink thickness, reflectance canbe computed by solving differential equations. Three forms of ink penetra-tion, uniform, linear, and exponential have been studied. A method is thenpresented for obtaining fundamental properties of the inks from spectral re-flectance measurements, like the scattering- and absorption-power of inks, inklayer thickness, and ink mixing scheme for the generation of secondary colors.The method is further developed for modelling the ink penetration in printingsystems consisting of dye based liquid inks and plain paper. By combiningthe spectral reflectance measurements with theoretical simulations, quantitieslike the depth of ink penetration is determined. These quantities, in turn, areused to predict the spectral reflectance of prints. Simulated spectral reflectancevalues have been in fairly good agreement with experimental results. Modelsdealing with light scattering inside the substrate resulting in optical dot gainfor halftone printing, in the case of existing ink penetration, have been devel-oped for both mono- and multi-color printing. It is shown that the optical dotgain leads to higher color saturation than predications from Murray-Davis ap-proximation. Additionally, tentative studies for physical dot gain were made.Finally, an evaluation of the chromatic effects of the ink penetration for print-ing on office copy paper has been carried out based on both experimental dataand simulations. It is found that ink penetration has a dramatic impact onchroma and hue of the color, and the color saturation is significantly reduced

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by the ink penetration. Consequently, the capacity for color representation, orthe 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 peopleand in many ways.

First of all, I would like to express my sincere gratitude to my supervisorProfessor Bjorn Kruse for giving me the opportunity to pursue the study inhis group, sharing his broad and deep knowledge and experiences in GraphicArts. His suggestions, comments, and inspiration have sparked initiatives ofthe researches. His continuous efforts for establishing contacts with researchinstitutes and industries have been very helpful for promoting and improvingour work. His encouragement, appreciation, and sharp view of the subjects andworks have been particularly important. Words like “I trust you” have meanta lot.

Associate professor Reiner Lenz, has acted as co-supervisor in the last coupleof years. His questions, comments, criticisms, and discussions have been veryimportant inputs to the researches and the formulation of the dissertation. Hisenthusiasm and research style have been strong influence.

Senior researcher, Nils Pauler, in M-Real Research (Sweden), has been aparticularly important person outside of the university. He, together withProfessor Kruse, initiated collaboration in the studies of ink penetration. Hishelp in spectral reflectance measurement has been very important for havinga good start. His kindness and hospitality when I visited Ornskoldsvik havemade the research visits not only rewarding but also enjoyable. He and his teammember, Jerker Wagberg, have been wonderful people to collaborate with.

I would like to thank all group members, for creating an amicable and ac-tive research atmosphere, providing courses and holding interesting seminars.Thanks Arash, Daniel, Linh, Sasan, and Thanh for pleasant coffee breaks andfree talks, and interesting discussions of various topics, from football to uni-verse.

Special thanks to Professor Hans Agren at Royal Technology Institute(KTH) for inviting me to Sweden, and for very fruitful collaborations duringthe time when we were at Physics Department (IFM) of Linkoping University.

Many thanks to our research engineers, especially Sven Franzen, for main-taining the office- and Lab-systems. Thanks to our secretaries, especially So-phie 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 linguisticreading and valuable technical comments during the time he had to meet a fewdeadlines of himself. Thanks Dr. Sasan Gooran for a helpful proof reading.

I also wish to thank all my Chinese friends in Linkoping, Norrkoping, andother places, for their friendship and constant help, especially Fang Hong andLin Dan, Luo Yi and KeZhao, and QinZhong and ZhuangWei.

Thanks to Swedish Foundation for Strategic Research for financial support

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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 turbidmedia, 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 inmulti-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 penetra-tion 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 inmulti-chromatic tone production, J. Imaging. Sci. Tech., 45 (2001)198-204;

6. L. Yang and B. Kruse, Chromatic variation and color gamut re-duction due to ink penetration, in Proc. TAGA Conf., 399-407, May6-9, 2001, San Diego, CA;

7. L. yang, B. Kruse, and N. Pauler, Modelling ink penetration in ink-jet printing, in Proc. IS&T NIP17 Conf., 731-734, Oct. 2001, FortLauderdale, FL;

8. L. Yang, R. Lenz, and B. Kruse, Light scattering and ink pene-tration 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-jetprinting, in Proc. TAGA Conf., 255-265, Apr. 2002, Asheville, NC;

10. L. Yang, Modelling ink-jet printing: Does Kubelka-Munk theoryapply ?, in Proc. IS&T NIP18 Conf., 482-485, Sep. 2002, San Diego,CA;

11. L. Yang, Color reproduction of inkjet printing: model and sim-ulation, J. Opt. Soc. Am. A, 2003 (accepted for publication);

12. L. Yang, Determination for depth of ink penetration in ink-jetprinting, 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 11.1 Goal of the study . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Status of studies and our contribution . . . . . . . . . . . . . . 2

1.2.1 Extension to Kubelka-Munk theory . . . . . . . . . . . . 21.2.2 Evaluation of effects of ink penetration . . . . . . . . . . 31.2.3 Optical dot gain . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Structure of the dissertation . . . . . . . . . . . . . . . . . . . . 6

2 Paper 92.1 Structures and properties of paper . . . . . . . . . . . . . . . . 9

2.1.1 Fibres, fillers and coating . . . . . . . . . . . . . . . . . 92.1.2 Density and porosity . . . . . . . . . . . . . . . . . . . . 10

2.2 Optical properties and measurements . . . . . . . . . . . . . . . 122.2.1 Brightness, opacity and gloss . . . . . . . . . . . . . . . 122.2.2 Optical measurements . . . . . . . . . . . . . . . . . . . 14

2.3 Paper permeability and mechanism of ink penetration . . . . . 18

3 Ink-jet printers and inks 213.1 Ink-jet technologies . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 The continuous ink-jet . . . . . . . . . . . . . . . . . . . 223.1.2 Drop-on-demand ink-jet . . . . . . . . . . . . . . . . . . 23

3.2 Characteristics of ink-jet printers . . . . . . . . . . . . . . . . . 25

x Contents

3.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.2 HP970Cxi ink-jet printer . . . . . . . . . . . . . . . . . 27

3.3 Ink-jet ink technologies . . . . . . . . . . . . . . . . . . . . . . 273.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.2 Dye-based and pigment-based inks . . . . . . . . . . . . 28

4 Optical modelling: an overview 314.1 Radiative Transfer Theory . . . . . . . . . . . . . . . . . . . . . 314.2 Phase function . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Multi-flux theory . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Kubelka-Munk method . . . . . . . . . . . . . . . . . . . . . . . 344.5 Monte-Carlo simulation . . . . . . . . . . . . . . . . . . . . . . 35

5 Extended Kubelka-Munk theory and applications 395.1 Assumptions in Kubelka-Munk theory . . . . . . . . . . . . . . 395.2 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . 425.2.2 Boundary reflection . . . . . . . . . . . . . . . . . . . . 43

5.3 Models of ink penetration . . . . . . . . . . . . . . . . . . . . . 455.3.1 Uniform distribution . . . . . . . . . . . . . . . . . . . . 455.3.2 Linear distribution . . . . . . . . . . . . . . . . . . . . . 465.3.3 Exponential distribution . . . . . . . . . . . . . . . . . . 46

5.4 Solutions of the differential equations . . . . . . . . . . . . . . . 475.4.1 Uniform ink distribution . . . . . . . . . . . . . . . . . . 475.4.2 Linear ink distribution . . . . . . . . . . . . . . . . . . . 505.4.3 Exponential distribution . . . . . . . . . . . . . . . . . . 54

5.5 Simulations for uniform- and linear-ink distribution . . . . . . . 555.5.1 Convergency of the series expansion. . . . . . . . . . . . 555.5.2 Optical effects of ink penetration . . . . . . . . . . . . . 575.5.3 Correction for boundary reflection . . . . . . . . . . . . 605.5.4 Effect on color gamut . . . . . . . . . . . . . . . . . . . 61

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Characterization of inks and ink application 636.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.2 Experiment, data analysis and simulation . . . . . . . . . . . . 64

6.2.1 Samples and measurements . . . . . . . . . . . . . . . . 646.2.2 Data analysis and simulation . . . . . . . . . . . . . . . 65

6.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . 676.3.1 Spectral characteristics of the primary inks . . . . . . . 676.3.2 Spectral reflectance values and relative ink thicknesses of

the primary inks . . . . . . . . . . . . . . . . . . . . . . 68

Contents xi

6.3.3 Spectral reflectance values and relative ink thickness ofsecondary colors . . . . . . . . . . . . . . . . . . . . . . 70

6.4 Remarks for application of Kubelka-Munk theory . . . . . . . . 726.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7 Characterization of ink penetration 757.1 Optical properties of plain paper . . . . . . . . . . . . . . . . . 757.2 Assumptions and notations . . . . . . . . . . . . . . . . . . . . 78

7.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 787.2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.3 Simulation of print on office copy-paper . . . . . . . . . . . . . 807.3.1 Primary colors . . . . . . . . . . . . . . . . . . . . . . . 807.3.2 Secondary colors . . . . . . . . . . . . . . . . . . . . . . 83

7.4 Optical effect of ink penetration . . . . . . . . . . . . . . . . . . 847.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

8 Dot gain in black and white 898.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.1.1 Murray-Davis equation . . . . . . . . . . . . . . . . . . . 898.1.2 Yule-Nielsen equation . . . . . . . . . . . . . . . . . . . 908.1.3 Status of the studies . . . . . . . . . . . . . . . . . . . . 92

8.2 Model and methodology . . . . . . . . . . . . . . . . . . . . . . 938.2.1 Point spread function approach . . . . . . . . . . . . . . 938.2.2 Probability approach . . . . . . . . . . . . . . . . . . . . 968.2.3 Impacts of the optical dot gain . . . . . . . . . . . . . . 100

8.3 Overall dot gain of monochromatic colors . . . . . . . . . . . . 1018.3.1 A model for overall dot gain . . . . . . . . . . . . . . . . 1018.3.2 Simulation of the overall dot gain . . . . . . . . . . . . . 103

8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

9 Dot gain in color 1079.1 Reflectance of a multi-color image . . . . . . . . . . . . . . . . 1079.2 Optical dot gain in multi-color tone reproduction . . . . . . . . 1109.3 Simulation for multi-layer color image . . . . . . . . . . . . . . 111

9.3.1 Two inks of round dots: dot on dot . . . . . . . . . . . . 1129.3.2 Two inks of square dots: dot on dot . . . . . . . . . . . 1159.3.3 Two inks of round dots: random dot distribution . . . . 117

9.4 The effects of optical dot gain on color reproduction . . . . . . 1199.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

xii Contents

10 Chromatic effects of ink penetration 12110.1 Basics in colorimetry . . . . . . . . . . . . . . . . . . . . . . . . 121

10.1.1 CIEXY Z color space . . . . . . . . . . . . . . . . . . . 12110.1.2 Chromaticity diagram . . . . . . . . . . . . . . . . . . . 12210.1.3 CIELAB color space . . . . . . . . . . . . . . . . . . . 122

10.2 Evaluation of chromatic effects from experimental data . . . . . 12310.2.1 Parallel comparison of prints on two types of substrates 12310.2.2 Two-dimensional representations of chromatic effects . . 125

10.3 Evaluation in 3D color space: simulations . . . . . . . . . . . . 12910.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

11 Summary and future work 13311.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13311.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

12 Appendix 137A Mathematical derivation for Equation (6.4) . . . . . . . . . . . 137B Probability model for optical gain . . . . . . . . . . . . . . . . . 138

Bibliography 140

Chapter 1

Introduction

Ink-jet printing a is commercially young but rapidly developing printing tech-nology. Success in making printers of high print resolution, color capacity, yetvery affordable price has made ink-jet printers available not only for big com-panies, but also for private users and small business units. According to CapVentures [Ash01], of all the printing applications, 60% was printed on uncoatedstocks, like plain paper or office copy paper during year 2000. This figure willreach 90% by year the 2005. Studies of ink-jet printing on uncoated substratesis therefore of great importance not only from an academic perspective but alsofrom 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, andfinal printout are necessary. This thesis thus consists of various study phasesand goals.

One goal is to establish a general method that deals with various typesof ink penetration, uniform or non-uniform, whatsoever. Because of existingwidely diversified ink-paper combinations, mechanisms that are responsible forink-paper interaction differ from one combination to another.

A second goal is to characterize printer and optical properties of inks. It in-cludes information about scattering and absorption characteristics of the inks,the volume of the inks being printed, and color mixing schemes for the genera-tion of secondary colors. In the case of having ink-penetration, this informationserves as an input for further studies.

The underlining phenomenon of ink penetration is formation of a layer ofan 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 fun-damentals of how the ink-paper interaction affects color reproduction and tocharacterize optical effects of ink penetration. Basic quantities that character-ize ink-penetration, such as penetration depth, is indirectly determined fromexperimental spectral reflectance values. A model that takes into account inkand paper mixing is developed allowing prediction of spectral reflectance of realprints.

That ink penetration into an uncoated substrate impact strongly on colorrepresentation of the printed images is an experimentally known fact. One ofthe goals is therefore to evaluate the impact of ink penetration from experimen-tal 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 proba-bly has to be considered. Thus, model development for dot gain description,including optical dot gain and physical dot gain, is an important feature of ourstudy.

Modelling and simulation are important tools contributing to our under-standing, deeper insights into the problems. Moreover, they serve also to guideus 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 thework presented in this thesis. It also provides a brief description of our contri-bution as well.

1.2.1 Extension to Kubelka-Munk theory

The original theory of Kubelka-Munk (K-M) was developed for light propaga-tion in parallel colorant layers of infinite xy-extension [KM31, Kub48]. Thefundamental assumptions of the K-M theory are that the layer is uniform andthat light distribution inside the layer is completely diffused. From these as-sumptions, the light propagation in the layer was simplified into two diffuse lightfluxes through the layers, one proceeding upward and another simultaneouslydownward. After its introduction in the 1930’s, K-M theory was subjected toextensions by removal of some of the assumptions. Among others, the bound-ary reflection at the interface bewteen two adjacent media was introduced bySaunderson [Sau42], i.e, the well-known Saunderson correction. Kubelka him-self also made an attempt to extend the applicability of the theory to opticallyinhomogeneous samples [Kub54]. However, this extension was only applied toa special case of inhomogeneous media, in which the ratio of the absorption tothe scattering is constant.

1.2 Status of studies and our contribution 3

Recently, Emmel and Hersch introduced an elegant mathematical formula-tion of the Kubelka-Munk theory, based on matrices [Emm98, EH99]. Theyproposed also a mathematical framework unifying the K-M model with theNeugebauer model [EH00]. This allowed them to apply the K-M theory to ahalftone image. Therefore optical dot gain was studied. Very recently, Mouradextended 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 scatteringin the substrate and therefore the optical dot gain.

The object of our studies is mainly on optical performance of a layer con-sisting of non-uniform media concentration. This is a topic that has not beenexplored theoretically. One of the applications of the study is ink penetration,where the ink distribution inside the substrate may be nonuniform, dependingon the mechanism of ink penetration. Phenomenon of non-uniform ink pen-etration has been observed experimentally (cross section image) by means ofmicrotomy, in ink jet printing [GKOF02]. In Chapter 5, we work out a frame-work 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, appli-cations of the K-M theory to ink jet printing has substantiated the applicabilityof the theory. Explanations and discussions around these issues are given inChapter 6.

1.2.2 Evaluation of effects of ink penetration

Absorption of ink constituents by the substrate, or ink penetration is significantover a range of timescales, from the first stages of ink-transfer and ink-dryingby absorption, through to long-term stability [Voe52, Oit76, Str88, MK00].Studies of ink penetration related issues have long been important topics inoffset printing and cover a wide range of topics, such as print gloss and printdensity, print defects (unevenness and mottle), print through, separation of inkconstituents [SGS00, Rou02], etc.

Studies have so far mainly concentrated on understanding the mechanismsof ink penetration and developing materials for paper coating. Reported stud-ies of the optical and chromatic effects of ink penetration are few, even thoughsuch studies have recently intensified [McD02, PL02, NA02]. Among others,Bristow and Pauler proposed a method by which the depth of ink penetra-tion could be determined indirectly from spectral reflectance measurementdata [Bri87, Pau87], in offset printing. The method was based on K-M the-ory, the additivity assumption and a uniform ink distribution. In recent years,Pauler and his research group have been very active in modelling and simulatingink penetration in ink jet printing [PWE02a, PWE02b, PWE02c].

Experimental measurements and theoretical simulations for the optical andchromatic effects of ink penetration have not been conducted without difficulty.

4 Introduction

From the experimental side, the measurements may be made by comparingprints having ink penetration with those that dot not. Unfortunately, onecan not obtain prints with and without ink penetration (into the celluloseporous structure of the substrate) by using the same ink-substrate combination.Different types of substrate, like plain paper and high grade photo paper haveto be used when comparisons are made. For halftone images, differences incolor and optical dot gain characteristics for different substrates, will contributeto the color difference between the prints. Moreover, differences in surfacecharacteristics between the substrates used for the prints may lead to significantdifference in physical dot gain.

Theoretical simulations have the advantages that one can use exactly thesame ink-substrate combinations when the effects of ink penetration is evalu-ated. One has the possibility to manipulate the ink penetration by switching iton or off in the simulation. The underlining difficulty is to establish a theoret-ical 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 inkpenetration problems, even though more sophisticated theories like Multi-fluxRadiative 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 K-M model with spectral reflectance measurements. We present a systematicframework stretching from the characterization of inks and ink application,to the modelling and the simulation of ink penetration. It begins with thedetermination of the scattering and absorption characteristics of pure ink layersof primary colors, as well as thickness of the ink layers. The scattering andabsorption characteristics of the primary inks are then applied to determine thecolor mixing scheme for the generation of secondary colors. The applicationsserve not only as tests to the quality of data, but also to the applicability ofthe 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-papermixture). The additivity assumption is modified by considering the correlationbetween the light scattering (from the paper) and the absorption (from theinks). With the help of our model, the depth of ink penetration is indirectlydetermined from the measured reflectance values. Spectral reflectance of inkjet prints on office copy paper (in both primary- and secondary-colors) havefairly well been reproduced by the simulation (see Chapter 7, for details).

Evaluation of the impact of ink penetration is obtained from both exper-iment and simulation perspectives. Color difference between prints with andwithout ink penetration are represented in chromaticity diagrams (2D) andCIELAB color space (3D). It shows that ink penetration has a dramatic im-pact on the chroma of the print and even on the hue. This leads to a dramaticreduction of color gamut which is absolutely negative for color reproduction

1.2 Status of studies and our contribution 5

(see Chapter 10, for details).

1.2.3 Optical dot gain

The physical phenomenon behind the so-called optical dot gain, or Yule-Nielseneffect, is basically light scattering within the substrate paper. Light scatteringinside the substrate of a printed image is a very complex process. Studies ofthe 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 Nielsenfound that the optical dot gain can be well approximated by introducing anempirical 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 contri-bution from multiple internal reflections between upper and lower boundariesof the substrate [CY53]. The assumptions made in the Clapper and Yule modelare that the ink layer is uniform and that the light is completely scattered bybulk scattering. Complete light scattering is a physical approximation whenthe average lateral light scattering distance is much greater than the size of thehalftone element.

In 1978, Ruchdeschel and Hauser [RH78] provided a physical explanation forthe n factor and showed that 1 ≤ n ≤ 2, if only optical dot gain involved. n = 1and 2 represent two extremes for the light scattering, where n = 1 correspondsto no light scattering while n = 2 corresponds to complete scattering. Theoriginal intention of the Yule-Nielsen model was to involve the optical dotgain. Nevertheless, it has often been applied to cases where there was alsophysical dot gain involved. Unsurprisingly, they got n factor bigger than 2 andsometimes much bigger than 2 [BT96]. Consequently, the physical meaning ofsuch n factor is difficult to explain.

In the past 10 years, Kruse and Wedin [KW95], among many others, pro-posed an approach which was thoroughly studied and fully implemented byGustavson [Gus97a, Gus97b]. The approach simulated the light scatteringprocess from a fundamental level. It was based on direct numerical simulationof scattering events which depend on the optical properties of the materials,the halftone frequency and the halftone geometry. This approach is similar innature to the Monte-Carlo method that is briefly explained in Sec. 4.5. Statis-tics are recorded over a large number of light scattering events. From these theprobability of an event can be established. Arney [Arn97] and Hubler [Hub97]independently proposed similar models based on probability descriptions of thelight scattering. In their models, the light scattering inside the paper was de-scribed by the probabilities that a photon emerges from the inked and non-inkedareas. These probabilities depend on the positions where the photon enters thepaper and emerges from the paper, as one will see in Sec. 8.2.2. A point spreadfunction (PSF) is a different representation of light scattering. Using the PSF

6 Introduction

approach, Rogers [Rog97, Rog98a, Rog98b, Rog98c] presented a method deal-ing with the light scattering process. He proposed a matrix approach wherethe tristimulus values of a halftone image could be calculated as the trace ofa product of two matrices. So far, the studies have not provided any explicitexpression for the reflectance or tristimulus as a function of dot percentage(as it was given by the Neugebauer equation). Moreover, the studies have sofar been limited to the mono-color or black-and-white case and no studies formulti-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 expressionsfor reflectance and optical dot gain have been worked out, studies of the opticaldot gain have been extended to multi-color cases, and ink penetration has beenincluded in the model. Detailed descriptions can be found in Chapters 8 and9.

1.3 Structure of the dissertation

This thesis consists of 6 parts and 11 chapters. The contents of the presenta-tion 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 theoret-ical extension of Kubelka-Munk theory (Chapter 5). This provides us with ageneral description of ink penetration. It is then applied to determine ink char-acteristics, ink application, and ink penetration in full tone prints (Chapters6 and 7). Furthermore, the model and application are extended to halftonecases, where dot gain plays an important role (Chapters 8 and 9). Finally, thechromatic effects from the ink penetration and dot gain are evaluated with helpof experimental data, as well as simulations.

Each chapter is intended to be self consistent and sufficiently independentof all others, that a reader only interested in a given topic needs only viewthat particular chapter. However, the chapters are also arranged according toa 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. Chapter3 briefly describes ink jet technologies, inks, and printers used in the thesisstudy. Chapter 4 provides an overview of optical modelling and simulations.

Part II consists of a single chapter (Chapter 5). A theoretical frameworkis proposed as an extension of the Kubelka-Munk theory. The extension allowsone to deal with both uniform and nonuniform ink distribution in cases of inkpenetration. Expressions for reflectance and transmittance of three types ofink 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 7

patches and consists of two chapters, Chapters 6 and 7. In studying ink pene-tration, one actually deals with an ink-paper mixture. It is therefore essentialto know what kind of ink, in terms of its optical properties (scattering andabsorption), has been printed and how much ink has penetrated into the sub-strate. Chapter 6 aims at characterizing inks and ink application controlledby ink-jet printer, such as scattering- and absorption-power of the inks, theamount of ink printed onto a substrate, and color mixing scheme in generationof secondary colors. Results of Chapter 6 serve as input for Chapter 7 whereink penetration into plain paper is studied. Quantity alike depth of the inkpenetration has been determined from simulations.

Part IV presents models and simulations for halftone-image-related is-sues, such as optical dot gain in both monochromatic (Chapter 8) and multi-chromatic (Chapter 9) printing cases. Even overall dot gain (physical- plusoptical-dot gain) has been tentatively studied (Chapter 8). The model hasbeen applied to simulating optical dot gain for different dot shapes and lo-cations. It is found that the optical dot gain results in more saturated colorsensation. Applications of the model to overall dot gain is tested by applyingit to a digital image.

Part V consists of two chapters, Chapters 10 and 11. In Chapter 10, theimpact of the ink penetration on capacity for color representation was evaluatedfrom both experimental and simulation perspectives. The impact is representedin (2D) chromaticity diagram and 3D color gamut (in CIELAB color space).It is found that ink penetration has a dramatic impact on chroma and evenon hue, and leads to a dramatic reduction in color gamut. Chapter 11 gives asummary of the thesis.

Part VI consists of an Appendix and the Bibliography.

Chapter 2

Paper

2.1 Structures and properties of paper

In the Graphic Arts industries, paper is the most commonly used substratethat receives the inks and colorants to form images. The properties of thepaper substrate are important partly because the substrate is visible betweenthe printed areas, and partly because the substrate defines the background re-flectance for the ink layer. Moreover, optical properties, mechanical properties,permeability to liquids, and so on, directly affect the quality of the images andthe production practices. Paper-making is a multi-disciplinary subject involv-ing mechanics, physics, optics, chemistry, etc. Interested readers can find moredetailed descriptions elsewhere [NKP98, Ran82, Bri86, Bak97]. In this chapterwe give a brief overview of the structures and properties of paper which areeither 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 fibersare much longer than the thickness of the paper sheet, the network is more orless flattened out [Nor91] and therefore almost two-dimensional (in xy-plane).For paper-making fibers, the basis weight is about 5 − 10g/m2, which meansthat ordinary printing and writing papers consist of normally 5-20 “layers”of fibers [NKP98]. The two-dimensional (2D) structure governs many paperproperties, such as in-plane mechanical properties. The fibre network in paperis 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 onthe 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 meetthese demands. As a result chemicals and post paper making processes are nowcommonly applied. For example, by adding fillers such as kaolin clay or calciumcarbonate to the furnish one can increase the specific scattering strength andin turn improve the opacity. Additionally, a post paper-making process whichis usually called coating is employed in order to improve both mechanical andoptical 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 reviewsabout the relation between the addition of fillers, coating constituents and theproperties 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 threedimensional [Fay02]. The z-directional structure and material distribution af-fect paper properties such as bulk, bending stiffness, optical properties, andsurface roughness. The distribution of fines and fillers are particularly impor-tant.

Density and thickness are basic macroscopic characteristics of paper struc-ture. 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 paper-making 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 offibers but more strictly an aerogel. The network of fibers embraces and createsa network of pores, and paper is thus a two-phase system in which the pores orvoids between the fibers are an important part of the paper structure [Bri86].The 3D pore structure controls the density and optical properties directly, whilethe mechanical properties and dimensional stability are indirectly controlledthrough the relative bonded area [NKP98]. It is therefore useful to introducea term called porosity, φ, that is defined as the ratio of pore volume to totalvolume,

φ =V − VfV

= 1− ρ

ρf(2.2)

where V is the volume of the entire sheet, Vf is the volume occupied by thefibres, 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. Thevariation of φ is controlled by the paper-making furnish and its beating level.Mechanical pulps with stiff and bulky fibers usually give higher paper porositythan chemical pulps. The beating of chemical pulp decreases porosity since

12 Paper

fiber flexibility and collapse increase during beating. Therefore, a sandwichtype of layer structure using chemical pulps in the upper- and lower-surfacelayers and a mechanical pulp in the middle layer, will make paper with goodsurface properties (smoothness, printability, and bending stiffness) and highporosity at the same time. In practice, for example, newsprint often primarilyconsists of mechanical pulp while copy paper mostly of chemical pulp. Mixtureof mechanical and chemical pulps finds use especially in printing papers andmultiply boards [RNN98].

The pore size distribution is also influenced by operation such as calen-dering, the mean pore size becoming smaller as a result of such compressivetreatments. Additionally, measurements made, for example by immersing asheet in a low viscosity oil show that virtually all the pore volume is accessibleto the liquid and it can be assumed that there are no isolated inaccessible voidswithin the structure [Bri86].

Surface roughness is another important characteristic of paper. It influencesthe optical properties such as gloss. High roughness reduces the contact areabetween 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 inkpenetrating into the paper sheet. Ink penetration determines how much of thetransferred ink remains on the surface of paper. Small penetration gives highprint density. It has been observed that unevenness in offset printing comesfrom the spatial variation in surface roughness and ink penetration [Kaj89].Detailed studies on the ink penetration for ink jet printing will be presented inChapter 7.

2.2 Optical properties and measurements

Optical properties such as opacity, brightness, and gloss are important to usersof most paper and board grades. In order to successfully manufacture paperwith desired optical properties, it is important to understand the physical prin-ciples of sheet structure and composition that determine these characteristics.Measurements of these optical properties can in turn provide information forcharacterizing 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 thesheet, light spreads and scatters in all directions. Some light (Ibulk) eventuallyreflects 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 intensityto the incident intensity,

T =ItranI0

. (2.4)

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 nochange 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 colorand because the human eye perceives blue colors as brightness.

Opacity characterizes the ability of a single sheet to hide text or pictureson the back side of the sheet. Quantitatively, opacity is defined as a ratio ofreflectance 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 smoothtop surface. A glossy material is characterized by a high reflectance in thedirection of regular reflection or close to that direction. If the illuminationis white, the glossy reflection is normally colorless despite the color pigmentsunder the surface of the print [Gra01]. Every day experience tells us that thegloss of printed paper depends considerably on the illumination and detectionangles.

Gloss paper has a high specular reflectance that is closely related to thesurface smoothness or in other words, surface roughness of the paper. Thesurface roughness of paper can be sorted into three categories according to thein-plane resolution: Optical roughness at length scale < 1 µm; micro roughnessat 1−100 µm; and macro roughness at 0.1−1mm. Gloss is a combination of theeffects of micro roughness and optical roughness of the paper surface. Microroughness affects gloss because titled surface facets reflect light in differentdirections as shown in Fig. 2.3

The scale of optical surface roughness has the same magnitude as the wave-length of the light. Optical roughness therefore causes light diffraction. Thetotal reflection, Rt, from a surface of normally distributed height is [Mic84]

Rt = R0exp[−(4πσ)2

λ2] +R0

25π4

m2(σ

λ)4(∆Θ)2 (2.6)

14 Paper

where λ is the wavelength of light, σ the root mean square (RMS) surfaceroughness, m the mean gradient of the surface, R0 the reflectivity of fibers orcoating color, and ∆Θ the solid angle of measurement. The first term is thespecular component responsible for the gloss of paper. Clearly, as roughness,σ, increases the exponential function decays to zero and gloss vanishes. Incontrast, the diffuse reflectance increases with roughness as σ4.

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 choiceof pigments and binders, together with the average size and distribution of thecoating 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 Artsindustries. In general, the measurements need to be relevant to what the humansees. For a reflective sample, an image for example, the light that stimulateshuman vision depends not only on the optical properties of the image butalso on the illumination and observation geometries. International standardsspecify the procedures for the reflectance measurements including the spectral


characteristics of the incident light and standards for the calibration and lightcollection has been established by CIE and ISO.

broadband light source

mono chromatic

detector

45o

a) broadband light source

mono chromatic

detector

45o

b)

(a) (b)

mono− chromator

detector

light source

detector

monochromator

sample

gloss trap

c)

light source

light trap

detector

mono− chromator

sample

d)

(c) (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 nameof Spectrophotometer despite their differences in illumination (monochromaticor white) and illumination-detection geometries. On the market, there are dif-ferent types of instruments available depending on the needs of the application.Figure 2.4 shows some examples of schematically different implementations forillumination and detection geometries (taken from Ref [Ryd97]). They are alsorecommended by CIE for reflectance measurements and are usually noted inabbreviation as (0o/45o), (45o/0o), (0o/D) and (D/0o), respectively, accord-ing 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 thenormal of the sample, and (D/0o) stands for the sample being illuminated bydiffuse light (D) and being measured along the sample’s normal. In the fol-lowing, 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 stud-ies. More information about other geometric implementations may be foundin Ref. [Ryd97], and references therein.

Elrepho 2000 belongs to the (D/0o) category and is one of the most widelyused instruments in paper making industry. A typical implementation of theinstrument is shown in Fig. 2.5. The instrument consists of an integratingsphere with holes for illumination, sample, and detector. The inside surface ofthe sphere is covered with white pigments that ideally scatter light isotropically.Inside the sphere there are baffles whose surfaces are also covered by the whitepigments. Using such an arrangement, light from the light sources is scatteredonto the inside surface of the sphere and no light from the light sources canshine directly onto the sample.

detector

light light

sample

gloss trap

baffle

Figure 2.5: Instrument of D/0o geometry for measurement of reflectance.

The accuracy of which the illumination mimics the ideal diffuse light de-pends on the optical properties of the white pigments and the sizes of the holeson the sphere. Generally speaking, the smaller the holes the closer the illu-mination to ideal diffuse light. Nevertheless, too small holes may cause largererrors in the measurement. For example, too small holes for the sample mayresult in lower signal/noise ratio. Additionally, it may also lead to less statis-tically meaningful measurements when the characteristics of a relatively large


area, say reflectance of a piece of paper, is of interest (this is exactly the casefor paper-making and Graphic Arts industries). Therefore, the implementationof diffuse illumination into a real instrument is a compromise between manyfactors. More detailed descriptions of the (D/0o) geometry instrument may befound in the international standard ISO 2469 [24694]. It should be pointed outthat measurements using Elrepho2000 are very time consuming because eachpatch has to be manually positioned.

In Graphic Arts, (45o/0o) or (0o/45o) -observing geometry (Fig. 2.6) is rec-ommended by CIE and well applied in practice. The illumination unit consistsgenerally of a cone shape glass illuminated by up to three different unpolar-ized light sources. This setup leads to an approximately circular illuminationcondition which helps to diminish the measurement dependencies on the sam-ple orientation. The spectrophotometer Gretag MachbethTM Spectrolino usedin our measurements belongs to this category. In a reflectance measurement,the sample gets circularly illuminated from the top with the collimated fluxI0λ. By using this instrument in combination with the software called Spec-troChart [GM01], one can measure many patches automatically in sequence.This makes the measurements used for system calibration, such as color gamutmapping, much easier. Comparative measurements made by applying bothElrepho 2000 and Spectrolino have shown excellent agreement in spectral re-flectance values for plain paper [Pau01].

monochromator & detector


45o/0o 0o/45o


Figure 2.6: Instrument of 45o/0o and 0o/45o geometry for reflectance measurement.

An angle-resolved spectrophotometer operating with a collimated illumi-nation (laser) [Gra01] is one of the high grade instruments. With it one canmeasure the monochromatic reflectance of a sample at any illumination- anddetection-geometries, which is particularly useful when there is specular re-

18 Paper

flection involved. Nevertheless, it is still too early to generally implement suchmeasurements in industrial practice because of the limitation of the instrumentitself (monochromatic illumination and extremely time consuming for the spacesampling). A more serious obstacle, though, is the lack of a simple theoreticalmodel 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 requiresa specific level of permeability. Printing paper has medium permeability, filterpapers, facial tissues, and sanitary papers must have high permeability, and inmany packaging papers and so-called barrier papers, low permeability [Les98b].The permeability of paper is closely related to its porosity, φ. One usuallyassumes that the structure consists of ellipsoidal cavities connected throughnarrow channels. Comparing the ideal model system with measurements offluid penetration or fluid flow through paper determines the apparent pore sizedistribution [Les98b].

For graphic arts industry, liquid (water, oil, etc.) penetration into paper hastremendous impact on the quality of the print. Absorption of ink constituentsby paper is driven by thermodynamic interaction between the ink and the pa-per, by capillary forces and by chemical diffusion gradients. Capillary pressureis acknowledged to be the main driving force in the offset ink oil transport in atypical paper coating porous structure. With increased latex content diffusion-driven transport of ink chemicals into the latex counterpart of the coating layerbecomes important [Rou02]. The penetration can also be driven by the print-ing nip during the printing and converting processes. Therefore, theoreticalhandling of the penetration process is complicated. Moreover, experimentalverifications of theory are challenged by the small dimension and by the rele-vance of both short and long time-scale. In the following, we will only mentionsome theoretical background for capillary driven penetration process that maybe 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 anglebetween the liquid and the capillary wall as shown in Fig. 2.7. Accordingto the Young equation, penetration occurs when the contact angle is smallerthan 90o (f > 0). However, experiments showed that even if the contactangle 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 deceleratesin diverging parts [Lyn93].

θ Liquid (ink)

Air

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 externalpressure govern the penetration. The time dependence of the penetration isdescribed by the Lucas-Washburn equation,

h2 =r2t

4η(2γcosθ

r+ pE) (2.8)

where h is the thickness of the ink penetration, η the fluid viscosity, and pEthe external pressure difference. The first term is the penetration driven by thecapillary force. It says that the depth of penetration is proportional to squareroot of the capillary radius, but inversely proportional to the square root of theviscosity of the ink solvent.

In practice, the mechanism of the ink penetration depends significantly onthe printing process. For offset printing, it is the nip pressure that dominatesin the printing and converting processes. This is followed up by the capillarydriven penetration afterwards. Similar process occurs in Toner Fusing Printingprocess [Hwa99]. For an ink jet, the external pressure comes from the kineticimpact (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 emit-ted from nozzles of the printer directly to a specified position on a substrateto create an image. The operation of the ink-jet printer is easy to visualize: aprinthead scans the page in horizontal strips, using a motor assembly to moveit from left to right and back, as another motor assembly rolls the paper invertical steps (see Fig. 3.1). A strip of the image is printed, then the papermoves on, ready for the next strip. To speed things up, the printhead does notprint just a single row of pixels in each pass, but a vertical row of pixels at atime.

Figure 3.1: A four-color ink-jet printer. The printhead moves along the drum per-

pendicular 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 streaminto 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]. Afterthat, continuous efforts have been made to improve the reliability of drop for-mation, to reduce the size of the ink droplets, while at the same time to increasethe jetting speed etc. This continuous development has led to continuous im-provement of printed images [Le98]. The state of the art ink-jet technology cangenerate ink droplet as small as 2 pico-liter (pL) in volume. Today, the ink-jetprinters can produce images of photo-quality with reasonable printing speed.

Ink Jet Printing

Drop on Demand Continuous Ink Jet

Raster

Binary

Hertz

Airbrush

Electrostatic

Piezoelectric

Thermal DuPontIris Graphics Stork

Domino Imaje LinxVideojet

Scitex Videojet

LACSign Tech Vutex

Epson iTi

AprionBrotherDataproductsEpsonMIT/XaarSpectraTektronicsTriden

CanonHPLexmarkOlivettiXerox

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 awide range of potential applications. Fundamentally, the technologies for inkapplication 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 periodicsignals (not the printing signal) which lead to a constant ink droplet genera-

3.1 Ink-jet technologies 23

tion. The generated droplets are selectively charged, a feature controlled bythe printing signals. The charged droplets correspond to no print and are de-flected 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 (seeFig. 3.3). The advantage of the continuous ink-jet technology lies at its higherprinting 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 technologygenerates 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 becategorized into four major methods: thermal, piezoelectric, electrostatic, andacoustic. The first two are the dominant technologies for products on markettoday, while the other two are still in the developmental stage.

In a printhead of a thermal ink-jet, there is an electric heater which isattached to the ink chamber. Heat is transferred from the surface of the heaterto the ink. The heater is controlled by an electric current pulse. When thecurrent is on, the ink is superheated to the critical temperature for bubblenucleation (about 300 oC for water based ink). When nucleation occurs, awater 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 bubblebegins to collapse on the surface of the heater. Concurrently, with the bubblecollapse, the ink drop breaks off and accelerates toward the paper as shownin Fig. 3.4. Once the ink droplet is ejected, ink is refilled into the chamberand the process is ready to begin again. Because the ink droplet is essentiallygenerated by the growth and the collapse of the vapor bubble, the thermal


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 printheadalong with its semiconductor fabrication process allow printheads to be builtat low coat and with high packing density. These made the thermal ink-jettechnology the most successful one on the market today [Le98]. Moreover, thebubble ink-jet with a high printing resolution and color capacity is availableat a very affordable price. On the market, there are many vendors who haveadapted 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 platesbonded to the diaphragm (electrodes) as shown in Fig. 3.5. Similar to thebubble jet, the piezoelectric material is also controlled by a current pulse. Inresponse to the electric pulse, the piezo-ceramic plates deform in shape whichcauses the ink volume change in the chamber to generate a pressure wave thatpropagates toward the nozzle. Consequently, an ink droplet is ejected out.Depending on the mode of shape deformation of the piezoelectric plates, thereare 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-jetis 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 25

Figure 3.5: A shear mode piezoelectric ink-jet design [Le98].

that ink droplets are generated and ejected when they are used in imaging. Thistechnology eliminates the complexity of drop charging and deflection hardwareas well as the inherit unreliability of the ink recirculation systems required forthe continuous ink-jet technology. The drop-on-demand ink-jet technology isthe most common technology on the market today.

The trend in the development of ink-jet technology is toward jetting smallerdroplets for image quality, fast drop frequency and a higher number of nozzleson the printhead for print speed.

3.2 Characteristics of ink-jet printers

3.2.1 General

The performance of an ink-jet printer may be characterized by its printingspeed and resolution. The speed depends above all on the jetting frequency orthe time interval between two consecutive ink-jets. On an ordinary ink-jet, theprinthead takes about half a second to print a strip across a page. Since A4paper 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 dotneeds printing. Nevertheless, higher printing speed may also be achieved by


adapting bigger printheads with more nozzles which enables it to deliver nativeresolutions of up to 1200 dpi and print speeds approaching those of currentcolor laser printers: 3 to 4 pages per minute (ppm) in color, and 12 to 14 ppmin monochrome.

Table 3.1: Descriptions for some desk ink-jet printers

Canon S300Jet type drop size Number of nozzles Resolution (dpi)1 Speed (ppm)2

(picoliter) black color black color black colorThermal 5 320 128×3 600×600 2400×1200 11 7.5

Epson C60Piezo 4 144 48×3 up to 2880×720 12 8

HP 920cThermal N/A N/A N/A up to 2400×1200 9 7.5

Lexmark Z53Thermal N/A N/A N/A up to 2400×1200 16 8

1 - it consists generally of 3 categories, draft, normal and best. Only thehighest possible resolution of the printer has been listed here.2 - the speed of printing, pages per minute (ppm), decreases when higher reso-lution is chosen in the printing. Only the higher possible speed (draft) is listedhere.

Resolution of the print depends on the volume of the ink droplet. Thesmaller the ink droplet the higher the possible printing resolution. The volumeof the ink droplet is determined by the diameter of the nozzle as well as thewidth of the current pulse (in time). Therefore, to be able to manufacturethe printhead with a very fine nozzles is of great importance for achievinghigh printing quality. For example, for 10 pL droplet, HP DeskJet 890C colorprinthead has nozzle diameter of around 20 µm. Apart from the printhead,the substrate onto which the ink drops can also have great impact on theprinting resolution, due to the ink-substrate interaction, such as ink spreadingand penetration. Ink bleeding can actually have significant impact on theimage resolution. Finally, it should be borne in mind that print speeds mayvary, depending on the document, software program, and computer settings.Table 3.1 is a collection of technical descriptions for some ink-jet printers thathave been available on the market over recent years.

3.3 Ink-jet ink technologies 27

3.2.2 HP970Cxi ink-jet printer

In this dissertation, the test charts were printed by employing a HP970Cxiink-jet printer, which is of the drop-on-demand ink-jet design. It uses dye-based liquid inks (water resolution). The technological characteristics of thisprinter is summarized in Tab. 3.2.

Table 3.2: Description of HP970Cxi ink-jet.

Document type Draft Normal BestResolution (black-white, dpi∗) 300 x600 600 x 600 600 x 600

Resolution (color, dpi) 300 x 600 Color layering 2400 x 1200Print speed (black text, ppm) 12.0 6.5 4.7

Print speed (Mixed text/graphics) 10.0 5.0 3.1Print speed (Full page color) 2.9 0.6 0.3

Type of ink Water-based organic dyesPrinter command language HP PCL level 3 enhanced

∗dpi - dots per inch.

3.3 Ink-jet ink technologies

3.3.1 General

Inks are probably the most critical components in the ink-jet printing. Thedevelopment 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 printedimages, but they also determine the characteristics of the drop ejection and thereliability of the printing system.

The ink-jet inks consist normally of colorants, an ink vehicle, and additivematerials. The ink vehicle like water, oil, solvent, resin, etc., governs the dy-namic properties of ink distribution. The ink vehicle is the major componentof the ink, whose amount in percentage varies from 40 − 90% depending onthe ink type. The colorants are the materials that create color of the printedimage, whose amount lies between 1 − 10%. The rest of the ink is generallyreferred to as additives. These improve the chemical and physical properties ofthe ink, such as ink viscosity, adhesion strength, heat stability, cure rate (forlight inks), surface tension (for liquid ink), etc.

Ink-jet inks can be sorted into different groups based on the different per-spectives, as shown in Tab. 3.3. Ink vehicle is one perspective: ink-jet inksare usually divided into 4 groups. They are, aqueous-based, nonaqueous-based, phase-change, and reactive.


Aqueous and nonaqueous inks use water or other solvents as ink vehicleswhose drying mechanism depends on penetration and absorption of the receiv-ing media (substrate). When ordinary copy or plain paper is used, the inktogether with the ink-vehicle are absorbed by the porous material. The mixingof 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 tempera-ture. The ink is jetted out from the printhead as a molten liquid. When themolten ink drop hits the substrate surface, it solidifies immediately. The quicksolidification prevents the ink from spreading or penetrating the substrate, andensures 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 basesAqeous-Based Solution, Dispersed, Microemulsion

Nonaqeous-Based Oil, SolventPhase-change Liquid to Solid, Liquid to Gel

Reative UV Cured, Two partsCategory according to colorants

Organic Dyes Direct, Acid, Reactive, Disperse, SolventPolymeric 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, polymericdyes, and pigments. The organic dyes consist of organic dye molecules, whilethe polymeric dyes consist of dye polymers. The pigmented inks are mainlyinorganic 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 widelydebated 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 pigmentswhich are generally insoluble. Chemically, dyes exist in the ink as individualmolecules, while the pigments exist as clusters that consist of thousands ofcolorant molecules.

Generally speaking, the dye-based inks have superior color representationcapability or greater color gamut than the pigment-based inks. By printing on

3.3 Ink-jet ink technologies 29

high grade substrates, dye-based ink-jet printing can deliver images of com-patible quality as those produced by silver halide. The disadvantages of thedye-based inks are their relatively poor (long term) image performance whichincludes light fastness (light fading stability), dark storage stability, humidityfastness, and water fastness.

One explanation to poor image stability for images produced by dye-basedinks, is that the dye consists of individual molecules which are chemically lessstable in terms of light exposure, oxidation, and humidity. Being a clusterof many molecules, the pigment inks have greater resistance to the impactof the environment and therefore possess better light fastness and humidityfastness. In terms of dye design, chromophore chosen has a dominant impacton the spectral characteristics and color stability achieved. Additionally, theirphysical and chemical properties also have great impact on color stability.

Trying to achieve a high degree of light fading stability and large color gamutat the same time can pose a challenge for ink development due to the rarity ofpigments and dyes that have both of these desirable properties [ADT+]. Overthe years, it has been a hot topic for debate in ink chemistry studies and ink-jettechnologies. Hopefully, we are now seeing a rapid closing of the gap in colorand image performance, between pigments and dyes [IB01].

Chapter 4

Optical modelling: anoverview

Reflectance, such as, brightness, opacity, etc, characterize the paper sheet quan-tities, but not the general material properties of the paper. Optically, thefundamental events that govern reflection are light scattering and absorption.Therefore, quantities that parameterize the fundamental processes, such ascoefficients of scattering (σs) and absorption (σa), are fundamental materialproperties. 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 groupof models is based on Radiative Transfer Theory [Cha60]. Another group usesMonte-Carlo methods [Jam80, Rub81] to simulate light scattering and absorp-tion. In this chapter we highlight the fundamental concepts of these methodsas well as their application to Paper Optics and Graphic Arts.

4.1 Radiative Transfer Theory

Radiative Transfer Theory (RTT) based approaches start with solving integro-differential equations which describe light propagation in media. According toRTT, the radiance L(.r, u) (W ·m−2 · sr−1) of light at position .r travelling ina direction of unit vector u is decreased by absorption and scattering but isincreased by light that is scattered from u′ into the direction u. The radiativetransfer equation which describes this light interaction is [Cha60]

u · ∇L(.r, u) = −(σa + σs)L(.r, u) +σs4π

∫4π

q(u, u′)L(.r, u′)dω′ (4.1)

32 Optical modelling: an overview

where σa (m−1) is the absorption coefficient, σs (m−1) is the scattering co-efficient, dω′ is the differential solid angle, and q(u, u′) is a phase function.The total extinction coefficient, σt, is a sum of the absorption and scatteringcoefficients,

σt = σa + σs (4.2)

The phase function, q(u, u′), describes angular distribution of a single scat-tering event. If the integral of the phase function is normalized to equal tounity, i.e.

14π

∫4π

q(u, u′)dω′ = 1 (4.3)

then it is the probability density function of scattering from direction, u′ todirection u. Assume the directions of the incident and the scattered lightare u(θ, φ) and u′(θ′, φ′). It is reasonable to assume that the phase functiondepends only on the scattering angle Θ (cosΘ = u·u′) rather than the incomingor the outgoing angles, i.e.

q(u, u′) = q(cosΘ) (4.4)

wherecosΘ = cosθcosθ′ + sinθsinθ′cos(φ′ − φ) (4.5)

A complete description of light transfer requires knowledge of σa, σs, andq(u, u′). These quantities depend not only on the properties of the raw mate-rials, but also on their distribution (or the structure of the system). To obtainthese 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 onthe form of the phase function, q(cosΘ). Different phase functions have beenproposed to physically describe different types of scattering. Among the well-known are the Rayleigh phase function [Ray71], Mie phase functions [Mie08],and Henyey-Greenstein phase function [HG41]. These phase functions wereoriginally proposed for studying radiative transfer in atmospheric gaseous sys-tems and in the galaxy. Later, they found applications in other fields.

Mie theory describes light scattered by isolated spherical particles of arbi-trary size and refractive index. Taking particle size (radius r) and refractiveindex as input parameters, Mie theory calculates efficiency parameters, i.e.,scattering efficiency Qsca and absorption efficiency Qabs. The angular distri-bution of the scattered light, or the phase function, is calculated by

q(cosΘ) =i(Θ)

2πα2Qsca(4.6)

4.3 Multi-flux theory 33

where α = 2πr/λ is the particle size parameter relative to the wavelength ofthe light, i(θ) is called the Mie theory intensity function. Mie theory found itsoriginal 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 approxi-mation 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 Kubelka-Munk approach. Clearly, Θ = 0, π are two singular points for g = ±1, respec-tively. It is easy to see that when Θ → 0 and g → 1

limg→1

q(cosΘ → 1) = limg→1

(1 + g)(1− g)

√(1 + g2 − 2gcosΘ)

→ δ(Θ) (4.8)

Consulting the normalization condition given by Eq. (4.3), one can concludethat g = 1 corresponds to complete forward scattering. Similarly, g = −1corresponds to complete backward scattering. One of the greatest advantageof the Henyey-Greenstein phase function is its simple form under the Legendreexpansion,

q(cosΘ) = 1 + 3gcosΘ+ 5g2P2(cosΘ) + 7g3P3(cosΘ) + ... (4.9)

This makes it a popular choice in applications, for example, in studying ra-diative transfer in biological tissues (dermal and aortic tissues) [Pra88, Yoo88,CPW90]. Very recently, it has been even considered for application in simula-tions for light transfer in paper [Eds02].

4.3 Multi-flux theory

Mathematically, the radiative transfer equation (Eq. 4.1) has no analytic so-lutions except in a few special cases, such as q(cosΘ) = const. To solve theproblem with the help of computers, one must to divide the direction in spaceinto channels as shown in Fig. 4.1. All light travelling in a direction within a po-lar angle θ1 of the positive direction of the axis perpendicular to the boundaryplane is said to be in channel 1. All light travelling at a polar angle between θ1and θ2 is allocated to channel 2, etc. Such a discrete ordinate method is calledDiscrete-Ordinate-Method Radiative Transfer [Cha60, STWJ88] or Multi-fluxRadiative Transfer Method [MR71].


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 byauthors using 2, 3, or 4 channels [Sch05, MR71]. A general mathematicaltreatment using this coordinate discretion, was first developed byWick [Wic43].It was then thoroughly exploited by Chandrasekhar [Cha44] and applied to theproblem of radiative transfer.

Mudgett and Richards [MR71, MR72] reformulated this method in a morecomprehensive way and applied it to parallel layered media, such as paintfilm [Ric70]. Their work was well followed up by other authors in modellingand predicting the optical characteristics of paint films [JVS00, All73].

In principle, by applying the Multi-flux approach with a sufficient numberof channels, one can accurately solve the radiative transfer problem. Never-theless, the solution depends directly on the knowledge of the phase function.Therefore, finding the proper phase functions for different types of papers isessential for optical modelling and simulations.

4.4 Kubelka-Munk method

The Kubelka-Munk (K-M) approach is actually a two-flux version of the multi-flux method for solving the radiative transfer problem. Here the ordinate isonly divided up into an up- and a low-hemisphere by the bounding plane (paper

4.5 Monte-Carlo simulation 35

sheet). In the K-M approach, the light propagation depends on the K-M coeffi-cients of light absorption (k) and scattering (s). The fundamental assumptionof this approach is that light distribution in the media is ideally diffuse. It re-quires the media to be of little absorption, while at the same time of involvingstrong and angle-independent scattering (q(cosΘ) = 1). Indeed, a compari-son between the more accurate multi-flux calculation with the Kubelka-Munkapproximation reveals that the two-flux approximation gives excellent resultsprovided the absorption is small compared to scattering, and the optical thick-ness is greater than 5 [MR71]. A key factor for a successful application ofthis method lies at finding the so-called K-M absorption coefficients, k, whichdepends not only on the physical properties of a medium but also on light dis-tribution in the medium. Detailed discussions about this issue is presented inSecs. 5.2 and 7.2.

The K-M approach has probably been the most widely used approach inpaper-making and color-using industries since it was introduced more than70 years ago [KM31, Kub48]. The continued popularity of this approach isattributable to the simple analytical solutions. Nevertheless, the solutions pro-vide insights to the processes of the light transfer and can be used to predict thereflectance of the specimen with reasonable accuracy. In addition, the simpleprinciples involved in the theory are easily understood by the non-specialist. Anexcellent review about the advantages and disadvantages of the K-M methodcan 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, theanalytical solution that has widely been used, is only applicable to layeredmedia that has a uniform concentration along the ordinate z-axis within thelayer, and the layer has infinite horizontal extension in the xy-plane. When thedistribution is non-uniform one has to work directly with the integro-differentialequation. In the next chapter, we extend the K-M method to cope with thenon-uniform ink distribution and surface reflection.

4.5 Monte-Carlo simulation

The Monte-Carlo approach belongs to another category of radiative transfersimulations. It was first used by Fermi, Von Neumann and Ulam who developedit for the solution of problems related to neutron scattering during the devel-opment of the atomic bomb. The name Monte-Carlo is used since the methodis based on the selection of random numbers. In this sense it is related to thegambling 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 con-sisting of straight paths between points of interaction with the media, suchas fibres or fillers in paper and colorant particles in print. The Monte Carlo


method can be considered as a very general mathematical method to solve agreat variety of problems. In the following, we will only highlight the basicconcepts of this method for the simulation of light propagation. More detaileddescription of this method may be found in references [Jam80, Rub81, Gus97b].

θ

1 2

A

B

l1

l2

l3

C

D A’

B’

3 1’

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 ina 2D scene (Fig. 4.2). The length, li, of an undisturbed straight path of onephoton is a stochastic variable. Its statistical expectance value, lp, is called themean free path and is inversely proportional to the extinction coefficient, σt,i.e.

lp = limN→∞

∑Ni=1 liN

=1σt

(4.10)

When the photon hits the media, it will either be absorbed (like photon 2 at siteB′), or be scattered randomly. On average, the probability of a photon beingabsorbed is related to the relative strength of the absorption coefficient to thetotal 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 absorbedand then re-emitted with the same (elastic scattering) or different (inelasticscattering or Raman Scattering) energy in a different direction. The lattercase is out of the scope of this work. The direction of the re-emission is atrandom. In a 3D scenario, the direction is specified by a pair of angles, polarand azimuth angles, (θ, φ).

4.5 Monte-Carlo simulation 37

By applying a large number of photons or equivalently by repeating the onephoton process a large number of times, one can obtain statistically meaningfulquantities (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 pho-tons, and Ntran photons that have reached the opposite hemisphere (3′), thenreflectance (R) and transmittance (T ) can be computed as

R =Nref

Ntot(4.11)

T =Ntran

Ntot(4.12)

A useful concept that describes light propagation in the media is the pointspread function (PSF). If a photon hits the surface of a sheet of paper at .r, theprobability that the photon exits the paper at .r′ is described by the PSF anddenoted as p(.r, .r′). If the PSF of the paper is known, quantities like reflectanceand transmittance can easily be computed. Therefore, to obtain the PSF ismore essential in the Monte-Carlo simulation. Recently, Gustavson [Gus97b]developed methods for computing the PSF of paper and simulated the opticaldot gain in halftone prints.

Chapter 5

Extended Kubelka-Munktheory and applications

5.1 Assumptions in Kubelka-Munk theory

The original theory of Kubelka-Munk (K-M) was developed for light propaga-tion in parallel colorant layers of infinite xy-extension [KM31, Kub48]. Thefundamental assumptions of the K-M theory are that the layer is uniform andthat light distribution inside the layer is completely diffused. From these as-sumptions, light propagation in the layer was simplified into two diffuse lightfluxes through the layers, one proceeding upward and the other simultaneouslydownward. After its introduction in the 1930’s, K-M theory was extended byremoving some of the original assumptions. Among others, a correction to theboundary reflection at the interface of two adjacent media was introduced bySaunderson [Sau42], i.e., the well-known Saunderson correction. Kubelka him-self also made attempt to extend the applicability of the theory to opticallyinhomogeneous samples [Kub54]. However, this extension applied only to aspecial case of the inhomogeneous media, in which the ratio of the absorptionto 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 (perpen-dicular to the z-axis) whose size in the xy-extension is much larger thanits thickness. Edge effects are therefore negligible.

2. The sample is optically homogeneous. In the case of inhomogeneoussample, 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 theangle 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 andthe last 2 are about light propagation and light-medium interaction.

This chapter presents an extension of the K-M theory by completely re-moving the restriction of assumption No. 2. Such an extension allows one tostudy cases where the scattering and absorption coefficients are any functionsof 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 lightthat is reflected from the image. Therefore, it is natural to separate the lightthat is reflected from the image, from that goes to the image. An equivalentexpression is to divide the light that travels toward the upper hemisphere (re-flected from the image) as one group, and the light that travels toward thelower hemisphere as another. According to the assumption No. 4, the light ineither group is uniformly diffuse. Consequently, one only need trace the lightflux at (any) one direction in each hemisphere to obtain complete informationabout the light propagation. It is therefore convenient to choose the directionalong the z-axis and the inverse direction as representatives for light propagat-ing toward the upper and the lower hemispheres, respectively (see Fig. 5.1).Mathematically, one can then simplify the actual three-dimensional light prop-agation by considering one-dimensional streams of light intensities, Iup and Idn,

5.2 Differential equations 41

representing light travelling upwards and downwards, respectively (they corre-spond 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 andbackscattering. On the other hand, Iup is enhanced by the backscattering fromthe stream Idn and vice versa. From these arguments, Kubelka-Munk [KM31]derived the following expressions for the gradients of the light intensities,

dIdndz

= −(s+ k)Idn + sIup (5.1)

−dIupdz

= −(s+ k)Iup + sIdn (5.2)

Applying the K-M theory to a semi-infinite medium layer, one obtains thereflectance of the medium layer, when the surface reflection is omitted [Nob85],

R∞ = 1 +k

s−

√(k

s)2 + 2

k

s(5.3)

where k and s are called K-M coefficients. They are phenomenal descriptions tothe light absorption and scattering in the medium. Therefore, they are closelyrelated to the fundamental optical properties of the medium, the absorption(σa) and scattering (σs) coefficients per unit path length of the material. How-ever unlike (σa) and (σs) that depend solely on the physical properties of themedium, the K-M coefficients, k and s, depend also on the light distribution inthe medium and even illumination geometries [Ste55]. Physics behinds this isthat k and s do not describe the absorption and the scattering in any specificdirection but their averages over the upper or lower hemisphere. Theoreticalanalysis [Kub48, WH66, Kor69] has shown that, for the completely diffusedlight 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 lightdistribution in the medium is particularly important when K-M theory is ap-plied to a mixture of different media. Dye-based ink, for example, has littlescattering, in which the light propagates essentially along a straight path. How-ever when the ink mixes with paper materials, like ink penetration, the lightpropagates instead in a zigzag fashion in the ink-paper mixture, because ofvery strong scattering of the paper. Consequently, the light appears to havegreater probability to be absorbed if it passes through the same vertical depthof the mixture as that of the ink layer. This explains the experimental obser-vations [Foo39, BS76] that the k appears differently when the dye mixes withdifferent pulps (because of different scattering power of the pulps).

Mudgett and Richards [MR71] investigated the relation between k and sand σa and σs for an optically thick, weakly absorbing layer. By means of


the multi-flux method, they calculated accurate values of the reflectance of asemi-infinite film (R∞) and of a finite film of the same material over an idealblack backing. The values of k and s were calculated from these results. Theresults confirmed Eq. (5.4) and they found that to a good approximation:

s = 0.75σs (5.5)

r1

r0

Rg

I0 I

0R

Idn

Iup

z

D

0

medium layer

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 in-terface with air at z = D on top, while at z = 0 it is in optical contact witha backing of reflectance, Rg. If surface reflectance at the interfaces is r0 andr1, respectively, as shown in the figure, one may then obtain the followingboundary conditions at the z = D interface,

Ibdn(D) = I0(1− r0) + Ibup(D)r1 (5.6)

I0R = I0r0 + Ibup(D)(1− r1). (5.7)

At z = 0 there isIaup(0) = Iadn(0)Rg. (5.8)

In Eqs. (5.6-5.8), the superscripts, a and b, denote the corresponding valuesabove (a) and beneath (b) the interface.

5.2 Differential equations 43

The background of the boundary conditions are the continuation of thelight streams across the interface. For example, Eq. (5.6) reveals that the lightpropagating downwards (the term on the left of the equation) consists of twolight streams, i.e., the illumination that passes through the interface (first termon the right) and the internal reflected light at the interface (second term onthe right). Following such principles one can establish boundary conditions forsystems consisting of any number of media layers [YK00].

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Angle of incidence, α

Ref

lect

ance

, r

n2/n

1=1.5

r||

r

r⊥

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. Becauseof the discontinuity of refractive index at the boundaries between the adjacentmedia (air, ink, and paper, etc.), a portion of the incident light falling on theinterface will be reflected on the interface. For a given ray, the amount reflectedcan be computed according to Fresnel’s equations [Cha60],

r‖ =[cosα−

√(n2/n1)2 − sin2α

cosα+√(n2/n1)2 − sin2α

]2

(5.9)

r⊥ =[(n2/n1)2cosα−

√(n2/n1)2 − sin2α

(n2/n1)2cosα+√(n2/n1)2 − sin2α

]2

(5.10)


Clearly, the boundary reflection depends on the refractive indices of the media,the light polarization and the angle of incidence with respect to the interfacenormal (see Fig. 5.3). For unpolarized incident light, the reflectance, r, is asimple average of r‖ and r⊥:

r = (r‖ + r⊥)/2. (5.11)

For clarity of discussion, we denote the reflection from within the side ofhaving greater refractive index, the internal reflection, as r1. On the other sidewe have the external reflection denoted by r0.

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Relative index of refraction, n=n2/n

1

Bou

ndar

y re

flect

ance

r||

r⊥

r

r||

r r⊥

Internal

External

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 fromair 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 glassor polymeric materials bounded by air, the external boundary reflection is r0 =0.0919. One may compare this with the value of reflectance at perpendicularincidence, 0.040. For diffuse light incident on the boundary from within, thesample is partially reflected (internal reflection, r1). A fraction (1− 1/n2) willbe incident on the boundary at angles greater than the critical angle and will betotally reflected. The total internal boundary reflectance for diffusely incidentlight is given by [Nob85],

r1 = 1− 1n2

+r0n2. (5.12)

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 in-creases dramatically with n. For the external reflection, however, the increaseswith 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 bythermodynamic interaction between the ink and the paper, by capillary forcesand by chemical diffusion gradients. Capillary pressure is acknowledged to bethe main driving force in the offset ink oil transport in a typical paper coatingporous structure. With increased latex content, diffusion-driven transport ofink chemicals into the latex counterpart of the coating layer become impor-tant [Rou02]. Different mechanisms of ink penetration result in different formsof 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 isclosely related to the properties of the substrate paper, on the one hand, andthose of the ink on the other hand. In the following we will investigate threeforms of f(z).

5.3.1 Uniform distribution

The simplest case is of course, f(z) = 0, the ink penetrates uniformly into thepaper (between z = 0 and z = D), with density

ρ(z) = ρ0 (for 0 < z ≤ D) (5.14)

If the coefficients of absorption and scattering are kp, sp and ki, si, for pa-per and ink, respectively, the corresponding coefficients for the ink penetratedpaper may be expressed as,

k = kp + µki (5.15)s = sp + si (5.16)

where µ is a constant depending on the strength of the scattering of the paperand illumination geometries as discussed in Sec. 5.2. Detail discussions will be


given later and in Chapter 7 as well. Because µ is only a multiplier of ki, wecan continue the study for the case of µ = 1 without losing generality.

One shortcoming of this model seems to be the discontinuity in the inkdistribution at the lower interface (z = 0) of the penetrating ink layer. Nev-ertheless, it may still be a reasonable approximation to a system consistingof liquid ink and plain paper whose pore size between cellulose fibres is muchbigger than the size of the ink molecules or micro-ink cells (100A). In such anink-paper system the dominant mechanism of ink penetration is the capillaryforce. Additionally, the model reflects the fundamental fact that there existsa layer of an ink-paper mixture. Furthermore, it is the only case in which onecan 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 ofthe ink penetration, i.e.,

f(z) = ρ0/D (5.17)

orρ(z) = ρ0

z

D(0 ≤ z ≤ D) (5.18)

The coefficients of the absorption and scattering in the ink-penetrated papermay then be written as,

k = kp + kiz

D(5.19)

s = sp + siz

D(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 equationsbecome relatively more complicated, and no analytic solution in known anylonger. Fortunately, as discussed later one may find a good approximation inpolynomial form, and in some cases even be able to handle this model in ananalytical manner.

5.3.3 Exponential distribution

In the case of ink penetration driven by a concentration gradient, the variationof 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-penetratedpaper, ρ(D) = ρ0, we may write the distribution as

ρ = ρ0eα(z/D−1) (0 < z ≤ D) (5.22)

Thus, the absorption and scattering coefficients of the ink-penetrated paperread,

k = kp + kieα(z/D−1) (5.23)

s = sp + sieα(z/D−1) (5.24)

Compared to the uniform- and linear-ink penetration models, the exponentialmodel is most mathematically complex. In turn, it causes complexities insolving the differential equations that govern light propagation in the ink-papermixture.

5.4 Solutions of the differential equations

Equations (5.1) and (5.2) provide the general descriptions to light propagationin either direction (upward or downward). Substituting the expressions fork and s (Eqs. (5.15, 5.16), Eqs. (5.19, 5.20), and Eqs. (5.23, 5.24)) into thedifferential equations and solving these, one obtains the intensity of the lightreflected from the ink layer and then the reflectance of the prints with inkpenetration 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 penetratedpaper layer) with and without backing. In practice, the layer with backing (seeFig. 5.2) serves as a model for the print where the substrate paper is only par-tially penetrated by the ink and the remaining portion (bottom layer) acts asa backing to the top layer. The case of no backing may be considered as paperbeing fully penetrated by the ink. It is a freely suspended layer. Because trans-mittance measurements for prints often provide useful information, expressionsfor 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 differ-ential equations (Eqs. (5.1) and (5.2)) have the form,

Iup = a1eBz + a2e

−Bz (5.25)Idn = b1e

Bz + b2e−Bz (5.26)


where B is a function of s and k,

B(s, k) = 12(

s

A(s, k) −A(s, k)

s) (5.27)

and

A(s, k) = sR∞

= s+ k −√k2 + 2ks (5.28)

For simplicity, A(s, k) is hereafter denoted as A, if not otherwise stated. InK-M theory, the reflectance (R) is usually expressed as a function of s andR∞ [KM31, Pau87]. One advantage of replacing R∞ by A is that one avoidspossible 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,

b1 =Asa1 (5.29)

b2 =s

Aa2 (5.30)

Therefore, there exist only two undetermined coefficients, say a1 and a2, whichcan 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

a1es2−A2

2A D + a2e− s2−A2

2A D = I0(1− r0) + r1[Aa1

se

s2−A22A D +

sa2

A e−s2−A2

2A D]

(5.31)

IR = I0r0 + (1− r1)[Aa1

se

s2−A22A D +

sa2

A e−s2−A2

2A D]

(5.32)

Rg(a1 + a2) =Asa1 +

s

Aa2 (5.33)

By solving these equations, one obtains the expression for the reflectanceof the medium with backing (Rg) as

R = r0 +(1− r0)(1− r1)[s(A− sRg)e−

s2−A2A D −A(s−ARg)]

(A− sr1)(A− sRg)e−s2−A2

A D − (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− r1R∞). (5.35)

Clearly, R∞ is a special case of R∞ when the interface reflection, r0 = r1 = 0,or, are negligible.


r1

r0

I0 I

0R

Idn

Iup

D

0

medium layer

I0T

r1

r0

z

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 replacedby

Aa1

s+sa2

A = r1(a1 + a2). (5.36)

The reflectance of the medium layer, R, is therefore

R = r0 +(1− r0)(1− r1)[s(A− sr1)e−

s2−A2A D −A(s−Ar1)]

(A− sr1)2e−s2−A2

A D − (s−Ar1)2. (5.37)

For the freely suspended medium layer, one may obtain valuable informationabout the layer by measuring its transmittance. From the continuity of the lightstream (propagating downward), one may obtain an extra boundary conditionbeneath the (z = 0) interface,

I0T = (a1 + a2)(1− r1). (5.38)

Combining Eqs. (5.31), (5.36), and (5.38), one may derive the expressionfor the transmittance, T, for the freely suspended medium layer,

T =(1− r0)(1− r1)(s2 −A2)e−

s2−A22A D

(s− r1A)2 − (A− sr1)2e−s2−A2

A D(5.39)


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)Rge

−2kD

1− r1Rge−2kD(5.40)

for the layer with backing (Rg), and

R = r0 + (1− r0)(1− r1)r1e

−2kD

1− r21e−2kD

(5.41)

T = (1− r0)(1− r1)e−kD

1− r21e−2kD

(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 lin-ear ink penetration are more complicated. Generally speaking, these equationshave 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 inthe vicinity of z = D.

r1

r0

Rg

I0 I

0R

Idn I

up

z

D

0

medium layer

backing

Figure 5.6: A schematic diagram of a linear ink distribution (with backing).


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 + kiZ (5.44)s = sp + siZ (5.45)

where Z varies in a range between 0 and 1.Let

Iup =∞∑

n=0

anZn (5.46)

Idn =∞∑

m=0

bmZm (5.47)

and insert the series solutions into the differential equations. One gets thefollowing algebraic equations,

1D

∞∑m=1

mbmZm−1 = −[k + s]

∞∑m=0

bmZm + s

∞∑n=0

anZn (5.48)

− 1D

∞∑n=1

nanZn−1 = −[k + s]

∞∑n=0

anZn + s

∞∑m=0

bmZm (5.49)

Reflectance of the linear ink-penetration (with backing)

There is a standard way to mathematically solve Eqs. (5.48) and (5.49). Com-paring the terms of the same order (Zn, n = 0, 1, 2, ...) on both sides of theequations, one finds,

a1 = D[(kp + sp)a0 − spb0] (5.50)−b1 = D[(kp + sp)b0 − spa0] (5.51)

and for n ≥ 2

an =D

n[(kp + sp)an−1 + (ki + si)an−2 − spbn−1 − sibn−2] (5.52)

−bn =D

n[(kp + sp)bn−1 + (ki + si)bn−2 − span−1 − sian−2] (5.53)


These relations reveal two facts. First, there are only two undeterminedcoefficients, say a0 and b0. All the other coefficients an or bn (n ≥ 1) are func-tions of them. Second, both coefficients, an and bn, are inversely proportionalto the order of n (actually, an, bn ∝ 1/n!). Therefore, they decrease monoton-ically with increase n. Thus, it provides us with the possibility of truncatingthis series expansion up to a certain order of the expansion. Simulations tovarious mixtures of media have confirmed these observations as one can seein Fig. 5.8 and 5.9 in Section 5.5. More examples can be found in a previouspublication [YK00]. Because the variable, 0 ≤ Z ≤ 1, is independent of therange of ink penetration these conclusions are generally meaningful.

Imposing the boundary condition at Z = 0 (Eq. (5.8)), we have,

b0 = Rga0 (5.54)

Thus, there is only one undetermined coefficient remaining (say a0) which canbe determined by applying the boundary condition at Z = 1 interface. InsertingEqs. (5.46) and (5.47) into the boundary at Z = 1 (Eqs. (5.6) and (5.7)), onehas

∞∑n=1

an = I0(1− r0) + r1

∞∑n=1

bn (5.55)

I0R = I0r0 + (1− r1)∞∑

n=1

bn (5.56)

The reflectance of the linear ink penetration system is then obtained,

R = r0 + (1− r0)(1− r1)∑∞

m=0 bm∑∞m=0(am − bmr1)

(5.57)

If the series expansion is truncated in a order of M, the reflectance of penetrat-ing ink layer can be expressed as,

R = r0 + (1− r0)(1− r1)∑M

m=0 bm∑Mm=0(am − bmr1)

(5.58)

Simulations show that the computed reflectance, R, generally has good conver-gency (see Sec. 5.5 and Ref. [YK00]).

Without difficulty one can see, from Eqs. (5.50-5.54), that both an andbn are proportional to a0. It makes the fraction in Eq. (5.58) be actuallyindependent of a0. Thus, one can compute the reflectance simply by setting a0

an arbitrary value (say, a0=1).


r1

r0

I0 I

0R

Idn I

up

z

D

0

medium layer

r2

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 = r2a0 (5.59)

Even though the remaining expansion coefficients (an, bn, n ≥ 1) change upona0 and b0, the expression of the reflectance remains unchanged as given inEq. (5.58). It is worth noticing that the internal boundary reflection at theupper 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 byapplying the boundary condition at Z = 0,

I0T = a0(1− r2) (5.60)

It results in

T =a0(1− r0)(1− r2)∑M

m=0(am − bmr1)(5.61)

Although a0 exists explicitly in the expression, the transmittance, T , interest-ingly enough, is actually independent of a0 because am and bm in the denomi-nator are proportional to a0 as we have previously noticed.


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 exponentialpenetrating ink distribution. Strictly speaking, no analytical solution to theseequations is known.

Series expansions for s and k

Here we propose an approximate approach by expanding the exponential func-tional,

ρ = ρ0eα( z

D −1) = ρ0e−αeαZ (5.62)

into a Taylor series to a maximum order, Nc. Accordingly, the absorption andthe scattering coefficients can be expressed as,

k = kp + kie−α(1 + αZ +

α2

2!Z2 + ...+

αNc

Nc!ZNc) (5.63)

s = sp + sie−α(1 + αZ +

α2

2!Z2 + ...+

αNc

Nc!ZNc) (5.64)

where 0 ≤ Z ≤ 1. Evidently, the accuracy of the expansion depends on thequantity αNc/Nc!. As long as α � Nc, the difference between the exponentialfunctional and its Taylor expansion becomes negligible.

Reflectance and transmittance of the exponentially ink-penetratedpaper

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). Substitutingk 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(Y1a0 − Y2b0) (5.65)b1 = D(Y2a0 − Y1b0) (5.66)

and (for n ≥ 1)

an+1 =D

n+ 1[Y1an − Y2bn + Y3(

Ns∑m=1

αm

m!an−m)− Y4(

Ns∑m=1

αm

m!bn−m)]

(5.67)

bn+1 =D

n+ 1[Y2an − Y1bn + Y4(

Ns∑m=1

αm

m!an−m)− Y3(

Ns∑m=1

αm

m!bn−m)]

(5.68)

5.5 Simulations for uniform- and linear-ink distribution 55

where

Y1 = kp + kie−α + sp + sie

−α (5.69)Y2 = sp + sie

−α (5.70)Y3 = kie

−α + sie−α (5.71)

Y4 = sie−α (5.72)

and

Ns ={

n, if n ≤ Nc

Nc, if n ≥ Nc(5.73)

From Eqs. (5.65-5.68), we observe two facts similar to the case of a linear inkdistribution. 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 inthe 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 expo-nential ink distribution actually share the same forms as those of the linear inkdistribution (Eqs. (5.57) and (5.61)). It should be noted, however, that theprogressive 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 dis-

tribution

Rather than focusing the discussions on any specific paper or ink, we willstudy the effect of ink penetration (on the reflectance) in a general sense. Thediscussions concern not only applications, but also the different approaches. Inaddition, it is worth noticing that the quantities, k and s, and consequently Rand T , are wavelength dependent. For simplicity, but without losing generality,we limit the simulations to a monochromatic band. k and s are therefore treatedas constants.

5.5.1 Convergency of the series expansion.

To test the convergence of the series solution for the case of linear ink pene-tration, calculations for various combinations of (kp,sp) and (ki,si) have beencarried out. As expected from the theoretical analysis in Sec. 5.4.2, the cal-culated reflectance values demonstrate rapid convergence with respect to theorder of the expansion (see Fig. 5.8). Moreover, the coefficients {an} and {bn}


0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Order of expansion, n

Ref

lect

ance

, R

Kp:S

p:K

i:S

i=1:10:50:1

Kp:S

p:K

i:S

i=1:100:50:1

Kp:S

p:K

i:S

i=1:100:50:40

Figure 5.8: Variation of computed reflectance with respect to the order (n) of poly-

nomial expansion (linear ink distribution, absorption power of the ink kiD = 0.5).

0 5 10 15 20 250

2

4

6Variation of the expansion coefficients, a

n and b

n

a n

K_p:S_p:K_i:S_i=1:10:50:1K_p:S_p:K_i:S_i=1:100:50:1K_p:S_p:K_i:S_i=1:100:50:40

0 5 10 15 20 25−2

0

2

4

Order of expansion, n

b 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 kiD = 0.5).


also show good convergency (to zero, see Fig. 5.9). For example, when the depthof the ink penetration (D) corresponding to an absorption power kiD = 0.5, Rconverges up to 4 digit accuracy if the series expansion is truncated at ordern = 10. For a smaller depth of ink penetration, D, the series truncation can besufficient even at lower order which makes it possible to express the solutionin the handy form of a polynomial. In other words, an analytic expression forthe reflection is possible for a thin ink layer. For deep ink penetration, (D islarge), however, the order of expansion increases rapidly and a computer hasto be used. Furthermore, Fig 5.9 shows another fact that the coefficients, anand 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 forthe effect of the ink penetration, we have made comparative calculations intwo 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 layerbeing left over the substrate surface). In the calculations, the density of the inkdistribution 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 and5.6).

In the calculations, the clean substrate paper (or part of the paper withoutink penetration) was considered as a background reflector with reflectance,Rg = 0.85. Such reflectance requires the paper having the scattering powerabout 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 substratesare shown in Fig. 5.10, in the case of uniform ink penetration. The thickness ofthe 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, ab-sorption power of the ink, kiD. The following systematic behavior of computedreflectance and transmittance are observed.

• the reflectance of the print with ink penetration is generally larger thanthat without, while the transmittance is generally smaller;

• the deeper the ink penetration, the bigger the differences between thecalculated values;

• the difference gradually approaches a constant when the ink thicknessfurther increases.


• the transmittance, T, corresponding to the prints with and without inkpenetration are identical, at D = 0 and ∞, but different otherwise.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1Reflectance

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1Transmittance

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

Absorption power of ink, kiD

Ref

lect

ance

, R

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1Tra

nsm

ittan

ce, T

complete ink pene.

no ink pene.

a) kp:s

p:k

i:s

i=1:100:25:1

b) kp:s

p:k

i:s

i=1:40:25:1

c) kp:s

p:k

i:s

i=1:100:25:1

d) kp:s

p:k

i:s

i=1:40:25:1

no ink pene.

complete ink pene.

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 theoptical parameters kp, sp, ki and si and its optical performance. Comparingpanel a with b in Fig. 5.10, one finds that the scattering power of the paperplays a crucial role for light reflection, i.e., the stronger the scattering power ofthe paper, the more profound the optical effect of the ink penetration, whichis in line with experimental observations (Arney and Alber,1998). Similarobservation 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 theink reduces the effect of ink penetration on the reflectance and the transmit-tance.

It is natural to make a comparison between different models of ink penetra-tion and to investigate to what extent the optical characteristics of the layerdepend on the form of the ink distribution inside the paper (ink penetration).However, such a comparison between the uniform model and the linear modelis not a trivial task, because they are actually not directly comparable. To


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1


Ref

lect

ance

, R

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1Tra

nsm

ittan

ce, T

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1Effect of absorption power of ink

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

complete ink pene.

no ink pene.

a) kp:s

p:k

i:s

i=1:100:50:1

b) kp:s

p:k

i:s

i=1:100:5:1

c) kp:s

p:k

i:s

i=1:100:50:1

d) kp:s

p:k

i:s

i=1:100:5:1

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 inkpenetration (no ink layer being left on the surface of the paper), the linearmodel contains only half the amount of ink as that of the uniform model. Inother words, the same amount of ink in both models requires the ink in thelinear model to reach twice the depth of the uniform model. Comparisons byconsidering the same penetrating depth (Fig. 5.12a, b) and the same amountof the penetrating ink (Fig 5.12c, d) have been carried out. These are shownin Fig. 5.12.

Summarizing the discussions, a dilemma exists in trying to limit the ef-fects of ink penetration. A natural choice for getting rid of ink penetrationis to improve the surface properties of the substrate paper and the rheologicproperties of the ink as well. In the paper making industry, post processescalled surface modification can greatly reduce ink penetration. However, suchprocesses have significantly increased the price of the paper which has largelylimited its popularity in producing ordinary print, such as newspaper. Al-though applying the paper with weaker scattering power may reduce the effectof the ink penetration, it reduces the opacity of the paper at the same time andis therefore unacceptable. Nevertheless, the use of ink having a higher absorp-tion can reduce the effect of ink penetration (Fig. 5.11). It requires inventionof new types of colorants (dye or pigment). Hopefully, a better understanding


0 0.5 1 1.5 2 0

0.2

0.4

0.6

0.8

1

Ref

lect

ance

, R

kp:s

p:k

i:s

i=1:100:25:1

0 0.5 1 1.5 2 0

0.2

0.4

0.6

0.8

1

Ref

lect

ance

, R

0 0.5 1 1.5 2 0

0.2

0.4

0.6

0.8

1

Depth of ink penetration,D

Tra

nsm

ittan

ce, T

0 0.5 1 1.5 2 0

0.2

0.4

0.6

0.8

1


Tra

nsm

ittan

ce, T

a)

b)

c)

d)

Linear ink pene.

Uniform ink pene.

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 con-tribution 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 andpaper-ink mixture, respectively. This means that the external and internalboundary reflection is, r0 = 0.0919 and r1 = 0.5960, respectively, for diffuseillumination.

The computed reflectance with consideration of the boundary reflection issmaller than that without, and the difference is generally independent of thethickness of the ink layer. This agrees with intuition, the internal reflection pre-vents a significant amount of light from passing through the ink/air interface.On the other hand, the transmittance with the boundary reflection is smallerthan that without. Nevertheless, their difference gradually vanishes when thethickness 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 refractiveindex of the ink layer is close to unity. However, in the case of ink penetration,


the refractive index of the ink-paper mixture may be remarkably bigger thatthat of air. Therefore, boundary reflection must be taken into consideration.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1Effect of surface reflection correction

Ref

lect

ance

, R

with surface ref.without surface ref.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1


Tra

nsm

ittan

ce, T

with surface ref.without surface ref.

a) kp:s

p:k

i:s

i=1:100:25:1

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 penetrationaffects 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 illuminatingthe object, and x(λ) the tristimulus function. If we denote the reflectance forthe bare paper, as R0(λ), and Rnp(λ), for the print with no ink penetration atall, and Rcp(λ) for the case with complete ink penetration, their X-stimulus


values, are, respectively.

X0 =∫R0(λ)S(λ)x(λ)dλ (5.75)

Xnp =∫Rnp(λ)S(λ)x(λ)dλ (5.76)

Xcp =∫Rcp(λ)S(λ)x(λ)dλ (5.77)

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 dif-ferences from paper white to full tone, (X0 −Xnp) and (X0 −Xcp), representranges of color reproduction for prints without and with ink penetration, re-spectively. Therefore, the latter inequality means that the ink penetrationreduces the color gamut of color reproduction. This theoretical conclusion isin line with experimental observation.

5.6 Summary

In this chapter we present models and solutions for three types of ink pen-etration, uniform, linear and exponential, that may correspond to differentink-paper combinations. In addition to the uniform model whose differentialequations can be solved analytically, series solutions corresponding to the linearand exponential models have been worked out. Generally, rapid convergenceof the series expansions have been found from simulations. Theoretically, fora certain amount of printed ink commanded by a printer, the printed imagewith ink penetration has greater reflectance than that without. In other words,the image becomes less saturated in color because of ink penetration. Conse-quently, the capacity for color reproduction (color gamut) is reduced due toink penetration.

The intention of the studies was to highlight the method for finding solu-tions rather than providing universal answers to complicated ink penetrationas a whole. Additionally, one model may be appropriate to some ink-substratecombinations but not others. For example, for dye based liquid inks, uniformink penetration may be a proper approximation when the pore size of the sub-strate is much bigger than that of the micelles of the dye [MK00] (about 100A).It is therefore important analyzing the characteristics of the ink-substrate com-bination before applying the models.

Chapter 6

Characterization of inksand ink application

6.1 Introduction

Ink-jet is a non-impact dot-matrix printing technique in which droplets of inkare ejected from a small aperture directly to a specified position on a substrateto create an image [Le98]. An ink jet printing system may be divided into threecritical components, the printer, the inks, and the substrate [vBKZT01]. Brieflyspeaking, the printer acts as a distributer for the ink droplets and the substrateacts as a receiver. It is the ink droplet that results in the observed color byselectively reflecting (or more precisely, selectively absorbing and scattering)the illuminating light. These three components together control the qualitiesof the printed image. It is important to bear in mind that these componentsare strongly correlated with each other.

Studies on substrate related issues, such as optical dot gain (or Yule-Nielseneffect) and ink penetration, have recently been reported [Hub97, 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 ink-mixing for generating the secondary colors, and of course, the optical propertiesof the inks (scattering and absorption coefficients). Some of the printer- andthe ink-related issues can strongly correlate with the substrate related issues.For example, the depth of ink penetration is largely dependent on the ink vol-ume. In addition, the color rendition also depends significantly on the opticalproperties of the inks when there exists ink penetration. Therefore, characteri-zations of the output print in terms of ink distribution and volume, the schemeof ink-mixing, light absorption and light scattering are of essential importancein 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 ink-and printer-related issues. The goal of the study is to determine the followingquantities: the absorption and scattering properties, the thickness of printedink-layers when different ink-level specification in printing process is used or toquantify the ink-level specification of the printer; and, the color mixing schemefor generating secondary colors. This knowledge is important for future studieswhen there is ink penetration (print on ordinary paper) because one needs priorknowledge about the ink (in terms of its absorption and scattering properties)and how much ink has been printed. For example, different amounts of printingink is possibly related to the different depths of ink penetration. Additionally,this information can probably help to fingerprint the printer which may findapplications in identifying ink-jet counterfeiting. This is an new topic of re-search that is becoming more and more important in light of the continuousimprovement 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 absorptionpowers (sz, kz, i.e. products of the scattering and absorption coefficients, sand 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-layerof any thickness. By varying ink-volume specification in the printer drivingsoftware (commonly available from printer manufacturers), one can print theink-layer in different thicknesses. Finally, these coefficients are used to predictspectral 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 secondarycolors (red, green, and blue) were printed, with the secondary colors beingmixtures of the primary colors. In order to prevent the inks from penetratinginto the substrate, as in ordinary office copy-paper which significantly modifiesthe color, ink-jet transparency (3M CG3460) were used as substrates. There-fore, the sample consists of a macroscopically uniform ink-layer and a plasticsubstrate. By varying ink-level specification in the printer driving software,one can adjust the parameter relating to the ink volume (ink-level) and printsamples with up to 5 ink-levels (the ink-volume increases from ink-level 1 to5).

Measurements were carried out by applying a spectrometer of the 45o/0o

geometry, with the collimated illumination from the top of the sample, asdescribed 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 underthe samples.

6.2.2 Data analysis and simulation

According to the radiative transfer theory [Cha60] and Kubelka-Munk approx-imation, the spectral reflectance value (Rjq) of an ink-layer is a function ofits scattering and absorption powers, sq(λ)zjq and kq(λ)zjq, being products ofthe scattering (sq) and absorption (kq) coefficients (m−1) and thickness of theink-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 ofuniform media (Eq. (5.34)), the function in Eq. (6.1) may be expressed as

f(sz, kz) =s(A− sRg)e−

s2−A2A z −A(s−ARg)

A(A− sRg)e−s2−A2

A z − s(s−ARg)

=sz(Az − szRg)e−

(sz)2−(Az)2

Az −Az(sz −AzRg)

Az(Az − szRg)e−(sz)2−(Az)2

Az − sz(sz −AzRg)(6.2)

where Rg(λ) is the spectral reflectance of the bare substrate and

A = s+ k −√k2 + 2ks (6.3)

Therefore, by fitting to two sets of measured spectral reflectance values, one canobtain the scattering power of the ink-layer (sz) and Az. From Eq. (6.3) onecan 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 primaryinks at ink-level 3 (defined by the printer driving program), and another setof data (RII) was from twice printed samples with the same ink-level. For thelatter case the second round of printing was carried out 10 minutes later. Thesample was measured 24 hours after printing. Assuming that the ink thicknessof the latter is twice that of the former, we then find (see Appendix A fordetailed derivation)

(sz −AzRII)(Az − szRII)

=(sz −AzRI)2(Az − szRg)(sz −AzRg)(Az − szRI)2

(6.4)

Combining Eq. (6.4) with the following equation

RI =sz(Az − szRg)e−

(sz)2−(Az)2

Az −Az(sz −AzRg)

Az(Az − szRg)e−(sz)2−(Az)2

Az − sz(sz −AzRg)(6.5)


one can determine the scattering and absorption powers of the ink-layer, sz andkz (through Eq. (6.3)). Equations (6.4) and (6.5) may have more than one pairof solutions. Therefore, an extra constraint has been added to the solutions tominimize the difference between the simulated- and experimental-values,

∆ =∑λ

{[RI − f(sz, kz)]2 + [RII − f(2kz, 2sz)]2} (6.6)

When the scattering and absorption-power of the primary ink-layer of ink-level 3, sq(λ)z3q and kq(λ)z3q, are known, these values can then be utilizedto predict reflectance values of an ink-layer of any given ink thickness, αjq =zjq/z3q, by applying

Rjq(λ, αjqz3q) = f(αjqsq(λ)z3q, αjqkq(λ)z3q) (6.7)

where αjq (q = c,m, y, and j = 1 − 5) is the relative ink thickness of the ink-layer. On the other hand, Eq. (6.7) can be applied inversely, i.e. for knownspectral reflectance values, Rjq, ink thickness of the ink-layer, αjq, can beestimated by fitting Eq. (6.7) to the Rjq. In this way, we obtained the relativeink thicknesses of the samples printed with ink-levels, 1, 2, 4, and 5. Becausethe spectral reflectance values of each sample consist of reflectance values at31 wavelengths (400− 700nm), the agreement between the computed spectralreflectance values and the measured ones can serve as a quantitative test of thequality of the scattering and absorption values obtained from ink-level 3. Theagreement 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 theprinting engine.

After obtaining the scattering and the absorption powers of the primaryinks, one can compute the spectral reflectance values of the secondary colors byapplying the additivity assumption [Pau87]. For example, color red is composedof ink magenta and ink yellow. Its scattering and absorption powers, accordingto additivity, can be expressed as

srzjr = βrjmsmz3m + βr

jysyz3y (6.8)krzjr = βr

jmkmz3m + βrjykyz3y (6.9)

where sqz3q and kqz3q (q = m, y, and j = 1− 5) are the scattering and absorp-tion powers that were obtained from the primary inks (ink-level 3). Therefore,by substituting srzjr and krzjr into Eqs. (6.2), one can compute the spectralreflectance values of red, Rjr. Inversely, the contribution from the primary inksβtjq (t = r, g, b denote the secondary color and q = c,m, y the primary colors)

can be determined by fitting to the measured spectral reflectance values of thesecondary color (red). Observe that the quantities, βr

jm and βrjy, are defined in

6.3 Results and discussions 67

exactly the same way as for αjm and αjy in Eq. (6.7). Therefore, they are theprimary ink amounts (relative to z3m and z3y, respectively) that are needed forgenerating the secondary color, red.

6.3 Results and discussions

In this section we present experimental measurements together with our the-oretical analysis and simulations, for a dye-based ink-jet printing system. Asmentioned above, the spectral absorption and scattering powers (kz and sz) ofthe primary inks were obtained by fitting to experimental spectral reflectancevalues of samples printed with ink-level 3, according to the specification of theprinter driving software.

400 450 500 550 600 650 7000

1

2

3

Ink layer printed in ink level 3

Abs

orpt

ion

pow

er, k

z cyanmagentayellow

400 450 500 550 600 650 7005.8

6

6.2

Wavelength, λ(nm)

Sca

tterin

g po

wer

, sz

(10−

3 )

a)

b)

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) areshown in Fig. 6.1. As expected, the inks cyan, magenta and yellow show strongabsorption in the long-, middle-, and short-wavelength regions, respectively.


In other words, cyan has a transparent window in the short to middle wave-lengths, 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 valuesof about 0.006. This means that for dye-based ink-jets, the scattering powerof the printed ink-layer is practically negligible, which is favorable for creatingcolor of high saturation.

400 450 500 550 600 650 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavelengths, λ(nm)

Ref

lect

ance

,R

Simulation

Measurement

Ink level 1

Ink level 5

cyan

yellow magenta

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 thick-

nesses of the primary inks

The scattering and absorption powers (sqz3q and kqz3q) were obtained by fittingthe computed spectral reflectance values of samples printed with ink-level 3(correspondingly, ink thickness z3q, q = c,m, y), to corresponding experimentalvalues. It is worthwhile to quantitatively test their reliability. The test wasmade by employing the scattering and absorption powers to compute spectralreflectance 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 inkvolumes (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


as αjqsq(λ)z3q and αjqkq(λ)z3q, where the quantity, αjq, represents the relativeink thickness of the sample.

Table 6.1: Quality evaluation for the simulated spectra in terms of color difference

(∆E) compared to experimental values.

Color Difference, ∆Eink-level Cyan Magenta Yellow

1 0.3093 2.1339 1.66102 0.3647 1.4876 2.01683 0.2753 0.6321 1.54224 0.8851 3.1225 1.25225 1.5858 3.5155 0.5329

1 2 3 4 51

1.5

2

2.5

3

3.5

4

4.5

Ink level printer , j

Rel

ativ

e in

k th

ickn

ess,

α

cyanmagentayellow

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 themeasured ones (dots), of all the 5 (primary ink) levels, have been shown inFig. 6.2. For a simpler comparison between the simulation and the measure-ments, both the simulated and experimental spectra have been converted totheir color coordinates in CIELAB color space. Their differences in terms ofcolor appearance (∆E) are listed in Tab. 6.1. From Fig. 6.2, as well as Tab. 6.1,


one may conclude that the simulation is in fairly good agreement with experi-ment over the whole range of visible light (31 wavelength-sampling points overthis range in the measurements). This may be considered as a confirmation ofthe validity of the method and the reliability of the sq(λ), kq(λ) values. It isworth noting that the 5-ink-level specification in the printer driving programdoes not always mean 5 different printing ink-levels. For ink cyan there areindeed 5 different ink-levels, but in practice there are only 3 and 4 differentink-levels for yellow and magenta, respectively. The correlation between theactual 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 non-linearly with respect to the ink-level specification (for simplicity, the actualvolume of ink-level 1, of each color, has been set to unity).

1 2 3 4 50.5

1

1.5

2

2.5

3

3.5

4Composition of the 2nd colors from the primary inks

Printing ink levels, j

Rel

ativ

e in

k am

ount

, β

Blue Green

Red

cyan magenta

yellow

Figure 6.4: Ink composition of secondary colors of different printing ink-levels (j=1-

5). 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 thick-

ness of secondary colors

Because samples of secondary colors, red, green and blue, are obtained bymixing two of the three primary colors in the printing process, they may serve asexcellent examples for testing the additivity of the scattering and/or absorptionpowers as stated in Eqs. (6.8) and (6.9).


400 450 500 550 600 650 7000

1.0

2.0

3.0

Abs

orpt

ion

pow

er, k

z RedGreenBlue

400 450 500 550 600 650 7008.4

8.5

8.6

8.7

Wavelength, λ(nm)

Sca

tterin

g po

wer

, sz

(10−

3 )

a)

b)

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 secondarycolors, βt

jq (j = 1 − 5, t = r, g, b and q = c,m, y), have been determined andshown in Fig. 6.4. It shows that the relative amounts of the primary inksstrongly depend on the colors. The color red, for example, is formed by nearlyequal mixing of magenta with yellow. The color blue, however, is predominatedby cyan relative to magenta, and the color green is something in between. Inaddition, the relative amounts of the primary inks vary modestly from oneink-level to another, which results in similar hue but different color saturation.Furthermore, the amounts of the primary inks used to generate the secondarycolors are not a simple superposition of the amounts of the primary samplesprinted. Color blue of ink-level 4, for example, consists of cyan (3.8) andmagenta (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 ink-jet printing technique compared to traditional offset printing. This flexibilityprovides the printer manufacturer the possibility to achieve an optimum colortone, while at the same time avoiding too much ink being printed, which inturn avoids a serious print through (on plain paper) and a prolonged dryingprocess, etc. Finally, similar to the observation for primary colors, the inkamounts vary nonlinearly with respect to the printer specified ink-levels.

The scattering and absorption powers of the secondary colors (ink-level3) are shown in Fig. 6.5. Plots in Fig. 6.5 clearly show the existence of ab-sorption/window structures that match well with intuition. Referring to theabsorption/scattering characteristics of the primary colors (Fig. 6.1), one can


easily see correlations between these absorption/window structures of the sec-ondary colors and those of their primary components, for example color red,the strong absorbing band (400−600 nm) consists of two sub-absorbing bandsfrom magenta (480− 600 nm) and yellow (400− 480 nm), respectively. Natu-rally, the scattering power of each ink-layer is very weak and varies little withrespect to wavelength.

400 450 500 550 600 650 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavelength, λ(nm)

Ref

lect

ance

, R

Simulation

Measurement

Blue Green

Red

Figure 6.6: Simulated and measured spectral reflectance values of samples in sec-

ondary colors. The samples were printed with program specified ink-level 3.

The simulated spectral reflectance values together with the correspondingexperiment values of ink-level 3 are shown in Fig. 6.6. As can be seen, thesimulations agree fairly well with experimental data. This may imply that theadditivity 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 thecolorist in research and in industry since its introduction in the 1930’s. Overthe years, the K-M theory has been subjected to very close scrutiny. The resultis an appreciation of the limitations and strengths of the theory [Nob85]. Ofall the original assumptions made by Kubelka and Munk, that of uniformlydiffuse forward and reverse flux through the sample is the most possible sourceof imprecision, when it is applied to such a system as an ink-layer with strong

6.5 Summary 73

absorption. The reason behind this is that rays that propagate in differentdirections will be attenuated differently. For example, oblique rays will beattenuated more than vertical rays, which invalidates the assumption that wehave diffuse light everywhere in the medium. Despite such a limitation of thetheory, it remains the most widely applied theory in paper-making and printingindustries, because of its simplicity and above all because it works well in themajority of cases. It is therefore worthwhile to closer examine the consequenceswhen the conditions of the theory are not well satisfied.

As shown in Fig. 6.1, the light-ink interaction (or the ink spectrum) showsclear transparent- and absorption-band structures. For light whose wavelengthlies well away from the absorption band, the absorption is less important. Cor-respondingly, the light distribution for this band will probably remain diffuse ifthe original illumination is diffuse. Therefore, this portion of light can be prop-erly described by the K-M theory applied to human color vision. On the otherhand, the portion of the illumination whose wavelength lies in the absorptionband will mostly be filtered out by ink absorption after passing through anoptically thick ink-layer, even though the light may not completely satisfy theconditions 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 theproperties of the inks by means of spectral reflectance measurements. Themeasured data were analyzed with the help of theoretical simulations. Theprinted ink volume (equivalently, thickness of the printed ink-layer) of theprimary colors and their absorption and scattering characteristics were de-termined. Spectrally, the inks show clearly absorption- and transparency-bandstructures with respect to the wavelengths of the illumination. The scheme ofcolor composition for the generation of secondary colors (from the primary inks)was determined. In addition, the additivity assumption for obtaining the scat-tering and the absorption power of secondary colors from their primary colorcomponents has been tested and seems to hold well (with regard to computedspectral reflectance values of the secondary color). Simulations of the spectralreflectance values have been carried out for both primary and secondary colors.The simulations are in fairly good agreement with the measurements.

Chapter 7

Characterization of inkpenetration

When ink is printed on ordinary (plain) office copy-paper, it penetrates intothe substrate, while at the same time it spreads on the paper surface. Theink penetrated paper is an ink-paper mixture which appears significantly dif-ferent from either constituent, in particular the pure ink (with paper backing).This arises because of presence of the paper materials (fillers, fibres, etc) thatgenerally have strong scattering power. Ink penetration modifies the ink per-formance significantly and results in serious undesirable consequences for theprinted color.

In Chapter 6, we determined the optical properties of the inks (sqzjq andkqzjq) and the ink volume controlled by the printer (αjq and βt

jq, for primaryand secondary colors). The sub and super scripts, q = c,m, y and t = r, g, bdenote the primary and secondary colors, respectively, and j = 1−5 refers to theink levels or equivalently the ink volumes specified by the printer driver. Thisknowledge forms one fundamental as past study in ink penetration . Anotheraspect 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 brighten-ing materials which absorbs illuminated UV light and re-emits it as fluorescencewhose wavelength lies in the blue light region. To avoid complexities in the de-termination of scattering and absorption properties of paper, a UV filter wasused in the measurements in order to eliminate the UV light from the illumi-nation.

To determine the scattering and absorption powers of the paper, spD and

76 Characterization of ink penetration

400 450 500 550 600 650 7000.4

0.5

0.6

0.7

0.8

0.9

1A sheet of plain paper with white and black backing

Wavelengths, λ(nm)

Ref

lect

ance

, R

white backingblack backing

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.

kpD, two sets of spectral reflectance values are needed. Here sp and kp arethe scattering and absorption coefficients of the paper, and D, the thicknessof a single paper sheet. These were obtained by measuring a single sheet ofthe plain paper for two types of backings, black and white. We denote thespectral reflectance values of the white and black backings as, Rgw and Rgk,respectively. Correspondingly, the reflectance values of a sheet of paper havingwhite and black backings are, Rw and Rk, respectively (see Fig. 7.1). Accordingto Eq. (5.34), one has

Rw = r0 +(1− r0)(1− r1)[sp(Ap − spRgw)e

− s2p−A2p

ApD −Ap(sp −ApRgw)]

(Ap − spr1)(Ap − spRgw)e− s2p−A2

pAp

D − (sp −Apr1)(sp −ApRgw)(7.1)

Rk = r0 +(1− r0)(1− r1)[sp(Ap − spRgk)e

− s2p−A2p

ApD −Ap(sp −ApRgk)]

(Ap − spr1)(Ap − spRgk)e− s2p−A2

pAp

D − (sp −Apr1)(sp −ApRgk)(7.2)

whereAp = sp + kp −

√k2p + 2kpsp (7.3)

7.1 Optical properties of plain paper 77

Solving these equations numerically, one gets spD and ApD if the boundaryreflection values, r0 and r1 are known. From Eq. (7.3), one can then obtainthe scattering and absorption powers, spD and kpD. It has been commonlyassumed that the refractive index of plain paper, n, is not much larger thanunity [Mou02a]. However, simulations of clean paper and ink penetrated paperreveal the existence of boundary reflectance at the air/paper interface. Addi-tionally, the simulations show that, n = 1.2, fits the data best. This shouldnot be so surprising since the paper contains fibres and fillers, which may con-tribute to a greater dielectric constant and consequently, a greater refractiveindex compared to the air.

400 450 500 550 600 650 7000

0.1

0.2

0.3

0.4

Abs

orpt

ion

pow

er, k

pD

400 450 500 550 600 650 7002

4

6

8

10

Wavelengths, λ (nm)

Sca

tterin

g po

wer

, spD

a)

b)

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 isshown in Fig. 7.2. The paper has little absorption in green and red parts of thespectrum, but absorbs blue light somewhat. Nevertheless, the absorption is stillmuch weaker than that of inks as found in Chapter 6. On the other hand, thepaper has much a stronger scattering power than do the inks. The scatteringpower of the paper is nearly constant in the green to red light regions. It grad-ually increases from the green to the blue light regions, reaching its maximumat about λ = 430nm, and then decreases sharply for shorter wavelength. Thisobservation generally coincides with the fluorescence distribution observed bybi-spectral measurements [Mou02b] and may indicate the existence of residualfluorescence which survives from the UV filtering. A combination of increasingabsorption with decreasing scattering at the short blue region makes far weakerreflectivity in this wavelength region, if no brightening materials are added.


The possible failure in completely removing the fluorescence from the mea-sured spectral reflectance will cause errors in the determination of the scatteringand absorption powers of the paper. Consequently, these errors will result inerrors in the simulated spectral reflectance of the ink-paper mixture. Indeed,there exists a remarkable discrepancy between the simulations and the mea-surements 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. Forsimplicity and clarity of the description, assumptions and notations used inthis chapter have been summarized or defined as fellow.

7.2.1 Assumptions

1. Paper making materials, fibres, fillers, etc., are uniformly distributed inthe paper;

2. The ink concentration in the paper is uniform and independent of depthof ink-penetration;

3. No pure ink is left on the paper surface and consequently the depth ofink penetration is proportional to the amount of the printed ink;

4. Light becomes completely diffused once it enters the paper or ink-papermixture;

5. The scattering and absorption powers of the ink-paper mixture fulfillsthe 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 ina first attempt to simulate ink penetration. With the help of mathematicaltreatments for non-uniform ink penetration, developed in Chapter 5, thesemodel can readily be extended to non-uniform cases, if needed. Additionally,the second assumption may be a reasonable approximation for a combinationconsisting of dye-based liquid inks and plain paper. Because the size of thecellulose pores in the paper is generally much larger than that of the dye micro-cell (100A), ink can easily penetrate into the substrate, which results in littlegradient in ink concentration. On the other hand, the inks are completelyabsorbed by the substrate which forms the grounds for the third assumption.

Physical considerations behind assumption No. 4 are that the paper mate-rials have a very strong scattering power. The strong scattering power of thepaper 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 be-comes diffuse in the ink-paper mixture. Such a consideration is of fundamentalimportance in the current study, when the additivity assumption (assumptionNo. 5) is applied.

As explained in Sec. 5.2, quantity k is a phenomenological description ofthe light absorption in the medium, which not only depends on the physicalproperties 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 ink-layer was of 45o/0o geometry with collimated illumination from the top of thesample (see Sec. 2.2.2 for detailed description). Considering that the pure inkhas little scattering power (see Chapter 6), light propagates essentially througha straight path in the ink-layer. Correspondingly, the absorption power, kqzq,is responsible for the light extinction along that path. However, in the ink-paper mixture, the light (in both transparent and absorption bands) becomesscattered and propagates in a zigzag fashion. Consequently, light has greaterpossibility of being absorbed, if it passes the same vertical depth in the ink-paper mixture as that in the ink-layer. Therefore, the averaged absorptionpower of the ink-paper mixture becomes 2kqzq, if the light becomes completelydiffused in the layer. It was experimentally observed that the absorption (k)of colored paper is approximately twice that of the cellophane compared at agiven amount of dye [BS76].

The additivity assumption stated in assumption No. 5 has been confirmedvalid in color mixing (see Sec.6.3.3). As one will see late, it holds even for theink-paper mixture, if a factor of 2 is introduced into the absorption coefficientof 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 theprinter deriving program, j = 1− 5, and color q = c,m, y.

• αjq = zjq/z1q – the relative ink thickness to ink level 1.

• βtjq– the amount of the primary inks (q = c,m, y) used to form the sec-

ondary colors (t = r, g, b) of ink level j. For example, the ink thickness(volume) of color red (r) of ink level j reads

zjr = βrjmz1m + βr

jyz1y (j = 1− 5)

• D– the thickness of a single sheet of paper.

• djq–the thickness of ink penetration into the paper (in percentage of asingle sheet paper thickness).


• 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). Theextra subscript, p, denotes paper.

• kqp–the absorption power of ink-paper mixture.

Among these quantities, sq, kq, zjq (or αjq) and βtjq (j = 1−5, q = c,m, y, and

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 withboth the primary and the secondary colors, the secondary colors being mixturesof two of the three primary colors. By varying ink-level specification in theprinter 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 werecarried out by applying a spectrometer, which covers a spectral range of 380 to730 nm at a interval of 10 nm. A UV filter was employed in order to minimizethe impact of fluorescence.

7.3.1 Primary colors

Following the notations and assumptions listed in Sec. 7.2, the scattering andabsorption powers of the ink-paper mixture (color q and ink level j) may bewritten as

sqpDdjq = sqzjq + spDdjq (7.4)kqpDdjq = 2kqzjq + kpDdjq (7.5)

where q = c,m, y represents for the primary colors, and djq the depths of inkpenetration .

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 theassumption No. 3, may be written as,

djq = αjqd1q (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 thesimulation is, therefore, to determine the depth of ink penetration of a singleink level (say d1q) for each primary color. The depths of ink penetration ofother ink levels can be computed from Eq. (7.6).

In practice it is d3q, the depth of ink penetration of ink level 3, that isfirst determined instead of d1q. The determination of the d3q is very similar towhat was done in the determination of βt

jq (see Section 6.2.2). When the inkswere printed on foil they formed pure ink-layers on the foil surface. Becausethe refractive index difference between the air and the ink was negligible, nocorrection for boundary reflection was needed. Nevertheless, when the inks areprinted on the office copy-paper the inks actually go down into the substratepaper. Due to the refractive index discontinuity at the air/paper interface, theboundary reflection between the air and the paper (or ink penetrated paper)has to be considered. Therefore, the following formula that takes care of theboundary reflection (Eq. (5.34)) is used, i.e.

R = r0+(1− r0)(1− r1)[sqp(Aqp − sqpRg)e

− s2qp−A2qp

AqpDdjq −Aqp(sqp −AqpRg)]

(Aqp − sqpr1)(Aqp − sqpRg)e− s2qp−A2

qpAqp

Ddjq − (sqp −Aqpr1)(sqp −AqpRg)(7.7)

where r0 and r1 are the external and internal boundary reflection, Rg, thereflectance of the plain paper, and

Aqp = sqp + kqp −√k2qp + 2kqpsqp. (7.8)

Our simulations show that n = 1.2 provided the best fit to the measuredspectral reflectance for both printed and non-printed paper. The external andinternal 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 ofink level 3 are depicted in Fig. 7.3. Evidently, the scattering power of theink penetrated paper bears a general similarity to that of the bare paper. Onthe other hand, the absorption power of the ink-paper mixtures are of similarshape to that of the inks. These observations reflect the facts that the scatteringcharacteristics of the ink-paper mixture are predominated by the paper and theabsorption characteristics by the inks.

A comparison between the simulated and the measured spectral reflectancevalues 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 allthe colors, and over the entire visible spectrum. Color differences between themeasurements and the simulations are just about visible and ∆E = 3.28, 4.14,and 5.15, for cyan, magenta, and yellow, respectively. A remarkable discrep-ancy occurs for ink cyan at around λ = 460 nm. This may be a consequence of


400 450 500 550 600 650 7000

2

4

6Ink level=3

Abs

orpt

ion

pow

er, k

z cyanmagentayellow

400 450 500 550 600 650 7000

0.2

0.4

0.6

0.8


Sca

tterin

g po

wer

, sz

a)

b)

Figure 7.3: Scattering and absorption powers of the ink-paper mixture (ink level 3).

400 450 500 550 600 650 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Ink level=3


Ref

lect

ance

, R

Simulation

Measurement

cyan

yellow magenta

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 beingobserved from the bare paper. Bispectral fluorescence measurements made byMourad [Mou02b] revealed that, for brightened paper, the fluorescence has itspeak at about λ = 450 nm and the fluorescence had even remarkable contri-bution 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 UVfilter employed in the present measurements. Such studies may provide cluesleading to a better understanding of the discrepancies.

Closer observation of the spectra of all the primary colors reveals that theminima of the reflectance values are generally independent of the color and farfrom 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 inSec. 5.5.3, the possible explanation for the difference can lie with the boundaryreflection occurring at the boundary between the air and the ink penetratedpaper. Another possible origin lies with the strong scattering power of theink-paper mixture that contributes to the light reflection.

400 450 500 550 600 650 7000

1

2

3

4

5Ink level=3

Abs

orpt

ion

pow

er, k

z

400 450 500 550 600 650 7000.2

0.4

0.6

0.8

1

1.2


Sca

tterin

g po

wer

, sz

a)

b)

red

green blue

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, sqpdjq, kqpdjq, obtained from the primary col-ors (printed on the paper) is directly tested when they are applied to predict


the spectral reflectance of the secondary colors. For secondary colors, accordingto the assumption No. 5, the scattering and absorption powers of the ink-papermixture may be expressed as, (color red (r), for example)

srpDdjr = βrjmsmpDdjm + βr

jysypDdjy

= βrjmαjmDsmpd1m + βr

jyαjysypDd1y (7.9)krpDdjr = βr

jmkmpDdjm + βrjykypDdjy

= βrjmαjmkmpDd1m + βr

jyαjykypDd1y (7.10)

The scattering and absorption powers of the ink-paper mixture for the sec-ondary colors (ink level 3) are shown in Fig. 7.5. Similarly to the primarycolors, scattering is predominated by the paper, and absorption by the inks.

In Eqs. (7.9) and (7.10) all the parameters are known, namely βrjq and

αjq from the ink on foil (Chapter 6), and sqpd1q and kqpdjq from Sec. 7.3.1.Therefore, by applying srpdjr and krpdjr to Eq. (7.7) one can predict thespectral reflectance of the secondary colors printed on the paper. As there isno data-fitting involved in predicting the spectra for the secondary colors, acomparison between the predicted and the measured spectral values provides atest 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. Asseen, the prediction is in fairly good agreement with the experiment for all thesecondary colors and over the entire visible spectrum.

It is convenient to depict the depth of ink penetration in terms of a percent-age of a single sheet paper thickness. The depths of ink penetration for bothprimary 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 obtainedby fitting to the measured spectral reflectance data, all the rest were actuallycomputed 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 di-rectly compare prints with and without ink penetration by using the same typeof substrate. Unfortunately, there exists no substrate which allows for ink pen-etration 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 paperother than copy-paper is used. Such a modification to the paper surface canindeed reduce or even eliminate ink penetration into the cellulose structure.Nevertheless, this changes the (optical, fluid dynamic, etc.) properties of thepaper which essentially implies another type of substrate.

7.4 Optical effect of ink penetration 85

400 450 500 550 600 650 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Ink level=3


Ref

lect

ance

, RSimulation

Measurement

Figure 7.6: The spectral reflectance of the secondary colors printed on office copy-

paper (ink level 3).

1 2 3 4 50

5

10

15

Ink penetration into the plain paper

Ink

pene

trat

ion,

d (

%)

cyanmagentayellow

1 2 3 4 55

10

15

20

Ink level, j

redgreenblue

a)

b)

Figure 7.7: Depthes of ink penetration of the primary and secondary colors.


Although there is no direct experimentally possible comparison, it can beachieved with help of simulations. The comparison was made for print of inklevel 3 (the default ink level of the printer). Because the scattering and absorp-tion powers of the inks (sqz3q, kqz3q), and the ink-paper mixture (kqpdjq, kqpdjq)are known from our studies, the spectral reflectance values of prints on the of-fice copy-paper with and without ink penetration can be simulated accordingto Eq. (7.7). In the case of no ink penetration, boundary reflection between theair/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. Forconvenience of discussion, we refer to the band containing the local maximumas the transparent band and that containing the local minimum as the absorp-tion band. In the transparent band, the print for the case of no ink penetrationhas greater reflection compared to that with ink penetration. In contrast, inthe case of the absorption band, the print with ink penetration shows strongerreflection than that without. These observations reflect the collective contribu-tion 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 sub-strate backing (plain paper), while it consists of an ink-paper mixture layer andthe substrate backing in the case of having ink penetration. As is known theink-layer (no ink penetration) has little scattering power, the reflection in thetransparent band is essentially due to reflection from the substrate. Compara-tively, the layer of the ink-paper mixture has a much stronger scattering powerwhich blocks the light from reaching the substrate backing to some extent. Onthe other hand, the absorption power of the ink-paper mixture (there existsabsorption even in the transparent band of the ink) is twice as great as that ofthe pure ink-layer. These factors work together and result in weaker reflectionfor the case of the print with ink penetration. In the absorption band, on theother hand, the light is dramatically attenuated by absorption when it passesthrough 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 mixturedue to scattering of the paper materials which makes the ink-paper mixturemore reflective.

Ink penetration also has significant impact on the color appearance of theprint. Because the saturation of the color depends on the contrast between thepeaks of the transparent bands to the bottoms of the absorption bands, thecomparison suggests that ink penetration significantly reduces the saturation ofthe printed color. At the same time, it causes even color shift (hue variation)because of nonlinear modifications to the spectral reflectance values by inkpenetration. A quantitative measurement of the color difference induced by inkpenetration is given in Tab. 7.1. Clearly, ink penetration has significant effect

7.4 Optical effect of ink penetration 87

400 450 500 550 600 650 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Ink level=3


Ref

lect

ance

, RInk penetration

No ink penetration

cyan yellow

magenta

(a) Primary colors

400 450 500 550 600 650 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Ink level=3


Ref

lect

ance

, R

Ink penetration

No ink penetration

blue green

red

(b) Secondary colors

Figure 7.8: Comparison of colors printed on office copy-paper with and without ink

penetration.


on color reproduction. Detailed discussions about the impact of ink penetrationon color rendition and correspondingly, on their graphical representations willbe given in Chapter 10.

Table 7.1: Color differences (∆E1)) induced by ink penetration.

color cyan magenta yellow red green blue∆E 13.13 31.98 30.61 29.20 36.97 25.40

1) ∆E =√(∆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 acombination of dye-based liquid ink and office copy-paper is presented. Themodel uses optical properties (scattering and absorption) of the ink and thepaper as inputs to simulate the ink-paper mixture (ink penetration). A methodused for determining fundamental quantities like depth of ink penetration bycombining spectral reflectance measurements with simulation has been pro-posed and tested. Parallel comparison between prints with and without inkpenetration has been made and it shows that ink penetration causes significantcolor saturation reduction and hue shifting.

Chapter 8

Dot gain in black and white

8.1 Introduction

8.1.1 Murray-Davis equation

For a small area of a halftone image as shown in Fig. 8.1, the light detected bya sensor (a human or an instrument), is a mixture of the light reflected from thehalftone dots and from the paper between the dots. Assume that the intensityof illumination is I0. Under uniform illumination, it is natural to assume thatthe illumination onto the dots and the substrate is proportional to the frictionof area, σ and 1 − σ, and can be written as I0σ and I0(1 − σ), respectively.Subsequently, one can express the total light reflected from an area, Ir, as asum of these two parts, i.e.

Ir = I0Rg(1− σ) + I0R01σ (8.1)

where Rg and R01 are the reflectance of the substrate paper and that of solid

print, respectively. The superscript 0 in R01, denotes hereafter that the re-

flectance of the solid print differs from that of halftoned ink dots (R1). Ac-cording to the definition, the average reflectance of image area, R, is thereforewritten as

R = Rg(1− σ) +R01σ (8.2)

Equation (8.2) is usually called Murray-Davis (M-D) equation [Mur36]. Despitebeing simple in derivation, the Murray-Davis equation has historically been amilestone in Graphic Arts, because it initiated an era of modelling the printmathematically.

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) +X1σ (8.4)

where S(λ) is the spectral distribution of the illumination and x(λ) is the stim-ulus function. X0, X1, and X are the tristimulus values of the substrate, theink solid, and the halftone image, respectively. Equation (8.4) is a represen-tation of Eq. (8.2) in chromatic perspective and is known as the Neugebauerequation.

sensor

Σ1

Σ0

sensor

a)

b)

Paper

1

1’ 2

2’

1 2 1’

2’

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 thatit gave only a gross approximation to the experimental results. In 1951 Yuleand Nielsen [YN51] reported their study on discrepancies between the mea-surements and theoretical predictions (by the M-D equation). They found thatlight 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 amodification to the M-D equation, which is known as the Yule-Nielsen Equa-tion,

R = [(R01)

1nσ + (Rg)

1n (1− σ)]n (8.5)

The exponent n is usually called the Yule-Nielsen factor and can be obtainedby fitting to the experimental data (such as optical density). Unfortunately,the Yule-Nielsen equation provides only a better numerical approximation. Itprovides no physical insight into the real process. In 1978 Ruchdeschel andHauser [RH78] obtained an estimation of the exponent n from their point spreadfunction (PSF) analysis. Their study showed that 1 ≤ n ≤ 2, when only lightscattering 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 isused for data fitting. Due to light scattering within the substrate the printeddots appear bigger than they geometrically are. Consequently, the Yule-Nielseneffect is also referred to as optical dot gain and mere recently as tone valueincrease.

A similar modification to the Murray-Davis equation was also introducedto the Neugebauer equation by Pobboravsky and Pearson [PP72] in the 1970’s.

sensor

Σ1

Σ0

scensor

a)

b)

Paper

1 1’

2

2’

1 1’

2

2’

Figure 8.2: The Yule-Nielsen effect resulting from light scattering within the sub-

strate. 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


naturally with the printing/viewing process. Besides the optical dot gain, therealmost always co-exists another type of dot gain, physical dot gain, that takesinto account the dot extension due to mechanical and/or physical origins, suchas ink spreading, film exposure to plate, etc. Dot gain measurements take intoaccount both the physical and the optical dot gain. In other words an overalldot gain is measured. Moreover, as we shall see later on, the optical dot gainand the physical dot gain correlate with one another. Finally, the optical dotgain and physical dot gain depend not only on the print processes but also onthe materials used.

8.1.3 Status of the studies

Light scattering in substrate is a very complex process and has attracted con-stant research interest over a long period of time. Kruse and Wedin [KW95],among many others, proposed an approach which was thoroughly studied andfully implemented by Gustavson [Gus97a, Gus97b]. The approach simulatedthe light scattering process from a fundamental level. It was based on direct nu-merical simulation of scattering events which depend on the optical propertiesof the materials, the halftone frequency and the halftone geometry. This ap-proach is similar in nature to the Monte-Carlo method that is briefly explainedin 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] andHubler [Hub97] independently proposed similar models based on probabilitydescriptions of the light scattering. In their models, the light scattering insidethe paper was described by the probabilities that a photon emerges from theinked and non-inked areas. These probabilities depend on the positions wherethe photon enters the paper and emerges from the paper, as one will see inSec. 8.2.2. A point spread function (PSF) is a different representation of lightscattering. Using the PSF approach, Rogers [Rog97, Rog98a, Rog98b, Rog98c]presented a method dealing with the light scattering process. He proposeda matrix approach where the tristimulus values of a halftone image could becalculated as the trace of a product of two matrices.

So far the studies have not provided any explicit expression for the re-flectance or tristimulus as a function of dot percentage (as was given by theNeugebauer equation). Moreover, the studies were limited to mono-color, orblack and white case. Finally, the effect of ink penetration into the substratewas barely been touched upon. This chapter describes our studies in two ofthese three perspectives, a description of multi-color print is presented in Chap-ter 9.

In Chapter 5, we established a theoretical approach to account for effectsof ink penetration in which it was assumed that the substrate paper was uni-formly covered by ink layers. In this chapter we generalize this by investigat-ing 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) andink-penetration. In Section 8.3, we further extend the model to simulate theoverall dot gain (optical plus physical dot gain) and derive expressions for thereflectance and overall dot gain (and the corresponding tristimulus values) asfunctions of the dot coverage. Finally, the approach is further illustrated withapplication 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 thesubstrate 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 amulti-layer case will be given in Chapter 9. Also for simplicity, we assume thatthe ink layer has uniform thickness irrespective of wether or not there is inkpenetration 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 isan arbitrary position on the surface of the paper under the dots (Σ1) and −→r0an arbitrary position on the surface of the paper between the dots (Σ0). Nowwe consider an element light, I0dσ1, that strikes the dot at −→r1 . The flux of thelight detected at −→r0 , due to scattering of the incident light from −→r1 to −→r0 , maybe written as

d2J10 = p(−→r0 ,−→r1)TI0dσ0dσ1 (8.6)

where p(−→r0 ,−→r1) is the point spread function and, T , the overall transmittanceof the print. If the transmittance of the ink layer on the substrate is T1, thenfor no ink penetration, T = T1, and TI0dσ1 is thus the amount of light enteringthe substrate under the dot. When the ink penetrates partially (or even fully)into the substrate [YGK00, YLK01], then

T = T1γ (8.7)

where γ describes the optical effect of the ink penetration, defined as

γ =

√R′

g

Rg(8.8)

where Rg is the spectral reflectance of the clean substrate, modified into R′g

by the ink penetration. γ may be considered as an induced transmittance by


the ink penetration. Figure 8.3 sketches out this approximation. By using theoverall transmittance expressed in Eq. (8.7), there is no need to distinguishwether there is ink penetration or not. One thing that should be kept in mindis that γ depends not only on the ink but also on the substrate.

The point spread function is the probability of photons entering the paperunder the dots at position −→r1 and exiting from the paper between the dots atthe position −→r0 . It is worth noting to be noticed that ink penetration destroysthe 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], be-comes invalid.

T1

R0’

T1

γ

R0

R0’ =γ2R

0

(a) (b) Figure 8.3: An approximate treatment of the effect of ink penetration. (a) remaining

ink layer (transmittance T1) and ink penetrated substrate (reflectance, R′g); (b) the

ink penetrated substrate is approximated by introducing an extra ink layer (trans-

mittance γ) and the clean substrate (reflectance Rg), where R′g = γ2Rg.

Next, suppose an extended light source covering the area of the dots andsuch that the incident intensity, I0, is unchanged, while the paper betweenthe dots is not illuminated. Then, the flux of the light detected at −→r0 , due toscattering of the incident light from the dot is the integration of Eq. (8.6) overthe dots area (Σ1),

dJ10 = I0T

∫Σ1

p(−→r0 ,−→r1)dσ0dσ1 (8.10)

Therefore, ∫Σ1

p(−→r0 ,−→r1)dσ1, (8.11)

is the probability of the incident light entering the substrate under the dots andbeing scattered into position −→r0 . Finally, by performing the integration over


the whole area of the paper between dots (Σ0) one obtains the total amountof light detected at the paper between dots that is scattered from the incidentlight on the dots,

J10 = I0T

∫Σ1

∫Σ0

p(−→r0 ,−→r1)dσ1dσ0 (8.12)

The double integral ∫Σ1

∫Σ0

p(−→r0 ,−→r1)dσ1dσ0 (8.13)

is the overall probability of photons being scattered from Σ1 into Σ0, which istherefore 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.For example, in the first step we put the light source, I0dσ1, at −→r0 and thedetector at −→r1 , keeping other conditions unchanged. We then have

d2J01 = p(−→r1 ,−→r0)TI0dσ1dσ0 (8.14)

From the optical reciprocity one obtains the following relation,

p(−→r1 ,−→r0) = p(−→r0 ,−→r1) (8.15)

Then we haveJ10 = J01 (8.16)

This means that under uniform illumination of the halftone sample, the amountof the light being scattered from Σ1 (halftone dots) into Σ0 (bare paper) isequal 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 valuep of the integrated point spread function, defined as,

p =1

σ(1− σ)

∫Σ1

∫Σ0

p(−→r1 ,−→r0)dσ1dσ0 (8.17)

is available, then the scattered light J10 can be calculated from

J01 = J10 = I0Tpσ(1− σ) (8.18)

Evidently, p depends not only on the physical properties of the substrate paperand 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 thereforeexperimentally measurable.


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 denotedas P11 and P10. Similarly, for a photon entering the substrate in Σ0, P01 andP00 are defined as the probabilities that the photon leaves the surface of thesubstrate from Σ1 and Σ0, respectively. These probabilities fulfill the followingconstraint conditions [YLK01],

P11 + P10 = Rg (8.19)P01 + P00 = Rg (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 thesurface of the substrate at Σ1 and then returning to the air. This is exactlythe definition of the reflectance, Rg, from a probability perspective. A similarargument holds also for Eq. (8.20). It is worth noting that the summation ofthe conditional probabilities is normally smaller than unity because light maybe absorbed in the substrate or be transmitted through to another side of themedia.

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, thatthe reflectance measured from the paper between the dots, R0, and that fromthe dots, R1, respectively, are

R0 = Rg − pσ(1− T ) (8.23)R1 = T 2Rg + p(1− σ)T (1− T )] (8.24)

Clearly, the first terms on the right in Eqs. (8.23) and (8.24) correspondto the reflectance of the bare paper and the solid print, respectively, and thesecond terms to the light scattering. Consequently, the reflectance measuredfrom 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 re-flectance measured from the dots is not a constant either and is generally greaterthan that of the solid ink. Finally, the average reflectance of the halftone image,R, can be computed as

R = R0(1− σ) +R1σ

= RMD −∆Ropt (8.25)


whereRMD = Rg(1− σ) +RgT

2σ (8.26)

is the computed reflectance of the halftone sample under the Murray-Davisapproximation (excluding the light scattering), and

∆Ropt = (1− T )2pσ(1− σ) (8.27)

results from the light scattering inside the substrate paper. Recalling the factthat the overall transmittance, T , is the product of transmittance of the inklayer (T1) on the substrate and the ink penetration induced transmittance (γ),i.e. T = T1γ, 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.

0 20 40 60 80 1000.2

0.4

0.6

0.8

Ref

lect

ance

, R

0 20 40 60 80 1000

0.02

0.04

0.06

0.08

Dot percentage, σ (%)

Opt

ical

dot

gai

n, ∆

Rop

t

Rp

R

Ri

No ink penetration

Ink penetration

a)

b)

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 willbe wavelength dependent. Spectroscopically, ∆Ropt reaches its maximum in


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 measuredquantities (∆Ropt or T ) when white light illumination like D65 is used. Thespectral average of any quantity, A(λ), is defined as

< A >=∫A(λ)dλ∫dλ

In the following, the same symbol A will be used for both a spectrally dependentfunction and its spectrally average whenever there is no confusion.

Because ∆Ropt > 0, the true reflectance, R, is smaller than its Murray-Davisvalue, RMD, and the halftone image appears to be darker (more saturated incolor). Accordingly, it appears to have a larger dot coverage than predictedwhen light scattering is ignored. It is for this reason that this effect is known asoptical dot gain. If scattering is not modelled, then the measured reflectance Rseems to originate from a dot size, σ + ∆σopt, instead of the true dot size, σ.From R(σ) = RMD(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(1 + T )Rg

σ(1− σ) (8.28)

The quantity, ∆σopt, provides a phenomenological description of optical dotgain. However, due to its spectral dependence, it differs fundamentally fromany physical extension such as physical dot gain. Moreover, optical dot gaindiffers for different colors (inks).

Because ∆σopt and ∆Ropt are the geometric and optical representations ofthe same origin, we will not distinguish between them, and hereafter refer tothem both as optical dot gain. A similar convention applies to the physicaldot gain. Besides the spectral dependence, the optical dot gain, ∆σopt, isproportional to p. From the measured optical dot gain profile, one can possiblyestimate 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


probability, p, is essentially its reflectance, Rg. Then, the optical dot gain hasits maximum at σ = 50% (see Fig. 8.4b) and can be computed from

(∆Ropt)max = (1− T )2Rg/4 (8.30)

p = constant is usually called complete light scattering [Rog97], and theoptical dot gain can be computed from

∆σopt =(1− T )(1 + T )

σ(1− σ) (8.31)

which has a single maximum at σ = 50% and a symmetric form around themaximum. It is easy to prove that here the complete light scattering corre-sponds to the Yule-Nielsen model with the Yule-Nielsen factor of n = 2.

400 450 500 550 600 650 7000

0.2

0.4

0.6

0.8

1 Spectral dependence of the optical dot gain

Tra

nsm

ittan

ce, T

Tc

Tm

T

y

400 450 500 550 600 650 7000

0.1

0.2


Opt

ical

dot

gai

n, ∆

σ

∆σc

∆σm

∆σy

<∆σc>

<∆σm

><∆σ

y>

a)

b)

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 computedaccording to Eq. (8.31) is depicted in Fig. 8.5, in the case of σ = 0.4 and com-plete light scattering. The spectral transmittance values of ink cyan, magenta,and yellow obtained from Chapter 7 are shown in Fig. 8.5a. Clearly, ∆σoptshows a disctinct correlation with its spectral transmittance. In the figure, thespectrally averaged dot gain values, < ∆σopt >, of different colors have also


been included. Comparisons between the colors show that the light scatteringin the substrate results in the smallest spectrally averaged optical dot gain forink yellow, and the greatest spectrally averaged optical dot gain for ink cyan.

0 10 20 30 40 50 60 70 80 90 100

Dot percentage, σ (%)

X0−

X

X0−X

X0−X

MD

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 −XMD 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 imagescan 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 halftoneimage from its reflectance (Eq. (8.25)). For example,

X = XMD −∆Xopt (8.32)

where

XMD =∫RMD(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-negativityof ∆Xopt we find that for any tristimulus value X0,

X0 −X ≥ X0 −XMD (8.35)

This inequality is particularly interesting when X0 is the tristimulus value ofbare paper, because X0 −X stands for the range of variation of the tristimulusvalue 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 there-fore have the situation shown in Fig. 8.6, which demonstrates that, for any dotpercentage (except for σ = 0, 100%), the range of variation computed withincluding light scattering effect is bigger than that excluding this effect. Be-cause X0, Y0, Z0 stand for the white point of the substrate, in monochromaticcase the quantity, X0 − X together with Y0 − Y and Z0 − Z are direct mea-sures to the color saturation. Therefore, Eq. (8.35) implies that for any dotpercentage the printed image is viewed more saturated due to the effect of thelight 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 chromaticperspectives, respectively, and Eq. (8.35) is a direct derivation of them. Inmulti-chromatic tone reproduction, experimental observation [And97] and nu-merical simulation [Gus97a, Gus97b] have shown that light scattering actuallyleads 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 re-sulting from printing processes. Distortions originating from distortions of theprinted dots (shape and size) are generally termed physical dot gain. The phys-ical dot gain is closely related to the printing processes, physical and chemicalproperties of the printing materials (colorants and substrate etc), and printingenvironment. For ink-jet printing the physical dot gain comes mainly fromink-substrate interaction. The surface properties of the substrate play a dom-inant role for ink setting. In any printed image there almost always existssome physical dot gain. In addition, the reflective optical measurements to thehalftone images always contain both physical and optical dot gains. Therefore,the physical dot gain has to be considered before a meaningful comparisonbetween 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 afterprinting, due to the physical dot gain. According to Eq. (8.25) its effect on the


reflectance can be expressed as,

R(σ +∆σphy) = R(σ)−∆Rphy(σ)= RMD(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 ofthe optical dot gain, ∆Ropt, has been derived in Eq. (8.27), and the termcorresponding to the physical dot gain can be expressed as

∆Rphy(σ) = ([Rg(1− T 2) + (1− T )2p(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 dotgain, and it shows a correlation between the optical dot gain and the physicalone.

The physical dot gain, ∆σphy, is a function of the commanded dot per-centage (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 dotgain.

Recalling the expressions given by Eqs. (8.27), (8.38), and (8.39), the overalldot 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 poly-nomial expansion, such as,

Q(σ) = c0 + c1σ + c2σ2 (8.41)

where the expansion coefficients, ci, can be obtained by fitting to the experi-mental dot gain curves of the calibration patches.


8.3.2 Simulation of the overall dot gain

The calibration charts consist of 21 patches for each primary color. The com-manded dot percentage of the patches ranges from 0 to 100% at an intervalof 5%. The patches were created with an HP970Cxi ink-jet. Ink-jet films andoffice copy-papers were used as substrates. For comparative purpose, the samesettings in the printer driver interface have been adopted for the prints on bothsubstrates. These are print quality: best, substrate type: plain paper, andink volume: 3. The patches were measured with a spectrophotometer.

0 20 40 60 80 100−2

0

2

4

6

8 Settings: Plain paper, best quality, ink level 3

cyanmagentayellow

0 20 40 60 80 100−4

−2

0

2

4

Commanded dot percentage, σ (%)

Ove

rall

dot g

ain,

∆R

tot

a) plain paper

b) OH film

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 andon ink-jet film, respectively. As shown, the ∆Rtot varies irregularly with respectto the commanded dot percentages, for different colors. Moreover it is negativewhen the commanded dot percentage is greater than 70% (Fig. 8.7a) for cyanand yellow. This observation is in contradiction to the common experiencesabout the dot gain. One explanation might be that the printer has invokedthe built-in software designed for reducing physical dot gain. Because of itssmoother surface and less ink spreading, the correction due to the built-insoftware for plain paper setting, leads to an over correction to the print onthe film. Hence the controversial characteristics becomes more remarkable inFig. 8.7b.

In order to study the effect of the substrate settings, the test patches werealso printed by employing other OH film as substrate setting. The prints seems


0 20 40 60 80 1000

5

10

15

20

25

Settings: other OH film, best, ink level 3

0 20 40 60 80 1000

5

10

15

Commanded dot percentage, σ (%)

Ove

rall

dot g

ain,

∆R

tot

Simulation Measurement

a) plain paper

b) OH film

cyan

magenta

yellow

magenta

yellow

cyan

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 maybe 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 inFig. 8.8. Clearly, they are significantly greater than that with the plain papersetting (Fig. 8.7). The characteristics of the dot gain curves agree well withintuitive understanding of dot gain. The magnitudes of the overall dot gaindemonstrate a clear substrate dependence. Print on plain paper shows strongerdot gain than that of the ink jet film, mainly because of greater ink spreading onthe surface of the plain paper and, in turn, stronger physical dot gain. On theother hand, the dot gain characteristics suggest a remarkable ink dependence.Ink cyan and ink magenta have competitive dot gain while ink yellow hasonly about a half of their magnitudes. Finally, the dot gain curves are wellapproximated by the polynomial expansion given by Eq. (8.41) for prints onboth substrates.

The successful parameterization of the overall dot gain characteristics forthe printer-substrate combination provides us with possibilities for controllingand improving qualities of printed images. Furthermore, the parameterizationcan greatly reduce the computing time involved in dot gain correction (sort ofcolor management) compared to that of using a look-up table (LUT). As anexample 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


(a) Print of the original image

(b) Print of the corrected image

Figure 8.9: Paradise bird flower printed from original and corrected images.


was 58 minutes when the LUT was used. This was reduced to 25 seconds withthe polynomial parameterization. Additionally, compared to the print of theoriginal 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. Reflectanceproperties of a print were described by four types of parameters, γ for ink pene-tration, p for light scattering, Ti for transmittance of the remaining ink layer onthe paper, and Rg for reflectance of the bare substrate. Due to light scatteringinside the substrate, the reflectance becomes a nonlinear function of the com-manded dot area. Moreover, the model predicts that the ink penetration leadsto a decrease in optical dot gain and that light scattering in a paper results ina 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 beused 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 thescattering properties as described by p.

Analysis of the overall dot gain was made. It was shown that the overalldot gain can be parameterized by polynomials. Application to a digital imagehelped 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 be-cause of its complexity and possibly also because of the lack of theoreticalguidance. From a theoretical point of view, the model that developed for themono-chromatic case in Chapter 8, can be extended in a straight forward fash-ion to the multi-chromatic tone reproduction case. By such an extension onemay 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 transmit-tances of the ink layers are, TI and TII . The image can be divided up to 4chromatically distinct regions, denoted Σ0 through Σ3, corresponding to white(Σ0), primary (Σ1,Σ3) and secondary (Σ2) color, respectively. If the dot per-centages 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

σ0 = (1− a)(1− b)σ1 = a(1− b)σ2 = 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 giventhat it enters the substrate at Σi. If the image is uniformly illuminated by lightof 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 = I0TiTjPijσi (i, j = 0, .., 3) (9.2)

where the Ti are the combined transmittance values describing the transmit-tance of the ink layer(s) and ink penetration of the region Σi, as defined inEq. (8.7). For example,

T0 = 1 (9.3)T1 = TI (9.4)T2 = TITII (9.5)T3 = TII (9.6)

Similar to a monolayer system (see Sec. 8.2.2), it is easy to show that theprobabilities Pij (j = 0−3) are constrained by the reflectance of the substrate,Rg, thus,

3∑j=0

Pij = Rg (i = 0, ..., 3) (9.7)

In addition, the probability, Pij , and its counterpart, Pji, fulfill the followingreciprocity relation,

Pijσi = Pjiσj (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 fromΣj . Thus, the total flux of the light outgoing from Σj may be expressed as

Jj =3∑

i=0

Jij

=3∑

i=0

I0TiTjPijσi (j = 0, ..., 3) (9.9)

Applying the constraint conditions and the correlation relation (Eqs. (9.7) and(9.8)), one can further write the flux as

Jj = I0T2j Rgσj − I0

3∑i=0,i =j

Tj(Tj − Ti)Pjiσj (j = 0, ..., 3) (9.10)

Accordingly, the reflectance of the Σj region is calculated from

Rj = T 2j Rg −

3∑i=0,i =j

Tj(Tj − Ti)Pji (j = 0, ..., 3) (9.11)

Thus, the regional reflectance Rj depends directly on the transmittance (Tj)of the ink layer. It depends also on differences of the transmittance betweenthe incident and exit regions, (Tj − Ti), and the probability of light transferbetween the two regions, Pji.

Knowing the reflectance, Rj , one can calculate the average reflectance ofthe image area,

R =3∑

j=0

Rjσj

=3∑

j=0

T 2j Rgσj −

3∑j=0

3∑i=j

Tj(Tj − Ti)Pjiσj

=3∑

j=0

T 2j Rgσj −

3∑j=0

3∑i<j

(Ti − Tj)2Pjiσj (9.12)

where the first term on the right is the reflectance of the halftone image calcu-lated according to the Murray-Davis assumption, and the second is due to thelight scattering that causes darker tone, namely the optical dot gain.

When ink penetration occurs, the transmittance, Ti, in Eq. (9.12) shouldbe replaced by Tiγi where

γ2i =

R0i

Rg(9.13)


and R0i 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 approachhas been given in Ref [YGK00].

9.2 Optical dot gain in multi-color tone repro-

duction

Equation. (9.12) can be generalized to images consisting of any number of inklayers or colors, that is,

R = RMD −∆Ropt (9.14)

where

RMD =N−1∑j=0

T 2j Rgσj (9.15)

is the reflectance of the halftone image under the Murray-Davis assumption,and

∆Ropt =N−1∑j=0

N−1∑i<j

(Ti − Tj)2Pjiσj (9.16)

is the term corresponding to the Yule-Nielsen effect, or the optical dot gain. InEqs. (9.15) and (9.16), N , represents the number of distinct color regions in theimage. For images consisting of 3 or 4 ink layers, N ≤ 23, or, N ≤ 24, respec-tively. From Eqs. (9.14)-(9.16) one can draw the conclusion that the opticaldot gain is a general phenomenon in multi-chromatic tone reproduction. Since∆Ropt is a non-negative quantity, Eq. (9.14) means that the real reflectanceof the halftone image, R, is smaller than that predicted by the Murray-Davisequation. In other words, the tone of the print becomes darker due to lightscattering. 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 distinctchromatic regions that are directly related to the color appearance of an im-age. Therefore, the term optical dot gain loses the intuitiveness it held in themonochromatic case, because the distinct regions are not necessarily biggeroptically than geometrically. For example, due to light scattering from Σ1

into Σ0, Σ1 appears to be extended towards Σ0 along the Σ0/Σ1 border (seeFig. 9.1b). However, Σ1 appears to be compressed due to light scattering fromΣ2 into Σ1 (i.e. the region of Σ1 close to Σ2 appears as if it is of secondarycolor). 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 theimage area as a whole rather than any individual chromatic region.

9.3 Simulation for multi-layer color image 111

9.3 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 opticaldot gain, ∆Ropt, depend on a set of probabilities Pij which are N(N − 1)/2 innumber, where N is the number of distinct color regions. For example, in thetwo inks case there are 3 pairs of probabilities, P01 and P10, P02 and P20, andP12 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 thesubstrate in the region, Σi, and then exits from Σj . By use of the point spreadfunction, Jij can also be written as,

Jij = I0TiTj

∫Σi

∫Σj

p(xi − xj , yi − yj)dσidσj (9.17)

Comparing Eq. (9.2) with Eq. (9.17) one gets

Pij =1σi

∫Σi

∫Σj

p(xi − xj , yi − yj)dσidσj (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 PSFis Gaussian,

p(xi − xj , yi − yj) = κe−[(xi−xj)2+(yi−yj)

2]/δ2(9.19)


the optical properties of the substrate are characterized by the Gaussian pa-rameter δ (κ is a normalization factor). This kind of PSF has been shown tofit the experimental data of Yule, et al., fairly well [RH78, YHA67]. As thevariables of the PSF, (xi−xj) and (yi−yj), are related to the relative positionsof regions Σi and Σj , Pij depends on the spatial distribution of the printed inkdots. Furthermore, the integrated value, Pij , depends on the size and shapeof the integration areas (Σi and Σj). Finally, the magnitude of the optical dotgain depends on the combined transmittance values of the related regions andtheir differences, (Ti−Tj)2, as can clearly be seen from Eq. (9.16). To examineto what extent these factors affect the computed reflectance values, simulationshave been carried out by applying a Gaussian type of PSF to images printedwith two inks.

020

4060

80100 0

2040

6080

100

0

0.2

0.4

0.6

0.8

1

b (%)a (%)

RM

D

Figure 9.3: Computed RMD, 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 contains3 × 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 9.3and 9.4 demonstrate computed reflectance values (RMD) under the Murray-Davis assumption, and the optical dot gain (∆Ropt). The printed image consistsof four ink spots, white, two primary colors (substrate covered by either ink


1 or 2), and one secondary color. The transmittance values correspondingto these regions are shown in Eq. (9.3)-(9.6). Because the reflectance valuecomputed according to the Murray-Davis model, RMD, is a bilinear functionof the dot percentages, a and b, it has a roof like structure with maxima alongthe diagonal a = b (see Fig. 9.3).

020

4060

80100

020

4060

80100

0.02

0.04

0.02

a (%)b (%)

∆ R

opt

δ=0.07Lc

020

4060

80100

020

4060

801000

0.02

0.04

0.06

a (%)b (%)∆

Rop

t

δ=0.12Lc

(a) (b)

020

4060

80100

020

4060

80100

0

0.05

0.1

a (%)b (%)

∆ R

opt

δ=0.20Lc

020

4060

80100

020

4060

80100

0

0.05

0.1

0.15

a (%)b (%)

∆ R

opt

δ=∞

(c) (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 valuesof the Gaussian parameter, δ. For generality, δ is measured by the length (orwidth) 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 lightscattering, 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, whenthe Gaussian parameter, δ, gets larger, or equivalently the PSF becomesbroader and more flat.

3. The magnitude of ∆Ropt becomes greater when δ is larger (Note the


different scales used in the sub-figures).

The appearance of local maxima is a hybrid consequence of the differencein transmittance values between adjacent regions, and the effective extensionof 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 theradius of the area covered by ink 2 (Σ2), | −→r2 |, increases from | −→r2 |= 0 to| −→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)2P01σ0

+ 2(T1 − T2)2P12σ1

+ 2(1− T2)2P02σ0 (9.20)

r2

r1 Σ

2 Σ

1 Σ

0

Light

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 andΣ1, the second from that between Σ1 and Σ2, and the third from that betweenΣ0 and Σ2. As we have assumed that | −→r1 | is fixed in the current consideration,the first term remains constant. Considering a photon that enters the substrateat (x, y) in Σ2 (region producing secondary color), the PSF that describes theprobability of finding the photon at a point (x′, y′) becomes very small when2√(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 regionmarked by dotted line circles, in Fig 9.5), there is a significant probability of


finding it in the adjacent region Σ1. In other words, the main contribution tothe second term comes from photons that hit the region between the dottedcircle lines. At the same time, there is little chance of the photon exiting thesubstrate from the non-inked region (Σ0), i.e., the third term is negligible, if| −→r2 |�| −→r1 | and δ �| −→r1 − −→r2 |. However, the third term grows when | −→r2 |approaches | −→r1 | (or a → b). Considering the fact that ∆Ropt is proportionalto the quantity (Ti − Tj)2 (see Eq. (9.20)) which has the biggest value in thethird term (i.e. (1 − T1)2 ≥ (T1 − T2)2), ∆Ropt grows at a quicker rate when| −→r2 | gets close to | −→r1 | (| −→r2 |→| −→r1 |). However, if | −→r2 | continue toincreases beyond | −→r1 |, ie., | −→r2 |≥| −→r1 |, ∆Ropt falls again. Thus, ∆Ropt

reaches its maximum when a = b. This explanation is consistent with observedfact No. 2 mentioned above. When δ gets larger, the PSF becomes broaderand more flat. Correspondingly, the area marked by the dotted circle lines(in Fig. 9.5) becomes wider and therefore the local maxima of ∆Ropt becomebroader and (relatively) less prominent, even though its absolute magnitudeincreases. Because of the greater probability of photon entering the substratein one region and exiting from the other (or even others), in the case of havinga larger Gaussian parameter, δ, ∆Ropt becomes greater. An extreme case iswhen δ → ∞. In this case, the PSF becomes constant (and therefore Pij aswell) over the whole paper. It means that the photon has equal probability ofbeing found anywhere on the paper, no matter where the photon enters thepaper. Therefore, the photon is said to be “completely scattered” [Rog97].Consequently, the local maxima disappear and only a global maximum is builtup (see Fig. 9.4d). As shown in Fig. 9.4, the location of the global maximummoves 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 rounddots in halftoning [ZVL+03]. It is therefore, necessary to study shape depen-dence of the optical dot gain. For this purpose, simulations of images printedwith square dots have been carried out. Unlike a print with round dots, thecolor appearance of a print with square dots depends not only on the areasof 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 forma-tion of the square dots, where each square within the solid lines represents asquare dot with area, a or b. Figure 9.6b corresponds to the case where thereis a screen angle between the dots. Naturally, to clarify the dependence of theYule-Nielsen effect (∆Ropt) on the dot shapes, comparisons with the imagesprinted with round dots will be made. For easier comparison, identical Gaus-sian parameter, δ = 0.12Lc, has been chosen in the simulations, where Lc isthe length of a halftone cell, as defined before. Furthermore, dependence of theYule-Nielsen effect on the angle between the screen lines will also be explored


by choosing different α values.

Σ2

Σ1

Σ0

Σ1

Σ2

(a) (b)

in coming light

in coming light

out going light

out going light

α

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, respec-tively. Compared to the print with round dots under the same light scatteringparameter (δ = 0.12Lc, Fig. 9.4b), little difference has been observed whenα = 0. However, ∆Ropt appears remarkably different when α = 15o and thedifferences become even more significant when the screening angle, α, furtherincreases. The differences can be summarized into the following,

1. The local maxima along the diagonal a = b, which is prominent in theround dots case, becomes broader and less well defined.

2. The global maximum, which is a sharp peak in the case of round dotsbecomes a broad plateau in the square dots case (where α �= 0).

Since the point spread function has a limited effective extension (charac-terized by the Gaussian parameter δ) as shown in the Fig. 9.6a, there is nosignificant probability that a photon is scattered from one region (say Σ2) intoanother (say Σ1), unless the photon hits the substrate at a point close enoughto the border of the incident region (marked by dotted square lines). In thecase 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 happenonly when the two ink dots have similar areas (b ≈ a). Consequently, it formsnarrow local maxima along the diagonal a = b, as shown in Fig. 9.7a. However,


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 pho-ton has a better chance to be scattered from Σ2 into Σ0, and vice versa, eventhough 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 simulationsalso show that the quantities of computed ∆Ropt decrease as the screen angle,α, increases. Therefore, increasing the screen angle will possibly be helpful inreducing the optical dot gain.

020

4060

80100

020

4060

80100

0

0.05

0.1

a (%)b (%)

∆ R

opt

α=0o

020

4060

80100

020

4060

801000

0.02

0.04

0.06

a (%)b (%)

∆ R

opt

α=15o

(a) (b)

020

4060

80100

020

4060

801000

0.02

0.04

a (%)b (%)

∆ R

opt

α=30o

020

4060

80100

020

4060

801000

0.02

0.04

a (%)b (%)

∆ R

opt

α=45o

(c) (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 dueto difficulties in registration. Additionally, to avoid artifacts, such as Moire,different screen angles are used when the different primary inks are printed.This kind of dot arrangement leads approximately to a random dot overlapdistribution that can well be described by the DeMichel equations [De’24]. For


a 2-color print, the ink coverages of different chromatic areas, are given inEq. (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 ofcomparison, the same Gaussian parameters (δ = 0.12Lc), as in the case of doton dot, have been chosen in the simulations. Figure 9.8a depicts the computedreflectance, under the Murray-Davis approximation. Similar to the case of doton dot, the reflectance decreases bi-linearly with respect to the primary inkpercentages, a, b. Nevertheless, the difference is also remarkable, i.e. thereexists no evident local maximum along the diagonal (a = b) in the random dotdistribution.

0 20

40 60

80 100

0 20

40 60

80 100

0

0.2

0.4

0.6

0.8

1

a (%)b (%)

RM

D

0 20

40 60

80 100

0 20

4060

80 1000

0.01

0.02

a (%)b (%)

∆ R

opt

δ=0.12Lc

(a) (b)

0 20

40 60

80 100

0 20

4060

801000

0.01

0.02

0.03

0.04

a (%)b (%)

∆Rop

t

δ=0.2Lc

0 20

40 60

80 100

0 20

40 60

80 100

0

0.02

0.04

0.06

0.08

a (%)b (%)

∆Rop

t

δ=∞

(c) (d)

Figure 9.8: Computed reflectance values, two inks (cyan and magenta) randomly

overlap with each other. (a) Reflectance under the Murray-Davis approximation

(RMD); (b)-(d) Computed optical dot gain (∆Ropt) with different Gaussian parame-

ters δ, Lc is the length of a halftone cell.

The reflectance resulting from light scattering in the substrate or the opticaldot gain, ∆Ropt, is computed and depicted in Fig. 9.8b-d. In contrast to thecase of a dot on dot (see Fig. 9.4) where the ∆Ropt has its narrow (local andglobal) maximum along the diagonal, here the ∆Ropt has rather a flat plateaudue to the diversity of the overlap between the dots. In addition, there existsa 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 doton dot has the greatest possibility of dot overlapping between the colors, thisobservation implies that the optical dot gain can be reduced by reducing dotoverlap between the primary colors. This is in line with our observation fromthe square dot case, i.e., the optical dot gain decreases as the difference betweenthe screening angles increases (for square dots).

9.4 The effects of optical dot gain on color re-

production

The discussion about the effects of the optical dot gain on the color appearanceof printed images can readily be extended to multi-color images. According tothe definition, one can compute the tristimulus values X,Y,Z of the halftoneimage, from its reflectance, R, (see Eq. (9.14)),

X = XMD −∆Xopt (9.21)

whereXMD =

∫RMD(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-chromatictone reproduction, which allows us to analyze properties of images printed withany number of inks, and in any halftone scheme. By applying a Gaussian type ofpoint spread function (PSF) the optical dot gain has been simulated for imagesprinted with 2 inks of different dot geometries (round and square dots), differentdot locations (dot-on-dot, random dot distribution), and different screen angles.The optical dot gain shows a strong dependence on the optical properties ofthe substrate and the inks, and on the geometric distribution of the printeddots (shape, size, locations, and relative orientation of the dots). The presentmodel is independent of the halftone scheme, and it is therefore applicable toimages produced with any kind of halftone algorithm.

Chapter 10

Chromatic effects of inkpenetration

In this chapter, we evaluate the chromatic effects of ink penetration. The eval-uation 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 dataof prints on plain paper and on photo-quality paper, respectively, in terms oftheir color representations. The second comparison (Sec. 10.3) is based on sim-ulations of prints with and without ink penetration. Each approach has itsadvantages and disadvantages.

10.1 Basics in colorimetry

To make the following easier for readers, fundamentals in Colorimetry is brieflypresented.

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. Accordingto 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 func-tions, respectively. In this thesis, measurements were carried out under D65

illumination and at 10o viewing geometry. k is a normalization factor and isdefined as

k =100∫

S(λ)y(λ)dλ(10.2)

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 chromatic-ity diagram was developed. The transformation from the tristimulus values tothe chromaticity coordinates is accomplished through a normalization that re-moves the luminance information, i.e.

x =X

X + Y + Z

y =Y

X + Y + Z(10.3)

z =Z

X + Y + Z

Clearly, there are only two dimensions of information in chromaticity coordi-nates, the third chromaticity coordinate, can always be obtained from the othertwo (for example, z = 1 − (x + y)). Therefore, the color is represented in a2-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 tristim-ulus values normalized to the white point of the illumination (specified byX0, Y0, Z0) are greater than 0.008856 [Fai98].

L∗ = 116(Y

Y0)1/3 − 16

a∗ = 500[(X

X0)1/3 − (

Y

Y0)1/3] (10.4)

b∗ = 200[(Y

Y0)1/3 − (

Z

Z0)1/3]

where L∗ represents lightness, a∗ approximates redness-greenness, and b∗ ap-proximates yellowness-blueness. Furthermore, chroma (C∗

ab) and hue (hab) can

10.2 Evaluation of chromatic effects from experimental data 123

be computed from the CIELAB coordinates.

C∗ab =

√a∗2 + b∗2 (10.5)

hab = tan−1(b∗/a∗) (10.6)

10.2 Evaluation of chromatic effects from ex-

perimental 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 typeof substrates, one has ink penetration, another not. To accomplish such acomparison 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 toprint on different types of substrates, such as plain (office copy) paper for printhaving ink penetration and photo quality paper for not having ink penetration.

10.2.1 Parallel comparison of prints on two types of sub-

strates

When different substrates are used for printing, it seems to be reasonable tocompare the color of the test patches in parallel, i.e., to compare patches of thesame (commanded) ink percentage but on different substrates. Nevertheless,two things at least should be bore in mind when the evaluation ink penetrationis made. First, different substrates have different optical properties (spectralreflectance, point spread function etc.) which will affect the color of the prints,and optical dot gain characteristics. Secondly, substrates have different surfaceproperties that will influence the ink distribution on (even in) the substrates.This will, for example, result in different physical dot gain, when halftone im-ages are printed. To some extent, color differences resulting purely from the dif-ferent substrate colors may be minimized by choosing the substrates of havingsimilar color under certain illumination condition (metamerism). Despite thesepotential effects, the chromatic effect of ink penetration can still be evaluated,at least qualitatively, by such a parallel comparison, because ink penetrationhas by far much stronger impact on the printed color when dye-based inks areprinted on ordinary office copy paper.

In the present study, plain paper (StoraEnso, 80g/m2) and photo gloss pa-per (Hewlett Packard, 175g/m2) were chosen as substrates. Their tristimulusvalues were, (84.03, 88.78, 94.75) and (86.49, 91.16, 96.10), respectively. Theircolor difference equals ∆E = 1.32, which is therefore hardly noticeable, even


though they are evidently different in gloss. Test patches of primary and sec-ondary colors were created by printing on these substrates. The commandedink percentages for each color range from 0 to 100% at a step of 5%. Thetristimulus values in these charts were measured by employing a spectropho-tometer.

Table 10.1: Color differences (∆E1)) between the photo gloss paper and the office

copy-paper before and after printing (solid patches only).

color paper white cyan magenta yellow black red green blue∆E 1.32 12.37 19.30 23.67 13.44 24.92 22.59 20.75

1) ∆E =√(∆L∗)2 + (∆a∗)2 + (∆b∗)2

0.1 0.2 0.3 0.4 0.5 0.6 0.70.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6Dependence of color gamut on substrates (PL)

x

y

photo paper

plain paper

G

Y

R

M

B

C

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 sub-

strates 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 sub-strates, have been collected in Tab. 10.1. As shown, the color difference betweenthe photo gloss paper and the plain paper before printing is hardly noticeable(∆E = 1.32). However after printing with full tone colors, their color differencedramatically increases to ∆E = 12.3− 25, depending on the printed colors. As


only solid printed patches were compared, the color differences arise mainlyfrom ink penetration.

−60 −40 −20 0 20 40 60 80 −60

−40

−20

0

20

40

60

80

100

Chromatic effect of ink penetration (plain paper setting)

a*

b*

plain paper

photo paperY

R

M

B C

G

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 ef-

fects

Chromaticity diagrams for halftone patches on different substrates have beenplotted in Fig. 10.1. To explicitly illustrate the color difference between printson photo gloss paper and prints on the plain paper (office copy paper), colors(noted with triangles) corresponding to patches of the same commanded inkpercentage but on different substrates have been connected by solid lines in pairin the figure. Moreover, areas possibly covered by prints on the photo glosspaper as well as by those on the copy paper are marked with solid and dashedlines, respectively. As these areas represent the possible color that can beproduced by printing on the substrates, the photo gloss paper has significantlygreater capacity to represent color than the plain paper.

Chromatic effects of ink penetration can be further examined in terms ofchroma and hue of the colors. Figure 10.2 is a 2-dimensional representationof the CIELAB color space. The colors corresponding to the same ink andsubstrate but different ink percentages have been joined up with solid lines and


dashed lines, respectively, for prints on the photo gloss paper and on the copypaper. 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 differsignificantly in both chroma and hue, in mid to dark tone colors.

• prints on the photo gloss paper produce colors of significantly greaterchroma.

These observations may be explained as the following. First as the inkcoverage increases, the white light reflected from the non-printed substratedecreases and the color becomes more saturated. Second the hue variationis partly due to the non-linear transformation from CIEXY Z to CIELABcolor system (Eq. 10.4) and partly due to dot gain. Under the Murray-Davisassumption, the tristimulus values vary linearly with the ink percentage. How-ever, existence of the optical dot gain leads to nonlinearity between XY Z andthe dot percentage, as known from Chapter 8. This, in turn, causes nonlinear-ity in the hue of the color. Third, because the two substrates have almost thesame color, curves corresponding to these substrates overlap each other whenthe ink coverage is small. However, the curves gradually separate when the inkcoverage increases, because ink penetration increases. Finally, ink penetrationforms an ink-porous mixture which has strong scattering-power and leads tothe 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 thatcan be delivered by the device. Color gamut of the system is usually obtainedby creating massive test patches with various combinations of ink percentages.The test patches are measured and their color coordinates represented in a colorspace (say CIELAB). The topology of the volume spanned by the measuredcolors 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 usersto adjust parameters that control the printing, compared to conventional offsetprinting. These flexibilities provide redundancy in mapping the color gamut forthe ink jet printing system. Figure 10.3 shows colors printed on the same typeof substrate (photo gloss paper) but using different substrate settings (plainpaper 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 colorgamut mapped from the test patches can be significantly different as shown inFig. 10.4.


−75 −50 −25 0 25 50 75

−50

−25

0

25

50

75

100

125

a*

b*

Photo paper setting

Plain paper setting

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.

−75 −50 −25 0 25 50 75−75

−50

−25

0

25

50

75

100

a*

b*

1

3

5

2

4

cyan

blue

magenta

red

yellow

green

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.


400 450 500 550 600 650 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavelengths, λ(nm)

Ref

lect

ance

, R Ink level 1

Ink level 5

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 in-creases from low to mid ink levels (1-3) and tends to be saturated when theink level (or ink volume) is further increased. Such a variation with respect tothe ink level builds up a hock type of structure for the primary colors (mostevidently seen from ink magenta, see Fig. 10.4). This phenomenon is a con-sequence of light absorption from the printed ink layer. As the ink has littlescattering power, light reflected from the print is essentially due to light re-flection from the substrate (white) paper. As known, the spectral reflectanceof 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 inklevel increases (from ink level 1 to 3), while the reflected light through thetransparent band decreases little. In other words, increasing absorption in theabsorption 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 throughthe transparent band. Because of existing absorption even in the transparentband, the chroma decreases if the ink level is further increased. Consequently,this leads to decrease in the chroma. This argument holds for other primarycolors, as well for the secondary colors. The variations in color composition ofdifferent ink levels (see Fig. 6.4), however, contribute to the irregularity in thesecondary colors.

10.3 Evaluation in 3D color space: simulations 129

10.3 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 noneed to use different types of substrates as is the case in the experimentalevaluation. Therefore, the differences in optical and physical dot gain resultingfrom using different types of substrates are avoided. In other words, the soleeffect of ink penetration can be studied.

To accurately map the color gamut of an ink-jet printing system, informa-tion like PSF (point spread function) of the substrate, ink spreading charac-teristics on the substrate, ink penetration of halftone dots, etc., are needed.To obtain such information requires carefully designed measurements, which isbeyond the scope of this thesis. Nevertheless, we made attempts to model inkpenetration in halftone images [YK01, YKP01]. This will be further developedin the future.

In the simulation, we took a simplified approach. The printer consists of 4inks, cyan (c), magenta (m), yellow (y), and black (k). The inks are distributedrandomly and the distinct colored areas described by the DeMichel equation,

aw = (1− c)(1−m)(1− y)(1− k) (10.7)ac = c(1−m)(1− y)(1− k) (10.8)am = (1− c)m(1− y)(1− k) (10.9)ay = (1− c)(1−m)y(1− k) (10.10)ab = cm(1− y)(1− k) (10.11)ag = c(1−m)y(1− k) (10.12)ar = (1− c)my(1− k) (10.13)ak1 = cmy(1− k) (10.14)ak2 = (1− c)(1−m)(1− y)k (10.15)ak3 = c(1−m)(1− y)k (10.16)ak4 = (1− c)m(1− y)k (10.17)ak5 = (1− c)(1−m)yk (10.18)ak5 = cm(1− y)k (10.19)ak7 = c(1−m)yk (10.20)ak8 = (1− c)myk (10.21)ak9 = cmyk (10.22)

The reflectance is then computed by applying superposition

R =∑

axRx (10.23)


where the subscript, x, denotes the colors in Eqs. (10.7)-(10.22), and Rx re-flectance of the colors. For the primary or secondary colors, Rx is approximatedby reflectance of its solid print, while the remainders (x = k1, ..., k9) are ap-proximated by reflectance of a solid black.

In the case of no ink penetration, Rx is computed by

Rx = T 2xRg (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 issimply the spectral reflectance value of the printed colors (solid print on plainpaper).

In the simulation, the commanded ink coverage of the primary inks rangesfrom 0 to 100 % at a step 4 %. To account for dot gain, the overall dot gaincharacteristics obtained (for plain paper) in Chapter 8 were applied in thesimulation.

Figure 10.6 depicts the simulated color gamut of printing with (inner vol-ume) and without (outer volume) consideration of ink penetration, in CIELABcolor space. Figure 10.6b is obtained from Fig. 10.6a by rotating it 180o aroundthe L∗ axis. As seen from these figures, the two color volumes have no pointsin common expect for the paper white. In other words, prints of the same inkcoverage (but different ink penetration status) have different color coordinates(lightness, chroma, and hue). Differences, that appear in the vicinity of solidblack is a consequence of increasing amount of pigmented black in ink compo-sition for the dark tone printing. It is observed that from light to mid tone, thegray is created by mixing the primary inks. However, for the dark tone gray(from about 65%), the pigmented black ink is gradually added, in increasingamounts as the tone values increase. Since pigment particles are much largerin size than the dye molecules, they do not penetrate much into the substrate(the dye compositions still do). Therefore, the simulations predict similar colorcoordinates for the solid black in both cases. Consequently, a nail-type struc-ture is formed in the vicinity of the black point. Furthermore, the evidentdifference between the inner and the outer volumes demonstrate the dramaticimpact of ink penetration on the capacity of the color reproduction or the colorgamut. It was shown experimentally that the color gamut of printing on highquality special paper can be up to 50% larger compared to prints on plainpaper [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 simu-lation. 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.


printed on two types of substrates, one with ink penetration and another with-out, have been compared. It is observed that ink penetration has significanteffect on the chroma and hue of the printed colors. Issues related to differ-ent optical and surface characteristics of the different substrates have beendiscussed.

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 showthat the color gamut of the printing system is dramatically reduced because ofink penetration.

Chapter 11

Summary and future work

11.1 Summary

An ink-jet printing system consists of three fundamental parts, inks, printingengine, and substrates. Creation of images is determined by underlying physicalprocesses, i.e. ink application and ink setting. These processes govern inkdistribution (dot size and shape), ink spreading (physical dot gain), and inkpenetration.

Visual appearance of printed images is determined by the underlying phys-ical phenomena, i.e., spectral light absorption and scattering by the substrate,the inks, and ink-substrate interaction. Being able to understand and charac-terize the printing processes and the consequences (physical phenomena) forimage color appearance are essential in order to improve image quality and torealize faithful image reproduction.

This thesis presents a few systematic methods for studying those issuesthat 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 formsof ink penetration (uniform and non-uniform ink distribution in the direc-tion perpendicular to the paper sheet). Expressions for spectral reflectanceof printed images as a function of scattering and absorption coefficients havebeen given.

Second, the thesis describes a method to characterize properties (scatter-ing and absorption powers) of the inks and printed ink volume, by combiningspectral reflectance measurements with simulations. The model has been suc-cessfully applied to simulate the measured spectral reflectance. It was shownto be capable of predicting spectral reflectance of primary and secondary colorink layers of any ink thickness.

134 Summary and future work

Third, the characteristics of the inks (scattering and absorption powers, inkvolume, etc.) are used for modelling ink penetration. Spectral reflectance ofsolid prints on plain paper has been well simulated, and a method to determinethe depth of ink penetration is presented. Moreover, effects of ink penetrationare studied and discussed in detail.

Fourth, a model describing light scattering in substrates subject to inkpenetration resulting in optical dot gain has been developed. The model isapplicable to both mono- and multi-chromatic images. It has been found thatoptical dot gain is a general phenomenon that exists in both mono- and multi-chromatic printed images. Theoretical analysis shows that optical dot gainmakes the color appear more saturated. Additionally, preliminary studies ofthe overall dot gain (physical plus optical dot gain) has been carried out. Thestudies show that the overall dot gain characteristics can be parameterized interms of polynomials. Application to a digital image has demonstrated thatsuch a parameterization can dramatically reduce the processing time in dotgain correction.

Finally, the chromatic effects of ink penetration has been evaluated usingexperimental data analysis and simulations. It has been found that ink pene-tration 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 be-comes much less saturated than that without ink penetration. Consequently,the color capacity or the color gamut of a printing system is dramatically re-duced by ink penetration.

11.2 Future work

Ink penetration or ink-paper interaction, in general, has a dramatic impact oncolor reproduction. It deserves considerably more study. On the other hand,the study of ink-paper interactions is a very complicated research topic in-volving experimental measurements and theoretical treatments. Both demandknowledge of physics, chemistry, fluid dynamics. etc.

In terms of follow-up work, the following topics are of interest. First, ap-plications of the models to different ink-paper combinations. Since ink-paperinteractions depend strongly on the properties of the ink and the paper, stud-ies of different ink-paper combinations will increase our knowledge and under-standing of the principal factors governing the quality of printed images. Asecond extension of the models and methodologies is to halftone images. Thisrequires careful characterization of non-uniform ink spreading and ink pen-etration mechanisms. For example, the depth of ink penetration under thecenter of an ink dot is possibly bigger than that under the edge of the dot. Athird followup can possibly be model testing and improvement. The modelsdeveloped 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 toink thicknesses z and 2z, respectively, can be expressed as

RI =s[A− sRg]e−

s2−(A)2

A z −A[s−ARg]

A[A− sRg]e−s2−(A)2

A z − s[s−ARg](A-1)

RII =s[A− sRg]e−

s2−(A)2

A 2z −A[s−ARg]

A[A− sRg]e−s2−(A)2

A 2z − s[s−ARg](A-2)

whereA = s+ k −

√k2 + 2ks (A-3)

From Eq. (A-1) and (A-2) one can derive the following relations

e−s2−A2

A z =(s−ARg)(A− sRI)(s−ARI)(A− sRg)

(A-4)

e−s2−A2

A 2z =(s−ARg)(A− sRII)(s−ARII)(A− sRg)

(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−ARII)

=(s−ARg)(A− sRI)2

(s−ARI)2(A− sRg)(A-6)

138 Appendix

Replacing s and k with sz and kz, one can readily obtain the relation given byEq. (6.4), i.e.

(Az − szRII)(sz −AzRII)

=(sz −AzRg)(Az − szRI)2

(sz −AzRI)2(Az − szRg)(A-7)

B Probability model for optical gain

Assume that the percentage of the paper under the dots is σ (Σ1) and the paperbetween the dots (Σ0) is (1−σ), respectively. If the intensity of irradiance ontothe whole system is I0, the flux of photons striking the dots (Σ1) and the paper(Σ0) areas are I0σ and I0(1− σ), respectively. Then, the flux Jmn of photonsscattered from Σm into Σn (m,n = 0, 1) is given by

J11 = I0T2σP11 (A-8)

J10 = I0TσP10 (A-9)J01 = I0T (1− σ)P01 (A-10)J00 = I0(1− σ)P00 (A-11)

Due to the optical reciprocity (see Sec. 8.2.1), there is

J10 = J01

= I0Tpσ(1− σ) (A-12)

Comparing Eq. (A-12) with Eqs. (A-9) and (A-10), one obtains

P01 = pσ (A-13)P10 = p(1− σ) (A-14)

andP10σ = P01(1− σ) (A-15)

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[TP10σ + P00(1− σ)] (A-16)

Consequently, the reflectance measured from the paper between the dots canbe calculated from

R0 =J0

I0(1− σ)= P00 + Tpσ (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, istherefore a summation of J01 and J11,

J1 = J11 + J01

= I0[T 2P11σ + TP01(1− σ)] (A-22)

and the corresponding reflectance reads

R1 =J1

I0σ

= T 2P11 + Tp(1− σ)= T 2Rg + pT (1− T )(1− σ) (A-23)

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