kubelka-munk theory in describing optical properties of paper - copy

1Introduction

2Kubelka-Munk model

Uvod

Model Kubelka-Munk

This review article is a general summary of the widelyused Kubelka-Munk model. Kubelka and Munk carried outa simplified analysis of the interaction of incoming lightwith a layer of material such as a layer of paint. The materialis assumed to be uniform, isotropic, non-fluorescent, non-glossy and the sample has to be illuminated by diffuse,monochromatic light.

Though real paper never completely satisfies all of theassumptions, and sometimes significantly deviates fromthem, the K-M model has been widely used in paperindustry for many years probably due to its explicit form,simple use and its acceptable prediction accuracy. It is usedfor multiple-scattering calculations in paper, papercoatings, printed paper, deinked paper (recycled paper andhand sheets), paint, plastic and textile. The theory has alsofound numerous applications in the paint and colorantindustry

Kubelka and Munk proposed, as Schuster had (thiswork was not known to Kubelka and Munk), a system ofdifferential equations based on the model of lightpropagation in dull colored layer that is parallel to a planesubstrate.

The K-M model is based on several assumptions:1. The medium (sample) is modeled as a plane layer of

finite thickness, but infinite width and length (infinitesheet approximation), so there are no boundary effects.

.

117

V. Dimbeg-Mali, . Barbari-Mikoevi, K. Itri

ISSN 1330-3651UDC/UDK 535.34/.36:676.017.55

KUBELKA-MUNK THEORY IN DESCRIBING OPTICAL PROPERTIES OF PAPER (I)

Vesna Dimbeg-Mali eljka Barbari-Mikoevi, Katarina Itri,

One of the greatest simplifications of the transfer equation is where all incident and scattered light is assumed to be perfectly diffuse, only two oppositedirections of light transport are considered, and the light intensity is assumed to vary along one axis only. Such two-flux approximations have been presented bymany authors, starting with a paper by Schuster [1]. One of the most famous two-flux approximations is the approximation presented by Kubelka and Munk [2]in 1931 and further developed by Kubelka [3,4]. Kubelka and Munk gave a comprehensive formulation and a treatment with a clear aim towards practicalmethods of measurement. It was quickly adapted for use by the papermaking industry [5-8] and has now been in widespread use for decades in the measurementand prediction of colour, brightness and opacity of paper sheets.

Keywords: absorption, Kubelka-Munk, paper, scattering

Most of the information reported here is taken from the references cited at the end of the articlewhich should be consulted for a more in-depth study.

Subject review

Jedno od najveih pojednostavnjenja jednadbe transfera pretpostavlja: upadno i raspreno zraenje je savreno difuzno, promatraju se samo dva suprotnasmjera prijenosa svjetlosti zapoeo je Schuster

up s jasnim ciljem ka praktinimmetodama mjerenja. Ta metoda je bila brzo prihvaena u papirnoj industriji u mjerenju i procjeni obojenja, svjetline i opacitetapri proizvodnji papira. Veina informacija u ovom lanku preuzeta je iz referenca iji se popis nalazi na kraju lanka i koje je potrebno konzultirati za detaljnijuanalizu.

, i intenzitet svjetlosti se mijenja samo du jedne osi. Takvu aproksimaciju dva smjera prijenosa svjetlosti [1], anastavili su Kubelka i Munk [2] 1931. a model je dalje usavrio Kubelka [3-4]. Kubelka i Munk su dali saeti oblik i prist

[5-8] i danas ima iroku primjenu

Kljune rijei: apsorpcija, Kubelka-Munk, papir, rasprenje

Pregledni lanak

Kubelka-Munk teorija u opisivanju optikih svojstava papira (I)

Kubelka-Munk teorija u opisivanju optikih svojstava papira (I)

Technical Gazette 18, (2011),1 117-124

2. A perfectly diffuse and homogeneous illuminationincident on the surface.

3. The only interactions of light with the medium arescattering and absorption; polarization andspontaneous emission (fluorescence) are ignored.

4. The medium is considered isotropic and homogeneous,containing optical heterogeneities (small compared tothe thickness of the layer) able to disperse light.

5. No external or internal surface reflections occur.6. Parameters and are constant whatever the thickness

of layer is.

Since the lateral extent of the medium is infinite, onlythe thickness direction is incorporated in the equations. Theincident, reflected and transmitted intensities are allassumed to be perfectly diffuse, and are assigned either oftwo directions: upwards or downwards.

The reflectance of the medium is denoted , and itstransmittance . The sample is placed in optical contact withan opaque substrate with a known reflectance , as shownin Fig. 1, and is split into a series of layers of equal thicknessd That layer receives a flux traveling downward and fluxtraveling upward, the reflectance being simply / (Fig. 1).

S K

RT

R

x. i jj i

g

Figure 1

Slika 1.

Light path of the Kubelka-Munk model in conjunctionwith the depth coordinate x

Put svjetlosti u Kubelka-Munk modelu u ovisnostio debljini sloja x

118

Kubelka- unk theory in describing optical properties of paper ( )M I

Tehni ki vjesnik ,18, 1(2011) 117-124

Passing from one layer to the next, these fluxes arechanged. Without scattering, the change in irradiance of theflux of light in the down direction would be d = daccording to the Beer-Lambert law. If is expressed in mm,the constant , is in units of 1/mm and is called anabsorption coefficient. The larger the value of , the greateris the probability that a photon will be absorbed.

Scattering also decreases the flux of light in the downdirection. Kubelka and Munk [1] suggested that thescattering phenomenon, like absorption, is a first orderphenomenon. is the Kubelka-Munk scattering coefficientand has the same units (inverse length) as the Kubelka-Munk absorption coefficient . The flux in down directionhas a positive value when the flux is going down.Accordingto Kubelka-Munk sign conventions, is the flux of light inthe up direction. The value of is positive when light isgoing up. The term describes that the up moving fluxis scattered to add to the down moving flux. The first orderdifferential equation is expanded to include two other terms:

I K xx

KK

S

K I

JJ

+SJdx


In practice, is the reflectance of a layer so thick to

completely hide the substrate, i.e. the limiting reflectancethat is not modified by any additional thickness of the samematerial [9]. Kubelka treated the same problem [3]as he hadwith Munk in 1931, but in his generalization the incidentbeam light comes to the colored layer under the angle 0[10].

There are other theories which describe in greater detailhow light interacts with material, but they have not been sowidely accepted as the simple K-M theory. It still has aspecial position not only in paper optics and in paperindustry [11-16], but also within the colorant branch whereit is used to calculate color pigment mixtures.

All the well known results of the K-M theory are listedin literature. Wyszecki and Styles [17] give solutions ofdifferential Eqs. (1) and (2) in terms of and . Thesolutions are complicated transcendental functions, but interms of only four parameters of the system:

R

R T

3Useful Solutions to the Kubelka-Munk DifferentialEquationsKorisna rjeenja Kubelka-Munk diferencijalnih jednadbi

3.1Some special cases of solutions of K-M differentialequationsNeki posebni sluajevi rjeenja K-M diferencijalnihjednadbi

d d d dI KI x SI x SJ x. (1)

A second differential equation describes the rate ofchange in the up moving flux:

d d d dJ KJ x SJ x SI x. ( )2

When passing through the layer, light that is absorbed islost, and some part of the light that is scattered changesdirection from upwards to downwards, or vice versa, and isexchanged between and Two coefficients and areintroduced to denote the amount of absorption and directionchanging scattering, respectively. The scattering andabsorption are both proportional to the intensities and to thethickness of the layer .

In a bulk material of infinite lateral extent, the lateraldimensions are assumed to be much larger than the meanfree paths, 1/ and 1/ , for absorption and scattering in thematerial. This means that no light leaks out of the edges ofthe material due to lateral scattering, so that the light flux inthe horizontal direction is ignored. (We will see later howlateral scattering has a significant impact on image tonereproduction by halftone processes.)

Equations (1) and (2) are two differential equationswith two flux terms, and . If we consider the value of =at the bottom of the material, then we can define atransmittance:

i j. K S

L

K S

I J I It

0I

IT .t (3)

Similarly, we can define a reflectance factor, , in termsof the up flux at the surface of the paper:

R

0I

JR .r ( )4

(

Kubelka and Munk solved these differential equationsto obtain analytical expressions for and associated to agiven wavelength in the case of an infinite ) or finite ( )

painted layer, in terms of the absorption and scatteringcoefficients and the reflectance of the support ( ):

R TR

R

R

g

1 1( )exp

( )1 1

( ) ( )exp

g

g

g g

R RR R SL R

R R RR

R R R SL RR R

. (5)

),,,(1 gRLKSfR and 2 ( ).gT f S ,K ,L,R (6)

The particular form of the functions, and , depend onthe boundary conditions of the system. The generalsolutions for reflectance and transmittance are:

f f1 2

)coth(

)coth(1

bSLbRa

bSLbaRR

g

g

(7)

)cosh()sinh( bSLbbSLa

bT .

( )8

With definition for and :a b

S

KSa ,

1.2 ab (9)

a) Rg=0 and S=0 , and K >0.

This is a transparent medium that absorbs light but doesnot scatter it. It is the "Beer-Lambert" case. Derivation ofthe limit of the Eq. (10) as approaches zero, converges tothe Beer-Lambert law.

S

119Technical Gazette ,18, 1(2011) 117-124


This is a Beer-Lambert system placed against aLambertian scattering reflector (ink on paper, photographicemulsion on paper).

Fig. 2 illustrates the sample arrangements used for thedefinition of quantities and .

An alternative method, especially when the amount ofmaterial is limited, is to measure reflectance over twodifferent backgrounds, white and black [18].

If density is homogeneous, the corresponding analysiscan be made where the thickness is replaced by thegrammage of the sheet. Van der Akker [5] showed that theK-M differential equations remain the same if the originalscattering and absorption coefficients and (unit m) arereplaced with the specific scattering and absorptioncoefficients and (unit m kg), and the thickness (unit m)is replaced with grammage (unit kg m ). The lightscattering power of a paper sheet is sometimes given asdimensionless product

The coefficients and can be determined for differentwavelengths or with different functions such as thebrightness function or /2 function (ISO 9416).Determinations of with function are common withinthe pulp industry since ( ) is a measure of the content ofchromophoric groups in the pulp. From Eq. (13) it is evidentthat depends on the relationship between and . High

values of the ratio / are required to attain the highestreflectivity values; have to be large and/or small, i.e. highlight scattering and low absorption.

The values of and are typically [18]:2 m /kg for coated and uncoated fine papers made from

bleached chemical pulps,3 6 m /kg for mechanical pulps and around 14 m /kgfor unbleached kraft pulp,

502 m /kg for filled and coated fine papers,20 40 m /kg for bleached and unbleached chemicalpulps,40 702 m /kgfor mechanical pulps.

R R

L

S K

s k L

w

s k

R YC

k R

k R

R k s

s ks k

k

k 0 and K > 0, and L (thickness) infinity.

This is the case of an infinite thickness of ascattering/absorbing material. It is a system that issufficiently thick to reflect the same amount of light with abackground of = 1 or = 0. This corresponds physicallyto a layer that is so thick that it is effectively opaque andtherefore hides the substrate completely. The thicknessrequired in order to approximate the complete hiding case isoften quite small. Most commercial paints are formulated tohave sufficient scattering and absorbing powers so that onlya thin coating will be needed to completely hide the color ofthe substrate.

In such systems, the following form of the Kubelka-Munk equation applies, so-called Kubelka-Munk function:

R Rg g

R

R

S

K

2

)1( 2(10)

The term reminds us that this is the reflectance when

the sample is infinitely thick (Fig. 2a). It is in fact ofequation (7) in the limit of .

Solving Eq. (10) for leads to another usefulexpression, in which and are defined as in Eq. (9):

R

R

L

R

a b

S

K

S

K

S

KbaRR

x2 .1lim

2

2

0

(11)

Since the thickness of the medium is infinite andtherefore irrelevant the only thing that matters are therelative proportions between absorption and scattering, andthe reflectance may be expressed as a function of only.

It is usual to use a black background with a reflectancevalue < 0,5 %, i.e. = 0. The reflectance of the papermeasured in that way is the reflectance factor over blackbaking = :

K/S

R R

R R

g g

0

0

01ln .1

1

RR

RRR

RR

L

S

(1 )2

Definitions of the quantities used in Eqs. (12) and(13)[19]:

reflectance of the layer (sheet, coating, ply) which hasbehind it a surface with a reflectance of

reflectance of the background behind the layer whosereflectance is being considered

reflectance of the layer with ideal black background,= 0

reflectivity = reflectance of the layer so thick thatfurther increase in thickness does not change thereflectance.

RR

R

R R

R

g

g

g0

R

RSK .

2

)1( 2(1 )3

Figure 2

Slika 2.

a) the reflectance factor for an opaque pad of paper,reflectivity R b) the reflectance factor for a single sheet over

a black background (R ) R .

a) faktor reflektancije za neprozirni kup papira,reflektivnosti R b) faktor reflektancije za jedan papir preko

crne podloge (R ) R .

;

;

g

g

=0

=0

0

0

d) Rg > 0, K = 0 and S > 0.

This is a medium that scatters light, but does not absorbit (Lambertian scattering reflector). This is particular casefor conservative scattering (zero absorption). For thatsystem the definitions for and (9) become:a b

1a and 0b (14)

and the reflectance is [20]:

)1(1

)1(

g

gg

RSL

RSLRR .

(15)

120 Tehni ki vjesnik ,18, 1(2011) 117-124

4Interactions of light with paperInterakcije svjetlosti s papirom

Paper is a complex structure consisting largely ofcellulose fibers, fillers and additives. The appearance of atypical sheet of paper is a function of its detailed structure,the presence and concentration of light absorbing groups(chromophores), the refractive indices of its components, itsgrammage and its surface reflective characteristics. Whenlight strikes the paper, part of it is reflected (no resonant typeof interaction) at fiber and pigments in the surface layer andinside the paper structure part of it is absorbed (ray 2)(resonant type of interaction) and the remainder part passesinto the air like transmitted light (no resonant type ofinteraction).

,

A common reflection classification is to distinguishebetween and . Diffuse reflectionis a light reflection that has no strong directional propertiesand might occur also from a totally opaque, highly reflectivebut rough surface. Thus the term "specular reflection" couldrefer to either a part of the first surface reflection, or all of it,depending on the topography of the surface. Since theinteraction of light with micro-rough surfaces is really ascattering phenomenon, the term is oftenused in optics literature [12].

Any interface between two media with differentrefractive indices will give rise to surface reflection (ray33 ), and this also concerns light that hits the paper surfacefrom within. Light that tries to exit from the paper mightactually be reflected and enter the paper again. If this effectis strong, it will likely have a considerable impact on thehalftone imaging properties of the medium [21]. For printson rough paper surfaces, this effect is not so prominent, butfor higher paper grades with a glossy and smooth coatinglayer it could be important to consider internal surfacereflections.

Kubelka and Munk first solved differential equations1 and 2 for the special case of a colorant layer with

opaque thickness 5). Analyzing the layer of a lesser

thickness the colorant appears translucent and the backingof reflectance starts to shine through. Kubelka and Munkgave a general solution for the reflectance of suchtranslucent layer, but they ignored any surface or internalreflections at the colorant boundaries caused by differentrefractive indices. Saunderson [22] presented aconventional adjustment (correction) of the estimate valueto the experimental results taking account of the ignoredinternal reflectance and transmittance factors at theinner top boundary surface:

diffuse specular reflection

surface scattering

R

R

R T

4.3Internal surface reflection

4.4Saunderson correction

Refleksija na drugoj granici sredstva (papira)

Saundersonova korekcija

'

( ) ( )(

g

i i

Kubelka- unk theory in describing optical properties of paper ( )M I V. Dimbeg-Mali, . Barbari-Mikoevi, K. Itri

Figure 3Slika 3.

Possible events for photons in a turbid medium (paper)Mogui dogaaji za fotone u zamuenom sredstvu (papiru)

4.1Direct and diffuse transmission

4.2Surface and bulk reflection

Direktna i difuzna transmisija

Povrinska refleksija i refleksija uzrokovana rasprenjem

In a more or less transparent medium, some portion ofthe incident photons might pass straight through and exit onthe opposite side. Paper is a strongly scattering medium, sothis is a rare event, and direct transmission (ray 11') can beneglected for any reasonably thick paper.

Paper is a turbid medium, but a paper sheet is also quitethin, and therefore translucent. The internal scattering oflight might well result in the photon leaving the paper at theopposite side from where it entered. There is a transmissionthrough a paper sheet, but that transmission is almost diffuse(ray 55').

Light is reflected at the interface between materialswith different refractive indices. The direction of specularlyreflected light depends on the incidence angle and theorientation of the surface normal. Since the surface of apaper is often rough, the specular surface reflection (ray 44')does not usually occur.

Paper is a translucent material, and multiple lightscattering in the bulk of the paper is really the main reasonfor most of its macroscopic optical properties, for examplethe fact that it is white. Light that enters the paper is typicallyscattered several times before it either exits the paper or isabsorbed. This reflection of scattered photons is bulkreflection (ray 66'). A photon that enters the paper at onepoint does not necessarily exit at the same point, although itis more probable that it exits close to the point of incidencethan far from it. This lateral spreading of light is quitedifferent from the local surface reflection.

KMi

KMiss

RR

RTTRR

1exp(16)

where and are the specular reflectance andtransmittance factors of the incident electromagneticradiation at the top surface of the layer.

As it is mentioned before, the paper is a complexstructure consisting of cellulose, fibers, fillers andadditives. The K-M equations can be used in description ofsuch paper in form of additivity principle:

R Ts s

5Additivity principle

5.1One layer paper

Princip adicije

Jednoslojni papir


Where the and coefficients indicate different scatteringand absorption coefficients for different ingredients andindicate the mass fraction of the particular component. Firstit is necessary to obtain values for and from Eqs. 12and 13) which correspond to the fiber component [23]. Forthat purpose unfilled paper sheet (fiber sample) has beenmade under the same experimental conditions as examinedpaper. So, a second set of handsheets (fiber + filler sample)has to be prepared with a known fraction of the chosenfiller. The values of the filler's scattering and absorptioncoefficient and can be calculated from [24]:

S K

x

S K

x

S K

i i

i

1 1

2

2 2

( )(

V. Dimbeg-Mali, . Barbari-Mikoevi, K. Itri Kubelka-Munk teorija u opisivanju optikih svojstava papira (I)

5.3Multi-layer paperVieslojni papir

Taking limes and rearrangement we obtain:n

3322111

xSxSxSxSSn

iiipaper (17)

3322111

xKxKxKxKKn

iiipaper (18)

122

2 )1(1

S ,xSx

S paper (19)

122

2 )1(1

K ,xKx

K paper ( )20

where

21 1 x .x ( )21

Addition of filler to the pulp does not always follow theadditivity rule because of the complexity of interactions.The filler light scattering effect is reduced when the fillercontent is increased, so cannot be treated as a constant.Light scattering is dependent on the structure and thestructure is changed when the filler is mixed with the pulp.Treatments such as calendering and wet pressing can alsochange the light scattering in the paper.

Kubelka published [3] a number of equations which areused for coated paper. For a system with layer 1 above layer2 it is possible to analyze how the light is reflected betweenthem. The result of the analysis was a number of newequations for transmittance and reflectance:

S2

5.2Two-layer paperDvoslojni papir

0201

2112 1 RR

TTT

( )22

0201

022

10112 1 RR

RTR .R

( )23

From the structure of the equations it is evident thattransmittance is a symmetrical process, i.e. thetransmittance is the same if the system with two layers isreversed ( = ), while the process of reflectance is notsymmetrical ( ). The following relationship is validfor a system of two layer sheets, i.e. two layer sheets over

1 two layer sheets [18]:

T T

R R

n

n

12 21

12 21

12)1(012

12)1(2

1201212

1

n

nn

RR

RTR .R ( )24

22 2012 021 12 012 021 12 012

12021 021 021

1 1

2 2

R R T R R T RR

R R R

(25)

with:

0121202112

121

R .RRR

T

(2 )6

A three layer system 123 is now easy to obtainconsidering the system as a two layer 12 placed on top oflayer 3 [11]. This simulation of three layers can be used tocalculate reflectivity for coated paper. For that purpose it isnecessary to know the absorption and scatteringcoefficients of the coating and of the substrate (paper) aswell as their weight and grammage.

The theory of Kubelka-Munk has found a wideacceptance for modeling the reflectance and transmittancebehavior of scattering dull materials having the samesurface characteristics over the whole area of examinedsubstrate. Still, halftone prints cannot be regarded asinfinitely wide colorant layers and the original concept failsto predict their measured color. K-M theory has to adapt foranalyzing the light scattering inside paper printed withhalftone. Halftone ink or toner prints consist ofmicroscopically varying transmittances and cannot beregard as an infinitely wide layer. Interaction of light with ahalftone printed paper is a very complex process andpossible events for a photon are shown in Fig. 4.

K S

6Interactions of light with a halftone printed paperInterakcija svjetlosti s otiskom na papiru

Figure 4

Slika 4.

Possible events for photons in a halftone print on the papersubstrate. Arrows marked by numbers 1-6 denote possible interactions

with paper as in Fig. 3., while arrows marked by letters A-F denoteinteractions caused by halftone print.

odgovaraju interakcijama fotona s rasterskim elementom.

Mogue putanje fotona za otisak na papiru. Strelice oznaenebrojevima od 1 do 6 odgovaraju moguim interakcijama fotona spapirom kao na Sl. 3., dok strelice oznaene sa slovima od A do F

There are no strongly specular surfaces in a normalprint, but specular reflection might happen at the surface ofthe ink (arrow B). A is a photon that is absorbed by the inklayer before reaching the paper substrate. The cause ofoptical dot gain is the type of event illustrated by arrow C,namely a photon that enters the paper at a point not coveredby ink, but is absorbed on its way out in the printed pattern

122 Tehni ki vjesnik ,18, 1(2011) 117-124

has introduced isotropic light scattering into a Kubelka-Munk oriented approach. He introduced a lateral analysis ofscattering light and presented a Kubelka-Munk basedapproach, which, by numerical evaluation, predicts thepoint transmittance and reflectance profile of an ideal print.His approach remains limited to a two-dimensional modeland ignores the influence of brightened paper or of internalreflections at the interfaces of the paper. Bergs model wasextended by Mourad in order to reduce the gap between themathematical description of the paper's point spreadfunction and the experimental results of simple reflectancemeasurements [48, 49]. This model analyzes the scatteringlight fluxes in a general three- dimensional space.

Arney and co-workers [50] discussed threeexperimental techniques for measuring light scattering andresolution characteristics of paper:a) direct measurements of lateral light scatter ( ) of

paper by microdensitrometric scans of an illuminatedknife edge

b) derivate from the K-M equations andexperimental measurements of paper reflectance

c) modeling of Yule-Nielsen effect of optical dot gain andfitting the model to experimental data.

The first of these techniques is the most difficult and theleast precise one, while the third presents the line screenmethod (ideal one-dimensional halftone) which is mucheasier to perform and can be done with reasonably highprecision. Reflectance measurements (the second of thesetechniques) are the easiest way of measurements andinstrumentation and they can be done in many laboratories.However, the K-M theory appears not to provide as exact anestimate of lateral spread as can be achieved by the linescreen technique. The possibility of poor correlationbetween the K-M theory and the line screen experiment isthe assumption of homogeneity in light scatter inherent inthe K-M model.

Oittinen [51] and Engeldrum and Pridham [52]suggested that K-M theory may provide a basis for an apriori derivation of the , and of paper,assuming that light scatters homogeneously in all directionsin the paper. Engeldrum derived of the paper by takingthe derivative of the K-M function for reflectance and thentaking the Hankel transform:

MTF

MTF

PSF LSF MTF

MTF

Kubelka- unk theory in describing optical properties of paper ( )M I

and might cause decreasing of bulk reflection. If the ink hasa low absorption, considerable amounts of light might passthrough the ink film without being absorbed, and the eventsindicated by arrows D, E and F also become important [20].This is for example the case in four color printing, where thecolored inks are rather transparent for many wavelengths[25].

The first optical model of tone reproduction in thehalftone process was the Murray-Davies equation [26]which was extended by Neugebauer [27]. These modelsdescribe a linear relationship between the reflectance of thehalftone image and the fractional dot area of the image thatis covered by halftone pattern, i.e. these models assume onlythe occurrence of a direct reflection on the surface which inreality is far from real (Fig. 4.). Therefore, lateral scatteringcannot be ignored. The entrance point of incident lightdiffers from its exit point which is well known as the Yule-Nielsen effect [28] (optical dot gain).

The work of Arney, Engeldrum and Zeng [29] hasshown that the dot gain depends both on the scatteringproperties of the paper and on the dots themselves. Theyconclude that the model should include two empiricalparameters instead of the oneYule-Nielsen parameter n.

P. Emmel and R. D. Hersch introduced unified model[30] and a new mathematical formulation based on matricesdescribing the light scattering and ink spreading in printing.This model generalized the K-M theory and unified it withthe Neugebauer model. The Saunderson correction, theClapper-Yule equation, the Murray-Davies relation, areparticular cases of that unified model.

Two theories commonly applied to describe the opticalbehavior of paper are the Kubelka-Munk theory and theLinear System theory. Kubelka-Munk theory describesreflectance and transmittance properties of scatteringmaterials in terms of the scattering and absorptioncoefficients, while the Linear System theory uses a pointspread function ( , ). ( , ) is a probability densityfunction that describes the probability of a photon returningto the surface of the paper at the location ( , ) away from thepoint of entry into the paper [31] The Fourier transform ofthe point spread function, ( ) is called themodulation transfer function of paper. Thecharacteristic lateral scattering of the paper is oftendescribed by either the function or function. Inliterature, several approaches of the point spread functionfor paper are known. Most of them determinedempirically [32] or assuming a specific type of function [33-39]. Yule and Nielsen [33] suggested that the LSFcharacteristic of paper is Gaussian and the corresponding

is also a Gaussian function, while Wakeshima [34]reported that PSF is an exponential function which wouldmake the a Bessel function and the a complexinverse power function.

Some of approaches are based on numericalsimulations [20, 40], microscopic reflectance measurements [41], image processing, multi-flux theory [42],modeling [43-46] or radiative diffusion [31].

Because of the negligence of lateral light streams, theoriginal concept of Kubelka-Munk is not adequate formodeling optical properties of halftone prints. In thisrespect, a recent improvement was made by Berg [47] who

6.1Lateral scatteringLateralno rasprenje(bono)

.

-

PSF x y PSF x y

x y

FFT PSFMTF

PSF MTF

MTF

MTF

LSF MTF


32 22

21

1 2( ) 1

2ln(1 )

j

j

RMTF

j jbSR

. (27)

This equation overestimates the influence of K in theof paper. Arney [53] suggests an model with

=0, because the majority of paper samples used to printhalftone images have low absorption coefficients. He has tomake empirical modification of that model by adding theempirical constant :

MTF MTFK

k0

014,5 k .eS

k LSp

(2 )8

Arney shows [53] that the K-M parameters and arenot sufficient descriptors of paper , even for paper withthe same background ( = 0) and negligible values of .This additional parameter may relate to the degree ofhomogeneity of the substrate. Engeldrum pointed out thatany model derived from the K-M theory implies the

S LMTF

R Kg


assumption that the scattering coefficient is homogenousthroughout the paper (coated papers are good example ofinhomogeneous substrates). In addition, directionalhomogeneity is assumed so that in the vertical direction isthe same as in the lateral direction, while the intrinsicdirectionality of paper suggests that vertical and lateralscattering distances are not the same.

The point spread function of brightened papers isaffected by the included fluorescent additives. The suppliedbrighteners absorb certain energy of the invisible radiations(ultraviolet part of electromagnetic spectrum). A specificamount of that energy is then released by radiativerelaxation (fluorescence spectrum) in the visible part ofelectromagnetic radiation. This technique compensates theyellowish of natural non-brightened papers. As thefluorescence emission happens in all space directions insidethe paper, it acts like a diffuse partial light source orconverter. Because of that the brightening effect amplifiesthe point spread function of the paper in the blue part ofvisible spectrum and must be taken into account whenanalyzing the .

As printed color layers behave like spectral light filters,they can reduce the energy of fluorescence emission of thepaper by absorbing a part of excitation spectrum of thefluorescent additives.

The Kubelka-Munk theory, though it remains the mostused in practice, has some disadvantages, like imprecisionin some cases (fluorescence problems for instance), initialassumptions that are too simplified. Real paper nevercompletely satisfies all those assumptions, but researchersare interested in finding a model to upgrade that theoryprobably due to its explicit form, simple use and itsacceptable prediction accuracy in many cases useful inpaper, paint and colorant industry.

To improve the results obtained by the use of the K-Mtheory, numerical methods are common nowadays whichsimulate individual photon paths (Monte Carlo simulationmodel, GRACE, DORT).

SS

PSF

6.2PSF correction for fluorescence

7Conclusion

8References

PSF korekcija za fluorescenciju

Literatura

Zakljuak

[1] Schuster, A. Radiation through a foggy atmosphere. //Astrophys. J. 21, 1(1905), 122.

[2] Kubelka, P.; Munk, F. Ein Beitrag zur Optik derFarbanstriche, Z. Tech. Phys. (Leipzig), 12(1931), 593601.

[3] P. Kubelka, New contributions to the optics intensely light-scattering materials. Part I // J. Opt. Soc. Am. 38(1948),448457.

[4] Kubelka, P. New contributions to the optics intensely light-scattering materials. Part II: Nonhomogeneous layers. // J.Opt. Soc.Am. 44(1954), 330335.

[5] Van den Akker, J. A. Scattering and absorption of light inpaper and other diffusing media. // Tappi J. 32(1949),498501.

,

,

,

V. Dimbeg-Mali, . Barbari-Mikoevi, K. Itri Kubelka-Munk teorija u opisivanju optikih svojstava papira (I)

[6]

,[7]

,[8]

,[9]

,

[10]

[11]

[12],

[13],

[14],

[15]

,[16]

;

o

, , 7 , .

,

,

,

,

,

,

,

,

ogy,

Stenius, . S. The Application of the Kubelka-Munk Theoryto the Diffuse Reflection of Light from paper, I, SvenskPapperstidning 54(1951), 663670.Stenius, . S. The Application of the Kubelka-Munk Theoryto the Diffuse Reflection of Light from paper, II, SvenskPapperstidning 54(1951), 701709.Stenius, . S. The Application of the Kubelka-Munk Theoryto the Diffuse Reflection of Light from paper, III, SvenskPapperstidning 56(1953), 607614.Vollz, H. G. Practical pigment testing with the aid of theKubelka- Munk theory, Progr. Organic Coatings 13(1985),153169.Philips-Invernizzi, B.; Dupont, D.; Caze, C. Bibliographicalreview for reflectance of diffusing media. // Society of Photo-Optical Instrumentation Engineers, 40, 6(2001), 10821092.Judd, D. B. Optical specification of light scattering materials.// Tech.Assoc. Papers. 21(1938), 474481.Adrian, A. P. New procedure for evaluating the opacifyingproperties of pigments.// Tech. Assoc. Papers. 27(1944),465471.Hillend, W. J.; Opacity problems in printing papers: Kubelka-Munk theory gives good, quick answers.// Tappi 49, 7 (1966),41A47A.Borch, J.; Scallan, A. M. A comparison between the opticalproperties of coated and loaded paper sheets.// Tappi 58,2(1975), 34135.Robinson, J. V. A summary of reflectance equations forapplication of the Kubelka-Munk theory of optical propertiesof paper. // Tappi 58, 10 (1975), 152153.Judd, D.; Wyszecki, G. Color in Business, Science andIndustry, J. Wiley & Sons, NewYork, 1975.

[17] Wyszecki, G.; Stiles, W. S. Color Science, 2nd ed., J. Wiley &Sons, NewYork, 2000.

[18] Pauler, N. Paper Optics, AB Lorentzen & Wettre, Sweden,2001.

[19] TAPPI. In TAPPI Test Methods, Interrelation of reflectance,R ; reflectivity, R TAPPI opacity, c ; scattering, s; and

absorption, k Standard practice T 1214 sp-98.// TAPPI Press(1998).

[20] Gustavson, S. Dot Gain in Colour Halftones, PhDDissertation No. 492, Dept. f Electrical Engineering,Linkoping University, Sweden, 1997.

[21] Clapper, F. R.; Yule, J. A. C. The effect of multiple internalreflections on the densities of half-tone prints on paper.//Journal of the Optical Society ofAmerica 43 (1953) 600

[22] Saunderson, J. L. Calculation of the Color PigmentedPlastics. // J. Opt. Soc.Am. 32(1942), 727736.

[23] Hubbe, M. A.; Pavlak, J. J.; Koukoulas, A. A. Paper'sappearance: a review, BioResources 3, 2(2008), 627665.

[24] Middleton, S. R.; Desmeules, J.; Scallan,A. M. The Kubelka-Munk coefficients of fillers. // J. Pulp Paper Sci. 20, 8(1994),231235.

[25] Clapper, F. R.; Yule, J. A. C. Reproduction of Color withHalftone Images, TAGAProceedings 7 (1955).

[26] Murray, A. Monochrome Reproduction in Photoengraving,Technical Report of the J. Franklin Institute 221 (1936).

[27] Neugebauer, H. E. J.: Die theoretischen Grundlagen desMehrfarbenbuchdrucks. // Zeitschrift fr wissenschaftlichePhotographie, Photophysik und Photochemie 364(1937),7389.

[28] Yule, J. A. C.; Nielsen, W. J. The penetration of light intopaper and its effect on halftone reproduction, TAGAProceedings 3 (1951), 6576.

[29] Arney, J. S.; Engeldrum, P. G. ; Zeng, H. An ExpandedMurray-Davies Model of Tone Reproduction in HalftoneImaging. // Journal of Imaging Science and Technology39(1995), 502508.

[30] Emmel, P.; Hersch, R. D. A Unified Model for ColorPrediction of Halftoned Prints. // J. Imaging Sci. Technol44(2000), 351359.

[31] Rogers, G. Optical dot gain in a halftone print. // J. ImagingSci. Technol. 41(1997), 643656.

0 0.89

Tehni ki vjesnik ,18, 1(2011) 117-124

[32] Ruckdeschel, F.; Hauser O. Yule-Nielsen Effect in Printing: aPhysicalAnalysis,Appl. Opt. 17(1978), 33763383.

[33] Yule, J. A. C. ; Howe, D. J. ; Altman, J. H. Tappi, (1967) 337-346.

[34] Wakeshima, H.; Kunishi, T. J. Opt. Soc.Amer. 58(1968), 272.[35] Oittinen, P.; Saareima, H. Paperi Ja Puu Paper & Timper 75

(1993).[36] Dainty, J. C.; Shaw, R. Image Science, Academic Press, New

York, 1974.[37] Fischer, G.; Rodriguez-Giles, J.; Scheuter, K. R. Ein

physikalisches Modell fur die Beschreibung vonLichtstreuprozessen, Die Farbe 30(1982), 199220.

[38] Rogers, G. Lateral Scattering Probabilities. // J. Imaging Sci.Technol. 42(1998), 336.

[39] Rogers, G. Effect of light scatter on halftone color. // J. Opt.Soc.Am.A 15(1998), 18131821.

[40] Yang, L.; Gooran, S. ; Kruse, B. Simulation of Optical DotGain in Multichromatic Tone Production. // J. Imaging Sci.Technol. 45 (2001), 198204.

[41] Shinichi, I. ; Norimichi, T.,Yoichi, M.Analyzing CTF of Printby MTF of Paper. // Journal of Imaging Science andTechnology, 42, 6(1998), 572576.

[42] Joshi, J. J.; Vaidya, D. B.; Shah, H. S. Application of Multi-Flux Theory Based on Mie Scattering to the Problem ofModeling the Optical Characteristics of Colored PigmentedPaint Films, Color Research and Application, 26, 3(2001),234245.

[43] Arney, J. S.;Arney, C. D. Modeling theYule-Nielsen HalftoneEffect. // J. Imaging Sci. Technol 40(1996), 233238.

[44] Pray, E. Kubelka-Munk Theory and Yule-Nielsen Effect onHalftones. // J. Imaging Sci. Technol 43(1999), 365370.

[45] Arney, J. S. A Probability Description of the Yule-NielsenEffect I. // J. Imaging Sci. Technol. 41 (1997), 633642.

[46] Arney J. S.; Katsube, M. A Probability Description of theYule-Nielsen Effect II: The Impact of Halftone Geometry.// J.Imaging Sci. Technol. 41(1997), 637642.

[47] Berg, F. Isotrope Lichtstreuung in Papier Neueberlegungen zur Kubelka-Munk Theorie, PhD thesis,Technische Hochschule Darmstadt, 1997.

[48] Mourad, M. S. Color Predicting Model for Electrophotographic Prints on Common Office Paper, PhD Dissertation No.40, Lausanne, 2002.

[49] Mourad, M. S.; Emmel, P.; Simon, K.; Hersch, R. D.Extending Kubelka-Munk s Theory with Lateral LightScattering, IS&Ts NIP 17: International Conference onDigital Printing Technologies, 2002 469473.

[50] Arney, J. S.; Arney, C. D.; Katsube, M. An MTF Analysis ofPapers. // J. Imaging Sci. Technol. 40(1996), 1925.

[51] Oittinen, P. Advances in Printing Science and Technology, W.H. Banks, Ed, Pentech Press, London, 16(1982), 121128.

[52] Engeldrum, P. G.; Pridham, B.Application of Turbid MediumTheory of Paper Spread Function Measurements, TAGAProceedings, 1995 339352.

[53] Arney, J. S.; Chauvin, J.; Nauman, J.; Anderson, P. G.Kubelka-Munk Theory and MTF of Paper. // J. Imaging Sci.Technol. 47(2003), 339345.

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Authors' addressesAdrese autora

Vesna Dimbeg-Mali, doc.eljka Barbari-Mikoevi, doc. dr. sc.Katarina Itri

dr. sc.

Faculty of Graphic ArtsUniversity of Zagreb

10000 Zagreb, [email protected]

Getaldieva 2

Kubelka- unk theory in describing optical properties of paper ( )M I V. Dimbeg-Mali, . Barbari-Mikoevi, K. Itri

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