Chapter 3: Physical models for color predictionread.pudn.com/downloads448/sourcecode/graph/texture_mapping/1… · 3.2 A few results from radiometry 3.3 Reﬂection and refraction

$Page 1: Chapter 3: Physical models for color predictionread.pudn.com/downloads448/sourcecode/graph/texture_mapping/1… · 3.2 A few results from radiometry 3.3 Reﬂection and refraction$
chapter three

Physical models for color prediction

Patrick Emmel

†

Clariant International

Contents

3.1 Introduction3.2 A few results from radiometry3.3 Reflection and refraction

3.3.1 Basic laws3.3.2 Interface reflection under diffuse light

3.4 Light absorption3.5 Light scattering

3.5.1 Rayleigh scattering3.5.2 Mie scattering3.5.3 Multiple scattering

3.6 Phenomenological models3.6.1 Radiative transfer3.6.2 Kubelka–Munk model (two-flux model) 3.6.3 Surface phenomena and Saunderson correction 3.6.4 Multichannel model

3.7 The fluorescence phenomenon3.7.1 Fluorescence: transparent layer 3.7.2 From a one-flux to a two-flux model for a reflective

substrate3.7.3 Spectral prediction for reflective fluorescent material3.7.4 Measuring the parameters of the fluorescence model

3.8 Models for halftoned samples 3.8.1 The Murray–Davis equation

† This work was done while the author was at the Ecole Polytechnique Fédérale de Lausanne(EPFL).

03 Page 173 Monday, November 18, 2002 8:13 AM

© 2003 by CRC Press LLC

174 Digital Color Imaging Handbook

3.8.2 The classical Neugebauer theory3.8.3 Extended Neugebauer theory3.8.4 The Yule–Nielsen equation3.8.5 The Clapper–Yule equation3.8.6 Advanced models3.8.7 The Monte-Carlo method

3.9 New mathematical framework for color prediction of halftones3.9.1 Some particular cases of interest3.9.2 Computing the area fractions and the scattering

probabilities3.10 Concluding remarksReferences.

3.1 Introduction

Numerous physical phenomena influence color: the light source, surfacereflection, light absorption, light scattering, reflection on the substrate, mul-tiple-internal reflections at the ink–air interface, and the combination ofseveral light-absorbing and light-scattering substances. In the particular caseof halftone prints, additional effects, such as the optical dot gain (also calledYule–Nielsen effect), must also be taken into account. This makes accuratecolor prediction very difficult. Until recently, the physical phenomenainvolved were described separately by several classical models: Lambert’slaw for diffuse light sources, the Fresnel reflection law, Beer’s absorptionlaw, the Saunderson correction for multiple internal reflections, and theKubelka–Munk model for absorbing and scattering media. The colors ofhalftone prints were predicted using other theories: the Murray–Davismodel, the Neugebauer model, the Yule–Nielsen model, and the Clap-per–Yule model for optical dot gain.

This chapter is based, however, on a new global approach that incorpo-rates all the physical contributing phenomena listed above into a single modelusing a mathematical framework based on matrices. Classical results (forexample, the Murray–Davis equation, the Clapper–Yule relation, or theKubelka–Munk model) correspond to particular cases of this model. Further-more, the model we present here predicts accurately the spectra of printedcolor samples (uniform or halftoned), and it can be used for any inks orcolorants, including the standard Cyan, Magenta, Yellow, and Black (CMYK)that are usually used in printing devices or any other nonstandard inks.

Throughout this chapter we will consider the entities , ,, . . . as spectra (transmission spectrum, reflection spectrum, density

spectrum, etc.). But, when the wavelength notation is omitted, we willconsider them as coefficients (transmission coefficient, reflection coefficient,density, etc.). We will consider these terms interchangeably as synonymsaccording to our best convenience.

We start our presentation with a few useful definitions from radiometrygiven in Section 3.2. Sections 3.3, 3.4, and 3.5 introduce the basic physical

T λ( ) R λ( )D λ( )

λ( )

-ColorBook Page 174 Wednesday, November 13, 2002 10:55 AM


Chapter three: Physical models for color prediction 175

laws of light reflection, light absorption, and light scattering, respectively.The complex interaction between light and matter forces us to use phenom-enological models, which are presented in Section 3.6. The particular caseof fluorescent media is discussed in Section 3.7. Traditional models used topredict the color of halftone prints are presented in Section 3.8. Finally, weexplain in Section 3.9 how all these models can be incorporated into a singlemodel for the prediction of halftones.

3.2 A few results from radiometry

Let us start with the definition of a few radiometric quantities and terms.

1

• A surface element of area receiving a light flux is said to beunder an

irradiance

(unit ).

(3.1)

• A surface element of area emitting a light flux has an

exitance

(unit ).

(3.2)

• A surface element of area is said to be of

radiance

(unit) if it emits a flux in a solid angle making an

angle with the normal to the surface.

(3.3)

• The

intensity

(unit ) of a light flux in a solid angle is defined by

(3.4)

• A light source whose radiance is constant in all space directionsis said to be

Lambertian

. The exitance of such a source is (see Reference 2).

• A diffuse reflector of constant radiance is said to be a

Lambertsurface

3

or a

Lambertian reflector

. The exitance of such a surfaceequals its irradiance , so and we have

(3.5)

da dφr

E W m 2–⋅

Edφr

da--------=

da dφe

M W m 2–⋅

Mdφe

da--------=

da LW sr 1– m 2–⋅ ⋅ d2φe dω

θ

Ld2φe

θcos ωdad--------------------------=

I W sr 1–⋅ φd dω

I dφdω-------=

LM M πL=

LM

E L E π⁄=

d2φe

θ ωdadcos-------------------------- E

π---=




This formula is called

Lambert’s cosine law

. Note that, in the literature,

4

Lam-bert’s cosine law is often presented in terms of intensity,

(3.6)

where is the area of the surface.Equation 3.6 implies that the

radiation pattern

of such a surface (i.e., thelocus of the extremities of the intensity vectors, also called

indicatrix

) is acircle (see Figure 3.1). Another useful result is Lambert’s cosine law for aconical light beam. The flux emitted in the direction given by the angle can be computed by considering the solid angle (seeFigure 3.2). By replacing in Equation 3.6, we deduce that a surface that receives from the upper hemisphere a total flux emits adiffuse flux whose angular distribution is given by

5

(3.7)

Natural light has a rather diffuse behavior in which rays do not have aprivileged orientation. Therefore, it is useful to define the term

diffuse irra-diation

.

• Let be an opening in an opaque plane (see Figure 3.3). The openingis said to be under a

diffuse irradiation

from the upper hemisphere if is a Lambertian source in the lower hemisphere. From the point

I θ( )ωd

dφe Eπ---A θcos I0 θcos= = =

A

I θ( )dωdφe E

π--- A θcos= =

A

θ

Figure 3.1 According to Lambert’s cosine law, the intensity I of the light emitted bya diffuse reflector of area A depends only on the cosine of the angle θ of observation(note that the radiation pattern of such a reflector is a circle).

θdω 2π θdθsin=

dω Aφr E sd

S∫=φe

1φr---- θ∂

∂φe 2 θ θcossin 2θsin= =

A

A




of view of the upper hemisphere, the surface is said to be a

Lambertian receiver

; its reception pattern or indicatrix is a circle.

Finally, light sources are classified by the way they produce light.

Incan-descent

sources produce light by thermal blackbody or near-blackbody radi-ation. All nonthermal light production is called

luminescence

. Hence, a

lumi-nescent medium

is a medium that produces light by any means except thermalexcitation. A common kind of luminescence is photoluminescence, which

dω 2π θdθsin=

θ

dθ

da

Figure 3.2 Solid angle that must be used when considering a conical light beamemitted by a Lambertian source or reflector.

I θ( )dωdφe E

π--- A θcos= =

A

Figure 3.3 Surface element A under diffuse illumination. If A is an opening in anopaque plane that separates the upper and the lower hemispheres, the surface ele-ment A is a Lambertian source in the lower hemisphere.

A




includes fluorescence and phosphorescence. A photoluminescent mediumproduces light when excited with photons. An extended list of forms ofluminescence is given in Reference 6.

3.3 Reflection and refractionThe term reflection refers to all interaction processes of light with matter inwhich the photons are sent back into the hemisphere of the incident light.We distinguish between two types of reflection: specular reflection, whichoccurs on smooth surfaces (where the irregularities are small compared tothe wavelength of the incident light), and diffuse reflection, which occurs onrough surfaces. A beam of light incident on a smooth surface is re-emittedas a well-defined beam, whereas, on a rough surface, it is re-emitted as amultitude of rays emerging in different directions. We define the reflectancespectrum (or reflection coefficient) as the ratio of the reflected light fluxto the incident light flux for a given wavelength .

The term refraction refers to the change of direction of a light beam whenentering a medium in which the speed of light is different. Therefore, wedefine the refractive index of a given medium as the ratio of the speed oflight in empty space to the speed of light in the medium. For example, therefractive index is for water, for crown glass, and

for rutile ( ).7 A medium that also attenuates electromagneticwaves has a complex refractive index , where is the attenu-ation index. For example, the complex refractive index is foraluminium, for iron, and for platinum.8Note that, in a metal, light is so intensely attenuated that it can penetrate toa depth of only a few hundred atoms.

3.3.1 Basic laws

Let us recall the basic laws of light refraction and light reflection. A lightbeam that, with an incidence angle , hits a refractive surface that separatestwo media of refractive indices and is partially reflected into the firstmedium and partially refracted into the second medium (see Figure 3.4).The incident beam, the reflected beam, and the refracted beam lie in thesame plane, called the plane of incidence. The reflected beam makes with thenormal to the surface the same angle as the incident beam, whereas the

R λ( )λ

n

n 1.33= n 1.52=n 2.907= TiO2

n̂ n 1 iκ+( )= κn̂ 1.44 i5.23+=

n̂ 1.51 i1.63+= n̂ 2.63 i3.54+=

θ1

n1 n2

θ1θ1n1

n2θ2

Figure 3.4 Reflection and refraction of a light beam at a refractive surface.

θ1




refracted beam makes with the normal an angle , which is related to by Snell’s law.

(3.8)

Note that, for , there is a critical angle abovewhich the incident beam is totally reflected.

The intensity of the reflected beam is calculated by considering twopolarized electromagnetic waves. One is polarized in parallel to the planeof incidence, and the other is polarized perpendicularly. It can be shown9

that the reflection coefficient for the parallel polarized wave and thereflection coefficient for the perpendicularly polarized wave are given by

and (3.9)

where = angle of incidence

= angle of refraction according to Snell’s law (see Equation 3.8 and Figure 3.4)

In the literature, these relations are known as the Fresnel relations. We denoteby the reflection coefficient of a beam propagating in a medium ofrefractive index , which has an incidence angle with the refractivesurface delimiting a medium of index . Because natural light can beconsidered as an equal mixture of both types of waves, its reflection coeffi-cient is the mean value of and ,

(3.10)

Note that Snell’s law is also valid for complex refractive indices , suchas the refractive indices of metals. In the particular case of a light beamhaving normal incidence on a medium of complex refractive index , thereflection coefficient is10

(3.11)

For non-normal incidence ( ), the angle is complex, meaningthat a phase change occurs on reflection. For example, a linearly polarizedlight is reflected as elliptically polarized light. The generalized Fresnel’sformulas for complex refractive indices are beyond the scope of this book,but a detailed presentation can be found in the literature.10

θ2 θ1

n1 θ1( )sin n2 θ2( )sin=

n1 n2> θ1max n2 n1⁄( )asin=

ra

re

ran2 θ1cos n1 θ2cos–n2 θ1cos n1 θ2cos+-----------------------------------------------

2

= ren1 θ1cos n2 θ2cos–n1 θ1cos n2 θ2cos+-----------------------------------------------

2

=

θ1

θ2

rn1 n2, θ1( )n1 θ1

n2

ra re

rn1 n2, θ1( )ra re+

2--------------=

n̂

n̂

r1 n̂, 0( ) n̂ 1–n̂ 1+------------

2=

θ1 0≠ θ2




3.3.2 Interface reflection under diffuse light

Let us now compute an average reflection coefficient for diffuse lightarriving on a plane refractive surface. Such an average is calculated byintegrating, over all directions, the product of the angular distribution ofdiffuse irradiation given by Equation 3.7 and the reflection coefficient of anatural light beam given by Equation 3.10.

(3.12)

This calculation was done by Judd11 in 1942 for a large number of refractiveindices. A typical result is the particular case of the surface reflection between air (whose refractive index is ) and a medium of refractiveindex n.

(3.13)

In the particular case of a plastic medium, we have . The com-putation of Equation 3.13 leads in this case to . This valueexpresses the average reflection coefficient under a perfect diffuse illumina-tion. When diffuse light crosses the refractive surface in the other direction,from a medium of refractive index to the air, the reflection occurs withinthe material medium, and it is therefore called the internal reflection ri.

(3.14)

The numerical result of the computation for is, in this case,. The numerical results for other refractive indices are given in

Table 3.1.Note that for diffuse light, internal reflection values are always much

higher than surface reflection values. When the first medium has a higherrefractive index than the second, there is a critical incidence angle, accordingto Snell’s law, above which light is totally reflected. This total reflection isresponsible for the high values of when .

3.4 Light absorptionThe term light absorption refers to all processes that reduce the intensity of alight beam when interacting with matter. We must distinguish between trueabsorption, where radiative energy is transformed into another kind of energy

rn1 n2,

rn1 n2, rn1 n2, θ( ) 2θsin⋅( ) θd0

π2---

∫=

rs

n1 1=

rs r1 n, θ( ) 2θsin⋅( ) θd0

π2---

∫=

n 1.5=rs 0.0918=

n

ri rn 1, θ( ) 2θsin⋅( ) θd0

π2---

∫=

n 1.5=ri 0.5963=

rn1 n2, n1 n2>




(thermal agitation energy, ionization energy, etc.), and apparent absorption,which is due to light scattering (see Section 3.5). To avoid confusion, we willuse the word extinction for all light intensity reducing processes and reservethe word absorption for true absorption.

The absorption mechanisms are different in gases, liquids, and solids.For gases and liquids, there are two main mechanisms. The first mechanismis a quantified change of the energy state of the molecules or the atoms,producing line spectra. The second mechanism is the dissociation of

Table 3.1 Reflection Coefficient at Normal Incidence r1,n(0), Surface Reflection Coefficient rs, and Internal Reflection Coefficient ri for Various Refractive Indices n

1.0 0.0000 0 0 1.3 0.0170 0.0611 0.44451.01 0.0000 0.0031 0.0228 1.31 0.0180 0.0627 0.45381.02 0.0001 0.0061 0.0446 1.32 0.0190 0.0643 0.46301.03 0.0002 0.0088 0.0657 1.33 0.0201 0.0659 0.47191.04 0.0004 0.0114 0.0860 1.34 0.0211 0.0675 0.48071.05 0.0006 0.0139 0.1056 1.35 0.0222 0.0691 0.48921.06 0.0008 0.0163 0.1245 1.36 0.0233 0.0706 0.49751.07 0.0011 0.0186 0.1428 1.37 0.0244 0.0722 0.50571.08 0.0015 0.0208 0.1605 1.38 0.0255 0.0737 0.51361.09 0.0019 0.0230 0.1777 1.39 0.0266 0.0753 0.52141.1 0.0023 0.0252 0.1943 1.4 0.0278 0.0768 0.52901.11 0.0027 0.0272 0.2105 1.41 0.0289 0.0783 0.53641.12 0.0032 0.0293 0.2261 1.42 0.0301 0.0799 0.54371.13 0.0037 0.0313 0.2413 1.43 0.0313 0.0814 0.55081.14 0.0043 0.0332 0.2561 1.44 0.0325 0.0829 0.55771.15 0.0049 0.0351 0.2704 1.45 0.0337 0.0844 0.56451.16 0.0055 0.0370 0.2843 1.46 0.0350 0.0859 0.57111.17 0.0061 0.0389 0.2979 1.47 0.0362 0.0873 0.57771.18 0.0068 0.0407 0.3110 1.48 0.0375 0.0888 0.58401.19 0.0075 0.0425 0.3238 1.49 0.0387 0.0903 0.59021.2 0.0083 0.0443 0.3363 1.5 0.0400 0.0918 0.59631.21 0.0090 0.0460 0.3484 1.51 0.0413 0.0932 0.60231.22 0.0098 0.0478 0.3602 1.52 0.0426 0.0947 0.60821.23 0.0106 0.0495 0.3717 1.53 0.0439 0.0962 0.61391.24 0.0115 0.0512 0.3829 1.54 0.0452 0.0976 0.61951.25 0.0123 0.0529 0.3939 1.55 0.0465 0.0991 0.62501.26 0.0132 0.0546 0.4045 1.56 0.0479 0.1005 0.63041.27 0.0141 0.0562 0.4149 1.57 0.0492 0.1020 0.63571.28 0.0151 0.0579 0.4250 1.58 0.0505 0.1034 0.64081.29 0.0160 0.0595 0.4348 1.59 0.0519 0.1048 0.6459

n r1 n, 0( ) rs ri n r1 n, 0( ) rs ri




molecules or ionization of atoms, which produces continuous spectra with anenergy threshold. In solids, the behavior depends on the arrangement of theatoms.12 The absorption of photons in isolators and ionic crystals inducesquantified changes of state that produce narrow band spectra. In semiconduc-tors, the electrons require a small amount of energy to jump over the energygap between the valence and the conduction band. Therefore, photons areabsorbed only if their energy is higher than the band gap. Such materialshave a continuous absorption spectrum with a threshold. In metals, theelectrons belonging to the conduction band move almost freely in the wholevolume, acting as a free electron gas. The absorption of photons is so strongover the whole spectrum that light is reflected. The reason is that incidentlight is an electromagnetic wave that induces an alternating electrical currentin the conducting material. According to Maxwell’s theory, this current re-emits light out of the metal. As far as alloys are concerned, there is no generalrule. Their behavior depends on their crystal structures.

The most widely known classical model for the absorption of light is theBeer–Lambert–Bouguer law (which is also called, by abuse of language, Beer’slaw).13 This model describes the intensity variation of a collimated light beamcrossing a medium that contains identical light-absorbing particles at a con-centration c. Let us consider an infinitely thin slice of thickness of thismedium (see Figure 3.5). The model relies on the assumption that the absorb-ing particles are independent. According to the Beer–Lambert–Bouguer law,the intensity variation of the light flux that crosses this slice is propor-tional to the concentration c, to the flux intensity of the light beam, andto the thickness of the slice. Hence, the flux of a collimated light beamthat crosses the infinitely thin layer varies as follows:

(3.15)

where = the light wavelength, and the proportionality coefficient = the molar decadic absorption coefficient (or, in short, absorption

coefficient) of the absorbing particle (unit: )

dx

φ φ + dφ

dx

Figure 3.5 Absorption of light by an infinitely thin layer containing light-absorbingparticles at a concentration c.

dφφ

dx

φd ε λ( )cφ 10( )dln x–=

λε λ( )

m2 mol 1–⋅




The absorption coefficient can be interpreted as the absorption cross-section area of a mole of particles. For particles of radius r, we have

(3.16)

where NA = = the Avogadro number

= the absorption efficiency factor of the particle

Relation 3.15 is a linear differential equation of the first order whosesolution is given by an exponential function. This kind of function will playa central role in our discussion. The integration of Equation 3.15 through alayer of thickness leads to

(3.17)

The transmittance spectrum (or in short transmittance) is then defined asthe ratio between the outcoming flux and the incoming flux .

(3.18)

The value corresponds to a transparent medium, whereas the value means that no light is transmitted, in which case the medium is

said to be opaque. Beer’s law is often expressed in a logarithmic form,

(3.19)

where is the (optical) density spectrum (or absorption spectrum), whichcorresponds to the transmittance . In the density scale, cor-responds to a transparent medium, and the values of increase loga-rithmically when the transparency decreases. The extreme case of a totallyopaque medium corresponds to an infinite density, . The trans-mission spectra and the corresponding density spectra of cyan, magenta,and yellow inks at various concentrations are given in Figures 3.6, 3.7, andFigure 3.8, respectively.

In a mixture of several different absorbing substances that do not inter-act, the density of the mixture equals the sum of the densities of the indi-vidual substances.

(3.20)

ε λ( )

ε λ( )N A

10( )ln-----------------πr2χabs λ( )=

6.022 1023⋅χabs λ( )

X

φ X( ) Xcε λ( ) 10ln–[ ]exp φ 0( )⋅ 10 Xcε λ( )[ ]– φ 0( )⋅= =

T λ( )φ X( ) φ 0( )

T λ( ) φ X( )φ 0( )----------- Xcε λ( ) 10ln–[ ]exp 10 Xcε λ( )[ ]–= = =

T λ( ) 1=T λ( ) 0=

D λ( ) T λ( )10log– X c ε λ( )⋅ ⋅= =

D λ( )T λ( ) D λ( ) 0=

D λ( )

D λ( ) ∞=

D λ( ) Dj λ( )j

∑=




This equation can also be written using the transmittances of the individualsubstances. The total transmittance equals the product of all transmittances.

(3.21)

In practice, Equation 3.21 allows us to compute the transmittance of themixture of purely light-absorbing inks.

Finally, let us calculate the average density of an infinitely thinslice of an absorbing medium under diffuse illumination (see Figure 3.9). Weknow the angular distribution of the diffuse light flux from Equation 3.7,and Beer’s law gives the absorption in the direction , which equals

Cyan

450 500 550 600 650 700nm

0.5

1

1.5

2

2.5D

c = 0.5

c = 1.0

c = 1.5

c = 2.0

c = 2.5

450 500 550 600 650 700nm

0.2

0.4

0.6

0.8

1

T

c = 0.5

c = 1.0

c = 1.5

c = 2.0c = 2.5

Figure 3.6 Transmission and density spectra of a cyan ink at various concentrations c.

T λ( ) T j λ( )j

∏=

D λ( )

θ

xdθcos

------------ ε λ( ) c⋅ ⋅




Hence, the average density can be computed by integrating over alldirections.

(3.22)

This shows that the optical density under diffuse illumination is twice thedensity observed for a collimated light beam. This fundamental result givesus a generalization of Beer’s law for diffuse light.

(3.23)

Magenta

450 500 550 600 650 700nm

0.5

1

1.5

2

2.5D

c = 0.5

c = 1.0

c = 1.5

c = 2.0

c = 2.5

450 500 550 600 650 700nm

0.2

0.4

0.6

0.8

1

T

c = 0.5c = 1.0c = 1.5c = 2.0c = 2.5

Figure 3.7 Transmission and density spectra of a magenta ink at various concentra-tions c.

D λ( )

D λ( )xd

θcos------------ ε λ( ) c 2θsin⋅ ⋅ ⋅

θd0

π2---

∫2 xd⋅ ε λ( ) c⋅ ⋅( ) θsin θd

0

π2---

∫2 xd⋅ ε λ( ) c⋅ ⋅

=

=

=

φd 2ε λ( )cφ 10ln d⋅ x–=




3.5 Light scatteringThe term light scattering refers to all physical processes that move photonsapart in different directions. This phenomenon is often caused by local vari-ations of the refractive index within a heterogeneous medium. Other scat-tering processes, as for instance the Raman effect and the Brillouin scattering,also change the wavelength (i.e., the energy) of the incident photon, butthese phenomena are rare in nature.

In this chapter, we will be interested in scattering caused by small par-ticles that are dispersed in a homogeneous medium. Let us consider a thinslice of thickness of this scattering medium (see Figure 3.10). The varia-tion of the collimated light flux that crosses this slice is proportional tothe flux intensity of the light beam and to the thickness of the slice.

(3.24)

Yellow

450 500 550 600 650 700nm

0.5

1

1.5

2

2.5D

c = 0.5

c = 1.0

c = 1.5

c = 2.0

450 500 550 600 650 700nm

0.2

0.4

0.6

0.8

1

T

c = 0.5c = 1.0c = 1.5c = 2.0c = 2.5

Figure 3.8 Transmission and density spectra of a yellow ink at various concentra-tions c.

xdφd

φ xd

dφ β λ( )φdx–=




where = light wavelength

= scattering coefficient of the medium

By analogy with the absorption phenomenon (see Section 3.4), we introducethe molar decadic scattering extinction coefficient (or, in short, scattering coeffi-cient) (unit, ), which can be interpreted as the scatteringcross-section area of a mole of particles of radius r.

(3.25)

(3.26)

dxθcos

------------

θ

dx

Figure 3.9 The average absorption of an infinitely thin slice under diffuse illumina-tion is related to the average path of the light in the medium.

φ φ + dφ

dx

Figure 3.10 Light scattered by an infinitely thin layer containing light-scatteringparticles at a concentration c.

λβ λ( )

σ λ( ) m2 mol 1–⋅

β λ( ) σ λ( )c 10( )ln=

σ λ( ) N A

10( )ln-----------------πr2χsc λ( )=




where = concentration of the particles

NA = = Avogadro number

= scattering efficiency factor of the particle

Note that the total extinction coefficient of a particle that has botha scattering and an absorbing behavior corresponds to the sum of the scat-tering and absorbing cross-section areas (Equations 3.16 and 3.26).

(3.27)

Remark: Sections 3.5.1 and 3.5.2 can be skipped or browsed rapidly for a firstreading and revisited later as required.

3.5.1 Rayleigh scattering

The Rayleigh scattering theory applies to independent scattering of particlesthat are about ten times smaller than the wavelength of the incident light.In 1871, Lord Rayleigh established the following equation, which gives ,the intensity of the light scattered in a direction having an angle with thedirection of the incident light beam:

(3.28)

where = intensity of the incident collimated light beam

= its wavelength in empty space

= = permittivity of empty space (unit, )

= distance at which the intensity is measured

= polarizability of the medium

N = = number of particles per unit volume

The scattering is rotationally symmetrical about the incident light beam. Thedetailed calculation can be found in the literature.14

The polarizability of a medium of permittivity containingparticles of refractive index and of volume is given by15

(3.29)

The scattering coefficient is calculated by integrating Equation 3.28over a sphere of radius .

c

6.022 1023⋅χsc λ( )

εT λ( )

εT λ( ) ε λ( ) σ λ( )+ πr2χabs λ( ) πr2χsc λ( )+[ ]N A

10( )ln-----------------⋅= =

Iθθ

Iθπ2

L2ε02λ0

4----------------

Nα2 1 θ2cos+2

---------------------- I0=

I0

λ0

ε0 8.842 10 12–⋅ F m 1–⋅

L Iθ

αc N A⋅

α εm nm2 ε0=

n v

α 3εmn2 nm

2–

n2 2nm2+

---------------------v⋅=

β λ( )L 1=




(3.30)

Using Equation 3.26, we can also deduce the scattering efficiency factor.

(3.31)

where = wavelength in the medium of refractive index

= scattering efficiency factor of the particle

Note that is proportional to , which means that blue light ismore strongly scattered than red light. Sunlight scattered in the atmosphereis mostly blue, which explains why the sky is blue. At sunrise and at sunset,the light from the sun has to traverse a thicker atmospheric layer than atnoon, so most of the blue light is scattered, and the remaining unscatteredlight is mostly red.

3.5.2 Mie scattering

The Mie scattering theory16 is a generalization of the Rayleigh theory, whichpredicts the scattering behavior of a medium of refractive index contain-ing particles of radius and of refractive index . This theoryassumes the absence of multiple scattering. In practice, this means that thedistance between two particles is greater than . The Mie scattering is alsorotationally symmetrical about the incident light beam. The intensity ofthe light scattered in the direction making an angle with the direction ofthe incident light beam, as derived from Maxwell’s equations, is given by17

(3.32)

where = wavelength of the incident light in the medium of refractive

index

= number of particles per unit of volume

= distance at which the intensity is measured

The coefficients and are defined by the following series:

β λ( )1I0---- Iθ ωd∫°

8π3

3-------- 1

ε02λ0

4----------

Nα2 24π3

λm4

-----------n2 nm

2–

n2 2nm2+

---------------------

2

Nv2⋅= = =

χsc λ( )

χsc λ( )83---

n2 nm2–

n2 2nm2+

---------------------

22πrλm---------

4

⋅ ⋅=

λm nm

χsc λ( )

β λ( ) λm4–

nm

r n̂ n 1 iκ+( )=

3rIθ

θ

IθNL2-----

λm

2π------

2 S1

2 S22+

2------------------------------ I0=

λm

nm

N

L Iθ

S1 S2




(3.33)

The angular functions and are defined by recurrence as follows:

(3.34)

where the first functions are

(3.35)

The coefficients and are combinations of Bessel and Hankel func-tions. To simplify their mathematical expressions, let us introduce the fol-lowing variables:

(3.36)

Furthermore, let us define the functions and .

(3.37)

where = Bessel function of the first kind

= Bessel function of the second kind

= a Hankel function

The properties of the Legendre polynomial, of the Bessel functions, and ofthe Hankel function can be found in most handbooks of mathematics.18 UsingEquations 3.36 and 3.37, the coefficients and can be written as follows:

S12l 1+

l l 1+( )----------------- al πl θcos( ) bl τl θcos( )⋅+⋅[ ]

l 1=

∞

∑=

S22l 1+

l l 1+( )----------------- al τl θcos( ) bl πl θcos( )⋅+⋅[ ]

l 1=

∞

∑=

πl τl

πl θcos( )2l 1–l 1–

-------------- θcos πl 1– θcos( )⋅ ll 1–----------πl 2– θcos( )–=

τl θcos( ) l θcos πl θcos( ) l 1+( )πl 1– θcos( )–⋅=

π0 θcos( ) 0=

π1 θcos( ) 1=

τ1 θcos( ) θcos=

π2 θcos( ) 3 θcos=

τ2 θcos( ) 3 2θcos=

al bl

m n̂nm------ ,= γ 2πr

λm---------= , and δ mγ=

Ψl ξl

Ψl x( ) πx2

------ J⋅l 1

2---+

x( )=

ξl x( ) πx2

------ H⋅l 1

2---+

1( )

x( ) πx2

------ Jl 1

2---+

x( ) iYl 1

2---+

x( )+⋅= =

Jl 1

2---+

Yl 1

2---+

Hl 12---+

1( )

al bl




(3.38)

Note that and are complex if the particles attenuate electromagneticwaves ( ).

For small ( ) and nonattenuating particles ( ), the series and can be limited to the terms , , and b1.

(3.39)

Note that corresponds to the contribution of the Rayleigh scattering. Ifwe neglect all terms beyond in Equation 3.33, we obtain the Rayleighscattering Equation 3.28.

The scattering coefficient and the absorption coefficient areobtained by integrating Equation 3.32 over a sphere of radius . Thiscalculation is tedious, as the coefficients and are very complex. There-fore, the scattering efficiency factor and the absorption efficiencyfactor for many different values of and have been calculatedin the past, and they can be found in tables listed in the literature.19 Usingthe approximation for small and nonabsorbing particles (taking intoaccount the terms , , and ), the scattering efficiency factor ofthe particle is

(3.40)

Note that the first term in this series corresponds to the efficiency factor ofthe Rayleigh scattering that we have already seen in Equation 3.31. For largerparticles, further terms in the series and must be taken into account.Nowadays, computers allow us to calculate all coefficients numerically.

It is found that scattering is proportional to in the Rayleigh region( ) and that, with further increase of , it tends to become proportionalto , i.e., wavelength independent. Therefore, the light scattered by largeparticles (e.g., smoke particles) is white. Furthermore, the forward scatteringbecomes greater than the backward scattering with increasing . Figure 3.11shows the scattering diagram of spherical gold particles for different radii.The Mie theory successfully predicts the spectra of colloidal suspensions,metallic suspensions, and atmospheric dust.

alΨl γ( )Ψl ' δ( ) mΨl δ( )Ψl ' γ( )–ξl γ( )Ψl ' δ( ) mΨl δ( )ξl ' γ( )–--------------------------------------------------------------=

blmΨl γ( )Ψl ' δ( ) Ψl δ( )Ψl ' γ( )–mξl γ( )Ψl ' δ( ) Ψl δ( )ξl ' γ( )–--------------------------------------------------------------=

S1 S2

κ 0≠γ 0.8< κ 0= S1

S2 a1 a2

a123--- m2 1–

m2 2+----------------

γ 3= , a21–

15------ m2 1–

2m2 3+-------------------

γ 5= , b1145------ m2 1–( )γ 5–=

a1

a1

σ λ( ) ε λ( )L 1=

al bl

χsc λ( )χabs λ( ) m γ

a1 a2 b1 χsc λ( )

χsc λ( )83---γ 4 m2 1–

m2 2+----------------

2

1 65--- m2 1–

m2 2+----------------

γ 2 …+ +=

S1 S2

λm4–

γ 1« γλm

0

γ




3.5.3 Multiple scattering

In the framework of the Rayleigh and the Mie theory, we assumed that thescattering particles are independent; i.e., the light scattered by one particledoes not interact with other particles. With diminishing distance betweenthe particles or increasing thickness of the medium, this assumption nolonger holds, and single scattering gives way to multiple scattering. It canbe shown that, for a sufficient number of particles, regardless of the scatteringlaw used, an isotropic distribution ultimately arises.20 With multiple scatter-ing, the characteristic properties of single scattering disappear more or lessrapidly according to the given conditions.

-0.1 -0.05 0.05 0.1

-0.06

-0.04

-0.02

0.02

0.04

0.06

r=40 nm

-2 -1 1 2

-1

-0.5

0.5

1

r=80 nm

-2 2 4 6 8 10

-4

-3

-2

-1

1

2

3

4r=120 nm

Figure 3.11 Scattering diagram according to Mie for spherical gold particles( , , ) with radii (a) ( Ray-leigh region), (b) ( ), and (c) ( ). The unit of theaxes is .

λm 550 nm= nm 1.33= n̂ 0.57 i2.45+= r 40 nm= γ 1«r 80 nm= γ 1≈ r 120 nm= γ 1>

λm 2π( )⁄( )2

(a)

(b)

(c)




3.6 Phenomenological modelsThere is no general quantitative solution to the problem of multiple scatter-ing for large particles ( ) that are tightly packed. In such cases,phase relations and interferences arise among the scattered beams. Therefore,new approaches based on phenomenological theories had to be developed.Throughout Section 3.6, the wavelength designation is dropped to sim-plify the notation, but , , , , , , , , , and are functions ofwavelength.

3.6.1 Radiative transfer

The astronomer S. Chandrasekhar established in 1947 the radiative transferequation,21 which corresponds to an energy balance. This equation describesthe intensity change of a light beam of given wavelength along a pathof length within a medium of density and total extinction coefficient

.

(3.41)

where corresponds to the source function characterizing a lightsource. In the particular case of a nonluminescent medium, is a scatteringfunction defined as

(3.42)

The function is called the phase function. It gives the amount ofintensity that is scattered into a solid angle of direction if a beamof radiation in the solid angle strikes a mass element ofthe medium (see Figure 3.12). Note that the phase function is normalized asfollows:

(3.43)

where = fraction of light lost from an incident beam due to scattering

Term is the albedo of the medium. The simplest example of phase func-tion is in the case of isotropic scattering. Another case ofinterest is Rayleigh’s phase function, which corresponds to the angular termof Equation 3.28.

2πr λm⁄ 1≥

λ( )I ρ εT i j K S ρg Rg R

Idds ρ

εT

sddI 10( )ln εTρI– ρj+=

j 10( )ln εT( )⁄j

j θ ϕ,( )10( )ln εT

4π---------------------- p θ ϕ θ' ϕ',;,( )I θ' ϕ',( ) θ' θ'dsin ϕ'd

0

2π

∫0

π

∫=

p θ ϕ θ' ϕ',;,( )ωd θ ϕ,( )

dω' θ' θ'dsin ϕ'd=

14π------ p θ ϕ θ' ϕ',;,( ) θ θdsin ϕd∫∫ ϖ0 1≤=

ϖ0

ϖ0

p θ ϕ θ' ϕ',;,( ) 1=




(3.44)

where is the angle between the direction of the incident radiation in and the direction of the solid angle . In general, the phase function canbe expanded as a series in Legendre polynomials of the form

(3.45)

where = the Legendre polynomial of degree

= a constant

Combining Equations 3.41 and 3.42 leads to an integro-differential equa-tion that is difficult to solve. Nevertheless, solutions could be calculated ina few particular cases, such as for isotropic scattering in a medium made ofparallel planes.21

The radiative transfer equation is very powerful, but it requires a tediousmathematical treatment. Therefore, simplified versions of this theory areused in practice, e.g., the Kubelka–Munk model and the multichannel model,which are presented in the following sections.

3.6.2 Kubelka–Munk model (two-flux model)

Let us consider a reflector made of a reflecting substrate of reflectance inoptical contact with a light-absorbing and light-scattering medium of thick-

θ’dω’

θ

ϕϕ’

dω

Idωdφ=

x

y

z

Figure 3.12 The phase function gives the amount of intensity that is scattered intoa solid angle of direction if a beam of radiation in the solid angle

strikes a mass element of the medium.dω θ ϕ,( )

dω' θ' θ'dsin ϕ'd=

p Θcos( )32--- 1 Θ2cos+

2-------------------------

=

Θ dω'ωd

p Θcos( ) ϖlPl Θcos( )l 0=

∞

∑=

Pl l

ϖl

ρg




ness (see Figure 3.13). The scattering is assumed to have an isotropicdistribution, as it results from multiple scattering (see Section 3.5.3). In 1931,Kubelka and Munk22 proposed a reflection model based on two diffuse lightfluxes: oriented downward and oriented upward.

Let us analyze the variation of these fluxes when they cross a layer ofinfinitesimal thickness dx. The axis is oriented upward, and the origin isset at the top of the substrate. Let be the phenomenological absorptioncoefficient corresponding to the fraction of the light flux absorbed by theinfinitesimal layer. Let be the phenomenological scattering coefficientcorresponding to the fraction of the light flux that is scattered backward bythe infinitesimal layer.

We first analyze the variation of when it crosses the layer. The flux is reduced due to absorption within the infinitesimal layer by an amount

, and the backscattering further reduces the flux by an amount. However, the flux is increased by the light that is backscattered

when the flux crosses the same layer: dx. Putting these elementstogether leads to the following equation:

(3.46)

The same analysis performed for the flux leads to a similar relation(notice the orientations along the vertical -axis).

(3.47)

Note that, in a transparent medium, equals 0, and differential Equations3.46 and 3.47 lead to Beer’s law for diffuse light (Equation 3.23).

X

i(x)

j(x)

dx

X

Substrate of reflectance ρg

Figure 3.13 Light-absorbing and light-scattering medium of thickness X that is inoptical contact with a substrate of reflectance ρg. The medium is divided into parallellayers of infinitesimal thickness dx. Note that two fluxes are considered: i(x), whichis oriented downward, and j(x), which is oriented upward.

i x( ) j x( )

x K

S

j x( )j x( )K j x( )dxS j x( )dx j x( )

i x( ) Si x( )

d j x( )dx

------------ K S+( ) j x( )– Si x( )+=

i x( )x

di x( )dx

------------ K S+( )i x( ) S j x( )–=

S




Equations 3.46 and 3.47 together form a system of linear differentialequations that describes the variation of and when they cross aninfinitesimal layer of thickness dx.

(3.48)

Kubelka and Munk solved Equation 3.48 using a traditional calculationmethod.22 Here, we propose a more modern approach based on matrixalgebra. The system in Equation 3.48 can be written in matrix form as follows:

(3.49)

This kind of matrix differential equation has a well-known solution, whichis given by the exponential of the matrix.23 By integrating the equationbetween and , we obtain

(3.50)

where , , , = elements of the matrix exponential

, = intensities of the fluxes and j at x = 0

Note that the exponential of a matrix is defined by the following powerseries:

(3.51)

From Equation 3.50 and the boundary condition , we canderive by algebraic manipulations24,25 all the well-known results of theKubelka–Munk theory that are listed in the literature.26 The most importantresult is the hyperbolic solution of the Kubelka–Munk model,

i x( ) j x( )

di x( )dx

------------ K S+( )i x( ) S j x( )–=

d j x( )dx

------------ K S+( ) j x( )– Si x( )+=

di x( )dx

------------

d j x( )dx

------------

K S+ S–

S K S+( )–

i x( )

j x( )⋅=

x 0= x X=

i X( )

j X( )

K S+ S–

S K S+( )–X 0–( )

exp i 0( )

j 0( )⋅

t uv w

i 0( )

j 0( )⋅

=

=

t u v w

i 0( ) j 0( ) i

M

M( )exp M( )l

l!-----------

l 0=

∞

∑=

j 0( ) ρg i 0( )⋅=




(3.52)

where and . The ratio is called thebody (or true) reflectance27 of the analyzed sample. To characterize the mediumalone, practitioners use the reflectance of a medium thatis so thick that further increase in thickness fails to change its reflectance. Inother words, if an additional layer of thickness is put on top of such amedium, we have . According to Equation 3.50,we have

(3.53)

From a mathematical point of view, this means that the vector isan eigenvector28 of the matrix given in Equation 3.50, and that

is the corresponding eigenvalue. This observation per-mits us to obtain by calculating the eigenvectors of this matrix. Bysolving the characteristic polynomial of the matrix,29 it can be shown thatthis matrix has two eigenvalues,

(3.54)

that are associated with the following eigenvectors, respectively:

(3.55)

Being a reflectance value, must be in the range between 0 and 1. Butbecause the second component of is outside the range , the solutiongiven by must be discarded, and is simply the second component ofthe eigenvector .

ρ j X( )i X( )----------

v ρg w⋅+t ρg u⋅+-----------------------

1 ρg a b bSX( )coth⋅–( )⋅–a ρg b bSX( )coth⋅+–

-------------------------------------------------------------------= = =

a S K+( ) S⁄= b a2 1–= ρ j X( ) i X( )⁄=

ρ∞ j X∞( ) i X∞( )⁄=

Xρ∞ j X X∞+( ) i X X∞+( )⁄=

i X X∞+( )

j X X∞+( )

t uv w

i X∞( )

j X∞( )⋅=

i X X∞+( ) 1ρ∞

t uv w

i X∞( ) 1ρ∞

⋅=

1 ρ∞,[ ]

α i X X∞+( ) i X∞( )⁄=ρ∞

α1w t+( ) w t–( )2 4uv+–

2--------------------------------------------------------------- K2 2KS+–= =

α2w t+( ) w t–( )2 4uv++

2--------------------------------------------------------------- K2 2KS+= =

V1

1

w t–( ) w t–( )2 4uv+–2u

---------------------------------------------------------------= V2

1

w t–( ) w t–( )2 4uv++2u

---------------------------------------------------------------=

ρ∞V1 0 1,[ ]

V1 ρ∞V2




(3.56)

This result is often presented in a more compact form known as theKubelka–Munk function.

(3.57)

Other important results are the reflectance of a layer with ideal blackbackground ( ), and the reflectance of a layer with ideal whitebackground ( ).

(3.58)

(3.59)

Traditionally, , , and the thickness of the medium are used todetermine the coefficients and S. In a first step, we extract from Equa-tions 3.58 and 3.59.

(3.60)

Once is known, is obtained immediately. In a second step, weextract from Equation 3.58.

(3.61)

Finally, from the definition of ,

(3.62)

Note that the phenomenological coefficients and can be related tothe fundamental optical properties introduced previously. In Section 3.4, wegeneralized Beer’s law for diffuse light (see Equation 3.23), which is equiv-alent to . Therefore, we have (see Equation 3.16)

(3.63)

ρ∞w t–( ) w t–( )2 4uv++

2u--------------------------------------------------------------- 1 K

S---- K2

S2------ 2K

S----+–+= =

KS----

1 ρ∞–( )2

2ρ∞----------------------=

ρ0

ρg 0= ρ1

ρg 1=

ρ0vt--- 1

a b bSX( )coth+----------------------------------------= =

ρ1v w+t u+------------- 1 a b bSX( )coth–( )–

a b bSX( )coth+------------------------------------------------------= =

ρ0 ρ1 XK a

a 12--- 1

ρ1 1–ρ0

--------------– =

a b a2 1–=S

S 1bX-------

1 aρ0–bρ0

-----------------acoth=

a

K S a 1–( )=

K S

dφ Kφdx–=

K 2 10ln ε⋅ λ( ) c⋅ 2N A c π⋅ r⋅ 2 χabs λ( )⋅= =




A similar calculation allows us to relate to the scattering coefficient .Because the scattering in the medium is assumed to have an isotropic dis-tribution, the scattering coefficient must be divided by two, because accounts only for backward-scattered light. Hence, we obtain from Equations3.25 and 3.26,

(3.64)

3.6.3 Surface phenomena and Saunderson correction

In the Kubelka–Munk theory, the diffuse reflector is modeled by a light-absorbing and light-scattering medium in optical contact with a substratethat is supposed to be a Lambertian30 reflector of reflectance . In a mediumhaving a refractive index different from that of air, surface reflection andmultiple internal reflections occur31 as shown in Figure 3.14. As a conse-quence, the reflectances prevailing in a medium of refractive index candiffer greatly from the reflectances measured at its surface. Traditionally, thisis taken into account by applying the Saunderson correction32 to the com-puted spectrum. In this section, we write the Saunderson correction in matrixform, to be applied to Equation 3.50.

Let us denote by the incident flux on the external surface of the paperand by the flux emerging from the paper. Let be the fraction of diffuselight reflected by the air-medium interface (external surface of the reflector),and let be the fraction of diffuse light reflected internally by theair–medium interface (internal surface of the medium); see Figure 3.15.According to Equations 3.13 and 3.14, the values of and depend onlyon the refractive index of the medium. Judd11 has computed their numer-ical values for a large number of refractive indices (see Table 3.1).

The balance of the fluxes at the air–medium interface, as shown inFigure 3.15, leads to the following system of equations for i(X), the incidentflux below the air–medium interface, and for j, the emerging flux above theair–medium interface:

S σ λ( )

S

S 2 10ln σ λ( )2

----------⋅ c⋅ N A c πr2 χsc λ( )⋅ ⋅ ⋅= =

ρg

n

Air

Substrate: Diffuse reflector

Medium ofrefractive index n

Figure 3.14 Surface reflection and multiple internal reflections caused by the inter-face between the air and the medium.

n

ij rs

ri

rs ri

n




(3.65)

Assuming that the refractive index of the medium is constant over thewhole visible range of wavelengths, and are also constant. Hence,Equation 3.65 can be written in the following matrix form:

(3.66)

We call the matrix in Equation 3.66 the Saunderson correction matrix. Notethat this correction matrix can be generalized for any interface between amedium of refractive index and a medium of refractive index . Accord-ing to Section 3.3.2, the values of and are then given by

and (3.67)

The Saunderson correction is obtained by combining Equations 3.66 and 3.50.

(3.68)

1 rs–( )i

j X( )

i X( )

1 ri–( ) j X( )i

Interface

ri j X( )

rsiAir

Medium ofrefractiveindex n

j

Figure 3.15 External and internal reflections of the upward and downward fluxeson the air–medium interface.

i X( ) 1 rs–( )i ri j X( )+=

j rs i 1 ri–( ) j X( )+=

rs ri

i j

11 rs–-------------

ri–1 rs–-------------

rs

1 rs–------------- 1 ri

rsri

1 rs–-------------––

i X( )

j X( )=

n1 n2

rs ri

rs rn1 n2, θ( ) 2θsin⋅( ) θd0

π2---

∫= ri rn2 n1, θ( ) 2θsin⋅( ) θd0

π2---

∫=

ij

11 rs–-------------

ri–1 rs–-------------

rs

1 rs–------------- 1 ri

rsri

1 rs–-------------––

K S+ S–

S K S+( )–X

exp i 0( )

j 0( )⋅ ⋅ t' u'

v' w'

i 0( )

j 0( )⋅= =




We denote the elements of the product matrix by , , , and . Thesecoefficients and the boundary condition allow the calculationof the reflectance R.

(3.69)

This equation allows us to compute the reflectance under diffuse lightillumination of a light-absorbing and light-scattering medium in opticalcontact with a substrate of known reflectance . If we develop the productin Equation 3.68 algebraically, we obtain the famous Saunderson correctedreflection formula.33

(3.70)

where ρ = = the body reflectance given by the Kubelka–Munk model

Figure 3.16 shows the reflectance and the body reflectance of a cyan sample.In the graphic arts, most measuring instruments use a 45°/0° measuring

geometry wherein the incident light beam is collimated with an incidenceof 45°, and the detector is placed at an angle of 0° (see Figure 3.17). This set-up prevents the light reflected specularly from entering the detector, hence

. Furthermore, in the particular case of a nonscattering medium( ), the refracted light beam, with the normal to the surface, forms anangle of , where is the refractive index of the medium.The entering collimated light beam follows within the medium a path oflength , which is shorter than the average path of length followed by diffuse light (see Section 3.4). Because the detector is at an angleof 0°, only the light emerging with an angle of 0° is detected. This emerginglight beam follows in the medium a path of length , which is also shorter

t' u' v' w'j 0( ) ρg i 0( )⋅=

R ji--

v' ρg w'⋅+t' ρg u'⋅+-------------------------= =

ρg

R rs1 rs–( ) 1 ri–( )ρ

1 riρ–---------------------------------------+=

j X( ) i X( )⁄

R

1

0.4

0.2

0.6

0.8

450 500 550 600 650 700 nm

Figure 3.16 Body reflection spectrum (dashed line) and reflection spectrum (contin-uous line) of a cyan sample.

rs 0=S 0=

α 1 n 2( )⁄[ ]asin= n

X αcos( )⁄ 2X

X




than the average path length of followed by diffuse light. Because thetotal path length of the light beam within the medium is shorter, the absorp-tion of the light beam within the medium is not the same as for diffuse light.Therefore, the Saunderson correction matrix in Equation 3.66 must be mod-ified as follows so as to take the 45°/0° geometry into account:34

(3.71)

Considering the particular case of a medium with refractive index n = 1.5,we have . The modified Saunderson correctionmatrix leads, after developing Equation 3.69, to the Williams–Clapper equa-tion,35

(3.72)

Note that is the reflectance of the substrate within the medium ofrefractive index n. The surface phenomena do not allow us to measure directly. Let be the reflectance of the substrate measured in air withoutthe medium on top of it. If the substrate has the same refractive index asthe medium, is deduced from by deriving the following formula fromEquation 3.70:

(3.73)

In practice, the substrate is not always available without the coatingmedium on top of it. This happens, for example, in the case of the high-

45˚

Collimated Detector

light source

Air

Medium

Diffusereflector

X

Figure 3.17 Path followed in the medium by the collimated light beam producedby a measuring instrument having a 45˚/0˚ measuring geometry.

2X

ij

12 αcos---------------- 1–

KX⋅exp ri1

2 αcos---------------- 1–

KX⋅exp–

0 1 ri–( ) KX2

--------exp

i X( )

j X( )=

1 2 αcos( )⁄ 1–( ) 0.44–=

R1 ri–( )ρg 1.06K– X[ ]exp

1 riρg 2– KX[ ]exp–--------------------------------------------------------------=

ρg

ρg

Rg

nρg Rg

ρg1

ri1 rs–( ) 1 ri–( )

Rg rs–-----------------------------------+

---------------------------------------------=




quality paper used in graphic arts, where the fiber substrate is coated withan ink-absorbing layer of refractive index n. Assuming that the coating istransparent, we can deduce from Equation 3.73 by replacing with themeasured reflectance of the paper.

Note that the matrix formulation of Equation 3.68 gives a better overviewof the modeled system. Instead of using several functions nested within eachother, the analyzed sample is simply modeled by the product of two matrices.

3.6.4 Multichannel model

The two-flux model proposed by Kubelka and Munk corresponds to a sig-nificant simplification of the radiative transfer equation (see Section 3.6.1).To improve the quality of the prediction, Mudget and Richards in 1971proposed an intermediate model by considering a larger number of lightfluxes.36 Each flux propagates in a different fraction of space called a channel(see Figure 3.18). Therefore, this theory is called the multichannel model or themultiple flux theory. In this context, the radiative transfer equation corre-sponds to a model that considers an infinite number of fluxes.

The multichannel model considers fluxes; fluxes denoted areoriented upward, and fluxes denoted are oriented downward. Let usdenote as the absorption coefficient in the th channel, and as thescattering coefficient from the pth channel into the lth channel. The scatteringcoefficients are computed by using a scattering model, as for instancethe Mie model presented in Section 3.5.2.

As in the Kubelka–Munk model, we analyze the variation of each fluxwhen it is traversing an infinitesimal layer of thickness . This gives us

linear differential equations of the first order with variables. Theseequations can be written in matrix form as follows:

ρg Rg

θ2

θ1

Figure 3.18 In the multichannel model the whole space is subdivided into 2m chan-nels where the lth channel corresponds to the space between the cone of angle and the cone of angle .

θl 1–

θl

2m m jl

m il

Kp p Sp l,

Sp l,

dx2m 2m




(3.74)

The sign inversion in the first rows occurs because the axis isoriented upward. The pth element on the diagonal of the matrix correspondsto the attenuation of the pth light flux. This attenuation is caused by the lightabsorption in the pth channel and by the scattering of light from the pthchannel into all other channels,

The off-diagonal element corresponds to the light received by the flux from the channel p.

The solution of Equation 3.74 is also given by the exponential of thematrix. As in the Kubelka–Munk model, the boundary conditions at thesurface of the substrate define the relations between the fluxes and thefluxes . An extended Saunderson correction matrix that allows us to

xdd

i1

:im

jm 1+

:j2m

K1 S1 l,l 1≠∑+

… S– m 1, Sm 1 1,+– … S2m 1,–

: : : :

S– 1 m, … Km Sm l,l m≠∑+

Sm 1 m,+– … S2m m,–

S1 m 1+, … Sm m 1+, Km 1+ Sm 1+ l,l m 1+≠∑+

– … S2m m 1+,

: : : :

S1 2m, … Sm 2m, Sm 1 2m,+ … K2m S2m l,l 2m≠∑+

–

i1

:im

jm 1+

:j2m

⋅

=

m x

Kp

Sp l,l p≠∑

Sp l,l

jl 0( )ip 0( )




predict the reflectance with a higher accuracy can be also defined. Thecomplete treatment is beyond the scope of this chapter, but the mathematicalprocedure is the same as presented in the Sections 3.6.2 and 3.6.3. Note thatpractitioners in the paint industry normally use the four-flux theory.37

3.7 The fluorescence phenomenonLet us first recall the basic principles of molecular fluorescence.38 We considera theoretical molecule having two electronic energy states, (ground state)and (excited state). Each electronic state has several vibrational states(see Figure 3.19). Incident polychromatic light (photons) excites the mole-cules that are in state and makes them temporarily populate the excitedvibrational states of (Figure 3.19a).

A vibrational excited state has an average lifetime of only s. Themolecule rapidly loses its vibrational energy and goes down to the electronicenergy state . This relaxation process is nonradiative, and it is caused bythe collisions with other molecules to which the vibrational energy is trans-ferred. This induces a slight increase of the temperature of the medium. Theexcited state has a lifetime varying between and s. Now, thereare two ways for the molecule to give up its excess energy. One of them iscalled internal conversion, a nonradiative relaxation for which the mechanismis not fully understood. The transition occurs between and the uppervibrational state of (Figure 3.19b), and the lost energy raises the temper-ature of the medium. The other possible relaxation process is fluorescence.It takes place by emitting a photon of energy corresponding to the transitionbetween and a vibrational state of (Figure 3.19c). The remaining excessenergy with respect to is lost by vibrational relaxation. To quantify thenumber of photons emitted by fluorescence, the quantum yield is introducedas the rate of absorbed photons that are released by radiative relaxation.

E0

E1

0123

0123

E0

E1

0123

0123

E0

E1 0

123

0123

E0

E1

(a) Absorption (b) Nonradiativerelaxation

(c) Fluorescence

ResonanceLine

Figure 3.19 The energy level diagram of (a) absorption, (b) nonradiative relaxation,and (c) fluorescent emission.

E0

E1

10 15–

E1

E1 10 6– 10 9–

E1

E0

E1 E0

E0




The wavelength band of absorbed radiation that is responsible for theexcitation of the molecules is called the excitation spectrum. This spectrumconsists of lines whose wavelengths correspond to the energy differencesbetween excited vibrational states of and the ground electronic state (according to the energy difference produced by the absorption of aphoton of wavelength : , where is Planck’s constant and

is the speed of light). The fluorescence emission spectrum (or fluorescencespectrum), on its part, consists of lines that correspond to the energy differ-ences between the electronic level and the vibrational states of . Themultitude of lines in both spectra is difficult to resolve and makes them looklike continuous spectra. Note that the fluorescence spectrum is made up oflines of lower energy than the absorption spectrum. This wavelength shiftbetween the absorption band and the fluorescence band is called the Stokesshift. A particular case in which the absorbed photon has the same energyas the one re-emitted by fluorescence is called the resonance line.

The shape of the fluorescence emission spectrum does not depend onthe spectrum of the absorbed light, but on the probability of the transitionbetween the excited state and the vibrational states of . Often, thefluorescence spectrum looks like a mirror image of the excitation spectrum(Figure 3.25);39 this is due to the fact that the differences between vibrationalstates are about the same in ground and excited states.

Experience shows that fluorescence is favored in rigid molecules thatcontain aromatic rings.40 This can easily be understood, as a rigid moleculehas a lower possibility of relaxing by a nonradiative process. In fact, thelower the probability of nonradiative relaxation, the higher the quantumyield. Hence, a rise in the medium’s viscosity induces a higher fluorescence.In the particular case of inks, the liquid substance fluoresces less than thedried-up printed ink, whose molecules have less degrees of freedom. On theother hand, a rise of the ambient temperature implies a higher probabilityof nonradiative relaxation due to collisions with other molecules, and a dropin fluorescence is observed.

The fluorescence spectrum is measured with a fluorescence spectrome-ter.41 A sample of the unknown fluorescent substance is excited with a mono-chromatic light beam whose wavelength is within the excitation band of themolecule. The emitted light is analyzed, and the resulting spectrum is thefluorescence spectrum. Its amplitude is maximal when the wavelength ofthe incident light corresponds to the maximum absorption of the fluorescentmolecule. We denote by the normalized fluorescence spectrum whoseintegral equals 1. A method for determining the quantum yield is describedin Section 3.7.4.

At high concentrations, the behavior of the fluorescent substance is nolonger linear. The absorption is too large, and no light can pass through tocause excitation. Temperature, dissolved oxygen, and impurities reduce thequantum yield; therefore, they also reduce the fluorescence. This phenome-non is called quenching. In our model, we will suppose that no quenchingoccurs.

E1 E0

E∆λ E∆ hc( ) λ⁄= h

c

E1 E0

E1 E0

f λ( )




3.7.1 Fluorescence: transparent layer

To establish a mathematical formula that predicts the behavior of a trans-parent medium containing fluorescent molecules, we consider a slice ofthickness . We denote by the absorption coefficient of the fluorescentmolecules, by their concentration, and by their quantum yield in thismedium. In the model for transparent media,42 only the positive directionof propagation is taken into account (see Figure 3.20).

The intensity variation of the light emerging in the positive directionhas two components. The first, , is due to the light that has beenabsorbed. As we have already seen in Equation 3.23, for diffuse light, thisabsorption is twice43 the value given by Beer’s law.

(3.75)

The second component, , is the light emitted by fluorescence.The fluorescent molecules emit a fraction of the photons absorbed in theexcitation band and spread them over the whole emission band definedby the normalized fluorescence spectrum . Due to the fact that fluores-cent emission is made in all directions of space, only one-half of the photonsgo into the positive direction. Hence, the quantum yield must be divided bytwo. The second component is therefore given by

(3.76)

The integral between square brackets multiplied by equals theamount of absorbed energy. Equation 3.76 leads to the following differentialform, which is an extension of Beer’s law for diffuse light and fluorescentmedia:

dx ε λ( )c Q

φ φ + φ d

dx

Positive direction

Figure 3.20 Absorption and emission in an infinitely thin fluorescent layer whichis irradiated by a diffuse light flux φ.

dφ dφ1 λ( )

dφ1 λ x,( ) 2 10ln cε λ( )φ λ x,( )dx–=

dφ2 λ x,( )Q

∆f λ( )

dφ2 λ x,( )

dφ2 λ x,( ) 2 10ln cQ2---- f λ( ) ε µ( )φ µ x,( ) µd

∆∫ dx⋅=

dx




(3.77)

This can be simplified due to the fact that we work with a finite number ofwavelength bands whose widths are , so the integral is replaced by afinite sum. The new relation is given in Equation 3.78, where the index runs through the wavelength bands.

(3.78)

Writing Equation 3.78 for each of the bands leads to a system of lineardifferential equations with constant coefficients that can be put into matrixform. If we denote , we obtain Equation 3.79.

(3.79)

dφ λ x,( ) 2 10ln cε λ( )φ λ x,( )dx–

2 10ln cQ2---- f λ( ) ε µ( )φ µ x,( ) µd

∆∫ dx⋅+

=

λ∆i

dφ λi q,( ) 2 10ln c ε λ( )– φ λi x,( )dx

2 10ln c Q2---- f λi( ) ε λ j( )φ λ j x,( ) λ∆

j ∆∈∑⋅+ dx

=

Fi j, ε λ j( ) f λi( ) λQ 2⁄∆=

dφ λ1 x,( )dx

-----------------------

.

.

.dφ λi x,( )

dx----------------------

.

.

.dφ λn x,( )

dx-----------------------

2c 10ln–

ε λ1( ) 0 . . . . . . 0F– 2 1, . .. . . .. . . .

F– i 1, . F– i j, . ε λi( ) . .. . . . .. . . . .. . . 0

F– n 1, . F– n j, . . . . F– n n 1–, ε λn( )

φ λ1 x,( )...

φ λ j x,( )...

φ λn x,( )

⋅=




The fact that the emitted photon has less energy than the absorbed oneimplies that for ; hence the matrix is triangular.

The solution of equations such as Equation 3.79 has already been inves-tigated by mathematicians.23 Systems of differential equations whose generalexpression is (where is the constant square matrix ofEquation 3.79 and is the column vector containing , . . . , )admit as a solution, when is integrated between and ,

(3.80)

The vector is the spectrum of the incident light (light source), and is the spectrum of the light emerging from a slice of thickness of

the fluorescent medium. The exponential of the matrix is defined asfollows:

(3.81)

where

(3.82)

We will call the fluorescence density matrix. The transmission spec-trum resulting from the combined action of fluorescence and absorp-tion can be computed for each wavelength using the expression =

, where and are, respectively, components of and . Note that an accurate prediction requires measuring ,

which can be significantly different from a standard illuminant (seeFigure 3.21).

The solution given by Equation 3.80 is a generalization of Beer’s law;for a purely absorbing substance when no fluorescence is present, the matrix

consists of the terms on the diagonal and of zeros anywhereelse. This simplification of Equation 3.80 leads to Equation 3.23, the absorp-tion equation for diffuse light.43

Fi j, 0= λ j λi≥

dΦ dx⁄ c– M Φ⋅= MΦ φ λ1 x,( ) φ λn x,( )

x 0 X

Φ X( ) M– cX( )exp Φ 0( )⋅=

Φ 0( )Φ X( ) X

M– cX

M– cX( )exp M– cX( )i

i!----------------------

i 0=

∞

∑=

M 2 10ln

ε λ1( ) 0 . . . . . . 0F– 2 1, . .. . . .. . . .

F– i 1, . F– i j, . ε λi( ) . .. . . . .. . . . .. . . 0

F– n 1, . F– n j, . . . . Fn n 1–,– ε λn( )

=

MT λ( )

λ T λ( )φ λ X,( ) φ λ 0,( )⁄ φ λ X,( ) φ λ 0,( )Φ X( ) Φ 0( ) Φ 0( )

M 2 10ln ε λi( )⋅




This approach can be extended to cases involving two or more fluores-cent substances but, for this end, we must distinguish between several pos-sible cases. Let and be two different substances whose fluorescencedensity matrices are and and whose respective concentrations are

and . Hence,

• If light goes first through a layer of thickness of substance andthen through a layer of thickness of substance , we have,

(3.83)

• If light goes first through a layer of thickness of substance andthen through a layer of thickness of substance , we have,

(3.84)

• If light goes through a layer of thickness consisting of a mixtureof the substances and , we have,

(3.85)

If either substance or is fluorescent, the matrices and do notnecessarily commute, so the resulting transmittance spectrum may be dif-ferent in each of these three cases. As an example, let us consider the caseconsisting of a yellow filter and of a fluorescent yellow filter that absorbsblue light between 400 and 500 nm and emits green light between 500 and

450 500 550 600 650 700nm0

0.25

0.5

0.75

1

1.25

1.5

1.75

Figure 3.21 Relative radiance spectrum of the tungsten light source of a spectro-photometer. It was measured by mounting the radiometer at the position of thesample holder. Note that this spectrum is significantly different from the standardilluminant A.

A BMA MB

cA cB

XA AXB B

Φ A B,( ) MB– cBXB( )exp MA– cAXA( )exp Φ 0( )⋅ ⋅=

XB BXA A

Φ B A,( ) MA– cAXA( )exp MB– cBXB( )exp Φ⋅ 0( )⋅=

XA B

Φ A B,( ) X MAcA MB+ cB( )–( )exp Φ 0( )⋅=

A B MA MB




600 nm (see Figure 3.22). If white light goes first through the yellow filterand then through the fluorescent yellow filter (see Figure 3.22A), the bluelight is absorbed by the yellow filter, and it cannot cause fluorescence in thefluorescent yellow filter. But, if white light goes first through the fluorescentyellow filter and then through the yellow filter (see Figure 3.22B), green lightis produced by fluorescence in the first filter. This green light is not absorbedby the second filter, hence the resulting spectrum is different from that ofthe first case.

3.7.2 From a one-flux to a two-flux model for a reflective substrate

In the present fluorescence model,44 we consider a fluorescent medium inoptical contact with a substrate that is a diffuse reflector (see Figure 3.23).This reflector is supposed to be Lambertian.30 Like in the Kubelka–Munkanalysis, we consider an upward flux and a downward flux goingthrough an infinitely thin layer of the medium, which contains a fluorescentsubstance at concentration (see Figure 3.24). The positive direction of thevariable is oriented upward, and its origin is at the bottom of the fluores-cent medium. To simplify the equations used in this section, let us introducethe column vector , whose components are the intensities of the upwardflux at various wavelength, and the column vector i, whose components arethe intensities of the downward flux.

1

nm500 600400 700

1

nm500 600400 700

White light source Yellow filter Fluorescent yellow filter

1

nm500 600400 700

1

nm500 600400 700

1

nm500 600400 700

White light source Yellow filterFluorescent yellow filter

1

nm500 600400 700

Spectrumof the light

Spectrumof the light

FluorescenceFluorescence

(B)

(A)

Figure 3.22 The noncommutativity of a yellow filter and a fluorescent yellow filter.

j i

cx

j




,

We also write the fluorescence density matrix of the ink as the differencebetween a diagonal matrix representing the absorption and a strictlylower triangular matrix representing the fluorescent emission.

Fluorescent

Substrate:

Air

Diffuse light fluxes

i j

X

0

Interface

Diffuse Reflector

Medium

Figure 3.23 Model of a fluorescent reflector made of a fluorescent medium in opticalcontact with a diffuse reflector. This model describes well a high-quality paper madeof an ink-absorbing layer in optical contact with a diffuse white reflector. The arrowsrepresent diffuse light fluxes (light is coming from all directions of one hemispherewith an angular distribution corresponding to that of a Lambert surface).

i

i + di

dx

j

j + djx

Figure 3.24 Absorption and emission in an infinitely thin layer of the transparentmedium containing a fluorescent substance.

j

j λ1( )...

j λn( )

= i

i λ1( )...

i λn( )

=

MA

F

M = A – F

2 10ε λ1( ) 0

˙ ⋅ 0 ε λn( )

0 0

˙ ⋅ Fi j, 0

–

ln=




(3.86)

Considering first the vector of upward flux j, its variation has two components. The first one is the absorption and fluorescent emissioncaused by the upward flux which is, according to the results of Section 3.7.1,

. The second component is the fluorescence caused by the down-ward flux i, which is emitted in the upward direction, . Hence, thechange of j is

(3.87)

The same reasoning is applied to the downward flux and leads to asimilar equation. Note that the downward orientation of i introduces achange of sign. By combining the equations obtained for j and i, we get asystem of differential equations whose matrix form is

(3.88)

Equation 3.88 is a linear differential equation of the first order withconstant coefficients. When is integrated between and , it admits asolution23 that is given by the matrix exponential

(3.89)

where and are, respectively, the spectra of the downward and ofthe upward flux at vertical location . The matrix exponential is defined asfollows:

(3.90)

2 10ε λ1( ) 0

˙ ⋅ F– i j, ε λn( )

ln=

d dx⁄( )j x( )

c– Mj x( )cFi x( )

1c---

xdd j x( )⋅ Fi x( ) M– j x( )=

1c---

xdd i x( )

j x( )⋅ M F–

F M–

i x( )

j x( )⋅=

x 0 X

i X( )

j X( )

M F–

F M–cX

exp i 0( )

j 0( )⋅=

i X( ) j X( )X

M F–

F M–cX

exp

M F–

F M–cX

i

i!------------------------------------

i 0=

∞

∑=




At the bottom of the fluorescent medium, the spectrum of the upwardflux is linked with the spectrum of the downward flux by therelation

(3.91)

where is the reflection matrix of the substrate.45 For pure reflectors, thismatrix is diagonal, and the coefficients on the diagonal are the body reflec-tances of the different wavelength bands. If the reflecting substrate containsfluorescent substances (as, for instance, optical brighteners), the matrix is triangular. Note that commercial bispectral spectrofluorimeters can beused to measure the matrix (see Section 3.7.4).

3.7.3 Spectral prediction for reflective fluorescent material

The reflective fluorescent material made of a diffusely reflecting substratewith a fluorescent coating is modeled by means of three matrices: the Saun-derson correction matrix (Equation 3.66); the matrix exponential (Equation3.89), which models the fluorescent medium; and the reflection matrix of the substrate (Equation 3.91).

By multiplying the Saunderson correction matrix with the matrix expo-nential, we obtain the following relation:

(3.92)

where T, U, V, W = matricesI = the identity matrix

Thanks to Equation 3.91, it is possible to express the vector as a functionof .

(3.93)

Because the multiplication of matrices is not commutative, the order ofthe terms in Equation 3.93 must be respected. This corresponds to the factthat superposed fluorescent layers do not commute, as we have already seenin Section 3.7.1.

If the fluorescent material is illuminated by a diffuse light source of spec-trum , the spectrum of the diffuse reflected light is . The reflectance spec-trum is computed by dividing the components of by the components of .

j 0( ) i 0( )

j 0( ) Rg i 0( )⋅=

Rg

Rg

Rg

Rg

ij

11 rs–-------------I

ri–1 rs–-------------I

rs

1 rs–-------------I 1 ri–

rirs

1 rs–-------------–

I

M F–

F M–cX

exp i 0( )

j 0( )⋅ ⋅ T U

V Wi 0( )

j 0( )⋅= =

ji

j V W Rg⋅+( ) T U Rg⋅+( ) 1– i⋅ ⋅=

i jj i




(3.94)

Figure 3.25 shows the absorption spectrum, the fluorescence spectrum,and the reflectance spectrum of a fluorescent yellow ink printed on paper.

As in Section 7.1, this approach can be extended to cases involving twoor more fluorescent substances. Once again, we must distinguish amongseveral possible cases. Suppose and are two different substances whose

R λ( ) j λ( )i λ( )---------=

450 500 550 600 650 700nm

0.20.40.60.8

11.21.4

R Reflection Spectrum

450 500 550 600 650 700nm

0.5

1

1.5

2

2.5fx100 Fluorescence Spectrum

450 500 550 600 650 700nm

0.1

0.2

0.3

0.4

0.5

0.6

D Absorption Spectrum

Figure 3.25 Absorption spectrum , normalized fluorescence spec-trum , and reflection spectrum of a fluorescent yellow ink printed on paper.In this particular case, the excitation spectrum and the absorption spectrum areidentical. The quantum yield of the yellow ink is Q = 0.7. The paper consists of atransparent coating of refractive index n = 1.5 in optical contact with a diffuselyreflecting substrate without optical brighteners. The spectrum of the light source ofthe measuring instrument is given in Figure 3.21. The measured reflection spectrum(continuous line) is well predicted by the model (dotted line). The dashed line showsthe prediction result when only absorption is taken into account.

D λ( ) 2 c ε λ( )=f λ( ) R λ( )

A B




fluorescence density matrices are and , andwhose respective concentrations are and . As a result,

• If a layer of thickness of substance is on top of a layer ofthickness of substance B, we have,

(3.95)

• If a layer of thickness of substance is on top of a layer ofthickness of substance A, we have,

(3.96)

• If light goes through a layer of thickness consisting of a mixtureof the substances and , we have,

(3.97)

If one of the substances or is fluorescent, the matrices and do not necessarily commute, so the resulting reflectance spectrum may bedifferent in each of these three cases. As an example, let us consider thecase consisting of a yellow filter and of a fluorescent yellow filter thatabsorbs blue light between 400 and 500 nm and emits green light between500 and 600 nm (see Figure 3.26). If the yellow filter is superposed on topof the fluorescent yellow filter (see Figure 3.26A), the blue light is absorbedby the yellow filter, and it cannot cause fluorescence in the resulting spec-trum. But if the fluorescent yellow filter is superposed on top of the yellowfilter (see Figure 3.26B), green light is produced by fluorescence in theresulting spectrum.

3.7.4 Measuring the parameters of the fluorescence model

To compute the fluorescence density matrix , four elements have to bedetermined: the excitation spectrum, the absorption coefficient , thenormalized fluorescence function , and the quantum yield . (Notethat contains discrete values of the functions and .)

Because the dye concentration is unknown, it is impossible to deter-mine the absorption coefficient . However, according to Equation 3.19,

MA AA FA–= MB AB FB–=cA cB

XA AXB

i X( )

j X( )

MA FA–

FA MA–cAXA

exp MB FB–

FB MB–cBXB

exp i 0( )

j 0( )⋅ ⋅=

XB BXA

i X( )

j X( )

MB FB–

FB MB–cBXB

exp MA FA–

FA MA–cAXA

exp i 0( )

j 0( )⋅ ⋅=

XA B

i X( )

j X( )

MA FA–

FA MA–cA

MB FB–

FB MB–cB+

X

exp i 0( )

j 0( )⋅=

A B MA MB

Mε λ( )

f λ( ) QM ε λ( ) f λ( )

cε λ( )




the density spectrum and the absorption coefficient are propor-tional and the proportionality factor is the dye surface density . Notethat each non-zero element of the fluorescence density matrix contains afactor (see Equation 3.79). Because, in Equation 3.80, is multipliedby q, each occurrence of is multiplied by q, and this product equals thedensity . Hence, for a given sample of density , we do not needthe actual values of and of , and we can work relatively to the densityspectrum of a reference sample so that

(3.98)

where is the proportionality factor between and .The excitation spectrum is determined in a two-step procedure. To avoid

deviations due to self-absorption, the fluorescence measurement must beperformed on a sample whose maximal density is smaller than 0.1 over thewhole spectrum ( ). This means that light emitted by fluorescenceis not reabsorbed by another molecule of the sample. At first, the wholedensity spectrum of our sample is measured with a spectrophotom-eter. This instrument uses a monochromatic collimated light beam that goes

1

nm500 600400 700

White light source

Yellow filterFluorescent yellow filter

1

nm500 600400 700

Fluorescence

Diffuse reflector

1

nm500 600400 700

1

nm500 600400 700

White light source

Fluorescent yellow filterYellow filter

Diffuse reflector

(A)

(B)

Figure 3.26 The noncommutativity of a superposition of a yellow filter and a fluo-rescent yellow filter covering a diffuse reflector.

D λ( ) ε λ( )q cX=M

ε λi( ) Mε λi( )

D λi( ) D λi( )q ε λi( )

D' λi( )

D λi( ) qε λ( ) q'D' λi( )= =

q' D' λi( ) D λi( )

D λ( ) 0.1≤

D λ( )




through the transparent sample before reaching a light detector. Becauseonly a small fraction of the fluoresced light passes through the entrance slitof the detector, the deviation induced by the fluorescent emission can beneglected.

In the second step, the location of the excitation spectrum within thedensity spectrum is determined. This can be done once we have an a prioriknowledge of the approximate position of the fluorescence spectrum (forexample, by a preliminary measurement using a fluorescence spectrome-ter).41 This device has two monochromators; the first one is used to generatea monochromatic light beam that excites the sample, and the second mono-chromator is used to analyze the light emitted by the sample. In our presentmeasurement, the second monochromator is set to a fixed wavelength thatis supposed to be within the fluorescence spectrum (the a priori knowledge).The first monochromator sweeps the whole spectrum, and the intensity ofthe emitted light is recorded. This provides the excitation spectrum and itslocation.

To determine the normalized fluorescence function, a fluorescence spec-trometer is needed. The sample is excited with a monochromatic light beamwhose wavelength corresponds to the maximum absorption in the excitationspectrum. Because the shape of the fluorescence emission spectrum does notdepend on the excitation wavelength (see Section 3.7), the normalized fluo-rescence function is easy to compute by dividing the measured fluorescencespectrum by its integral value, which is proportional to the number of flu-oresced photons.

Once the excitation spectrum and the fluorescence function have beenmeasured, the quantum yield is determined using a method described inthe literature.46 This method is based on a measurement made relatively toa standard fluorescent substance of known quantum yield. To be reliable,the location of the excitation spectrum and the location of the fluorescencespectrum of the standard substance must correspond to those of our sample.Based on these criteria, the standard substance is chosen from tables givenin the literature.47

The quantum yield of the unknown substance is given by46

(3.99)

In this equation, the subscript u stands for unknown and the subscript s forstandard. is the absorption at the excitation wavelength, and is thequantum yield. The refractive indices of the solvent of the standard fluo-rescent substance ( ) and of the medium of the unknown fluorescentsubstance ( ) are also taken into account. The variable is proportionalto the total number of photons emitted by fluorescence. This value is com-puted by integrating the spectrum emitted by fluorescence during theexperiment.

QuAsFun2

AuFsn02

----------------- Qs⋅=

A Q

n0

n F




Within the excitation spectrum, we must select a single wavelength thatgives the highest possible fluorescence in both the standard substance andour sample. These two substances are excited using the same fluorescencespectrometer at the selected wavelength, and the spectrum of the fluorescedlight is measured. By integrating the fluorescence spectra of the standardsubstance and of our unknown sample, we get the respective number ofphotons, and , emitted by fluorescence. Because the excitation spectraof both substances are known, we have the respective absorption factors and at the selected excitation wavelength. Finally, the quantum yield of our sample is calculated using Equation 3.99. Note that three values mustbe found in the literature: the quantum yield of the standard substance,46

the refraction index of the medium containing it, and the refraction index of our sample’s medium.

This experimental determination of the quantum yield is rather difficultto perform. Therefore, it is often preferred to estimate the quantum yield byusing a best-fit method applied on a test sample. This is an iterative process.First we give a start value, then we compute the reflectance spectrum ofthe test sample and compare the result with the measured spectrum. If thefluorescence is underestimated, we increase Q; otherwise, we decrease it.The computation is then redone with the new value of Q. This iterativeprocess stops when the square of the difference between the computed andthe measured spectra is minimal. Note that this estimation method reducesthe number of experiments to be performed, but it no longer guarantees thatthe real physical quantum yield is used.

Because the refractive index of the fluorescent medium is known fromthe literature,48 the internal and external reflection and can be computedusing Judd’s method (see Equations 3.13 and 3.14 in Section 3.3.2). Thereflection matrix of the substrate is measured using the two monochro-mator method described by Donaldson,45 using barium sulfate ( ) asthe white reference. Note that this measurement must be performed on anidentical sample without the fluorescent coating. Sometimes, only the coatedsubstrate without fluorescent substances in the coating is available, e.g., inthe case of coated paper. In this particular case, the measured reflectionmatrix corresponds to the matrix product.

(3.100)

This relation can be solved for the matrix as follows:

(3.101)

Note that the multiplication of matrices is not commutative, so the order ofthe terms in Equation 3.101 must be respected.

Fs Fu

As

Au Qu

Qs

n0

n

Q

nri rs

Rg

Ba SO4

R

Rrs

1 rs–-------------I 1 ri–

rirs

1 rs–-------------–

Rg⋅+ 1

1 rs–-------------I

ri

1 rs–------------- Rg⋅–

1–

⋅=

Rg

Rg 1 ri– rs–( )I ri 1 rs–( )R+[ ] 1– R rsI–( )⋅=




3.8 Models for halftoned samplesMost printing devices are only bilevel, meaning that they are capable ofprinting ink only at a certain fixed density or leaving the substrate unprinted,but they cannot produce intermediate ink densities. In such devices, thevisual impression of intermediate tone levels is usually obtained by meansof the halftoning technique, i.e., by breaking the original continuous-toneimage into small dots whose area coverage varies depending on the tonelevel. Halftoning is also used for most color printing devices, where each ofthe inks [usually cyan (C), magenta (M), yellow (Y), and often black (K)] isonly bilevel. This gives to the eye, when looking from a sufficient distance,an illusion of a full range of intermediate color levels, although the printingdevice is only bilevel. In this section, we focus our discussion on predictingthe reflectance of halftoned samples, where dyes (or pigments) are no longeruniformly distributed over the entire surface.

3.8.1 The Murray–Davis equation

Let us consider a surface of unit area, and let be the reflectance spec-trum of a solid sample, i.e., a sample whose surface is fully covered with anink layer of constant density. The reflectance spectrum of the bare substrateis denoted . The total reflectance spectrum of a halftoned samplehaving a fraction of area covered with ink ( ) is given by thefollowing weighted sum (see Figure 3.27):

Rs λ( )

Rg λ( ) R λ( )a 0 a 1≤ ≤

0.2 0.4 0.6 0.8 1a

0.2

0.4

0.6

0.8

1R

Figure 3.27 Reflectance of halftoned samples having a fraction a ofarea covered with black ink (continuous line). The Murray–Davis model assumes alinear behavior (dotted line), whereas the Clapper–Yule equation predicts a nonlinearbehavior caused by the light scattering in the substrate (dashed line).

R λ = 550 nm( )




(3.102)

This relation is often written in a different way using the reflection densityspectrum . In this case, Equation 3.102 is called the Mur-ray–Davis equation,49

(3.103)

This equation is sometimes given in an alternative form that is useful forconverting densities into area coverage.

(3.104)

3.8.2 The classical Neugebauer theory

In 1937, Neugebauer proposed a method for predicting the spectra of half-toned color prints produced by the superposition of cyan, magenta, andyellow dot-screens.50 In traditional printing, a dot-screen is a regular latticeof dots that are ordered in parallel rows along two perpendicular axes. Thedots have variable sizes so as to produce the correct halftone levels. Further-more, the cyan, magenta, and yellow screens are mutually rotated by 30° or60° to avoid moiré patterns.51 Neugebauer observed, under the microscope,that such a halftone print was in fact a mosaic of eight colors, which corre-spond to the possible overlaps of the cyan, magenta, and yellow inks:white (= no ink), cyan, magenta, yellow, red, green, blue, and black (see Table3.2). These colors are called Neugebauer primaries. Neugebauer based hismodel on the assumption that the dots in the different screens are almostindependent of each other. This assumption, attributed to Demichel,52 is,however, only approximately true in traditional color printing.53

To explain Neugebauer’s method, let c, m, and be the fractions of areacovered by the cyan ink, the magenta ink, and the yellow ink, respectively.From a statistical point of view, c, m, and can also be interpreted as theprobabilities for a given point to be covered by one of the three inks. Hence,the probability for a given point to be white, i.e., not covered by any ink,equals (no cyan ink) times (no magenta ink) times (noyellow ink). By a similar reasoning, we deduce the fraction of area occupiedby the eight Neugebauer primaries as shown in Table 3.2. The reflectancespectrum of the halftoned color print is then given by the following Neuge-bauer equation:

(3.105)

R λ( ) 1 a–( )Rg λ( ) aRs λ( )+=

D λ( ) R10 λ( )log–=

D λ( ) 1 a–( )10Dg λ( )–

a10Ds λ( )–

+[ ]10log–=

a 1 10D– λ( ) Dg λ( )–

–

1 10Ds λ( )– Dg λ( )–

–-----------------------------------=

23

y

y

1 c– 1 m– 1 y–

R λ( ) ajRj λ( )j 1=

8

∑=




This is a simple extension of Equation 3.102. Note that Equation 3.105 is apolynomial of degree three for the dot area triplet .

The classical Neugebauer equation leads to color prediction errors ofabout in CIELAB. Several attempts have been made to improvethe Neugebauer model.54 One of the most important improvements is thecellular Neugebauer method proposed in 1992 by Heuberger et al.55 TheCMY color space is subdivided into rectangular cells. The reflectance spectra

of the samples corresponding to the corners of the cells are measured.By means of Equation 3.105, the new reflectance spectra ( ) ofeight equivalent primaries according to Table 3.2 are computed for each cell.The reflectance spectrum of a new given color defined by the dot areatriplet is now computed in a two-step process. First, we find thecell of the CMY color space to which it belongs; then, we assign the dot areatriplet into the traditional Neugebauer equation with the equiva-lent reflectances that we have computed for the cell.56 Note that thiscorresponds to a polynomial interpolation of degree three within each cell.Using this improved model, the average prediction error drops to in CIELAB when the color space is subdivided into cells; however,this requires measuring samples. The main drawback of this cel-lular method lies, indeed, in the large number of samples that must bemeasured.

3.8.3 Extended Neugebauer theory

In some printing processes, the number of inks is greater than three, andeach ink may have density levels ( ). The Neugebauer theory can beeasily generalized to such cases by considering each of the possible ink

Table 3.2 Fraction of Area Occupied by the Eight Primaries of the Neugebauer Model

Primary Ink Combination Reflectance Fraction of Area

White —

Cyan Cyan

Magenta Magenta

Yellow Yellow

Red Magenta, yellow

Green Cyan, yellow

Blue Cyan, magenta

Black Cyan, magenta, yellow

R1 λ( ) a1 1 c–( ) 1 m–( ) 1 y–( )=

R2 λ( ) a2 c 1 m–( ) 1 y–( )=

R3 λ( ) a3 1 c–( )m 1 y–( )=

R4 λ( ) a4 1 c–( ) 1 m–( )y=

R5 λ( ) a5 1 c–( )my=

R6 λ( ) a6 c 1 m–( )y=

R7 λ( ) a7 cm 1 y–( )=

R8 λ( ) a8 cmy=

c m y, ,( )

E∆ 10=

R λ( )Rj˜ λ( ) 1 j 8≤ ≤

R' λ( )c' m' y', ,( )

c' m' y', ,( )Rj˜ λ( )

E∆ 3=43 64=

53 125=

km m 2≥

mk




superpositions as a Neugebauer primary. The generalized Neugebauer equa-tion thus obtained is:

with (3.106)

where = reflectance of the Neugebauer primary

= fraction of area it occupies

Note that, if the superposed layers are not independent of each other, theparameters cannot be calculated as in the classical Neugebauer model.

3.8.4 The Yule–Nielsen equation

Yule and Nielsen pointed out that light does not emerge from the substrateat the point where it entered. This is a consequence of the light scattering inthe substrate. Therefore, a photon that penetrates the substrate in an areawithout ink may emerge in an inked area, and vice versa. As a consequenceof this exchange of photons, the fraction of area obtained from Equation3.104 (the Murray–Davis equation) is greater than the real area covered byink. This phenomenon is called optical dot gain or the Yule–Nielsen effect.

To improve the prediction of the reflection density of a halftonedprint, in 1951, Yule and Nielsen suggested the following correction to Equa-tion 3.103:

(3.107)

where = reflectance density of the substrate

= fraction of area covered by the ink whose solid reflectance density is

= an empirical correction factor called the Yule–Nielsen factor

Factor n must be determined experimentally and depends on the opticalproperties of the substrate. In the literature, Equation 3.107 is called theYule–Nielsen equation.57,58 Note that, in the particular case of , Equation3.107 gives the Murray–Davis equation (Equation 3.103). The generalizationof Equation 3.107 for Neugebauer primaries was suggested by Vig-giano.59 In the literature, this generalization is called the n-modified Neuge-bauer equation, and it can be written as follows:

R λ( ) ajRj λ( )j 1=

mk

∑= ajj

∑ 1=

Rj λ( ) j

a j

a j

a

D λ( )

D λ( ) n 1 a–( )10Dg λ( )

n-------------–

a10Ds λ( )

n------------–

+10log–=

Dg λ( )

a

Ds λ( )

n

n 1=

mk




(3.108)

3.8.5 The Clapper–Yule equation

After formulating the Yule–Nielsen equation, Yule worked with Clapper todevelop an accurate model for halftone prints, based on a theoretical analysistaking into account surface-reflection, multiple-scattering, internal-reflec-tion, and ink transmission.60 In this model, light is reflected many timesinternally by the air–medium interface and by the substrate. A fraction oflight emerges at each reflection cycle, and the total reflectance is the sum ofall those fractions.

Let us denote as the surface reflection, as the internal reflection, as the body reflectance of the substrate, and as the transmittance

spectrum of the ink under diffuse light. A light beam that strikes the surfaceof a halftone print is partially reflected and partially transmitted into themedium (see Figure 3.28). The reflected fraction is given by the surfacereflection . The transmitted fraction has two components. The first com-ponent enters the medium through the unprinted area; thesecond component enters the medium through the ink oftransmittance . Therefore, the irradiance in the substrate resulting fromthe entering light is . The light is assumed to betotally scattered within the substrate of body reflectance . The emerginglight emerging is again attenuated by a factor as a result ofthe ink pattern, and by a factor because of the internal reflection.The first emergence of light is given by . The

R λ( )[ ]1n---

aj Rj λ( )[ ]1n---

j 1=

mk

∑=

rs ri

ρg T λ( )

Substrate

Air

Ink

a

rs

1 rs–( ) 1 a–( )

1 rs–( )aT λ( )

T 0 ρgWaT2 λ( )ri

T 0 ρgW 1 a–( )ri

T 0 ρgW 1 ri–( )aT λ( )

T 0 ρgW 1 ri–( ) 1 a–( )

T 0

W ri 1 a– aT2 λ( )+( )ρg[ ]

n 1–=

Figure 3.28 In the Clapper–Yule model, fractions of light emerge at each reflectioncycle.

rs

1 rs–( ) 1 a–( )1 rs–( )aT λ( )

T λ( )T0 1 rs–( ) 1 a– aT λ( )+( )=

ρg

1 a– aT λ( )+( )1 r– i( )

1 rs–( ) 1 ri–( ) 1 a– aT λ( )+[ ]2ρg




internally reflected light suffers one further change. The fraction thatattempted to emerge through an inked area must pass through the ink asecond time, so its intensity must be multiplied by . The light that re-enters the substrate after the internal reflection is then given by

. This sequence of events contin-ues until the remaining light is negligible. The emerging fractions of lightare as follows (see Figure 3.28):

• Surface reflection,

• First emergence,

• Second emergence,

• Third emergence,

• . . . th emergence,

The sum of this geometrical series is the reflectance of the halftone print (seeFigure 3.27). This leads to the following Clapper–Yule equation, which waspublished in 1953:60

(3.109)

3.8.6 Advanced models

The Yule–Nielsen effect has a large impact on the color produced by halftoneprints. Intensive investigations have been made so as to relate the empiricalparameter of the Yule–Nielsen equation to physical quantities. The result-ing theories model the light scattering in the substrate by a point spreadfunction (PSF) , which expresses the density of probability for a pho-

T λ( )

1 rs–( ) 1 a–( ) aT λ( )+[ ]ρg ri 1 a– aT2 λ( )+( )[ ]

rs

1 rs–( ) 1 ri–( ) 1 a– aT λ( )+[ ]2ρg

1 rs–( ) 1 ri–( ) 1 a– aT λ( )+[ ]2ρg ri 1 a– aT2 λ( )+( )ρg[ ]

1 rs–( ) 1 ri–( ) 1 a– aT λ( )+[ ]2ρg ri 1 a– aT2 λ( )+( )ρg[ ]2

n

1 rs–( ) 1 ri–( ) 1 a– aT λ( )+[ ]2ρg ri 1 a– aT2 λ( )+( )ρg[ ]n 1–

R λ( ) rsρg 1 rs–( ) 1 ri–( ) 1 a– aT λ( )+( )2

1 ρgri 1 a– aT2 λ( )+( )–---------------------------------------------------------------------------------+=

n

P x y,( )




ton entering the substrate at location to emerge at the location .The light reflected at location of a halftone print is then given by:

(3.110)

where = convolution operator

= body reflectance of the substrate

= surface reflection

= transmittance at location

If there is ink in this location, then ; otherwise, .The wavelength designation is dropped to simplify the notation, but

, , and are functions of wavelength. The reflectance of the whole halftone print is the spatial average of . Note that

the multiple internal reflections are accounted by the PSF .In 1978, Ruckdeschel and Hauser derived the empirical Yule–Nielsen

factor from the PSF and the period of the halftone screen.61 They assumeda Gaussian PSF,

(3.111)

where is a characteristic scattering length of the photon in the substrate. According to their calculations, the Yule–Nielsen factor is given by the

following relation:

(3.112)

where = period of the screen

Note that the value of approaches 1 as the substrate approaches a specularsurface ( , Murray–Davis model; see Section 3.8.1) and approaches 2as the substrate becomes a perfect diffuser ( , Clapper–Yule model,see Section 3.8.5). In 1997, Rogers showed that the characteristic scatteringlength is related to two physical parameters:62 the absorption in the sub-strate and the optical thickness of the substrate. If there is no absorption inthe substrate, increases without bound as the optical thickness of thesubstrate tends to infinity. If absorption occurs, however, the scatteringlength reaches a limit.

0 0,( ) x y,( )x y,( )

R x y,( ) rs 1 rs–( )T+ x y,( )P x y,( )*T x y,( )ρg

rs 1 rs–( )T+ x y,( )ρg P x x'– y y'–,( )T x' y',( ) x'd y'd∫∫=

=

*

ρg

rs

T x y,( ) x y,( )

T x y,( ) T λ( )= T x y,( ) 1=λ( )

P x y,( ) T x y,( ) R x y,( )R λ( ) R x y,( )

P x y,( )

n

P x y,( )1

πσ2--------- x2 y2+

σ2----------------

–exp=

σ

n 2 πσL---–

exp–≈

L

nσ 0=

σ ∞=

σ

σ

σ




Further investigations made by Rogers showed that the PSF is a seriesof convolutions whose terms are the contributions of the multiple internalreflections occurring in the substrate.63

(3.113)

where = internal reflection

= internal point spread function (internal PSF), which does not take multiple internal reflections into account

Note that, for substrates having a low internal reflection , we have. On a macroscopic scale, the PSF shown in Equation

3.113 induces Yule–Nielsen factors that are greater than 2. SuchYule–Nielsen factors are often found in practice, and they are not explainedby the simple PSF given by Equation 3.111.

The internal PSF derives from the radiative transfer equation (see Section3.6.1), but its analytical form, which is a series in MacDonald functions (alsocalled modified Bessel functions), is cumbersome.62 According to Gustav-son’s studies,64 the internal PSF is closely approximated by a function that has a circular symmetry ( ) and a strong radial decay.

(3.114)

where controls the radial extent of the internal PSF. In practice, iscomputed from the light profile measured across an optically sharp edgebetween an inked and a non-inked area.65,66

In 1997, Arney proposed a probabilistic approach that is less complexthan the PSF convolution.67,68 He introduced the scattering probability for a photon that enters the substrate through a region covered by theNeugebauer primary of transmittance , to emerge through a regioncovered by the Neugebauer primary of transmittance . The reflectance

of a halftone print is then given by

(3.115)

where = reflectance of the substrate

= fraction of area covered by the Neugebauer primary

P x y,( ) 1 Γi+( )[ p x y,( ) p x y,( )+( )* T2 x y,( )p x y,( )( ) ρgri( )=

+ p x y,( )* T2 x y,( )p x y,( )( )* T2 x y,( )p x y,( )( ) ρgri( )2 …]+

ri

p x y,( )

ri

P x y,( ) p x y,( )≈ P x y,( )n

p r( )r x2 y2+=

p r( )1

2πdr------------ r

d---–exp=

d d

δi j,

j T j

i Ti

R λ( )

R λ( ) Rg aiTi ajT jδi j,j

∑

i∑=

Rg

ai i




In the particular case of traditional halftone screens, Arney showed that thescattering probabilities are given by the following empirical relations:

(3.116)

where = an empirical parameter.Arney suggested that is related to the characteristic scattering distance

and the period of the screen by , where is anexperimentally determined proportionality coefficient. Note that the scatter-ing probabilities can also be computed from the PSF convolution.

3.8.7 The Monte-Carlo method

The reflectance of a medium that is inhomogeneous or anisotropic (e.g.,biological tissue) can be computed by a Monte-Carlo simulation.

69

Themedium is subdivided into volume elements called

voxels

. Each voxel isassociated with an absorption coefficient and a scattering phase function (seeSection 3.6.1).

The computer casts a virtual ray of unit intensity on the voxels. Whenthis ray enters a new voxel, its new intensity is computed from the absorptioncoefficient of the voxel, and a random number is generated to decide, accord-ing to the scattering phase function, in which direction the ray should bescattered. The process is iterated until the ray leaves the voxels or until theintensity of the ray drops below some predefined threshold. The reflectanceof the medium is deduced from the results of a large number of simulations.

3.9 New mathematical framework for color prediction of halftones

The Kubelka–Munk model presented in the Section 3.6.2 assumes that thecoating medium is uniform, i.e., that the same amount of dye is everywhere.In halftoned prints, this is no longer true, because ink is not applied uni-formly over the whole surface. A photon can penetrate the printed mediathrough an inked region and leave the printed media through a non-inkedregion, or vice versa (see the Yule–Nielsen effect in Section 3.8.4).

In this section, we generalize the models presented in Sections 3.6, 3.7,and 3.8, and we incorporate them into a new mathematical framework basedon matrices.

70

For the sake of simplicity, we consider only two Neugebauerprimaries: inked and non-inked. In case of colored samples, more primariesmust be considered. Furthermore, because the ink layer is very thin (less than10 µm), we assume that the exchange of photons between inked and non-

di j,

d j j, 1 1 aj�( ) 1 1 aj�( )w� 1 ajw�( )+[ ]�=

di j, 1 d j j,�( ) ai

1 aj�-------------Ë ¯

Ê �=ÓÔÌÔÏ

ww

s L w 1 a s L§( )�[ ]exp�= a

di j,

03 Page 228 Monday, November 18, 2002 8:16 AM



inked areas takes place only in the substrate. We also assume that the inklayer behaves according to the Kubelka–Munk model described previously.

Let us now consider such a surface having only two different inkinglevels. As in the Kubelka–Munk model, we define for each inking level twolight fluxes: , which is oriented downward, and , which is orientedupward. The index takes the value 0 for the non-inked region and 1 forthe inked region (see Figure 3.29). Note that we drop the wavelength desig-nation to simplify the notation, but , , as well as , , , and ,are all functions of wavelength.

The matrix Equation 3.49 can be extended to take several inking levelsinto account. Let us denote as this extended block matrix. For twoinking levels, the equation can be written as follows:

(3.117)

where , , , and are, respectively, the absorption and scatteringcoefficients of the non-inked medium and the inked medium. By integratingEquation 3.117 between and , we get

(3.118)

The definition of the matrix exponential is given in Equation 3.51.

ik jk

k

Substrate:

Interface

Infinitely

j0

i0

diffusereflector

thin layer

Air

j1

i1

Figure 3.29 A schematic model of the printed surface. On top of the substrate, eachsurface element is considered to be a uniform layer which behaves according to theKubelka–Munk model.

λ( ) T ρg ik jk Sk Kk

MKM

xdd

i0 x( )

j0 x( )

i1 x( )

j1 x( )

MKM

i0 x( )

j0 x( )

i1 x( )

j1 x( )

⋅

K0 S0+ S0– 0 0S0 K0 S0+( )– 0 00 0 K1 S1+ S1–

0 0 S1 K1 S1+( )–

i0 x( )

j0 x( )

i1 x( )

j1 x( )

⋅= =

K0 S0 K1 S1

x 0= x X=

i0 X( )

j0 X( )

i1 X( )

j1 X( )

MKM X⋅( )exp

i0 0( )

j0 0( )

i1 0( )

j1 0( )

⋅=




To take into consideration the multiple internal reflections, the Saunder-son correction must also be applied here. Note that, in our case, the ink isinside the medium and not on top of it. Hence, the interface between the airand the ink-absorbing medium is the same in non-inked regions as in inkedregions. Therefore, from Equation 3.66, we can directly derive the resultingSaunderson correction matrix .

(3.119)

The key to our model lies in the way optical dot gain is expressedmathematically. Because we assume that the exchange of photons takes placeonly in the substrate, the optical dot gain affects only the boundary condi-tions at . This implies that the upward-oriented fluxes and depend on both downward-oriented fluxes , and the body reflec-tance of the substrate. This can be written in a general way in matrixform as follows:

(3.120)

where the coefficient represents the overall probability of a photonentering through a surface element having the inking level to emerge froma surface element having the inking level u. Note that the probability is takenthroughout the full sample area. This probabilistic approach was introducedby Arney (see Section 3.8.6). Because we deal with probabilities, the sum ofthe coefficients belonging to the same line of the matrix in Equation3.120 must equal 1. The computation of the scattering probabilities willbe addressed in Section 3.9.2.

Now we can put all elements together and write the matrix equation ofour new prediction model. By combining Equations 3.118 through 3.120 weobtain

(3.121)

MSC

i0

j0

i1

j1

MSC

i0 X( )

j0 X( )

i1 X( )

j1 X( )

⋅

11 rs–-------------

ri–1 rs–------------- 0 0

rs

1 rs–------------- 1 ri

rsri

1 rs–-------------––

0 0

0 0 11 rs–-------------

ri–1 rs–-------------

0 0rs

1 rs–------------- 1 ri

rsri

1 rs–-------------––

i0 X( )

j0 X( )

i1 X( )

j1 X( )

⋅= =

x 0= j0 0( ) j1 0( )i0 0( ) i1 0( )

ρg

j0 0( )

j1 0( )ρg

δ0 0, δ0 1,

δ1 0, δ1 1,

i0 0( )

i1 0( )⋅ ⋅=

δu v,v

δu v,δu v,

i0

j0

i1

j1

MSC MKM X⋅( )exp

1 0 0 00 δ0 0, 0 δ0 1,

0 0 1 00 δ1 0, 0 δ1 1,

i0 0( )

ρgi0 0( )

i1 0( )

ρgi1 0( )

⋅ ⋅ ⋅=




The first matrix of Equation 3.121 represents the Saunderson correction,the second matrix corresponds to the Kubelka–Munk modeling of the ink-absorbing layer, and the third matrix models the light scattering in thesubstrate.

Computing the emerging fluxes and as functions of the incidentfluxes and requires rearranging the lines and columns of the matrices.To keep the block structure of the matrices, we introduce a change of basismatrix as shown in Equation 3.122. Note that this particular change of basismatrix is its own inverse. Furthermore, the last vector of Equation 3.121 iswritten in Equation 3.122 as the product of a matrix by a two-dimen-sional vector.

(3.122)

After computing the matrix products in Equation 3.122, we get a matrix that can be split into two matrices. The first matrix relates thevector to , and the second matrix relates to

. By multiplying the second matrix by the inverse of the firstmatrix, we derive a relation that expresses the emerging fluxes and aslinear functions of the incident fluxes and .

Because the incident light has the same intensity on inked and non-inkedregions, we have . Let be the inked fraction of area and

be the non-inked fraction of area. As in the Neugebauer model(see Equation 3.106), the reflectance spectrum of the whole surface isgiven by the weighted sum of the emerging light divided by the incidentlight, where the weights are the area coverages of the various primaries.Hence, the final result is given by

(3.123)

j0 j1

i0 i1

4 2×

i0

i1

j0

j1

1 0 0 00 0 1 0

0 1 0 00 0 0 1

MSC MKM X⋅( )exp⋅ ⋅=

1 0 0 00 0 1 0

0 1 0 00 0 0 1

1–1 0 0 00 1 0 00 0 δ0 0, δ0 1,

0 0 δ1 0, δ1 1,

1 00 1ρg 00 ρg

i0 0( )

i1 0( )⋅ ⋅ ⋅ ⋅

4 2×2 2×

i0 i1[ , ] i0 0( ) i1 0( )[ , ] j0 j1[ , ]i0 0( ) i1 0( )[ , ]

j0 j1

i0 i1

i0 i1 i= = a1

a0 1 a1–=R λ( )

R λ( )

a0 a1

j0

j1

⋅

a0 a1

i0

i1

⋅

------------------------------1 a1–( ) j0 a1 j1+

i--------------------------------------= =




3.9.1 Some particular cases of interest

Let us consider the particular case in which the average lateral light scatter-ing distance is large compared to the size of the halftoning element. This isthe assumption of complete scattering. In this case, for any inking level ,the probability equals the fraction of area occupied by the inkinglevel u,

and (3.124)

By introducing the relations of Equation 3.124 in Equation 3.122 andassuming that , , , we obtain from Equation 3.123 thewell-known Clapper–Yule relation (see Equation 3.109),

(3.125)

where . Note that this derivation requires the help of amathematics software package.

In another particular case, lateral light scattering can be neglected.Hence, the probability of a photon being scattered in a region with a differentinking level equals 0. This implies that and for . Inother words, the second to last matrix of Equation 3.122 is an identity matrix.In this case, assuming , , , , leads to theMurray–Davis relation (see Equation 3.102),

(3.126)

where . Note that because and .In the case of a fluorescent ink or of a fluorescent substrate, each element

of the matrices in Equation 3.122 must be replaced by a matrix. Let us denote as the fluorescence density matrix of the fluorescent ink as defined in

Equation 3.86, and as the reflection matrix. Furthermore, let us denote as the identity matrix, which has the same dimension as the fluorescencedensity matrix and the reflection matrix .

In the Kubelka–Munk matrix , the sum has to be replacedby M, and has to be replaced by the matrix defined in Equation 3.86.In the same way, and must be replaced by matrices. If the ink-absorbing medium is nonfluorescent, the scalar values and aremultiplied by the identity matrix . The body reflectance is replaced bythe reflection matrix introduced in Section 3.7.2. In the Saunderson cor-rection matrix , each element is replaced by its scalar value multipliedby . The same kind of substitution must be done in the change of basis

vδu v, au

δ0 0, δ1 0, a0 1 a1–= = = δ0 1, δ1 1, a1= =

S0 0= S1 0= K0 0=

R λ( ) rsρg 1 rs–( ) 1 ri–( ) 1 a1– a1T λ( )+( )2

1 ρgri 1 a1– a1T2 λ( )+( )–-------------------------------------------------------------------------------------+=

T K1X–[ ]exp=

δu u, 1= δu v, 0= u v≠

S0 0= S1 0= K0 0= ri 0= rs 0=

R λ( ) Rg 1 a1–( ) a1T2+[ ]=

T K1X–[ ]exp= Rg ρg= ri 0= rs 0=

MRg I

M Rg

MKM K1 S1+S1 F

K0 S0+ S0

K0 S0+ S0

I ρg

Rg

MSC

I




matrix, its inverse matrix, and the matrix containing the scattering probabil-ities . Note that , , , and are replaced by vectors whosenumber of components equals the number of columns of the matrix M.Hence, the vector equals the spectrum of the incident lightsource, and equals the spectrum of the reflected light.

3.9.2 Computing the area fractions and the scattering probabilities

The area fractions and the scattering probabilities are computed bya numerical simulation.70 High-resolution grids model the printed surface,one grid being used for each ink. The value of a grid point corresponds tothe local amount of a given dye (see Figure 3.30C). In the particular case ofinkjet printing, the density profile of an isolated ink impact, which wasmeasured under the microscope, can be approximated by a parabolic func-tion.71 A single dot is modeled as a stamp (see Figure 3.30B).

Digital printing systems use halftoning or dithering algorithms to deter-mine whether a given location on the printed surface must be covered by adot. To simulate accurately a given printing system, the same halftoning ordithering algorithm must be used to provide the locations of all printed dots(Figure 3.30A). Wherever a dot is printed, the model is stamped at the cor-responding location on the high-resolution grid. In the particular case ofinkjet printing, stamp overlapping is additive.

In a color print using inks, halftoned ink layers are used. The inkcombination covering a surface element at position is given by the setof values of the grid points in the superposed high-resolutiongrids. The area covered by a given combination of inks is estimated bycounting the number of grid points having the same set of values. Thefraction of area is determined by counting the number of grid points thatbelong to the same inking level u.

The light-scattering process can be seen as an exchange of photonsbetween a grid point and its neighbors. As we saw in Section 3.8.6, it can bemodeled by an internal point spread function that expresses thedensity of probability for a photon entering at location to emerge atlocation . The discrete form of the internal PSF gives the probabilityfor an entering photon to emerge from another grid point. The functionsuggested by Gustavson64 is a good approximation of the internal PSF (seeEquation 3.114). The scattering probability equals the weighted sumover the whole grid of points having the inking level with a neighborhaving an inking level . The weights of the neighbors are given by ourdiscrete internal PSF.

3.10 Concluding remarksBy using a global approach, all classical color prediction models were unifiedwithin a mathematical framework based on matrices. This matrix frameworkprovides a new insight into color prediction by modeling a reflective surface

δu v, i j i0 0( ) i1 0( )

i0 i1 i= =1 a1–( )j0 a1j1+

au δu v,

k kx y,( )

k x y,( ) kk

kau

p x y,( )0 0,( )

x y,( )

δu v,u

v



234 D

igital Color Im

aging Handbook

(C) Stamp of a single dot(B) Printed surface simulated on a high resolution grid

(A) Locations of printed ink drops

1 3 4 6 7 6 4 3 2

6 8 9 11 13 13 14 13 14 11 10 8 4 1

2 7 12 15 17 17 19 20 22 22 19 18 17 14 12 7 2

5 10 15 18 22 23 25 27 28 27 28 27 25 23 22 19 15 10 5

4 10 17 20 24 28 30 32 32 33 34 33 33 31 30 29 24 20 16 10 3

7 9 16 22 24 29 33 36 37 38 40 40 40 38 37 36 32 29 25 22 16 10 5

2 10 16 22 25 30 35 37 41 43 43 44 45 44 44 43 41 37 35 30 26 21 16 11 2

1 7 15 21 25 31 35 38 42 45 47 48 50 49 50 47 47 45 42 39 35 31 26 20 14 8

5 12 18 24 29 34 39 42 46 48 50 52 53 53 53 52 50 48 46 42 39 35 29 24 18 12 5

8 15 23 28 33 38 42 46 50 51 54 55 56 56 56 55 54 51 49 46 42 37 32 28 23 14 8

1 9 17 23 31 36 41 45 48 51 54 57 58 58 59 59 58 57 54 51 49 45 41 36 31 23 17 9 1

2 11 18 26 31 36 42 47 50 54 57 58 60 60 61 60 60 58 57 54 50 47 41 36 32 25 19 11 3

4 13 20 26 33 38 44 48 52 55 57 60 61 62 63 62 61 60 57 55 52 48 43 38 33 26 19 13 3

6 12 21 28 34 40 45 49 53 55 59 60 62 63 63 63 62 61 59 55 53 49 45 39 34 27 21 13 6

6 14 22 28 34 40 44 49 53 56 59 61 62 63 64 63 63 61 59 56 53 49 44 40 34 27 21 14 5

7 13 20 27 34 40 44 49 53 56 58 60 62 63 63 63 62 61 59 56 52 50 44 40 34 27 20 13 6

4 13 19 27 32 39 43 47 52 55 58 59 61 62 62 62 62 60 58 55 52 48 43 38 33 27 19 12 3

3 11 18 25 31 37 42 46 51 54 57 58 60 61 61 61 60 58 57 54 51 47 42 37 32 25 19 11 2

1 10 17 23 30 36 41 45 48 52 54 56 57 58 59 59 57 57 54 51 48 45 41 36 31 24 17 9 2

7 14 23 29 33 38 41 46 50 51 53 55 56 56 56 55 54 51 49 46 41 37 33 28 23 14 7

5 12 18 24 29 35 39 41 46 48 51 52 52 53 53 52 50 48 46 42 38 36 29 24 18 11 5

8 15 21 26 30 35 39 43 46 47 47 49 50 49 47 46 45 42 38 35 30 25 20 15 7 1

2 8 16 22 26 31 35 38 41 43 44 44 45 44 43 43 40 37 35 30 25 20 16 10 3

6 10 16 20 26 30 33 36 37 39 40 40 39 39 37 36 32 29 26 20 16 9 5

4 9 16 21 24 29 30 32 32 34 34 34 32 31 30 29 24 21 16 10 4

4 9 15 18 22 24 25 27 28 28 28 26 26 23 22 18 14 9 6

2 8 12 15 17 17 19 21 21 21 19 18 17 15 12 8 3

1 7 8 9 11 13 12 14 13 12 11 9 8 5 1

1 2 4 6 6 5 4 3 1

Figure 3.30 A high-resolution grid used for modeling the printed surface. The value of a grid point corresponds to the local amount of dye.

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by three matrices: the Saunderson correction matrix, the Kubelka–Munkmatrix, and the light-scattering matrix. This approach also allows us topredict colors with a higher accuracy because a larger number of physicalphenomena are taken into account.

However, this generalized approach does not provide the ultimateanswer to all color prediction needs. Several important physical phenomenaare not yet taken into account; and, effects induced by the surface roughness,metallic pigments, or pearlescent pigments cannot be predicted. Neverthe-less, the unified framework is more powerful than a collection of separateclassical models put side by side, because it also provides solutions fordifficult cases such as, for example, halftones printed with fluorescent inksor printing with a large number of nonstandard inks. In most cases, solutionsare found by considering larger matrices. The difficulty is simply turned intomore work for the computer.

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Chapter 3: Physical models for color predictionread.pudn.com/downloads448/sourcecode/graph/texture_mapping/1… · 3.2 A few results from radiometry 3.3 Reﬂection and refraction