Light Scattering and Ink Penetration Effects on Tone Reproduction

Li Yang, Bjorn¨ Kruse and Reiner Lenz Institute of Science and Technology, Linkping University, S-601 74 Norrkoping,¨ Sweden

Abstract Based on a PSF approach, Rogers [7, 8] proposed a matrix We develop a framework to account for the effects of light approach where the tristimulus values of a halftone image scattering (or Yule-Nielsen effect) and ink penetration on are calculated as the trace of a product of two matrices. the reﬂectance and tristimulus values of a halftone sam- In addition, Arney et. al. [9, 10] extended the probability ple. We derive explicit expressions for reﬂectance values model which was originally introduced by Huntsman [11]

to account for the optical dot gain. Similar model has also

Ê Ê Ê Ô of ink dots ( i ), bare paper ( ), the halftone image ( )

been reported by H¬ubler [12]. Nevertheless the effect of f and the optical dot gain ¡ as functions of properties of materials (paper and ink) and their bilateral interaction. It ink penetration was approximated roughly which some- is predicted that light scattering extends the color gamut of times leads to results not matching expectations [13]. the printed images. The present approach and conclusions Very recently we established a theoretical approach to are appliable to both AM and FM halftoning schemes. account for the effect of ink penetration in the system that the substrate paper has been uniformly covered by ink lay- ers [14]. In the present paper we moved a step forward to 1. Introduction study halftone images where light scattering exists coinci- dently with ink penetration. Optical and rheological properties of ink, substrate paper and their mutual interaction play important roles in color reproduction. The attempt to explain the optical and color 2. Model and Methodology appearences of the printed image in terms of these proper- The basic geometry used is shown in Fig. 1 where the sur- ties led in the late 30’s to the Murray-Davis formula [1] and face of the substrate paper has been divided into two sets,

the Neugebauer Equations [2]. These equations were soon ¦

½ the paper under the dots (or ink penetrated paper), and,

found to be inaccurate in describing experimental results. ¦

¾ the bare paper. For simplicity, only the case of a single In 1951 Yule and Nielsen [3] interpreted the discrepancy layer of dots is analyzed. Extension to a multilayer sys- as the result of light penetration and scattering in the paper tem is straight forward but tedious and therefore will be substrate, which is now known as the Yule-Nielsen effect. reported elsewhere. Also for simplicity, we assume that This led to a modiﬁcation of the Murray-Davis formula the ink layer has uniform thickness.

which was replaced by

½ ½

Ò Ò

Ò 2.1. Point Spread Function Approach

Ê =[Ê ´½ f µ] f · Ê

Ô (1)

Á d ½

Consider an element light source ¼ that strikes the dot

! !

The exponent Ò is usually obtained by ﬁtting the experi-

Ö ¾ ¦ Ö ¾ ¦

½ ¾ ¾

at ½ , the ﬂux of light detected at due to !

mental data (such as optical density). In 1978 Ruchdeschel Ö

scattering of the incident light from ½ may be written as

and Hauser [4] obtained an estimation of the exponent Ò in

! !

d Â = Ô´ Ö ; Ö µÌÁ d d

½ ¾ ¼ ½ ¾

terms of point spread function (PSF) describing the scat- ½¾ (2)

Ò ¾

tering of light in paper. The study showed that ½ .

Ì ÌÁ d ½ The transmittance of the ink is and ¼ is thus the

However there is experimental evidence showing many ex- amount of light entering the substrate under the dot. The

! !

ceptions where Ò . In addition, both theoretical and

Ô´ Ö Ö µ ½

PSF, , ¾ , is the probability that photons enter the pa-

experimental studies have pointed out that the exponent Ò Ö

per under the dot at position ½ and exit from the bare

f>5¼±

itself may depend on f , especially in the case of . Ö

paper at position ¾ . The total amount of light that enetrs

Recently, the Yule-Nielsen effect has been further stud-

¦ ¦ ¾

½ and is scattered into may be written as: Z

ied using numerical, probability approach and analytical Z

! !

Ô´ Ö ; Ö µd d = Á Ì

methods. Gustavson [5, 6] investigated the Yule-Nielsen Â

½ ¾ ½ ¾ ¼

½¾ (3)

¦ ¦ ¾ effect by direct numerical simulation of the scattering events. ½

The double integral in Eq. (3) is the overall probability 2.2. Probability Approach

¦ ¦ ¾ of photons scattered from ½ into . This is therefore Light propagartion can be formulated in another way ie. a measure of the Yule-Nielsen effect.

by the so-called probability approach. If a photon enters ¦

the surface of the paper under the dot ( ½ in), we deﬁne È

½½ as the probability that it returns the surface under the ¦

dots ( ½ out. Note it is not necessarily to be the same dot È

as it enters), and ½¾ as the probability that it leaves the ¦

surface of the paper between the dots ( ¾ out). Similarly

È È ¾¾

we deﬁne ¾½ and as propabilities that a photon en- ¦

ters the surface from paper between the dots ( ¾ in) and ¦

then exits the surface under the dots ( ½ out) and between ¦

the dots ( ¾ out), respectively. The probabilities fulﬁl the

following constrain conditions[15]

È · È = Ê ½¾

½½ (9)

È · È = Ê ¾¾

¾½ (10)

Ô ¼

Figure 1: Overview over the halftone image. The surface is sub- where Ê is the reﬂectance of the paper under the dots (ink

¦ ½

divided into two groups: points under ink dots ( ) and between penetrated paper) and Ê that of bare paper. In the case of

¼ ¼

Ê Ê

dots ( ) no ink penetration, there is = .

Ô i

If the percentages of ink dots and the bare paper are f

f µ

and ´½ , respectively, and if the intensity of irradiance Á

onto the whole system is ¼ , the ﬂux of photons striking

If one exchanges the position of the lighting source

¦ ¦ Á f Á ´½ f µ

¾ ¼ ¼

the ½ and areas are and , respectively.

Á d ½

with that of the detector by puting the light source ¼

Â ¦ i

Then the ﬂux ij of photons entering andthenleav-

! !

Ö Ö ½

at ¾ and puting the detector at , and keeps other condi-

¦ i; j =½; ¾

ing j ( ) may be expressed as,

tions unchanged. Then one gets

Â = Ì Á fÈ

¼ ½½

½½ (11)

! !

d Â = Ô´ Ö ; Ö µÌÁ d d

¾ ½ ¼ ½ ¾

¾½ (4)

Â = ÌÁ fÈ

¼ ½¾

½¾ (12)

= ÌÁ ´½ f µÈ

From the optical reciprocity one obtains the following re- Â

¼ ¾½ ¾½ (13)

lation,

Â = Á ´½ f µÈ

¼ ¾¾

¾¾ (14)

! ! ! !

Ô´ Ö ; Ö µ=Ô´ Ö ; Ö µ

¾ ¾ ½ ½ (5) Here the ink layer is approximated as a ﬁlter with trans-

mittance Ì . Comparing Eqs. (12, 13) with Eq. (7), one can and therefore

obtain the the following expressions,

Â = Â ¾½

½¾ (6)

È = Ôf

¾½ (15)

È = Ô´½ f µ

It means that under the uniform illumination onto the whole ½¾ (16)

halftone sample, the amount of light being scattered from

È È ¾½

Evidently probabilities ½¾ correlates with by,

¦ ¦ ¾

½ (halftone dots) into (bare paper) is equal to that of

¦ ¦

¾ ½

´½ f µ=È f

the light scattered from into . Eq. (3) can be further È ½¾ ¾½ (17) expressed as,

The total ﬂux of photons emerging from the bare paper

Â = Â = Á Ì Ôf ´½ f µ

Â Â

¾½ ½¾ ¼ i

(7) Ô and from ink dots may be written as

Â = Á [ÌÈ f · È ´½ f µ]

Ô ¼ ½¾ ¾¾

where Ô is the mean value of the integrated PSF deﬁned as (18)

Z Z

Â = Á Ì [ÌÈ f · È ´½ f µ]

¼ ½½ ¾½

i (19)

! !

Ô = Ô´ Ö ; Ö µd d

¾ ½ ¾

½ (8)

f ´½ f µ

i ¦

¦ Correspondingly the reﬂectance values of the dots and

½ ¾ Ê

the paper between dots Ô can be calculated by,

Evidently Ô depends not only on the physical properties of

Â Ô

the substrate paper and the ink, but also on the geometric ¼

Ê = = Ê Ôf ´½ Ì µ

Ô (20)

´½ f µÁ

and spatial distribution of the dots. As shown later on, ¼

is closely related with the optical gain, and are therefore Â

Ê = Ô´½ f µ´½ Ì µ] = Ì [Ê Ì ·

i (21)

experimentally measurable. Ô

fÁ ¼

2262

¼ ¼

= Ê =Ê ¡Ê>¼ Ê

where describes the effect of ink penetra- Because the true reﬂectance is smaller than its

i Ê

tion upon the reﬂectance of the substrate paper. Because Murray-Davis value ÅD and the halftone image appears

of stronger absorption from the ink penetrated paper, is to have larger dot coverage than predicted when scatter-

generally smaller than unit. The value of Ô depends on ing is ignored. It is why this effect is known as optical

the size and the spatial distribution of the printed dots and dot gain. If scattering is not modeled then the measured

f ·¡f

the optical properties of the materials involved. Thus the reﬂectance Ê seems to originate from a dot size

Ô f: Ê ´f µ=Ê ´f ·¡f µ

knowledge of is of critical importance in predicting and instead of the true dot size From ÅD

Ê ¡f

reproducing the desired reﬂectance. The variable Ô does one can then obtain the optical dot gain, , as the func-

Ê Ê

not depend on and therefore or i should be used tion of the optical properties of the materials and ink pen-

when parameters must be ﬁtted to experimental data. etration:

Ôf ´½ f µ ´½ Ì µ ¡Ê

= f =

¡ (26)

¼ ¾ ¼ ¾

´½ Ì µ Ê ´½ Ì µ

3. Discussion and Examples Ê

Ô Ô

The equations (20) and (21) describe how the reﬂectance From the measured optical dot gain proﬁle, one can there-

values depend on the material properties and geometry. fore estimate Ô and obtain valuable information about the They reveal the following important facts: PSF.

The maximum of the optical dot gain can be obtained

Ê Ê i 1. Ô and are no longer constants as they were as- from equation,

sumed in Murray-Davis Equation, as soon as light

f ´½ f µ·Ô´½ ¾f µ= ¼

scattering has to be taken into account. Ô (27)

¼ ¼

= dÔ=df Ô = ¼ where Ô .For , the optical gain has a

2. In the case where Ô is independent on the dot cov- = 5¼±

single maximum at f and has a symmetric proﬁle

f Ê Ê f Ô erage , i and vary linearly with .Inthe

around the maximum.

Ê Ê f i other words, the non-linearity of Ô and with

We now illustrate the inﬂuence of the form of Ô on the

provides information about the f -dependence of the reﬂectance functions and the optical dot gain. In the ﬁrst

light scattering effect (or Ô).

= Ê :

example we consider Ô In the case of no ink pene-

Ô =½

3. Because the analysis was not restricted to any spe- tration ( ), this corresponds to the Yule-Nielsen model =¾

ciﬁc type of halftoning, all of expressions and con- with the exponent Ò . In this case (see Fig. 2a), the re-

Ê Ê f Ô clusions are appliable to both AM and FM halfton- ﬂectances i and vary linearly with the dot area .The

ing schemes. mean reﬂectance Ê of the whole image, on the other hand,

varies quadratically. The calculated optical dot gain has a = 5¼± 3.1. Reﬂectance of a Halftone Image and Optical Dot parabolic proﬁle with a single maximum at f .In

Gain the second example we adopt

¼ Ñ ½ Ñ

Ô = Ê f ´½ f µ µ ´½ (28)

The reﬂectance of the halftone sample is given by Ô

Ñ = ¼:7 Ê

with . Now all reﬂectance, especially Ô and

Ê = Ê f · Ê ´½ f µ Ô

i (22) Ê

i , are highly nonlinear functions of the dot coverage (see

Fig. 3a). Their behavior is characteristic for typical AM

dÔ=df = ¼ which is a quadratic function of f if as can

halftone images as can be seen from the measurements[9].

Ô=df 6= ¼

be seen from Equations (20) and (21). For d ,

Ê f

The sharp decrease of Ô as increases shows that the f

this is no longer the case and the relation between Ê and È

probability ( ¾¾ ) of a photon entering and then exiting depends on the form of Ô.

from the bare paper decreases dramatically when the area

Ê Ê i Substituting the expressions of Ô and (Equations (20) and (21)) into Equation (22), one gets of the bare paper becomes relatively much smaller than

the area of the ink dots. Accordingly, the maximum of the

Ê = Ê ¡Ê

ÅD (23) computed optical dot gain has been shifted downwards to =¿7± f .

where The dependence of the reﬂectances on the ink penetra-

¾ ¼

¼ tion has also been shown in Fig. 3 which we will explore

Ê = Ê Ì f · Ê ´½ f µ

ÅD (24) Ô i further in the next subsection. We use these two examples is the reﬂectance of the halftone sample without light scat- only to illustrate the power and ﬂexibility of the model de- tering (i.e. the Murray-Davis value). Light scattering in- veloped above. In a real application one can either deter-

side the substrate paper is described by mine Ô by ﬁtting the measured data (such as a optical dot

gain proﬁle) or compute it by integrating the PSF over the

Ê =´½ Ì µ Ôf ´½ f µ ¡ (25) halftone pattern using Equation (8).

2273

¡f Ê ; Ê ; Ê i 3.2. Optical Effects of Ink Penetration Ô and the optical dot gain, , with and without considering the effect of ink penetration. In the case

Assume the thickness and transmittance of the printed ink

Ì =½:¿Ì ¼

of ink penetration, we have assumed that i and

d Ì

¼ ¼

layer are and ¼ , respectively, and the reﬂectance of the

= Ê =Ê =¼:8

. As shown in Fig. 3, all the reﬂectance

i ¼

paper is Ê . The penetration of a part of the printed ink Ô values increase when ink penetration is taken into account

into the substrate paper has two optical effects: the trans- (dotted lines). The largest relative increase is obtained for Ì

mittance of the pure ink layer, i , becomes bigger due to Ê

the reﬂectance of the ink dots, i . Due to light scattering

the thinner ink layer. On the other hand, the reﬂectance of the ink penetration also changes the reﬂectance of the bare ¼

the ink penetrated paper, Ê , becomes smaller due to the

i Ê

paper, Ô . Ink penetration also decreases the optical dot

strong absorption of the penetrating ink. We recently [14] gain. Ink penetration also decreases the optical dot gain.

¾ ¼ ¾ ¼

Ê Ì Ê

showed that Ì .

¼ Ô

i i To illustrate the inﬂuence of this effect on the properties of the halftone image, we computed the reﬂectances, 0.9 no ink pene. 0.8 0.9 with ink pene. 0.7 0.8 0.6 R 0.7 p 0.5 0.6 R 0.4 0.5 R p Reflectance 0.3 0.4

Reflectance 0.2 0.3 R R i 0.1 0.2 R i 0 0.1 0 20 40 60 80 100 Dot percentage, f (%) 0 0 20 40 60 80 100 Dot percentage, f (%) (a) Computed reﬂectance

(a) Computed reﬂectance 9 no ink pene. 8 with ink pene. 18 7 16 6 14 f (%) ∆ 5 12 f (%) ∆ 4 10 3 8 Optical dot gain,

6 2 Optical dot gain, 4 1

0 2 0 20 40 60 80 100 Dot percentage, f (%) 0 0 20 40 60 80 100 Dot percentage, f (%) (b) Computed optical dot gain

Figure 3 Computed reﬂectance and optical dot gain with Ô given

(b) Computed optical dot gain :

Ê = ¼:87;Ñ = ¼:7

in Equation (28), where Ô . Solid lines -

Ô =

Ì = Ì = ¼:¾; = ½ ¼

Figure 2: Computed reﬂectance and optical dot gain with no ink penetration, i ; dot lines - with

Ê =¼:87 Ì =¼:¾

Ì =½:¿Ì ; =¼:8

Ô ¼ and consideration of ink penetration, i

An explanation is probably that the absorption of the ink This holds especially for the the tristimulus values of the Ï

penetrated paper decreases the probability of photon ex- paper (i.e. ¼ is the tristimulus values of the bare pa-

¦ ¦ Ï Ï

¾ ¼ change between the areas ½ and . The dependence of per). Condisering the fact that means the range of the optical dot gain proﬁle on the ink penetration could be color presentation of the image, we can therefore draw the

used to obtain estimates of material properties such as Ô conclusion that the light scattering leads to a larger color

and . gamut in halftone printing. This is in line with results of numerical simulations reported in [5, 6]. 3.3. Investigation of Color Printing 4. Acknowledgement Finally we sketch how the methodology developed above can be used in the investigation of tone reproduction. Simi-

This work was supported by the Surface Science and Print-

Ï = lar to Equation (22) we get the tristimulus values Ï ( ing Program of the Swedish Foundation for Strategic Re-

X; Y ; Z ) of the halftone sample

search and Reiner Lenz was supported by Ceniit.

Ï = Ï f · Ï ´½ f µ Ô i (29)

where References Z

[1] A. Murray, Technical Report of J. Franklin Institute

Ï = Ê ´µË ´µÛ ´µd

221, 721 (1936).

Ï = Ê ´µË ´µ Û ´µd i i [2] H. Neugebauer, “Die theoretischen Grundlagen des

Z Mehrfarbenbuchdrucks,” Z. Tech. Phys. 36, 75Ð89

Û ´µd Ï = Ê ´µË ´µ Ô Ô (1937).

[3] J. Yule and W. Nielsen, “The penetration of light into

´µ To simplify notation we use Û to represent one of the

paper and its effect on halftone reproduction,” TAGA

Ý Þ: tristimulus functions Ü; or In addition we use nota-

Proceeding 3, 65Ð76 (1951).

Ê ´µ;Ê ´µ Ê ´µ Ô tions, i and to denote explicitly their dependence on the wavelength of the illuminating light. [4] F. Ruckdeschel and O. Hauser, “Yule-Nielsen Ef- Equation (29) has a similar structure to that of Neugebauer- fect in Printing: a Physical Analysis,” Appl. Opt. 17,

Equations. However the interpretation is rather different. 3376Ð3383 (1978).

Ï Ï f f i

Here both Ô and depend on because of the - Ê

Ê [5] S. Gustavson,“Dot Gain in Color Halftones”, Ph.D i dependence from Ô and . Therefore the linearity assumption underlying the original Neugebauer Equations [2] Disertation No.492,1997, Dept. of Electrical Engi-

Ï neering, Link¬oping University, Sweden. does no longer hold. Moreover Ô is not only a function of the color coordinates of the paper since scattering from [6] S. Gustavson, “Color gamut of halftone reproduc- the ink-area has to be taken into account. tion,” J. Imaging. Sci. Technol. 41, 283Ð290 (1997). Following the approach used in Equation (23) the in- ﬂuence of the light scattering on the tristimulus values can [7] G. Rogers, “Effect of light scatter on halftone color,”

be described by J. Opt. Soc. Am. A 15, 1813Ð1821 (1998).

Ï = Ï ¡Ï ÅD (30) [8] G. Rogers, “Neugebauer revisited: random dots in where halftone screening,” Color Res. Appl. 23, 104Ð113

Z (1998).

Ï = Ê ´µË ´µÛ ´µd ÅD ÅD (31) [9] J. Arney, “A Probability Description of the Yule- Nielsen Effect, I,” J. Imaging. Sci. Technol. 41, 633Ð is the contribution from light following the Murray-Davis’ 636 (1997). assumption, and

Z [10] J. Arney and M. Katsube, “A Probability Description

Ï = ¡Ê ´µË ´µÛ ´µd ¡ (32) of the Yule-Nielsen Effect II: The Impact of Halftone Geometry,” J. Imaging. Sci. Technol. 41, 637Ð642

corresponds to the Yule-Nielsen effect. From the non- (1997).

¡Ï Ï

negativity of we ﬁnd that for any tristimulus value ¼ we have: [11] J. Huntsman, “A new model of dot gain and its appli-

cation to a multilayer color proof,” J. Imaging. Tech-

Ï Ï Ï Ï

¼ ÅD ¼ (33) nol. 13, 136Ð145 (1987).

[12] A. Hübler, In IS&T’s NIP-13: International Conference on [15] L. Yang, R. Lenz, and B. Kruse, “Light Scattering and Ink Digital Printing Technologies, (TREK INCORPORATED, Penetration Effects on Tone Reproduction,” J. Opt. Soc. New York, 1997). Am. A, (submitted).

[13] J. Arney and M. Alber, “Optical Effects of Ink Spread and Biography Penetration on Halftone Printed by Thermal Ink Jet,” J. Imaging Sci. Technol. 42, 331Ð334 (1998). Li Yang is a PhD student in Media Group at Department of Science and Technology, Campus Norrköping, Linköping [14] L. Yang and B. Kruse, “Ink penetration and its effect on University. He had physics in background. His research printing,” In IS&T/SPIE’s 12th Annual International interest is in color reproduction of printing in general and Symposium, (2000), in press. the mechanism of YuleÐNielsen effect and ink-penetration in particular.

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