IS&T©s 2000 PICS Conference IS&T©s 2000 PICS Conference Copyright 2000, IS&T 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) i 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 ¾ ! ! d  = 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 ! ! ¾ 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  ½ ¾ ½ ¾ ¼ ½¾ (3) ¦ ¦ ¾ effect by direct numerical simulation of the scattering events. ½ 2251 IS&T©s 2000 PICS Conference IS&T©s 2000 PICS Conference Copyright 2000, IS&T 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] ¼ È · È = Ê ½¾ ½½ (9) i ¼ 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 i ¼ ¦ ½ divided into two groups: points under ink dots ( ) and between penetrated paper) and Ê that of bare paper. In the case of Ô ¼ ¼ ¦ ¾ 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  ¦ 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 ¾  = Ì Á fÈ ¼ ½½ ½½ (11) ¾ ! ! d  = Ô´ Ö ; Ö µÌÁ d d ¾ ½ ¼ ½ ¾ ¾½ (4)  = ÌÁ fÈ ¼ ½¾ ½¾ (12) = ÌÁ ´½ f µÈ From the optical reciprocity one obtains the following re-  ¼ ¾½ ¾½ (13) lation,  = 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,  =  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  =  = Á Ì Ôf ´½ f µ   ¾½ ½¾ ¼ i (7) Ô and from ink dots may be written as  = 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  = Á Ì [ÌÈ f · È ´½ f µ] ¼ ½½ ¾½ i (19) ½ ! ! Ô = Ô´ Ö ; Ö µd d ¾ ½ ¾ ½ (8) f ´½ f µ Ê 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) Ô Ô ´½ f µÁ and spatial distribution of the dots. As shown later on, ¼ is closely related with the optical gain, and are therefore  i ¼ Ê = Ô´½ f µ´½ Ì µ] = Ì [­Ê Ì · i (21) experimentally measurable. Ô fÁ ¼ 2262 IS&T©s 2000 PICS Conference IS&T©s 2000 PICS Conference Copyright 2000, IS&T ¼ ¼ = Ê =Ê ¡Ê>¼ Ê where ­ describes the effect of ink penetra- Because the true reflectance is smaller than its Ô 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: ¾ 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.