modeling ink spreading for color prediction · color prediction method 2 in order to predict the...

237

JOURNAL OF IMAGING SCIENCE AND TECHNOLOGY® • Volume 46, Number 3, May/June 2002

Original manuscript received December 12, 2000

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der the microscope. This approach lead to accurate colorpredictions but it has two major drawbacks: firstly, thesimulated dot shapes are not realistic; secondly, thenumber of different configurations and the number ofenlargement coefficients increases dramatically with thenumber of ink layers.

Therefore, a new investigation method is proposed. Itis based on an impact model whose shape changes as afunction of the configuration of the neighboring impactsand of the state of the surface. Because the number ofconfigurations is high, Pólya’s counting theory is usedto find a reduced set of cases whose analysis allows usto deduce the spreading in all other configurations. Bybuilding samples with this reduced set, standard mea-suring instruments can be used to estimate the inkspreading. This information will be used by our colorprediction model in order to compute accurately thespectra of printed samples.

The Impact ModelIn our color prediction software,2 high resolution gridsmodel the printed surface. One grid is used for each ink.The value of a grid point corresponds to the local amountof a given dye (see Fig. 2). The density profile of an iso-lated ink impact was measured under a microscope andapproximated by a parabolic function.4 The resulting inkimpact model (see Fig. 2) is used as a stamp. Whereveran ink drop hits the surface of the printed media, theimpact model is stamped at the same location on thehigh resolution grid. Stamp overlapping is additive.

In a color print using n inks, the ink combinationcovering a surface element at position (x, y) is given bythe set of n values of the grid points (x, y) in the nsuperposed high resolution grids. The area covered by

Modeling Ink Spreading for Color Prediction

Patrick Emmel▲

Clariant International, Masterbatches Division, Muttenz, Switzerland

Roger David Hersch▲

Ecole Polytechnique Fédérale de Lausanne (EPFL), Département d’Informatique, Laboratoire de Systèmes Périphériques, Lausanne, Switzerland

This study aims at modeling ink spreading in order to improve the prediction of the reflection spectra of three ink color prints.Ink spreading is a kind of dot gain which causes significant color deviations in ink jet printing. We have developed an inkspreading model which requires the consideration of only a limited number of cases. Using a combinatorial approach based onPólya’s counting theory, we determine a small set of ink drop configurations which allows us to deduce the ink spreading in allother cases. This improves the estimation of the area covered by each ink combination that is crucial in color prediction models.In a previous study, we developed a unified color prediction model. This model, augmented by the ink spreading model, predictsaccurately the reflection spectra of halftoned samples printed on various ink jet printers. For each printer, the reflection spectraof 125 samples uniformly distributed in the CMY color cube were computed. The average prediction error between measured andpredicted spectra is about ∆Ε = 2.5 in CIELAB. Such a model simplifies the calibration of ink jet printers, as well as theirrecalibrations when ink or paper is changed.

Journal of Imaging Science and Technology 46: 237–246 (2002)

IntroductionIn previous publications,1,2 we presented a color predic-tion model for halftoned samples which is based on anew mathematical formulation of the Kubelka–Munkequations. This new model requires, like Neugebauer-based methods,3 an estimation of the area covered byeach ink density combination. This estimation needs tobe computed by simulating the printing process.

In ink jet printing, the superposition of ink drops causesa significant dot gain, i.e., a change of the covered area(see Fig. 1). When ink drops are printed one over anotheror just partially overlap, an ink spreading process takesplace. This phenomenon results from a complex physicalinteraction between the ink drops and the surface of theprinted media. Changing the inks or the paper modifiesthe magnitude of the spreading and induces color pre-diction errors ranging up to ∆Ε = 20 in CIELAB.

According to observations made under the microscope,the spreading depends on the state of the surface (“wet”or “dry”) and the configuration of the neighboring im-pacts. In our previous study,2 we analyzed samples madeof two ink layers, and deduced a set of empirical rulesdescribing the enlargement of the impact of an ink drop.For each ink–paper combination twelve enlargementcoefficients were estimated by observing samples un-

238 Journal of Imaging Science and Technology® Emmel and Hersch

a given combination of n inks is estimated by countingthe number of surface elements having the same set ofn values.

In ink jet printing, the shape of an impact highly de-pends on the configuration of the neighboring ink dropimpacts and on the state of the surface. Therefore weneed to extend the traditional circular or elliptic im-pact model5 to more sophisticated shapes.

Most ink jet printers use a hexagonal grid when print-ing in color. Hence, each impact has six neighbors.Therefore, the circumference of an impact is param-eterized by six vectors having a common origin at theimpact center. Each vector is oriented to the directionof the midpoint between two neighboring impacts (seeFig. 3). Let us denote ri(1 ≤ i ≤ 6) the length of the vec-tor which is oriented in the direction given by the angleθi = π/6 + (i – 1) · π/3. The shape of the impact is ap-proximated by a parametric curve that joins the verti-ces defined by the six vectors.

Let us denote ρ = r(θ) the equation in polar coordi-nates of this parametric curve. Because we assume thata neighbor influences only locally the shape of the im-pact, the parametric curve depends only on ri and ri+1

when θi ≤ θ ≤ θi+1. In order to get realistic impact shapes,we interpolate the values of the radius ρ between ri andri+1 by using a polynomial of degree three:

ρ = r(θ) = 2(ri – ri+1)t3 – 3(ri – ri+1)t2 + ri (1)

where

t i= −

−36π

θ π

and

π π θ π π6

13 6 3

+ − ⋅ ≤ ≤ + ⋅( ) .i i

Note that

r i ri

π π6 3

+ ⋅

= ,

and that the derivative

drd

iθ

π π6 3

0+ ⋅

= .

These properties guarantee the continuity of r(θ) and ofits first derivative.

We assume that the density D at the location definedby the polar coordinates (ρ, θ) (0 ≤ ρ ≤ r(θ)) is given by:

D DrM( , )( )

ρ θ ρθ

= ⋅ −

12

(2)

where DM is the density at the center of the impact. Notethat a circular impact (r(θ) constant) has a parabolicdensity profile as observed in our previous study.4

The amount of dye remains constant during thespreading process, only the spatial distribution ischanged. Therefore, the maximal density DM at the cen-ter of the impact must decrease when the impact is en-larged. By integrating Eq. (2) over the area occupied bythe impact, we get the total amount of dye within theimpact (see Appendix). Because this amount must re-main constant, we get the following equation for DM:

D Dr

r r r rM

ii i i i

= ⋅∑ + ⋅ −

=+ +

002

1

61 1

2

613

140( )

(3)

(a) (b)Figure 1. Microscopic views of a halftoned cyan sample (a) and of a green halftoned sample (b). The green sample (b) is made ofthe same cyan layer as (a) and covered with a uniform yellow layer. Note the enlargement of the cyan clusters in (b).

Cyan clusterPaper Green clusterYellow ink

Modeling Ink Spreading for Color Prediction Vol. 46, No. 3, May/June 2002 239

each impact has six neighbors whose centers are the cor-ners of a regular hexagon. In a three-ink color print, eachpixel of the printed surface is in one of the following fourstates: no ink, covered with one ink drop, covered withtwo ink drops or covered with three ink drops. Becausean impact has six neighbors, there are 46 = 4096 possibleneighbor configurations. On the considered center point,a new ink drop is printed on a surface that is in one ofthe following three states: dry (no ink), covered with oneink drop or covered with two ink drops. Therefore, thereare 3 × 46 = 12288 configurations to consider.

Among these 12288 configurations, many are equiva-lent by reflection or by rotation. Note that there are

Figure 2. High resolution grid modeling the printed surface.The value of a grid point corresponds to the local amount ofdye. The density profile of an isolated ink impact is parabolic.

Figure 3. Six vectors define the circumference of the impact.The dashed circles indicate the locations of neighboring impacts.

where r0 and D0 are respectively the radius and the maxi-mal density at the center of an isolated circular impactwhich did not spread.

This new impact model allows us to simulate thespreading by changing the six ri coefficients accordingto the configuration of the neighbors and the state ofthe surface.

Pólya CountingIn this section, we derive the number of non-equiva-

lent ink-drop configurations with the help of Pólya’s“counting theory.” (Readers interested mainly in the fi-nal ink spreading model may skip over this section.)Pólya’s counting requires three steps: first, defining thegroup of symmetries acting on the set of corners, sec-ond, factorizing each permutation into cycles, and third,calculating the cycle index polynomial.

As we pointed out in the previous section, most ink jetprinters use a hexagonal grid when printing in color. So

Figure 4. There are six rotations and six reflections that bringa regular hexagon onto itself.


twelve symmetries that bring a regular hexagon ontoitself: six rotations and six reflections (see Fig. 4). Byconsidering only non-equivalent configurations, thenumber of cases to analyze can be reduced.

At the beginning of the century, the mathematicianGeorge Pólya developed a theory for this kind of count-ing problem.6 Later, his work was translated into En-glish7 and became known in the mathematical literatureas “Pólya’s counting theory.” In the forthcoming discus-sion, we present only how to apply this theory to ourproblem. Good presentations of Pólya’s counting theorycan be found in textbooks.8

As mentioned above, there are twelve geometric mo-tions, six rotations and six reflections, which act as per-mutations of the corners of the regular hexagon. Let usdenote ρ0

6, ρ16, ρ2

6, ρ36, ρ4

6 and ρ56 the rotations by an angle

of θ = 0, θ = π/3, θ = 2π/3, θ = π, θ = 4π/3 and θ = 5π/3respectively. Furthermore, we denote τad, τbe, τcf, the re-flections whose axes are the lines (ad), (be), (cf) respec-tively, and τab, τbc, τcd, the reflections whose axes are themediators of the segments [ab], [bc], [cd] respectively.This set of twelve symmetries with the composition op-eration has a group structure:9 the unit element is ρ0

6,each symmetry has an inverse, and the composition oftwo symmetries is still a symmetry. Let us denote thisgroup D6. The group is acting on the set of corners ofthe hexagon.

By applying the same operation, one of the symmetrieslisted in Table I, iteratively to the hexagon, the cornersfollow cyclic trajectories. After a finite number of itera-tions, each corner gets back to the starting point. Forinstance, by applying iteratively the rotation ρ2

6 to thecorner (a) of Fig. 4, we get the cycle [a, c, e]: a is firstmoved to c, then to e, and finally comes back to a. Notethat for a given symmetry, an individual corner belongsonly to one cycle. A classical result from group theoryshows that each element of D6 acting on the set of cor-ners of the hexagon can be factorized into disjoint cycles.10

The factorizations of the elements of D6 are listed in thesecond column of Table I. The reflection τcf, for example,is factorized into two cycles of one corner ([c] and [f]) andtwo cycles of two corners ([a, e] and [b, d]).

Let the variable zi correspond to a cycle having i cor-ners. To each symmetry we associate a monomial thatis the product of the (zi)’s according to the cycle factor-ization as shown in the third column of Table I. Thereflection τcf is associated with z1· z1 · z2 · z2 = z2

1z22.

The sum of the monomials divided by the number ofthe symmetries of the group D6 is called the cycle index:

P z z z z z z z z z z1 2 3 6 1

66 3

223

12

221

122 2 4 3, , ,( ) = ⋅ + + + +( ) (4)

Let us denote k be the number of states in which acorner can be. According to Pólya’s theory,11 the numberof non-equivalent hexagons is:

P k k k k k k k k k, , ,( ) = ⋅ + + + +( )1

122 2 4 36 2 3 4 (5)

Therefore, the number of non-equivalent configura-tions of six neighboring ink drop impacts being in oneof k states is given by P(k, k, k, k). Furthermore, a newdrop is printed on a surface which is in one of (k – 1)states. So finally, the total number of non-equivalentconfigurations is:

N k P k k k k= − ⋅( ) ( , , , )1 (6)

In color prints using three inks, we have k = 4 states(no ink, one ink drop, two ink drops, three ink drops).Hence the number of non-equivalent configurations isN = 1290. In four-ink printing (k = 5), N rises to 6020.Note that for two inks k = 3 and N drops to 184.

Other geometric shapes, as for instance non-regularhexagons, are mapped onto themselves by other groupsof symmetries. But the same procedure can be appliedto find the number of non-equivalent configurations.

A Simplified Model of Ink SpreadingThe spreading process is a complex interaction betweenthe inks and the printed surface. It is strongly relatedto physical properties like wettability and solvent ab-sorption. Therefore the inks behave differently on ev-ery surface. Furthermore, the number of cases inthree-ink-printing (N = 1290) is too large for perform-ing exhaustive measurements.

We propose a simplified model of ink spreading. First,the geometry of the problem is simplified.12,13 The hexa-gon is subdivided into six triangles which share the cen-ter O of the hexagon as a common vertex (see Fig. 5).Second, the spreading in the direction of the mediatorof the segment [a, b] is supposed to depend only on thestate of the neighbors located in a and b, and the stateof the surface at the center O of the hexagon. This sim-plified geometry allows to define a group S of symme-tries acting on the segment [a, b], because only a and bplay equivalent roles in the triangle (a, O, b). The groupS has two elements: ρ0 the rotation by the null angle,and the reflection τab whose axis is the mediator of thesegment [a, b]. The cycle factorizations and the mono-mials of the group S are listed in Table II. The cycleindex is:

P z z z z1 2 1

22

12

,( ) = ⋅ +( ) (7)

Finally, the total number of non-equivalent configu-rations is given by combining Eq. (6) and Eq. (7):

N k P k k

k k k= − ⋅ = − ⋅ ⋅ +( ) ( , )

( ) ( )1

1 12

(8)

where k is the number of states of a neighbor.According to Eq. (8), in two-ink-printing (k = 3) we must

consider N = 12 cases; in three-ink-printing (k = 4), wemust consider N = 30 cases; and in four-ink-printing(k = 5), we must consider N = 60 cases.

TABLE I. Cycle Factorizations and Monomials of the GroupD6 Acting on the Corners of the Hexagon

Symmetry Cycle factorization Monomial

ρ06 [a], [b], [c], [d], [e], [f], z6

1

ρ16 [a,b,c,d,e,f] z6

ρ26 [a,c,e], [b,d,f] z2

3

ρ36 [a,d], [b,e], [c,f] z3

2

ρ46 [a,e,c], [b,f,d] z2

3

ρ56 [a,f,e,d,c,b] z6

τad [a], [d], [b,f], [c,e] z21z2

2

τbe [b], [e], [a,c], [d,f] z21z2

2

τcf [c], [f], [a,e], [b,d] z21z2

2

τab [a,b], [c,f], [d,e] z32

τbc [b,c], [a,d], [e,f] z32

τcd [c,d], [a,f], [b,e] z32


These sets of cases are small enough for performingexhaustive measurements in order to calibrate the inkspreading model. From each case, we can deduce thevalue of a coefficient ri , which is used by our previouslydefined impact model. The shape of a simulated impactdepends on six ri coefficients which are related to theconfiguration of the neighboring impacts and the stateof the surface.

Enumerating the ConfigurationsPólya’s counting theory gives the number of configura-tions that must be considered in order to calibrate theink spreading model. The configurations themselves arecomputed by other means. Powerful generating algo-rithms exist,14 but for the sake of simplicity, a naive sievemethod is used. A computer generates the list of all con-figurations. Considering the first configuration, allequivalent configurations are removed from the list. Thecomputer applies this procedure to the next configura-tion of the list until the end of the list is reached. The30 non-equivalent configurations for three-ink-printingare listed in Fig. 6.

Prediction Results and DiscussionThe new ink spreading model was combined with ourcolor prediction method2 in order to predict the spectraof two series of 125 samples uniformly distributed inthe CMY color space. Each series is a set of 5 × 25samples printed on five sheets of paper. The area cover-age of the yellow ink is constant for all samples printedon the same sheet, and varies from sheet to sheet. Thefirst series was printed on Epson Glossy Photo QualityPaper using an Epson Stylus-Color™ printer which isbased on a piezo electric technology,15 and the secondseries was printed on HP Photo Paper, using an HP-DJ560C printer which is based on conventional ther-mal ink jet techniques.16 All samples were produced witha clustered dither algorithm with 33 tone levels.

For both series, the radius r0 of an isolated circularimpact was measured accurately under the microscope.In the HP series, the superposed cyan, magenta and yel-low ink drop impacts have almost the same size, whereasin the Epson series, the radius of the yellow drop im-pact is 20% larger than the radius of the cyan drop im-pact, which is, in turn, is 30% larger than the radius ofthe magenta drop impact.

Figure 5. A simplified geometry: the hexagon is subdividedinto six triangles that share the center of the hexagon as acommon vertex. The spreading of the ink drop impact in thedirection of the mediator of [a, b] depends only on the state ofO, a, and b.

TABLE II. Cycle Factorizations and Monomials of the GroupS of Symmetries Acting on the Segment [a,b]

Symmetry Cycle factorization Monomial

ρ0 [a], [b] z12

τab [a,b] z2

Figure 6. List of the 30 non-equivalent configurations in athree-ink print. The dashed circles indicate the locations ofneighbors that are not covered with ink. The white disks cor-respond to one ink drop, the gray disks correspond to two inkdrops, and the black disks correspond to three ink drops. Thevectors indicate the orientation of the spreading. The thirdand sixth columns indicate which test-sample is used to deter-mine the spreading coefficient.


For each ink drop configuration, the correspondingradius vector length ri is estimated by indirect means.We observed that a given ink drop spreads only if thelocal amount of solvent is higher than the amount ofsolvent at the neighbor ’s location. Therefore we assumethat there is no spreading in the following configura-tions: 3, 4, 5, 6, 7, 8, 9, 10, 16, 17, 18, 19, 20, and 30 (seeFig. 6). Considering the other configurations we defineten test samples, each of which being a mosaic composedof a single pattern as shown in Fig. 7. Note that the dotgeometry of the test samples has been chosen to corre-spond to the dot geometry of the clustered halftoningalgorithm being used. For example, test samples (a) and(c) given in Fig. 7 correspond to the samples shown inFig. 1a and in Fig. 1b respectively. Each test sample isused to determine one or two ink spreading coefficientsr = ri/r0. In other words, a given test sample containsseveral occurences of one or two dot configurationswhose spreading coefficients are unknown. All testsamples are printed and measured. For each test sampleseveral simulation iterations are required to fit the pre-dicted and the measured reflection spectrum by vary-ing the values of the radius vectors lengths ri. The inkspreading coefficients r = ri/r0 computed for each seriesare given in Fig. 8. Note that the ink spreading coeffi-cients for the Epson series are given for the magentadrop impact.

Using our prediction model and its extension for fluo-rescent substances17 (the magenta ink of the Hewlett-Packard printer being fluorescent), we computed thespectra of 250 samples, and compared them with thespectra measured under a tungsten light source whoserelative radiance spectrum is given in Fig. 9. TheCIELAB values of all samples were computed for the 2°standard observer. The average colorimetric deviationbetween measured and predicted spectra of the Epsonseries and of the HP series are given in Table III andTable IV respectively. The results obtained for the Epsonseries are a little better than the results for the HP se-

Figure 7. List of the ten test patterns used to determine thespreading coefficients. The dashed circles indicate the loca-tions of neighbors that are not covered with ink. The whitedisks correspond to one ink drop, the gray disks correspond totwo ink drops, and the black disks correspond to three inkdrops. Each test-sample is used to determine one or two spread-ing coefficients whose numbers are given in the third column.Note that the spreading coefficients of the configurations 1, 2,11, 13, 14, 15 must be determined prior to 12, and that thespreading coefficients of the configurations 21, 23, 26, 27, 28,29 must be determinated prior to 22, 24, 25.

TABLE III. Prediction Results in CIELAB ∆E and ∆E94 for theEpson Series

Series ofconstantyellow ink Maximal Maximalpercentage ∆E

∆En

2∑ ∆E ∆E94

∆En

942∑ ∆E94

0% 1.67 1.97 3.58 0.95 1.14 2.4925% 1.81 1.96 3.11 1.10 1.22 2.3750% 1.95 2.14 4.13 1.24 1.34 2.4675% 2.91 3.02 5.59 1.85 1.97 3.56

100% 2.85 3.11 5.75 1.96 2.16 3.93

TABLE IV. Prediction Results in CIELAB ∆E and ∆E94 for theHP Series

Series ofconstantyellow ink Maximal Maximalpercentage ∆E

∆En

2∑ ∆E ∆E94

∆En

942∑ ∆E94

0% 2.07 2.34 4.70 0.96 1.06 1.8925% 2.57 2.85 4.85 1.19 1.28 2.6450% 2.59 2.90 5.48 1.34 1.52 3.6575% 3.27 3.49 6.85 1.97 2.13 4.13

100% 3.92 4.28 6.89 2.45 2.68 4.14


ries. This could be due to the stronger spreading of theHP inks.

In order to show how our ink spreading model im-proves color prediction, we consider the 25 samples ofthe series having a 100% coverage of yellow ink andwhich are printed on HP Photo Paper with an HPDJ560C printer. The prediction results correspondingto this series are given in the last row of Table III. Themeasured and computed spectra of the 25 samples areconverted into CIE-XYZ values that are shown in agraphical form in Fig. 10. Figure 10a shows the devia-tion between measured and computed colors taking inkspreading into account; and Fig. 10b shows the devia-tion between measured and computed colors withouttaking ink spreading into account. Without taking ink

Figure 8. Spreading of HP inks printed on HP paper and spreading of Epson inks printed on Epson paper. For each of the 30 non-equivalent configurations in a three-ink print, the spreading coefficients r = ri/r0 are respectively given in the columns entitledHP and Epson. The second and sixth columns show the dot configurations. The dashed circles indicate the locations of neighborsthat are not covered with ink. The white disks correspond to one ink drop, the grey disks correspond to two ink drops, and theblack disks correspond to three ink drops. The vectors indicate the orientation of the spreading.

spreading into account, the prediction error is higherthan ∆E = 10 in CIELAB for samples made of more thantwo inks, because printing ink drops one over anothersignificantly increases ink spreading. The spectra ofsamples made with only one ink are well predicted be-cause almost no ink spreading occurs.

The measured and predicted spectra of the cyan andgreen samples shown respectively in Fig. 1a and Fig.1b are given in Fig. 11a and Fig. 11b. Sample (b) is madeof the same cyan layer as sample (a) and covered with auniform yellow layer. In spite of the absence of absorp-tion of the yellow ink at 630 nm, the measured reflec-tion coefficient of sample (b) is lower than that of sample(a) at this wavelength (compare Fig. 11b and Fig. 11a).This difference is explained by ink spreading. The cyan


ink covers a larger area with a lower density, and there-fore produces a higher light absorption. Note that inthe case of sample (b), the prediction error is ∆E = 9.2in CIELAB if ink spreading is not taken into account.

ConclusionsWe introduce a new method for investigating inkspreading. The spreading process is modeled by enlarg-ing the drop impact according to the configuration ofits neighbors and the state of the surface. The numberof cases that must be analyzed is reduced to a smallset by using Pólya’s counting theory. In a three-ink-

printing process, only 30 cases must be considered in-stead of 3 × 46 = 12288.

The printing process is simulated by stamping impactsof different shapes on high resolution grids. Each shapeis determined by 6 radii. The value of each radius is esti-mated by fitting the reflection spectra of ten test samplesin several simulation iterations. This allows us to com-pute the relative areas occupied by the various ink com-binations in a three-ink process. We predict accuratelythe spectra of 250 samples produced by two differentprinters. The average prediction error is about ∆E = 2.5and the maximal error is less than ∆E = 7 in CIELAB.

Figure 9. Relative radiance spectrum of the tungsten light source.

0 20 40 60 80

X

0

20

40

60

80

Y

Yellow

Red

Green

(a) (b)Figure 10. Projection onto the XY plane within the CIE-XYZ color space of the deviations between computed and measuredcolors of 25 samples having a 100% coverage of yellow ink, and which are printed on HP photo paper using an HP DJ560Cprinter. Each dot indicates the measured color, and the other end of the segment indicates the computed color: (a) shows the colordeviation when ink spreading is taken into account;, (b) shows the color deviation when ink spreading is not taken into account.

1.75

0.25

550 650600 700


(a)

(b)

Figure 11. Measured spectra (continuous lines) and predicted spectra (dashed lines): (a) of the halftoned cyan sample shown inFig. 1a (25% cyan ink coverage), and (b) of the green sample shown in Fig. 1b (superposition of a cyan layer at 25% ink coverageand a yellow layer at 100% ink coverage). Note that the reflection coefficient of sample (b) is lower than that of sample (a) at 630nm in spite of the absence of absorption of the yellow ink at this wavelength.

In our view the proposed ink spreading model mayconsiderably increase the accuracy of existing advancedcolor prediction models.18 It may also be used for im-proved Neugebauer-based color prediction methods3

because they require an accurate estimation of the areacovered by each ink combination.

Appendix: Total Amount of Dye Within an Ink DropImpactThe density profile of an ink drop impact is given byEq. (2). The total amount T of dye within an ink dropimpact is obtained by integrating Eq. (2) over the areacovered by the impact:

T Dr

d dr

M= ⋅ −

∫∫ 00

2 2

1( )

( )

θπ ρθ

ρ ρ θ (9)

During the spreading process, the amount of dye Tremains constant. We can establish a relationship be-tween the maximal density D0 at the center of a drop

that did not spread, and the density DM at the center ofa drop that spread, by calculating T in both cases.

Within an isolated circular impact (r(θ) = r0) whichdid not spread (DM = D0), the total amount of dye is:

T D r= ⋅ ⋅π

2 0 02 (10)

Let us now consider an ink drop that spread. In thiscase, r(θ) is defined piecewise as shown in Eq. (1). Theintegral in Eq. (9) can be written:

T Dr

d di

rM

i

i= ⋅ −

=∑ ∫∫ +

1

6

0

21

1( )

( )

θ

θ

θ ρθ

ρ ρ θ (11)

where

θ π π

i i= + − ⋅6

13

( ) .

nm

nm


By integrating in Eq. (11) with respect to variable ρ, weget:

T D

rdM

i i

i==∑ ∫ +

1

6 21

4θ

θ θ θ[ ( )]. (12)

In the next step, the following change of variable isperformed: t = (3/π) (θ – π/6) – i. The integral (12) can bewritten:

T

Dr r t r r t r dtM

ii i i i i

=

⋅ − − − +[ ]=

+ +∑ ∫π12

2 31

6

0

11

31

2 2( ) ( ) . (13)

By integrating Eq. (13) with respect to the variable t,we get:

T

Dr r r rM

ii i i i= ⋅ + ⋅ −

=+ +∑π

1213

1401

6

1 12( ) (14)

Finally, by combining Eqs. (10) and (14) we obtainEq. (3).

Acknowledgment. We would like to thank Dr. RitaHofmann from Ilford Imaging for providing useful in-formation on ink jet paper, and the Swiss National Sci-ence Foundation (grant No. 21-54127.98) for supportingthe project.

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