EDIC RESEARCH PROPOSAL 1

Juxtaposed Halftoning based on Discrete LinesVahid BabaeiI&C, EPFL

Abstract—Device characterization is a critical prerequisite forhigh-quality color reproduction in color imaging systems [1]. Incharacterizing a printer, an important parameter to consider isthe halftoning method. The halftoning method itself is dependenton special factors that the end-product requires. When workingwith special inks whose superposition is not allowed, it is notpossible anymore to use the classical halftoning algorithms.Instead, goal-oriented halftoning methods are needed in whichcolor layers are printed side-by-side without superposition [2].Finally, we introduce the concept of discrete lines [3] and proposeto use this tool for dealing with the problem of juxtaposedhalftoning.

Index Terms—Color halftoning, color reproduction, discreteline, device characterization, Neugebauer model, fluorescent inkimages, juxtaposed halftoning, gamut mapping.

I. INTRODUCTION

D IGITAL color imaging has revolutionized the way wecreate, edit, and view images. Achieving consistent and

high-fidelity color reproduction in a color imaging systemrequires a deep understanding of the color characteristics ofthe various devices of the imaging system. The device we areparticularly interested in is the printer, whose availability ata low cost has greatly simplified the process of producingphysical copies of digital documents. As a result, digitalpublishing has grown to become a multi-billion dollar industry.

Proposal submitted to committee: June 28th, 2011; Candi-dacy exam date: July 5th, 2011; Candidacy exam committee:Pascal Fua, Roger D. Hersch, Sabine Susstrunk.

This research plan has been approved:

Date: ————————————

Doctoral candidate: ————————————(name and signature)

Thesis director: ————————————(name and signature)

Thesis co-director: ————————————(if applicable) (name and signature)

Doct. prog. director:————————————(R. Urbanke) (signature)

EDIC-ru/05.05.2009

Although the classical methods of color reproduction letus reproduce as accurate colors as the technology limitationallows, there are special cases which cant be addressed usingclassical halftoning and color prediction models. One suchexample is the creation of color images that are non-visibleunder daylight and visible under UV light. For this purpose,fluorescent inks should be employed which introduces newchallenges. The superposition of fluorescent inks results inquenching− a chemical effect which lowers the intensity ofthe emission spectrum and reduces the gamut of fluorescentinks, so that the traditional printing methods involving inksuperposition cannot be used. Juxtaposition of ink layers suchthat inks are printed side by side without overlapping is asolution to this problem.

In this write-up, I first give in Sections II, III, and IVa short summary of three selected references [1, 2, and 3].Section II is a short overview on device characterization.However, I mainly concentrate on detailed explanations andexamples related to our topic, i.e. printing. In section IIIthe problem of printing with fluorescent inks, with a focuson applied halftoning method, is studied. In Section IV thediscrete lines are introduced. Then, in Section V, I give a shortoverview of my proposed thesis plan: an efficient juxtaposedhalftoning algorithm based on the concept of discrete linesand its potential applications. This new research should openthe way to tackle many non-standard printing cases includingthe problem of printing with fluorescent inks. Furthermore,additional research work will be required for the extension ofthe existing color prediction models that are currently used forpredicting the spectral reflectance of classical halftones and fortheir adaptation to our new halftoning model.

II. DEVICE CHARACTERIZATION

The relationship between device-dependent and device-independent color representations for an imaging device isknown as characterization process [1]. Prior to characteri-zation, a calibration step is needed such that the device ismaintained with a fixed known color response. By calibration,one can ensure that the control values are kept at fixed nominalsettings. A common example is gray balanced calibrationwhereby equal amounts of device control values (e.g. R, Gand B for camera and C, M and Y for printer) yield device-independent responses that are neutral or gray.

Depending on the direction of the bridge built by the char-acterization function between device-dependent and device-independent space, two kinds of characterizations can bedefined. The forward characterization determines the deviceresponse to a given input, through describing the color char-acteristics of the device. The inverse characterization defines

EDIC RESEARCH PROPOSAL 2

the input to the device that is required to obtain a desiredresponse.

Although it is the inverse function which is used in thefinal imaging path to perform color correction to images,deriving the forward characterization function is more straight-forward. Having derived the forward characterization function,its inversion is a computational step which is derived throughanalytical or numerical methods.

There are two approaches to derive a forward characteri-zation function. One approach tries to establish a model de-scribing the physical process by which the device captures orrenders color. The parameters of the model are usually derivedwith a relatively small number of color samples. The secondis the empirical approach which derives the characterizationfunction using a relatively large set of color samples and sometype of mathematical fitting or interpolation techniques. Inorder to explain these approaches in more detail, we give heretwo examples, one model-based and the other empirical, ofthe forward characterization function of a printer.

A. Model-Based Printer Characterization

Several physics-based models have been developed to pre-dict the spectral response of a printer. Among them, theNeugebauer model is known for its simplicity and is the basefor some more advanced models. The Neugebauer model [4]is used to predict the spectral reflectance of a halftone print.Halftoning is a method to deal with the binary nature of theprinter, which allows for each dot only two possibilities, beingprinted or not. In a halftoning process, each primary colorantis rendered using a spatial pattern of dots. The impressionof intermediate levels is achieved by modulating the shape,size, and spatial frequency of the dots. The simplest caseconsists of a black-and-white halftone. At any given spatiallocation, we have two possible colorant combinations, blackor white. The reflectance of a halftone pattern is predicted bythe MurrayDavies equation,

R = (1− k)Pp + kPk (1)

where k is fractional area covered by the black dots and Pp andPk are reflectances of paper and black colorant, respectively.

The Neugebauer model is a straightforward extension of theMurray-Davies equation to color halftones. Since there are twopossibilities for each spatial location in binary printing, em-ploying N colorants results in 2N combinations of colorants.For example, the set of colorant combinations of C, M, Ycolorants is S = {P, C, M, Y, CM, MY, CY, CMY}, where Pdenotes paper white, C denotes solid cyan, CM denotes thecyan-magenta overprint, etc. For this case, the Neugebauermodel predicts the reflectance of the color halftone as aweighted average of the reflectances of the eight colorantcombinations

R(λ) =∑i∈S

wiPi(λ) (2)

where S is the aforementioned set of colorant combinations,Pi the spectral reflectance of the ith colorant combination(Neugebauer primary) and wi the relative area coverage of

the ith colorant combination, which is determined by thehalftoning method.

As can be seen in Eq. (2), a critical parameter whichhas a great impact on the accuracy of model is the weightsassociated to each primary. A common assumption is that thedot placements of the colorants are statistically independent.However, the reliability of this assumption is not always con-firmed. A halftone screen for which statistical independenceis often assumed is the rotated halftone screen configuration,where the screens for C, M, Y are placed at different angles,carefully selected to avoid moire artifacts. If a colorant isplaced at a particular spatial location independent of othercolorants being placed at the same location, this leads tothe Demichel equations according to which the correspondingweights of set S are given by [5]

wi ∈ {(1−c)(1−m)(1−y), c(1−m)(1−y),m(1−c)(1−y),

y(1− c)(1−m), cm(1− y), cy(1−m),my(1− c), cmy}(3)

Here, c,m, y are the fractional area coverages correspondingto digital values of C, M, Y, respectively.

An important phenomenon not modeled by the Neugebauerequations is the scattering of light within the paper. In thiscase, light that enters the paper through an area with nocolorant may leave the paper through an area that is coveredwith colorant, and vice versa. To account for this, Yule andNielsen proposed a simple correction to the Murray-Daviesmodel for a black and white print. The generalized form ofYule-Nielsen spectral Neugebauer prediction model is [5]

R(λ) = (∑i∈S

wiPi(λ)1/n)n (4)

where n is known as the Yule-Nielsen n-value. Usually, nis acquired using a set of color samples to minimize thedifference between measurements and predictions.

B. Empirical Printer Characterization

Empirical techniques generate a target of known device-dependent samples and after measuring their pairs in device-independent space, the characterization function is derivedvia data fitting or interpolation. Lattice-based interpolationis a prototype example of this class of techniques. In thecase of a CMY printer, a regular three-dimensional latticein CMY space is created. A full grid of training samplesas well as some test samples is generated and after printingthey are measured. Using the training samples, the device-dependent lattice and an irregular lattice in CIELAB spaceare obtained (Fig. 1). There is always a trade-off between thesize of the training set and the resulting accuracy. However,after a certain threshold, increasing the set size does notyield a significant reduction in the characterization error. Theforward characterization function is a set of three-dimensionallook-up-tables (LUTs) that map CMY to CIELAB space.To test the validity of the characterization function, usinga three-dimensional interpolation technique (e.g. tetrahedralinterpolation), one can find the corresponding location of a test

EDIC RESEARCH PROPOSAL 3

CMY sample in CIELAB space and compute ∆E between itsestimated and measured CIELAB coordinates.

The extension to CMYK printers is straightforward. How-ever, the lattice size (hence, the number of measurements)increase significantly and can result in a prohibitive largenumber.

III. PRINTING UNDER SPECIAL CONDITIONSThe characterization of a print process as described above

may no longer be valid under some special circumstances.Such circumstances may include (1) printing with custominks, i.e. inks other than cyan, magenta and yellow; (2)applying halftoning methods which dont respect Demichelequations (see Eq. (3)); (3) printing with non-transparentinks which cover the inks printed beneath; and (4) printingwith fluorescent inks whose superposition results in chemicaleffect known as quenching which lowers the intensity of theemission spectrum and reduces the gamut of fluorescent inks.Even if layers are printed dot-off-dot, they may have lateralsuperposition because of misregistration occurring for non-exact devices.

The work by Hersch et al [2] towards creating color imagesprinted with fluorescent inks, which are only visible under UVlight, is a good example showing how classical paradigms canbe changed under special conditions. We use this example inthe rest of this section to illustrate the problems encounteredwhen printing under special conditions. The particularity offluorescent inks is that they absorb energy in the UV wave-length range and reemit a part of it in the visible wavelengthrange.

The first challenge encountered when working with flu-orescent inks is the problem of paper. Traditionally, colorprinting relies on the fact that inks are printed on top of awhite paper. While the white paper acts both as white colorantand as a reflector which reflects the incoming light, underUV illumination non-printed paper areas are black and thereis no paper white anymore. Here, the role of the paper issimply to be a support for depositing the inks. Since thereis always a need for white colorant, the paper white needsto be replaced with another white colorant. For this purpose,all ink superpositions in all dot size combinations are printedand the achromatic colorant having the most intense emissionspectrum is selected as white colorant.

As mentioned earlier, fluorescent inks absorb light in theUV wavelength range and reemit part of it in the visiblewavelength range. Therefore, within the visible wavelengthrange, the color mixing is an additive process in contrast toclassical printing, where the subtractive mixing is in charge ofgenerating new colors.

The device characterization is needed to produce printedimages visible under UV light which have a high resemblancewith the original color images. Hence, it is necessary toestablish a common device-independent color space, e.g. theCIELAB space for fluorescent inks. The first question thatarises is what is the “white reference”. In a classical subtrac-tive system, the paper is considered as white. But here, thespectrum of the predefined white colorant multiplied by 2.5 isadopted as pseudo white stimulus. The multiplication is donebecause the generated white colorant has a maximal spectralintensity lower approximately 2.5 times than the maximalspectral intensity of the fluorescent blue ink.

In order to map the original colors into the reducedgamut of the fluorescent inks, first the gamut of fluorescentinks should be built. Deriving the color coordinates requiresprinting hundreds of patches and measuring them which istime consuming. An alternative approach is to establish andcalibrate a color prediction model which predicts the colorproduced with given surface coverages of the set of availablecolorants. Both the spectral Neugebauer and the Yule-Nielsenmodel (explained in Section II) are examined on 70 differentrepresentative fluorescent halftone test patches. Both of themyield a mean prediction error of ∆E94 = 3.5. In the case ofthe Yule-Nielsen model, the optimal Yule-Nielsen n-value isn = 0.92, a value close to 1. The Yule-Nielsen model witha value of one is identical to the Neugebauer model. Thisindicates that lateral propagation of light within the paperand multiple internal reflections (Fresnel reflections) at theboundaries between the paper surface and the air do not havea significant impact on the emitted spectra.

The gamut mapping consists in compressing the input gamutinto the limited gamut of the fluorescent inks. The propertiesexpected from an appropriate gamut mapping are to preservecolor continuity and smoothness. In other words, a continuouscolor wedge located in the original color gamut should bemapped such that the color wedge keeps the continuity in the

EDIC RESEARCH PROPOSAL 4

reduced target gamut. In addition, among different possiblemappings, preserving the original hues at least to a certainextent is preferred. Hence, colors outside the available targetgamut hues are mapped as desaturated “pseudo-grays”. Colorsinside the target gamut are preserved as close as possible tothe original colors.

The next step after gamut mapping is to determine the ap-propriate amounts of the fluorescent colorants which reproducethe mapped input color, when viewed under UV light. This isknown as color separating. Like the empirical characteriza-tion discussed in Section II but in the inverse direction, theequivalent tetrahedron of mapped input color in the devicespace is determined. Linear interpolation between tetrahedronvertices creates the correspondence between the mapped inputcolor device-independent value and the corresponding surfacecoverages of the contributing colorants.

Halftoning is the final step to render the intended fluo-rescent image. Classical color halftoning algorithms rely onmutually rotating the color screens of different colorants by30 degrees. As a result, partial overlappings of different layersare inevitable. As discussed earlier, superposing the fluorescentinks on top of each other as well as lateral overlaps becauseof misregistration should be avoided. Juxtaposed halftoning,in which colorants are printed side by side and “paper black”surrounds them is the solution to this problem proposed byHersch et al [2]. They created a library of screens suchthat there is a one-to-one correspondence between a possiblecombination of surface coverages and each screen. Havingestablished this library, halftoning an image consists in callingfor the proper member of the screen library according to colorspecifications of the target pixel. Following is a more detailedexplanation of the juxtaposed halftoning algorithm.

First, the base cells for a uniform distribution of colorantsc1, c2, c3, i.e. s1 = s2 = s3 = 1/3 is defined. Starting withsquare cells of side a, one may easily create a diagonallyoriented screen with a 3× 3 screen dot cell array, containingin one row the cells c1, c2 and c3, and in each successive rowthe same cells, but shifted by one position (Fig. 2). This screenis the base to be developed according to any input values forsurface coverages of three colorants. If there is a colorantwith a surface coverage larger than 1/3, its correspondingsquare is grown and the other colorant surface coverages areshrunk. Diagonally displacing of each dot cell array has theadvantage that a dot cell of a given colorant has the twoother colorants as two direct horizontal and vertical neighbors.Therefore, growing and shrinking the colorants is facilitatedand clustered-dots are formed.

To distribute the paper black evenly between colorant screendots, from the initial colorant surface coverages s1, s2, s3 aderived distribution s′1, s

′2, s

′3 covering the full screen surface

is first computed, such that these coverages cover the screenelement without leaving gaps. The unprinted surface partspaper is evenly distributed among the colorants, i.e.

spaper = 1− s1 − s2 − s3

s′1 = s1 +spaper

3, s′2 = s2 +

spaper3

, s′3 = s3 +spaper

3(5)

Fig. 2. A 3× 3 diagonally oriented cell array where s1 = s2 = s3 [2].

After designing the screen tiles according to derived colorantsurface coverages, each surface will be scaled down to 1 −spaper. This will ensure that the unprinted space is correctlyplaced around each colorant surface.

For each colorant, the surface coverages larger than 1/3are distributed between neighboring horizontal and verticalcells depending on the unprinted surface coverage ratio ofthe neighbors. Fig. 3 shows the case s′1 > 1/3, s′2 < 1/3,and s′3 < 1/3 where the colorant surface s′1 − 1/3 is spreadout over neighboring cells c2 and c3. Using simple geometricconsiderations, the width h of the horizontal and vertical bandsshould be computed such that the surface s′1−1/3 from cell c1spreads into horizontal and vertical neighboring cells c2 andc3, according to the ratio of 1/3−s′2 and 1/3−s′3. This leadsto a set of equations

s′12 = (s′3 −1

3)

1/3− s′22/3− s′2 − s′3

s′13 = (s′3 −1

3)

1/3− s′32/3− s′2 − s′3

w2 =√s′2, w3 =

√s′3

s′12 = h12(w2+√

1/3), s′13 = h13(w3+√

1/3), s′23 = 0 (6)

where the surfaces s′12 and s′13 represent the part of surfaces′1 spread into cells c2 and c3, respectively and h12 and h13are the necessary bandwidths to be computed to satisfy theequations. Since the sum of the surfaces of three adjacent cellsforms a unit surface, the corresponding nominal cell side isan = 1/3. When the surfaces of two colorants are larger than1/3, surfaces of both colorants spill out into one another. Witha similar set of equations, the corresponding bandwidths areobtained.

Fig. 3. Screen element surfaces growing both horizontally and verticallyover neighboring cells [2].

EDIC RESEARCH PROPOSAL 5

After partitioning the screen element according to the re-spective surface coverages s′1, s′2 and s′3, the paper blackis restored between the screen dots by scaling down eachpolygonal screen element shape so as to recover its originalsize.

To rasterize the screen elements having a discrete surfaceclose to the surface of their respective polygons, obliquepolygon borders are needed. When the rasterization is appliedon rectangles with horizontal and vertical edges, slightlyincreasing the surface coverage (width or height of colorantcells) does not lead to a slight increase of size in the generateddiscrete surfaces. In other words, there are only coarse stepsin the size of discrete surface element. As a solution, one mayrotate the initial quadratic screen cells of side a by a smallangle e.g. α = arctan 1/a. This leads to a smooth increaseof the size of the discrete element as edge lines of the celltranslates from zero to one unit in the x or respectively ydirection. Also in order to have the vertices of the discretesquares located on the grid, the side of the squares are scaledby s = (a2 + 1)1/2/a.

The members of the screen library with N + 1 intensitylevels are chosen by changing the surface coverages of threecolorants from 0 to 1 by 1/N intervals with the constraints1+s2+s3 ≤ 1. Paving the output image by an oblique screenis most efficiently done by the well-known Holladay algorithm[6]. This algorithm yields a rectangular cell such that pavingthe image with this cell is equivalent to tiling the plane withthe oblique screen. For a screen element having 3 dot cells,the corresponding rectangular tile comprises N = 3(a2 + 1).For memory requirement, a screen with a = 8, as an example,needs approximately 96 MB of memory. Fig. 4 shows a 2Dcolor wedge halftoned according to the described juxtaposedhalftoning algorithm.

Fig. 4. Color wedge halftoned according to juxtaposed halftoning algorithm[2].

IV. DISCRETE LINESThe proposed juxtaposed halftoning algorithm described in

the previous section, brings about a few limitations. First ofall, it is limited by three colorants, while in some applicationsusing more colorants is desired. Furthermore, the approach toproduce the screens is to rasterize the adjacent polygon whichmakes it difficult to reproduce the exact surface coverageson the screens. In addition, although for three colorants amemory of less than 100 MB is sufficient, when the numberof colorants increases, a huge amount of memory is required,

which excludes the idea of using a screen library. Usingdiscrete lines to create a new juxtaposed halftoning algorithmis a promising idea for tackling these problems.

A. Discrete Line Concept

The arithmetic definition of a discrete line introduced byRevellies is one of the most fundamental concepts of digitalgeometry [3]. It allows mastering the precise creation of linesof any desired thickness and orientation. A discrete line isdefined using a double Diophantine inequality as the set D ofpoints (x, y) in Z2 satisfying

D(a, b, γ, w) = {(x, y) ∈ Z2 | γ ≤ ax− by < γ + w} (7)

where all parameters a, b, γ and w are integers, a/b is theline slope, γ defines the affine offset of the discrete linewhich indicates its position in the plane, and w determinesits arithmetic thickness. A particular case of this definition isthe strictly 8-connected line with the arithmetical thicknessw equal to max(|a|, |b|) which is called a naive digital line.A particularly interesting property of a discrete line is itsb − periodicity. As it is shown in Fig. 5 for a given naiveline with parameters a and b, after b pixels in the horizontaldirection the same line segment is repeated.

Fig. 5. A discrete naive line with a slope equal to a/b = 4/7. As indicatedby the black pixels, it repeats itself each 7 pixels [3].

In order to show the usefulness of working with discreteline, we explain shortly the problem of combinatorially distinctsegments of given length in a digital line.

B. Combinatorial pieces in digital line

Studying the combinatorial pieces of a digital lineD(a, b, 0, b) consists in enumerating the number of geomet-rically distinct segments of a given length λ that can be takenout of D. For a given piece found in D we also want to findall the abscissas x such that the segment of length λ starting atx is equal to that piece. Fig. 6 shows two periods of the digitalline D(5, 17, 0, 17) and its combinatorial pieces of length 3.The classic way of dealing with the problem of combinatorialpieces of a digital line is to name directly the sequence of

EDIC RESEARCH PROPOSAL 6

differences of each two successive ordinates. By differentiatingthe y coordinates of each pixel from its left neighbor, one cancharacterize the structure of the digital line using a binarysequence (known as Sturm sequence) without ambiguity. Forexample, this sequence for the line D(5, 17, 0, 17) shown inFig. 6 is

00010010001001001

Although at first sight it seems the binary Sturm sequence isa good tool, it can become very complicated. As an alternative,Reveilles [3] proposed a very simple solution to the question ofcombinatorial pieces of a digital line using only the arithmeticof the two integers a and b. Besides the Sturm binary coding,there is an algebraic coding using the function

r(i) = {aib} (8)

where the curly bracket denotes the Euclidean remainder. Wecall r the remainder function of the digital line D; note that itonly depends on a and b. For example, the remainder functionof the line D(5, 17, 0, 17) as a sequence is

0, 5, 10, 15, 3, 8, 13, 1, 6, 11, 16, 4, 9, 14, 2, 7, 12

At a first glance, we see that there are 5 increasing subse-quences (0, 5, 10, 15), (3, 8, 13), (1, 6, 11, 16), (4, 9, 14) and(2, 7, 12) whose lengths are in one-to-one correspondence withthe horizontal steps of the line. Some other results describingthe advantage of using remainder function of line D(a, b, 0, b)are as follows:

- Each horizontal step starts at the integer abscissa i iff0 ≤ r(i) < a.

- The increasing subsequences of r(i) are equal to [b/a]or [b/a]+1. Where the square bracket denotes Euclideandivision.

- The first increasing subsequence is long, the last one isshort.

- The sequence of first values of the increasing sequencer′(i) is

r′(i) = {(a− { b

a})i

b} (9)

The local maxima of the subsequence r(i) indicates where theends of the horizontal steps are located. On the other hand,the upper bound of r(i) is the upper convex hull of the set ofpoints (i, r(i)). From these observations, Reveilles [3] showedthat there are λ combinatorially different segments of length

λ. The set of segments defined by the interval [m,n] wherem and n are two positive integers is a complete system whichrepresents the combinatorial pieces.

V. THESIS PLAN

We intend to extend the idea of juxtaposed halftoning toa new halftoning method which creates side-by-side laid outcolorant lines, with no theoretical limit on the number ofcolorants. We found out that the discrete line geometry is apromising tool for designing halftone screens such that eachcolorant is defined as an individual discrete line and eachline is juxtaposed in the screen. As a result, the screen is aparallelogram which should pave the output image, accordingto the well-known Holladay algorithm [6] (Fig. 7). When itis needed to generate long lines, one can take advantage ofthe b-periodicity of discrete lines. This is fast and also has thebenefit of robustness against accumulative error.

Fig. 7. A juxtaposed halftone screen with 7 different colorants. The discreteline slope is a/b = 4/7. The corresponding parallelogram screen is alsoshown.

For a black-and-white halftone, N screens correspondingto the N gray levels using discrete lines with thicknessescorresponding to each gray value are generated level by levelfrom 0 to 1 with N intervals, and stored into a LUT. To extendthe algorithm to multiple colorant layers, one might computethe corresponding halftone screen for each combination ofcolorants and build the associated LUT. As we mentioned,this results in a huge LUT and a very complex algorithm.To solve this problem, we try to compute the correspondinghalftone screen for any combination of colorants using the

EDIC RESEARCH PROPOSAL 7

LUT of a single colorant. Once the LUT of a single coloranthas been generated, the task of computing the screen of thesecond (and generally the nth) colorant can be accomplishedin constant time with a few simple operations. Since the goalis to juxtapose different layers side by side, every new colorantcan be produced by subtracting two screens of the single LUT.The first screen is one that corresponds to the summation of allpresent surface coverages and the second is that of all surfacecoverage minus the surface coverage of the target colorant.The idea is fairly similar to summed-area tables that are usedfor texture computation and real-time face detection.

However, the algorithm raises a number of challenges to besolved in the future. The fact that each screen is a combinationof a few lines does not give good clusters in the halftone, aproperty which is highly desired when working with deviceswith limited ability to print individual dots. Another problemis auto moire. Auto moire, also known as internal moire, isa phenomenon which occurs due to interference between thehalftone grid and the device grid, and it is different from thecommon moire effects between several screens. Since in ouralgorithm the same structure repeats periodically, auto moirein the form of a visually objectionable artifact may appear.

We intend to investigate the applicability of our method forseveral interesting areas. Printing with any combination andnumber of custom inks opens the way for entirely new printingstrategies. However, it also raises new challenges. An accuratecharacterization of a device which uses custom inks is verychallenging, mostly because of the effects of trapping and theinterreflection of light between ink layers. These effects bythemselves are a result of overprinting multiple inks usingclassical halftoning methods.

We believe that juxtaposed halftoning can be a solutionwhich, initially, frees the printing system from the ink trans-parency restriction. Besides from fluorescent inks, printingwith metallic or any other opaque inks where the superpositionof layers is not possible is a potentially interesting researcharea. Another important benefit of our method is that thesurface coverages are controlled. Traditionally, the validityof Demichel equations is taken for granted whereas it is notalways true [7].

A Further research subject will consist of creating newspectral prediction models which account for the newly definedcircumstances.

ACKNOWLEDGEMENT

I am grateful to Isaac Amidror for reading my write-up andfor helpful discussions. Also, I thank Ehsan Kazemi for hisassistance in preparing the LATEX document.

REFERENCES

[1] R. Bala, “Device Characterization”, in Digital Color ImagingHandbook, Ed. G. Sharma, CRC Press, pp. 269-382, 2003.

[2] R. D. Hersch, P. Donze, and S. Chosson, “Color images visible underUV light”, ACM. Trans. Graphics (Proc. ACM SIGGRAPH),vol. 22, no. 3, pp. 427-436, 2007.

[3] ] J. P. Reveills, “Combinatorial pieces in digital lines and planes”, SPIEV ision Geometry IV , 2573 pp. 23-34, 1995.

[4] W. Rhodes, “Fifty years of Neugebauer equations”, SPIE Vol. 1184:Neugebauer memorial seminar on colour reproduction, vol.1184, pp. 7-18, 1989.

[5] Yule, J. A. C., Principles of Color Reproduction: Applied toPhotomechanical Reproduction, Color Photography, and theInk, Paper, and Other Related Industries, John Wiley and Sons,New York, 1967.

[6] T. M. Holladay, “An optimum algorithm for halftone generation fordisplays and hard copies”, in Proc.SID, vol. 21, pp. 185-192, 1980.

[7] I. Amidror, R. D. Hersch, “Neugebasuer and Demichel: dependence andindependence in n-screen superpositions for colour printing”, Col ResAppl, vol. 25, no. 4, 2000.