Least-squares model-based halftoning - Image Processing ...webstaff.itn.liu.se/~sasgo26/Dig_Halftoning/Literature/Least_Square.pdf · 1102 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL

1102 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 8, NO. 8, AUGUST 1999

Least-Squares Model-Based HalftoningThrasyvoulos N. Pappas,Senior Member, IEEE,and David L. Neuhoff,Fellow, IEEE

Abstract—A least-squares model-based (LSMB) approach todigital halftoning is proposed. It exploits both a printer modeland a model for visual perception. It attempts to produce anoptimal halftoned reproduction, by minimizing the squared errorbetween the response of the cascade of the printer and visualmodels to the binary image and the response of the visualmodel to the original gray-scale image. It has been shown thatthe one-dimensional (1-D) least-squares problem, in which eachrow or column of the image is halftoned independently, can beimplemented using the Viterbi algorithm to obtain the globallyoptimal solution. Unfortunately, the Viterbi algorithm cannot beused in two dimensions. In this paper, the two-dimensional (2-D)least-squares solution is obtained by iterative techniques, whichare only guaranteed to produce a local optimum. Experimentsshow that LSMB halftoning produces better textures and higherspatial and gray-scale resolution than conventional techniques.We also show that the least-squares approach eliminates most ofthe problems associated with error diffusion. We investigate theperformance of the LSMB algorithms over a range of viewingdistances, or equivalently, printer resolutions. We also show thatthe LSMB approach gives us precise control of image sharpness.

Index Terms— Facsimile, halftoning, least-squares,model-based, printer, visual perception.

I. INTRODUCTION

DIGITAL halftoning is the process of generating a patternof binary pixels that create the illusion of a continuous-

tone image. The term spatial dithering is also used to refer tothis process. Digital halftoning is necessary for display of gray-scale images in media in which the direct rendition of graytones is impossible. The most common example is printing ofgray-scale images on paper. In this paper, we propose a least-squares model-based (LSMB) approach to digital halftoningthat exploits both a printer model and a model for visualperception to produce high quality images using standard laserprinters (typically 300 dpi).

Model-based halftoning can be especially useful in trans-mission of high quality documents using high fidelity gray-scale image encoders [1], [2]. In such cases, halftoning isperformed at the receiver just before printing. Apart fromcoding efficiency, this approach permits the halftoner to betuned to the individual printer, whose characteristics may vary

Manuscript received March 5, 1996; revised September 10, 1998. Portionsof this work were published in the Proceedings of the 7th Workshop onMultidimensional Signal Processing, Lake Placid, NY, September 1991, andin Proceedings of the SPIE, Human Vision, Visual Processing, and DigitalDisplay III, February 1992, vol. 1666, San Jose, CA. The associate editorcoordinating the review of this manuscript and approving it for publicationwas Dr. Ping Wah Wong.

T. N. Pappas is with Bell Laboratories, Lucent Technologies, Murray Hill,NJ 07974 USA (e-mail: [email protected]).

D. L. Neuhoff is with the Department of Electrical Engineering andComputer Science University of Michigan, Ann Arbor, MI 48109 USA.

Publisher Item Identifier S 1057-7149(99)06001-7.

considerably from those of other printers, for example, write-black versus write-white laser printers.

Halftoning relies on the fact that the eye acts as a lowpassfilter. More detailed models of visual perception have beendeveloped, and we show that it is possible to implementhalftoning methods that exploit them. The eye models we useare based on estimates of the spatial frequency sensitivity ofthe eye, e.g., by Mannos and Sakrison [3].

Halftoning also relies on the assumption that the surface areaof the black portion of a printed binary pattern is proportionalto the intended fraction of black dots (pixels) in the pattern.This means that the area occupied by each black dot is roughlythe same as the area occupied by each white dot. Thus, the“desired” shape for the black dots produced by a printer wouldbe squares, where is the dot spacing. However,most printers produce approximately circular dots [4]. Theirradius must be at least so that they are capable ofblackening a page entirely. Thus, black dots cover portionsof adjacent spaces, causing the perceived gray level to bedarker than intended. We refer to this fact as “dot overlap.”Moreover, most printers produce black dots that are larger thanthe minimal size, which further distorts the perceived graylevel. The most commonly used digital halftoning techniques,such as clustered ordered dither, protect against dot overlapby clustering black dots so the percentage effect on perceivedgray level is reduced. Unfortunately, such clustering constrainsthe spatial resolution of the perceived images and increases thelow-frequency artifacts. The proposed method exploits printerdistortions in order to increase both gray-scale and spatialresolution. A key element in such a method is an accurateprinter model.

Several authors have previously proposed model-basedhalftoning approaches. Allebach [5] proposed halftoning soas to minimize a visual model based distortion measure,but did not give specific methods for doing so. He alsosuggested the use of a display model but did not givea specific one. Anastassiou [6] proposed halftoning so asto minimize a frequency weighted squared error betweenthe binary and original images. The latter is equivalent toour formulation except that we incorporate a printer modeland permit nonlinearity in the eye model. Anastassiou andKollias [6], [7] also described a neural network techniquethat approximately minimized the frequency weighted squarederror. Roetling and Holladay [8] proposed a dot-overlap printermodel, like the one used in this paper and [9], [10], but usedit only to modify ordered dither so that it results in a lineargray scale. Allebach [11] also proposed a modification ofordered dither that takes into account dot overlap to improvethe gray scale of the printed images. However, ordered dither

1057–7149/99$10.00 1999 IEEE

PAPPAS AND NEUHOFF: HALFTONING 1103

is a highly constrained method that cannot achieve the highspatial resolution, increased gray-scale resolution, and visuallypleasant textures of the proposed model-based approach, nor ofthe modified error diffusion mentioned in the next paragraph.Another technique that takes into account dot overlap wasproposed by Pryoret al. [12]. Its performance is also inferiorto the proposed method.

In [9] and [10] it was shown that printer models, especiallythe dot-overlap model, can be used to modify error diffusionto account for printer distortions. The resultingmodified errordiffusion(MED) algorithm provides high quality reproductionswith reasonable complexity. However, error diffusion does notmake use of an explicit eye model, and this results in somewell known artifacts and asymmetries. The first to propose theuse of a dot-overlap model to account for printer distortionsin error diffusion was Stucki [13], [14]. In [10], it was foundthat Stucki’s algorithm is more efficient computationally,while the MED algorithm has better performance. Recently,Rosenberg [15] used our dot-overlap printer model to derivea gray-scale transformation that can be applied to the imagebefore halftoning to compensate for printer distortions. Thisapproach works only when the printer introduces a monotonicnonlinearity, and cannot match gray levels as precisely as themodel-based approaches examined here and in [10].

Finally, in [16] and [17], it was shown that the parame-ters of sliding-window printer models can be identified frommeasurements of the reflectance of various test patterns. Thesewere then used in the MED and LSMB approaches.

The LSMB halftoning approach attempts to produce anoptimal halftoned reproduction, by minimizing the squarederror between the output of the cascade of the printer andvisual models in response to the binary image and the outputof the visual model in response to the original gray-scaleimage. In [18] and [19], we showed that the one-dimensional(1-D) least-squares problem, in which each row or columnof the image is halftoned independently, can be implementedwith the Viterbi algorithm. The Viterbi algorithm provides anefficient way to search the solution space and leads to a globaloptimum in a finite number of steps. Unfortunately, the Viterbialgorithm cannot be used in two dimensions.1 In this paper,the two-dimensional (2-D) least squares solution is obtainedby iterative techniques, which are only guaranteed to producea local optimum. The visual quality of the resulting halftoneimages depends on the starting point and the optimizationstrategy. Experiments show that LSMB halftoning can produceimages with high gray-scale and spatial resolution and pleasingtextures. We show that the least-squares approach eliminatesmost of the problems associated with error diffusion. However,despite these problems, the performance of the MED algorithmis remarkably good. As we will see, MED provides thebest starting points for the LSMB algorithms which mostlypreserve its texture. In fact, when one takes into considerationboth computational complexity and image quality, the MEDalgorithm provides the best performance. The main valueof the LSMB approach is as a tool for learning how wellhalftoning can do under various circumstances. It gives us

1Wong [20] used the Viterbi algorithm to solve a 2-D halftoning problembut the solution is not optimal.

the ability to study how the optimal halftoning patterns varywith eye and printer models. For example, we investigatethe performance of the LSMB algorithms over a range ofviewing distances (or equivalently, printer resolutions2). Asthe viewing distance increases, the cutoff frequency of theeye model decreases, which means it is less sensitive tothe “coarseness” of the halftone texture. Thus, more coarselytextured halftone patterns can be considered. This raises thequestion of whether more coarsely textured patterns actuallyare needed for optimality.3 If so, then the resulting halftoneswill depend on the viewing distance, while otherwise, theywill be fairly robust. We also show that the LSMB approachgives us precise control of image sharpness.

The error criterion of the LSMB approach provides ameasure of halftone image quality that takes into accountthe characteristics of both the display device (printer) andthe human visual system and can be used to evaluate theperformance of any halftoning technique. Our results indicatethat the LSMB error metric agrees remarkably well withvisual evaluations of image quality but we also point out itslimitations. Overall, it provides a powerful tool for algorithmdevelopment and evaluation.

In addition to [6], the LSMB approach has been recentlypursued by several authors. The LSMB approach, includingboth eye and printer models, was presented by the presentauthors in [22] and [23]. In similar efforts, Mulligan andAhumada [24] assumed perfect printing, and Analoui andAllebach [25] considered a different display model. Carraraetal. [26] considered variations of the scheme presented in [25]and also adopted our dot-overlap printer model, while Flohretal. [27] investigated computational issues. Zakhoret al. [28],[29] also assumed perfect printing, but the performance of theapproach they proposed is inferior to the above techniquesbecause it breaks the image into small blocks and halftonesthem independently. Another approach that breaks the imageinto blocks and uses an eye model and perfect printingwas presented by Vander Kamet al. [30]. Finally, a relatedtechnique using Markov random fields was presented by Geistet al. [31].

The paper is organized as follows. Section II discusseseye models. The printer models are presented in Section III.Section IV presents the least-squares model-based halftoningapproach. The experimental results are presented in Section Vand the conclusions are summarized in Section VI.

II. M ODELS OF VISUAL PERCEPTION

As mentioned in the introduction, halftoning works becausethe eye acts as a spatial lowpass filter. In this section we reviewthe spatial frequency characteristics of the eye and introduce amore detailed model that will be exploited by the least-squaresapproach of Section IV. A similar discussion can be found in[18] and [19]. Parts of that discussion are repeated here forcompleteness.

2Strictly speaking, changing the viewing distance is equivalent to changingthe printer resolution only if the characteristics of the printed dots are notaffected when the printer resolution is changed.

3This question was also addressed in [21].


Numerous researchers have estimated the spatial frequencysensitivity of the eye, often called the modulation transferfunction (MTF). Typical of such is the estimate that Mannosand Sakrison [3] found to be good for predicting the subjectivequality of coded images

(1)

where is in cycles/degree. This MTF is plotted in Fig. 1in dotted line. As indicated by (1), the eye is most sensitiveto frequencies around 8 cycles/degree (others have variouslyestimated the peak sensitivity to lie between 3 and 10 cy-cles/degree [32, p. 55]). The decrease in sensitivity at higherfrequencies is generally ascribed to the optical characteristicsof the eye, (e.g., pupil size) [33]. The decrease in sensitivityat low frequencies accounts for the “illusion of simultaneouscontrast” (a region with a certain gray level appears darkerwhen surrounded by a lighter gray level than when surroundedby a darker) and for the Mach-band effect (when two regionswith different gray levels meet at an edge, the eye perceivesa light band on the light side of the edge and a dark band onthe dark side of the edge).

The eye is more sensitive to horizontal or vertical sinusoidalpatterns than to diagonal ones [34]. Specifically, it is leastsensitive to 45 sinusoids, with the difference being about 0.6dB at 10 cycles/degree and about 3 dB at 30 cycles/degree.This difference is not considered to be large, but it is usedto good effect by the “Classical” halftoning technique [35],which produces patterns with a diagonal structure.

Besides the MTF, many other characteristics of the humanvisual system have been studied. A recent, comprehensivereference is [36]. Based on such characteristics, a varietyof visual models have been proposed for use in variousimage processing tasks. The simplest, arguably, include justa filter, for example the filter of (1). The next simplestinclude also a memoryless nonlinearity, followed by a filter.Such nonlinearities account for Weber’s law, which says thatthe smallest noticeable change in intensity is proportional tointensity. Most commonly it is represented as a logarithm orpower law. More complex models include, for example, a filterbefore the nonlinearity or a bank of filters.

In many cases, practical considerations dictate a finiteimpulse response (FIR) filter. Indeed, for the least-squareshalftoning of Section IV, we require a discrete-space modelof the form

(2)where the ’s are samples of the image, the ’s are themodel outputs (upon which cognition is based), andis somesliding-windowfunction ( is a nonnegative integer). Such amodel can easily incorporate a memoryless nonlinearity andan FIR filter. We experimented with models of the form

(3)

where is a memoryless nonlinearity, is the impulseresponse of an FIR filter, anddenotes convolution. One wayto obtain a 2-D filter is as a separable combination of 1-D FIR filters. Fig. 1 shows (in solid lines) the frequency and

Fig. 1. Spatial frequency sensitivity of the eye (according to Mannos andSakrison and according to Daly) and frequency and impulse response of the1-D best-fit model (for printer resolution of 300 dpi and viewing distance of30 in).

impulse responses of a 1-D FIR approximation to the MTF of(1) [18], [19]. Note that the impulse response is specified indegrees of visual angle. The spacing of the dots is given by

radians degrees

(4)where is the printer resolution in dpi and is the viewingdistance. The sampling frequency is . The filter ofFig. 1 was designed for a printer resolution of 300 dpi and aviewing distance of 30 in.

Note that the frequency response of the filter of Fig. 1 doesnot decrease at low frequencies, that is, it does not modelthe Mach-band effect. This would require a considerablylonger FIR filter which would increase the complexity of thealgorithms we consider. Moreover, as we will see in Section V,higher order filters, that do capture the Mach-band effect, havevery little effect in the resulting images. In any case, it isthe lowpass characteristic of the eye that is necessary forhalftoning to work. The fact that the eye is not sensitive tovery low frequencies does not have to be used. Note also thatthe halftoning algorithm should not compensate for the Mach-band effect, i.e., the eye should perceive the same Mach-bandeffect from the halftoned images as it does from the originalcontinuous-tone image.

Similar FIR filters can be designed for different printerresolutions and viewing distances. Figs. 2 and 3 show theimpulse and frequency responses of a sequence of such filtersdesigned to match the (flattened) Mannos and Sakrison MTFfor samples taken at 300 dpi and viewing distances of 12, 24,and 36 in. Since the sampling is coarser at shorter distances,the order of the FIR filter is lower, and more importantly, theapproximation is cruder. We will refer to these filters as the1-D best-fit models.


Fig. 2. Impulse response of eye models at different viewing distances.

We have found that the impulse responses of the 1-Deye filters we designed above are well approximated by aGaussian shape with appropriate standard deviation. In fact,as was pointed out in [37], at 300 dpi and a viewing distanceof 30 in, the impulse response of the 1-D eye filter and aGaussian filter with are virtuallyidentical. This is convenient because one can simply scalethe standard deviation to obtain Gaussian filters for differentviewing distances. Equivalently, one can sample (at multiplesof ) a Gaussian with standard deviation fixed at .Note that the 2-D Gaussian filters are circularly symmetric.In addition to the 1-D best-fit models, Figs. 2 and 3 showthe impulse and frequency responses of such a sequence of(truncated) Gaussian filters that match the (flattened) Mannosand Sakrison MTF for samples taken at 300 dpi and viewingdistances of 12, 24, and 36 in. At 12 in, the Gaussian filterdoes not match the Mannos and Sakrison MTF as well. Thisis due to aliasing; the sampling is too coarse at this viewingdistance. A better approximation can be obtained by using theseparable combination of the 1-D best-fit models. However,our experiments with the techniques of Section IV indicatethat the two filters result in very similar halftone images.

Circularly symmetric Gaussian filters were also used in [24].(They were truncated to 5 5 with standard deviations rangingfrom 0.2–1.6 pixels.) Finally, Gaussian filters were used in thevoid and cluster method for blue-noise screen design [38]. Thestandard deviation of the filter was chosen to be 1.5 pixels,which interestingly, corresponds to a viewing distance of 30in at 300 dpi.

Fig. 3. Frequency response of eye models at different viewing distances.

Fig. 4. Actual dot overlap.

The frequency response of the eye filters we consider hereis also very close to the MTF proposed by Daly [39], [40].This can be seen in Fig. 1, where Daly’s MTF is plotted as adashed line. Daly’s model was used in [25]–[27].4 Daly’s MTFalso models the variations in the visual MTF as a function ofviewing angle (pattern orientation). The FIR filter of (3) couldbe designed to account for such variations. However, unlike[41], our experiments with the techniques of Section IV didnot show a significant effect on the resulting halftone patternswhen a filter with angular dependence is used. Thus, the

4Note that Daly’s model (like Mannos and Sakrison’s) is specified in thefrequency domain; the FIR filter specification in the spatial domain wasobtained by the inverse 2-D fast Fourier transform (FFT) of the sampledfrequency domain function.


Fig. 5. Definition of�, �, and for printer model.

circularly symmetric Gaussian filters seem to be a reasonablesimplification. In the remainder of this paper we will usethe Gaussian approximation because it is easier to adapt todifferent viewing conditions. We will assume that the printerresolution is fixed and will consider a sequence of eye filterscorresponding to different viewing distances. Our approachis also appropriate for a fixed viewing distance and variousprinter resolutions.

Finally in [18] and [19], we considered a nonlinearity in theform of a power law, and found that it offers no significantimprovements. As we will see in Section V, the same is truein this case.

III. PRINTER MODELS

In this section, we review the printer models that weredescribed in [9], [10]. The printer models are independent ofthe characteristics of the human visual system. Our work isoriented to high resolution laser printers. We have used “write-black” laser printers with 300 dpi resolution (such as the HPLaserJet II) as our test vehicles. However, the discussion andmethods are intended to apply directly, or be adaptable, to awide variety of printers.

To a first approximation, such printers are capable ofproducing black spots (usually called dots) on a piece of paper,at any and all sites of a Cartesian grid with horizontal andvertical spacing of in. The reciprocal of is generallyreferred to as theprinter resolution in dpi. The printer iscontrolled by an binary array , whereindicates that a black dot is to be placed atsite whichis located inches from the left and inches from the topof the image, and indicates that the site is to remainwhite.

As discussed in the introduction, printers produce black dotsthat are more round than square. Fig. 4 illustrates the mostelementary distortion introduced by most printers: their dotsare larger than the minimal covering size, as if “ink spreading”occurred.5 A variety of other distortions are present in actualprinters, caused by the heat finishing, reflections of light withinthe paper, and other phenomena. As a result, the gray levelproduced by the printer in the vicinity of site dependsin some complicated way on and neighboring bits. Thepurpose of a printer model is to predict this dependence. In[9] and [10], we proposed a discrete space printer model, theinput of which is the binary array and the output an array

5Note that laser printers use toner, not ink, as colorant. For historical reasonswe sometimes still refer to toner as ink.

Fig. 6. Dot-overlap model with� = 0:33, � = 0:029, and = 0:098.

of gray-scale values, where represents the darknessof site . Specifically, the model has the form

(5)

where denotes the bits in some neighborhood ofand is some function thereof. This discrete space modeloffers considerable computational advantages. The underlyingassumptions of the above discrete space model can be mademore concrete when one considers a continuous space versionof the model, as was done in [10].

For the least-squares approach, it is essential thatbeentirely determined by the bits in a finite window around ,e.g., a 3 3 window. In this case, the possible values ofcan be listed in a table, e.g. with elements. The individualelements of this table might be derived from a detailed physicalunderstanding of the various phenomena affecting gray levelor from measurements of the gray level due to various dotpatterns. An example of a printer model based on the firstapproach is given below. Alternative printer models based onthe second approach are explored in [16] and [17].

A simple printer model that accounts for the “dot-overlap”distortion illustrated in Fig. 4 is thecircular dot-overlapmodel[9], [10]

ifif (6)

where the window consists of and its eight neigh-bors, is the number of horizontally and vertically neigh-boring dots that are black, is the number of diagonallyneighboring dots that are black and not adjacent to anyhorizontally or vertically neighboring black dot, and is thenumber of pairs of neighboring black dots in which one is ahorizontal neighbor and the other is a vertical neighbor, andwhere , , and are the ratios of the areas of the shadedregions shown in Fig. 5 to .

The parameters, , and can be expressed in terms of theratio of the actual dot radius to the ideal dot radius [9],[10]. Note that the model is not linear in the input bits, whichis due to the fact that the paper saturates at black intensity. Fora printer with the ideal dot size, , the minimum value,and , , and . For the HP printer, ourexperience indicates that , which results in ,

, and . Fig. 6 illustrates how the dotpattern in Fig. 4 is modeled with these values.


Fig. 7. Printer and visual perceptual models.

In [9] and [10], we showed that (assuming that the aboveprinter model is valid) dot overlap can be exploited to obtainmore gray levels than could be obtained without it. Moreover,as we will show in the next sections, this can be done withoutsacrificing spatial resolution.

IV. L EAST-SQUARES MODEL-BASED HALFTONING

As mentioned earlier, the least-squares model-based(LSMB) halftoning approach attempts to minimize the squarederror between the output of the cascade of the printer andvisual models in response to the binary image and the outputof the visual model in response to the original gray-scaleimage. The least-squares approach is noncausal. That is, thedecisions at any point in the image depend on “past” as wellas “future” decisions. In standard error diffusion, the decisionsat any point in the image depend on the “past” only. It is thisnoncausality of the least-squares approach that gives it thefreedom to make sharp transitions and track edges better thanpreviously described approaches.

The 2-D least-squares halftoning problem can be formulatedas follows. Suppose we are given a gray-scale image ,with , and , where is thewidth of the printable lines in dots and is the number ofprintable lines. We assume that the image has been sampledso there is one pixel per dot to be generated.6 The gray level

of each pixel varies from 0 white to 1 black. We arealso given a printer model with the sliding-window form of(5), and an eye model of the form specified in Section II witha memoryless nonlinearity followed by an FIR filter withimpulse response . In the LSMB approach we seek thebinary approximation that minimizes the squared error

(7)

where, as illustrated in Fig. 7

(8)

(9)

(10)

6When the number of samples of a given image is different from thenumber of dots to be generated, interpolation is necessary.Bilinear andsplineinterpolation can produce an image of any size. However, for sampling rateconversions by a rational factor, the best results are obtained by anexpanderfollowed by an equiripple lowpass FIR filter [42 pp. 105–109].

Fig. 8. Original test images.

and indicates convolution. The boundary conditions are

for (11)

for (12)

These boundary conditions assume that no ink (toner) is placedoutside the image borders.

Note that we have allowed different impulse responses,for the eye filters corresponding to the continuous-tone

and halftone images. We found that when we do not filter thegray-scale image, the resulting halftone images are sharper.7

This was suggested by Anastassiou in [6]. In fact, as we willsee in the next section, one can consider a whole range offilters that result in different amounts of sharpening.

In principle, the optimal solution can be obtained by anexhaustive search over all possible binary patterns for theentire image. This approach is not computationally feasible,however. The number of possible patterns for animage is , (e.g., just for a 16 16 image).Thus, the 2-D least-squares solution is obtained by iterativeoptimization techniques. Such techniques find a solution thatis only a local optimum. They assume that an initial estimateof the binary image is given. This could be a trivialimage, e.g., a constant or random image, or the output ofany halftoning algorithm including the MED [9], [10] andthe 1-D LSMB algorithm [18]. The visual quality of theresulting halftone images depends on the starting point andthe optimization strategy.

A simple iterative scheme, initially presented in [22] and[23], updates the values of the binary image one pixel at atime, in the following way. Given an initial estimate , forevery image site (in some fixed or random order, usually

7In effect, we are prefiltering the continuous-tone image with the inverseof the eye filter, i.e., a highpass filter.


(a) (b) (c)

Fig. 9. Details of 300 dpi printouts (magnified by 3). (a) Classical screen. (b) MP-MED with FS filter. (c) MP-MED with JJN filter.

a raster scan), we find the binary value that minimizes thesquared error

(13)

where is a neighborhood of the point . If theeye filter is FIR and the neighborhood is chosen largeenough,8 then choosing to minimize is equivalentto choosing to minimize the overall error . In otherwords, when the eye filter is FIR, the binary value of eachpixel affects only the model outputs in its neighborhood,and thus, the error need only be computed locally. Here weshould mention that the value of should not be computedfrom scratch each time; the value of never changes andonly a few terms in the convolution sum that determinesmust be updated (those that involve and its neighbors).An iteration is complete when the minimization is performedonce at each image site. The number of iterations depends onthe starting point and the effective filter width. Usually, thewider the eye filter, the more the required iterations. We willrefer to the above optimization procedure as thetoggle-onlyscheme,because in comparison to schemes described later, theonly permitted change at each image site is toggling thevalue of . As we will see in the next section, when thetoggle-only scheme is used, the dependence on the startingpoint is strong.

A similar scheme was proposed by Mulligan and Ahumadain [24]. In [6], Anastassiou proposed a similar optimizationprocedure using neural nets. Note that, even though the erroris computed locally, there is a strong dependence on theneighboring pixels that can propagate over large areas. Hencebreaking up the image into blocks and optimizing themindependently (as in [28] and [30]) could result in significantdeviations from optimality.

Selecting a trivial neighborhood that consists of justthe point , reduces the amount of computation but is not

8Let Sh be the support of the eye filterh andSp be the support of theprinter model. ThenBi; j must be at least as large as�Sh � Sp, where�denotes dilation and�Sh is the transpose of the setSh.

TABLE IPER PIXEL SQUARED ERROR FORDIFFERENT HALFTONING TECHNIQUES

FOR 300 DPI PRINTER RESOLUTION AND VIEWING DISTANCE OF 30 IN

guaranteed to reduce the overall error, especially for narroweye filters. For wider eye filters, i.e., at 300 dpi and viewingdistance of 30 in, the results are almost as good as thoseobtained with the large neighborhood [23]. In the remainderof this paper, however, we will assume that the neighborhood

is large enough to guarantee that minimizing isequivalent to minimizing , as discussed above.

In order to obtain lower errors and better results, we con-sidered a variation of the above iterative technique, whereby,each minimization is performed over a small set of adjacentpixels. That is, given an initial estimate , for each imagesite , we find the binary values in a neighborhood

of that minimize the squared error ,as expressed in (13).9 However, the amount of computationincreases exponentially with the number of pixels in . Inorder to control the amount of computation, Carraraet al.[26], suggested testing only a small subset of all possiblebinary patterns in the neighborhood . Specifically, theychose to be a 3 3 window surrounding , andtested only ten of the possible binary patterns: the currentpattern as is, the pattern formed by toggling the value of

, and the eight patterns formed by swapping the valueof with the value of one of its eight neighbors in .

9Again, Bi; j must be large enough to guarantee that minimizingEi; j isequivalent to minimizingE .


(a) (b)

Fig. 10. LSMB halftoning (toggle-only scheme) at 300 dpi (magnified by 3). (a) Random start. (b) MP-MED with JJN filter start.

(a) (b) (c)

Fig. 11. LSMB halftoning (toggle-only scheme) for different viewing distances at 300 dpi (magnified by 3). (a) One ft (MP-MED with FS filter start).(b) Two ft (MP-MED with JJN filter start). (c) Three ft (MP-MED with JJN filter start).

This scheme is called thetoggle/swap scheme.Analoui andAllebach [25] considered swaps only. Flohret al. [27] showedthat the toggle/swap scheme produces higher quality imagesthan the toggle-only scheme. Our results presented in the nextsection are consistent with this. They also indicate weakerdependences on the starting point.

As we pointed out above, the images produced using theabove methods are only local minima of the least-squaresproblem. There could in fact be several local minima for agiven image, depending on the starting point, as we showin the next section. More sophisticated (and computationallyintensive) schemes use simulated annealing [24], [37] but havenot yet shown any significant improvements in image quality.

Finally, in an attempt to minimize the dependence on thestarting point, we tried the following “progressive” scheme:we used the LSMB estimate at a given viewing distance as astarting point for the next higher viewing distance, starting at12 in with 6-in increments. We used the multipass MED10

with a Floyd–Steinberg (FS) filter as the original starting

10The multipass MED will be defined in the next section.

point, and the simple iterative scheme with toggle/swap ateach viewing distance. Even though this scheme requires aconsiderable amount of computation, we use it in an attemptto find alternative minima and to investigate the sensitivity ofthe results to the optimization strategy.

The error criterion of the LSMB algorithm can be usedto evaluate the performance of any halftoning technique (seethe next section and [37]). It provides a measure of halftoneimage quality that takes into account the characteristics of boththe display device (printer) and the human visual system. TheLSMB image quality metriccan be computed over the wholeimage or over a small segment of the image to provide alocalized measure of performance. Thus, we can define

(14)

where and are given by (9) and (10) and is a subsetof the image containing pixels. Usually does not includethe boundary pixels to avoid biases due to edge artifacts. In


(a) (b) (c) (d) (e) (f)

Fig. 12. MP-MED and LSMB (toggle-only scheme) halftoning at 300 dpi (magnified by 3). (a) MP-MED with FS filter, (b) MP-MED with JJN filter.(c) MP-MED with JJN filter with random perturbations. (d) LSMB at 1 ft (MP-MED with FS filter start). (e) LSMB at 2 ft (MP-MED with JJN filterstart). (f) LSMB at 4 ft (MP-MED with JJN filter with random perturbations start).

the next section, this error criterion is used to compare othermethods to LSMB halftoning.

V. EXPERIMENTAL RESULTS

We tested the LSMB approach on several images and com-pared it to other schemes. Fig. 8 shows two of those images.

The resolution of the Lena image is 512 512 pixels andthe resolution of the double gray-scale “Ramp” is 78880pixels. The gray-scale resolution of the original images is 8bits/pixel. The images in Figs. 9–14 are magnified details (bya factor of three) of the halftone images as they would beprinted on a 300 dpi laser printer. They were printed using


simulated dot overlap with at one third of the printerresolution.

We evaluate the performance of the LSMB technique interms of 1) spatial resolution, (i.e., sharpness), 2) texture, (i.e.,visibility of halftone patterns), 3) gray-scale resolution, (i.e.,number of perceived gray levels), and 4) gray-scale distortionof the halftoned images. As we saw in [19], some of theseattributes can be interdependent.

Fig. 9(a) shows a magnified detail of one of the test imageshalftoned using a clustered ordered dither scheme (“classical”),and Fig. 9(b) and (c) show the image halftoned with themodified (to compensate for dot-overlap) error diffusion algo-rithm (MED) [9], [10], using the FS and Jarvis–Judice–Ninke(JJN) filters, respectively. The figure shows the results of themultipassversion of the modified error diffusion algorithm(MP-MED) which was introduced in [16] and [17]. As wasshown in [16] and [17], the MED algorithm produces imagesthat are slightly (almost imperceptibly) darker than the originalcontinuous-tone image, while the multipass MED algorithmproduces the correct gray level. The values of the LSMBimage quality metric for these techniques are given in Table I.(Note, that the table includes data from two images that are notshown in Fig. 8.) Observe that “classical” screening, which isthe worst visually, results in the highest errors. The bias ofthe MED algorithm is reflected in significantly higher errorvalues than the corresponding values for the multipass MEDalgorithm.

The performance of the LSMB method depends stronglyon the choices of printer and eye models. As we will seeshortly, it also depends on the starting point and the optimiza-tion strategy. We evaluated the results of the method withthe following “baseline” choices: circular dot-overlap printermodel with , a Gaussian eye filter for a viewingdistance of 30 in at 300 dpi, and no eye nonlinearity.11 Weused the toggle-only scheme, described in the previous section,that updates one pixel at a time. We also assumed that the twofilters and are equal. Fig. 10 shows the results ofthe LSMB approach with two different starting points. Thestarting point for Fig. 10(a) was a random binary image, andfor Fig. 10(b) the MP-MED algorithm with the JJN filter.Observe that there is a strong dependence on the starting point.The scheme tends to preserve the original texture for the MP-MED starting point, while the random start results in patternsthat look considerably grainier. Table I shows that the LSMBalgorithm with MP-MED starting point results in the lowesterror; the error for the random start is significantly higher.Thus, the values of the LSMB image quality metric agree withthe visual evaluation. Using a blue-noise screen as a startingpoint also results in higher errors and grainy images [37].In general, our experiments indicate that this scheme eitherpreserves the original texture or converges to grainy patternslike those of Fig. 10(a).

As was shown in [23], the LSMB algorithm results in grainyimages even when the (one-pass) MED output is used as astarting point. The reason for this is that, as we mentionedabove, the MED algorithm produces images that are slightly

11Our experiments indicate that, for the given set of images, the eyenonlinearity offers no significant improvements.

(a) (b) (c)

Fig. 13. LSMB halftoning (progressive scheme) for different viewing dis-tances at 300 dpi (magnified by 3). (a) One ft. (b) Two ft. (c) Four ft.

darker than the original continuous-tone image. As the iterativealgorithm tries to modify the binary image to restore thecorrect brightness, it destroys the error diffusion patterns.This problem is fixed by the multipass MED algorithm,which produces the correct gray level. Indeed, this was the


TABLE IIPER PIXEL SQUARED ERROR OF LSMB (TOGGLE-ONLY SCHEME) FOR DIFFERENT STARTING POINTS—LENA

TABLE IIIPER PIXEL SQUARED ERROR OF LSMB (TOGGLE-ONLY SCHEME) FOR DIFFERENT STARTING POINTS—BAR

original motivation for the introduction of the multipass MEDalgorithm by Dong in [16].

As indicated in the previous section, the number of iterationsdepends on the starting point. The MP-MED starting pointrequires the fewest iterations. This can be explained by the factthat the toggle-only algorithm tends to preserve the originalMP-MED texture.

We now consider variations in the eye filters to account fordifferent viewing distances. A similar experiment was carriedout by Mulligan and Ahumada in [24]. We chooseand consider eye models that correspond to viewing distancesof 1, 1.5, 2, 2.5, 3, 4, 5, and 6 ft at 300 dpi. Table II lists the perpixel squared errors for the whole range of viewing distancesand a number of different starting points. The lowest errorfor each column is shown in boldface characters. The tabledemonstrates that the toggle-only scheme is quite sensitive tothe starting point, i.e., there is considerable variation in theper pixel squared errors within each of the columns, as canbe seen by the standard deviation divided by the mean thatis listed at the bottom of each column. Observe that differentstarting points are tuned to different distance ranges. At shortviewing distances (1–1.5 ft), the FS starting point results in thelowest error, while at medium distances (2–3 ft), the JJN startresults in the lowest error. At even higher viewing distances(4–6 ft), the lowest error was obtained with the JJN filter withrandom weight perturbations [35]. Fig. 11 shows an imagehalftoned with LSMB using Gaussian eye filters ( )with standard deviations 0.6, 1.2, and 1.8 pixels, correspondingto viewing distances of 1, 2, and 3 ft (3, 6, and 9 after the

magnification). The starting points were those that result in thelowest error in Table II. As the viewing distance increases, oneexpects an increase in the number of perceived gray levels atthe expense of coarser textures (which become less visible)and reduced image detail. This is indeed what we observe.For example in (b) and (c) the nose has more gray tones,the overall texture is coarser, and feathers have less detail.However, since the results are only approximate, we cannotbe sure that a completely different texture will not producea lower error. In 1-D LSMB halftoning, where the optimalsolution can be obtained using the Viterbi algorithm, it wasshown that the optimal textures do get coarser as the viewingdistance increases [19]. The number of iterations required forconvergence increases with viewing distance from about 7to about 17, for all starting points with the exception of therandom start, which requires about twice as many iterations.

Fig. 12 shows a ramp halftoned with LSMB using Gaussianeye filters ( ) with standard deviations 0.6, 1.2, and 2.4pixels, corresponding to viewing distances of 1, 2, and 4 ft(3, 6, and 12 after the magnification). The starting points arealso shown in the figure and were those that result in thelowest error in Table III, which lists the per pixel squarederrors for the whole range of viewing distances and variousstarting points. Observe that for the given MP-MED startingpoints the algorithm preserves most of the texture patterns,each of which is matched to the viewing distance. Again,Table III demonstrates that different starting points are tunedto different distance ranges. Notice also the asymmetries inthe nearly white regions of the MP-MED images of Fig. 12:


TABLE IVPER PIXEL SQUARED ERROR OF LSMB (TOGGLE/SWAP SCHEME) FOR DIFFERENT STARTING POINTS—LENA

TABLE VPER PIXEL SQUARED ERROR OF LSMB (PROGRESSIVESCHEME SCHEME) FOR DIFFERENT STARTING POINTS—LENA

The slanting strings of dots are caused by the raster-scanprocessing. In addition, the slanting is different when the whiteend of the ramp is at the top or at the bottom of the image. Suchasymmetries are eliminated by the LSMB algorithm. Finally,notice that, as Mulligan and Ahumada [24] demonstrated ina similar experiment, when the viewing distance decreases(i.e., the eye filter becomes narrower), large white areas appearin the lighter parts of the image. This is because the LSMBmetric penalizes dots that are farther apart than the distanceover which the eye model can average. Similarly, the extentof the black areas in the darker parts of the image increaseswith decreasing viewing distance. Notice, however, that theblack areas are not as large as the white ones because of theasymmetry in the printer model.

We also tried the toggle/swap scheme [26]. As mentioned inthe previous section, Flohret al. [27] showed that this schemeproduces higher quality images than the toggle-only scheme.Our results also show that the toggle/swap scheme produceshalftone images of higher quality and lower error than thetoggle-only scheme, and indicate a weaker dependence onthe starting point for viewing distances of 1–2 ft. This isdemonstrated in Table IV, which lists the per pixel squarederrors for the Lena image for the whole range of viewingdistances and the different starting points. Notice the similarityof the elements within each of the first few columns of thetable, in comparison to the variability in the elements ofthe columns of Table II. The MP-MED starting points stillresult in the lowest error, however, with the FS filter slightlybetter at short viewing distances (1–1.5 ft) and the JJN filterslightly better at medium distances (2–3 ft). The differencesin visual quality with different starting points are small andagree with the metric predictions. At higher viewing distancesthe dependence on the starting point becomes stronger, whichindicates that the toggle/swap scheme is not as effective, inthe sense that it does not produce a result as close to thelocal optimum found by exhaustive search. Again, as the

viewing distance increases, the number of perceived graylevels increases at the expense of coarser textures and reducedimage detail, and the number of iterations for convergenceincreases. The numbers are comparable to the toggle-onlycase. Here, however, the random start does not require asmany iterations (only about 25% more than the other startingpoints).

In an attempt to challenge the conjecture that coarsertextures are necessary for an increase of perceived gray levels,we tried the progressive schemedescribed in the previoussection. That is, we used the LSMB estimate at a given viewingdistance as a starting point for the next viewing distance,starting at 12 in with 6-in increments. We used the MP-MEDwith the FS or JJN filter as the original starting point, andthe toggle/swap scheme at each viewing distance. The resultsare shown in Table V and Fig. 13. Observe that the errorsare lower than those for the earlier results, and the changes intexture with increased viewing distance are a lot less dramatic.However, one can still detect slightly coarser textures as theviewing distance increases. Clearly, the use of this progressivescheme has a significant effect on the resulting images, bothvisually and in terms of the error metric. In this case, thenumber of iterations for convergence decreases slightly withviewing distance from 8 to 4; however, the whole sequencehas to be computed.

We now show how we can control the amount of sharpeningby varying the eye filter corresponding to the continuous-tone image. Fig. 14 shows an image halftoned with LSMBusing a fixed Gaussian filter with standard deviation 1.5pixels, corresponding to a viewing distance of 2.5 ft (7.5ft after magnification), and different Gaussian filterswithstandard deviations 1.5, 0.9, and 0 pixels (the last filter is aunit impulse). As expected, the narrower the impulse responseof the filter , the sharper the halftone image. Thus, the choiceof the filter can be used to control the amount of smoothing(or sharpness) of the halftone images. In fact, the MED result


(a) (b) (c)

Fig. 14. LSMB halftoning (toggle/swap scheme) with different amounts of sharpening for viewing distance of 30 in at 300 dpi (shown magnified by 3).(a) h0

= h. (b) h0= Gaussian with� = 0:9 (corresponding to viewing distance of 18 in). (c)h0

= unit impulse.

of Fig. 9(c) is sharper than the LSMB result of Fig. 14(a),which indicates that error diffusion introduces some amountof sharpening. This last observation agrees with the findings ofEschbach and Knox [43], [44], who demonstrated the presenceof an inherent amount of sharpening in the standard errordiffusion algorithm, and showed that the amount of sharpeningintroduced by error diffusion can be controlled by the useof an input-dependent threshold. A more detailed study ofthe edge behavior of error diffusion can be found in [45].Another approach to increase the amount of sharpening inerror diffusion was presented in [46].

Finally, we tried using an eye filter (31st order FIR) thatcaptures the Mach-band effect, i.e., the filter response de-creases at low frequencies. We found that, if the two filters

and are not identical, then the halftone images come outblurred and unpleasant. For example, dropping the filteris equivalent to prefiltering the continuous-tone image withthe inverse of the filter , i.e., with a band-stop filter. Inother words, the algorithm tries to compensate for the Mach-band effect. Since there is no compensation for the Mach-bandeffect in the original continuous-tone image, the Mach-bandcompensated halftone image is unpleasant to the eye. On theother hand, when both images are filtered with the same filter,then the decrease at low frequencies has very little effect onthe halftone images.

Before we close this section, we would like to discuss theissue of computational complexity. The goal of this paper wasto develop an approach that can lead to the best possiblehalftone images for a given set of viewing conditions anddisplay parameters. Hence, we did not try to optimize thecomputational efficiency of the LSMB algorithms in anysignificant way. This issue was considered extensively byFlohr et al. [27], who examined various ways for speeding upthe LSMB algorithms, including table lookups and selectivepixel updates during each iteration. However, as an indicationof the amount of computation involved, we consider thetoggle-only scheme applied to the Lena image with the FSMP-MED starting point. The execution time on an SGI O2workstation ranged from about one minute for the shortest

viewing distance to about 20 m for the longest (or about 14–70s per iteration).

As we discussed in the previous section, the LSMB ap-proach provides a metric of halftone image quality that takesinto account the characteristics of both the display device(printer) and the human visual system. The results of thissection indicate that the LSMB metric predictions agree withvisual evaluations of image quality. However, it also hasserious limitations because it is very difficult to capture imagequality using a single number. For example, this metric isaffected by image sharpness, gray-scale resolution, textureartifacts, geometric and gray-scale distortions, etc. In addition,a halftone image that has been displaced by a few pixels mayhave higher error than an image that has been halftoned bya clearly inferior technique, even though the first image isvisually superior. Similarly, as we saw in this section, the errorincreases significantly with a small uniform shift in brightness,while the perceived quality is barely affected. One has tobe especially careful when using this metric to evaluate theperformance of techniques, like error diffusion, that introduceslight shifts in the image edges (to be more precise, it enhancesthe edges asymmetrically [45]) that are not objectionable tothe eye but affect the error metric. On the other hand, whenthis metric is used carefully, it agrees remarkably well withhuman perception and provides a powerful tool for algorithmdevelopment and evaluation.

VI. CONCLUSION

We presented a new least-squares model-based approachto digital halftoning. It exploits both a visual and a printermodel to produce an “optimal” halftoned reproduction. The2-D least-squares solution is obtained by iterative techniques.We showed that the visual quality of the resulting halftoneimages depends on the starting point and the optimizationstrategy. Our experiments demonstrate that LSMB halfton-ing can produce images with high gray-scale and spatialresolution and pleasing textures. We considered the perfor-mance of the LSMB algorithms over a range of viewing


distances. Overall the least-squares model-based approachoffers a substantial improvement over conventional halftoningtechniques, and also eliminates the problems associated witherror diffusion. Our results also indicate that error diffusionwith the Floyd–Steinberg filter is better suited for shorterviewing distances or lower display resolutions (typical CRTdisplays), while the Jarvis–Judice–Ninke filter is more suitablefor longer viewing distances or higher display resolutions (300dpi printers and higher). Finally, we showed that the LSMBapproach gives us precise control of image sharpness.

We showed that the error criterion of the LSMB approachprovides a measure of halftone image quality that takes intoaccount the characteristics of both the display device (printer)and the human visual system and can be used to evaluate theperformance of any halftoning technique. Our results indicatethat, despite its limitations, the LSMB error metric agreesremarkably well with visual evaluations of image qualityand provides a powerful tool for algorithm development andevaluation.

REFERENCES

[1] D. L. Neuhoff and T. N. Pappas, “Perceptual coding of images forhalftone display,” in Proc. ICASSP-91,Toronto, Ont., Canada, May1991, vol. 4, pp. 2797–2800.

[2] , “Perceptual coding of images for halftone display,”IEEE Trans.Image Processing,vol. 3, pp. 341–354, July 1994.

[3] J. L. Mannos and D. J. Sakrison, “The effects of a visual fidelity criterionon the encoding of images,”IEEE Trans. Inform. Theory,vol. IT-20,pp. 525–536, July 1974.

[4] Q. Lin, “Screen design for printing,” inProc. ICIP-95, Washington,DC, Oct. 1995, vol. II, pp. 331–334.

[5] J. P. Allebach, “Visual model-based algorithms for halftoning images,”Proc. SPIE, Image Quality,vol. 310, pp. 151–158, 1981.

[6] D. Anastassiou, “Error diffusion coding for A/D conversion,”IEEETrans. Circuits Syst.,vol. 36, pp. 1175–1186, Sept. 1989.

[7] D. Anastassiou and S. Kollias, “Digital image halftoning using neuralnetworks,” Proc. SPIE,vol. 1001, 1988.

[8] P. G. Roetling and T. M. Holladay, “Tone reproduction and screendesign for pictorial electrographic printing,”J. Appl. Phot. Eng.,vol.15, pp. 179–182, 1979.

[9] T. N. Pappas and D. L. Neuhoff, “Model-based halftoning,” inProc.SPIE, Human Vision, Visual Proc., and Digital Display II,San Jose,CA, Feb. 1991, vol. 1453, pp. 244–255.

[10] , “Printer models and error diffusion,”IEEE Trans. Image Pro-cessing,vol. 4, pp. 66–80, Jan. 1995.

[11] J. P. Allebach, “Binary display of images when spot size exceeds stepsize,” Appl. Opt.,vol. 19, pp. 2513–2519, Aug. 1980.

[12] R. W. Pryor, G. M. Cinque, and A. Rubinstein, “Bilevel displays—Anew approach,” inProc. SID,1978, vol. 19, pp. 127–131.

[13] P. Stucki, “MECCA—A multiple-error correcting computation algo-rithm for bilevel image hardcopy reproduction,” Res. Rep. RZ1060,IBM Res. Lab., Zurich, Switzerland, 1981.

[14] , “Advances in digital image processing for document repro-duction,” in VLSI Engineering,T. L. Kunii, Ed. Berlin, Germany:Springer-Verlag, 1984, pp. 256–302.

[15] C. J. Rosenberg, “Measurement-based verification of an electropho-tographic printer dot model for halftone algorithm dot correction,”in Proc. IS&T’s 8th Int. Cong. Adv. Non-Impact Printing Technology,Williamsburg, VA, Oct. 25–30 1992, pp. 286–291.

[16] C.-K. Dong, “Measurement of printer parameters for model-basedhalftoning,” B.S./M.S. thesis, Mass. Inst. Technol, Cambridge, MA, May1992.

[17] T. N. Pappas, C.-K. Dong, and D. L. Neuhoff, “Measurement of printerparameters for model-based halftoning,”J. Electron. Imag.,vol. 2, pp.193–204, July 1993.

[18] D. L. Neuhoff, T. N. Pappas, and N. Seshadri, “One-dimensional least-squares model-based halftoning,” inProc. ICASSP-92,San Francisco,CA, Mar. 1992, vol. 3, pp. 189–192.

[19] , “One-dimensional least-squares model-based halftoning,”J. Opt.Soc. Amer. A,vol. 14, pp. 1707–1723, Aug. 1997.

[20] P. W. Wong, “Entropy-constrained halftoning using multipath treecoding,” IEEE Trans. Image Processing,vol. 6, pp. 1567–1579, Nov.1997.

[21] H. Trontelj, J. E. Farrell, J. Wiseman, and J. Shu, “Optimal halftoningalgorithm depends on printing resolution,” inSID Dig. Technical Papers,Boston, MA, May 1992, pp. 749–752.

[22] T. N. Pappas, D. L. Neuhoff, and N. Seshadri, “Model-based halftoning,”in Proc. 7th Workshop on Multidimensional Signal Processing,LakePlacid, NY, Sept. 1991.

[23] T. N. Pappas and D. L. Neuhoff, “Least-squares model-based halfton-ing,” Proc. SPIE, Human Vision, Visual Proc., and Digital Display III,vol. 1666, pp. 165–176, Feb. 1992.

[24] J. B. Mulligan and A. J. Ahumada, Jr., “Principled halftoning based onmodels of human vision,”Proc. SPIE, Human Vision, Visual Proc., andDigital Display III, vol. 1666, pp. 109–121, Feb. 1992.

[25] M. Analoui and J. P. Allebach, “Model based halftoning using directbinary search,”Proc. SPIE, Human Vision, Visual Proc., and DigitalDisplay III, vol. 1666, pp. 96–108, Feb. 1992.

[26] D. A. Carrara, M. Analoui, and J. P. Allebach, “Recent progress indigital halftoning,” inProc. IS&T’s 8th Int. Cong. Advanced Non-ImpactPrinting Technology,Williamsburg, VA, Oct. 25–30, 1992 pp. 265–270.

[27] T. J. Flohr, C. B. Atkins, and J. P. Allebach, “Can DBS ever be apractical halftoning technique?,” inSID Dig. Technical Papers,Seattle,WA, May 1993, pp. 143–146.

[28] A. Zakhor, S. Lin, and F. Eskafi, “A new class of B/W and colorhalftoning algorithms,” inProc. ICASSP-91,Toronto, Ont., Canada, May1991, vol. 4, pp. 2801–2804.

[29] , “A new class of B/W halftoning algorithms,”IEEE Trans. ImageProcessing,vol. 2, pp. 499–509, Oct. 1993.

[30] R. A. Vander Kam, P. A. Chou, and R. M. Gray, “Performance evalua-tion of greedy search algorithms for image halftoning and compression,”in Proc. 27th Asilomar Conf. Signals, Systems, and Computers,1993,pp. 951–955.

[31] R. Geist, R. Reynolds, and D. Suggs, “A markovian framework fordigital halftoning,” ACM Trans. Graph.,vol. 12, pp. 136–159, Apr.1993.

[32] A. K. Jain, Fundamentals of Digital Image Processing.EnglewoodCliffs, NJ: Prentice-Hall, 1989.

[33] C. F. Hall and E. L. Hall, “A nonlinear model for the spatial charac-teristics of the human visual system,”IEEE Trans. Syst., Man, Cybern.,vol. SMC-7, pp. 162–170, Mar. 1977.

[34] F. W. Campbell, J. J. Kulikowski, and J. Levinson, “The effect oforientation on the visual resolution of gratings,”J. Physiol.,vol. 187,pp. 427–436, 1966.

[35] R. Ulichney, Digital Halftoning. Cambridge, MA: MIT Press, 1987.[36] B. A. Wandell,Foundations of Vision. Sunderland, MA: Sinauer, 1995.[37] M. A. Schulze and T. N. Pappas, “Blue noise and model-based halfton-

ing,” Proc. SPIE, Human Vision, Visual Proc., and Digital Display V,vol. 2179, pp. 182–194, Feb. 1994.

[38] R. Ulichney, “The void-and-cluster method for dither array generation,”Proc. SPIE, Human Vision, Visual Proc., and Digital Display IV,vol.1913, pp. 332–343, Feb. 1993.

[39] S. Daly, “Subroutine for the generation of a two dimensional humanvisual contrast sensitivity function,” Tech. Rep. 233203Y, EastmanKodak, Rochester, NY, 1987.

[40] J. Sullivan, L. Ray, and R. Miller, “Design of minimum visual modu-lation halftone patterns,”IEEE Trans. Syst., Man, Cybern..,vol. 21, pp.33–38, Jan./Feb. 1991.

[41] D. J. Lieberman and J. P. Allebach, “Digital halftoning using the directbinary search algorithm,” inProc. IST Int. Conf. High Technology,Chiba, Japan, Sept. 11–14, 1996, pp. 114–124.

[42] A. V. Oppenheim and R. W. Schafer,Discrete-Time Signal Processing.Englewood Cliffs, NJ: Prentice-Hall, 1989.

[43] R. Eschbach and K. Knox, “Error-diffusion algorithm with edge en-hancement,”J. Opt. Soc. Amer. A,vol. 8, pp. 1844–1850, Dec. 1991.

[44] K. T. Knox, “Error image in error diffusion,”Proc. SPIE, ImageProcessing Algorithms and Techniques III,vol. 1657, pp. 268–279, Feb.1992.

[45] Z. Fan and F. Li, “Edge behavior of error diffusion,” inProc. ICIP’95,Washington, DC, vol. III, pp. 113–116.

[46] S. Thurnhofer and S. K. Mitra, “Nonlinear detail enhancement of error-diffused images,”Proc. SPIE, Human Vision, Visual Proc., and DigitalDisplay V, vol. 2179, pp. 170–181, Feb. 1994.


Thrasyvoulos N. Pappas(M’87–SM’95) receivedthe S.B., S.M., and Ph.D. degrees in electricalengineering and computer science from the Mass-achusetts Institute of Technology, Cambridge, in1979, 1982, and 1987, respectively.

He has been a Member of Technical Staff atBell Laboratories, Murray Hill, NJ, since 1987. Hisresearch interests are in image and multidimensionalsignal processing. He has worked on analysis ofmedical images and algorithms for image and videosegmentation. He has also worked on numerical

analysis, optimization, and control. His recent work has been on perceptualimage and video compression, model-based halftoning, color printing, andvideo/audio integration for teleconferencing.

Dr. Pappas is an Associate Editor and Editor for Electronic Abstracts forthe IEEE TRANSACTIONS ON IMAGE PROCESSING. He is a Member of the IEEEImage and Multidimensional Signal Processing Technical Committee. He isalso co-chair for the Conference on Human Vision and Electronic Imaging,sponsored by SPIE and IS&T.

David L. Neuhoff (M’74–SM’83–F’94) was bornin Rockville Centre, NY, in 1948. He received theB.S.E. degree from Cornell University in 1970 andthe M.S. and Ph.D. degrees in electrical engineeringfrom Stanford University, Stanford, CA, in 1972 and1974, respectively.

In 1974, he joined the University of Michigan,Ann Arbor, MI, where he is now Professor ofelectrical engineering and computer science. From1984 to 1989, he was an Associate Chairman of theSystems Science and Engineering Division, Electri-

cal, Electronics, and Computer Science Department. He spent from September1989 through June 1990 and from January through May 1997 on sabbaticalat Bell Laboratories, Murray Hill, NJ. His research and teaching interests arein communications, information theory, and signal processing, especially datacompression, quantization, image and video coding, source-channel coding,Shannon theory, high-resolution quantization theory, data synchronization, andhalftoning.

Dr. Neuhoff was an Associate Editor for Source Coding for the IEEETRANSACTIONS ON INFORMATION THEORY, from 1986 to 1989. He served onthe Board of Governors of the IEEE Information Theory Society from 1988to 1990. He chaired the IEEE Southeastern Michigan Chapter of Division Iin 1978, co-chaired the 1986 IEEE International Symposium on InformationTheory, Ann Arbor, and served as tutorial chair for ICASSP’95.

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