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Printer Model and Least-Squares HalftoningUsing Genetic Algorithms

Chih-Ching Lai and Din-Chang Tseng*Institute of Computer Science and Information Engineering, National Centra

University, Chung-li, Taiwan 320

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Abstract

In this paper, a least-square model-based halftoning tnique using a genetic algorithm is proposed to proda halftone image by minimizing the perceived reflectadifference between the halftone image and its origiimage. We use a least-square criterion incorporating wthe property of the human visual system to measuredifference between the two images. The genetic arithm is used for investigating the complicated seaproblem. The standard halftoning techniques, sucherror diffusion and least-square halftoning, produce grlevel distortion because of dot-gain problem. In tstudy, we use a modified dot-overlap printer modelcompensate the gray-level distortion. The printer mocombines with a measurement-based algorithm to emate the print-dot radius and makes the propohalftoning approach adapted to a wide variety of prers and papers. Experiments show that the proposeproach produces more accurate gray levels than secommon-used halftoning methods produce.

Introduction

Digital halftoning refers to any algorithmic process thcreates the illusion of continuous-tone images from judicious arrangement of binary picture elements.1 Manyhalftoning techniques1–4 have been proposed to improvprinting quality such as, artifact, texture, or false-cotour removing; deblurring; and contrast enhancing. Hoever, only a few techniques have been proposedaccurate gray-level rendition.5–8 Accurate gray-level rendition means to minimize the perceived gray-level dference between a printed halftone image and its origimage. In halftone printing, we always assume that printed dot is square; however, most printers prod

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circular dots and the dot size is always larger thansquare size because of ink spread, wax compressionink absorption into papers. The increase in dot size termed as the physical dot gain as shown in Fig. 1. phenomenon causes the printed gray levels to be dathan the expectation. In this report, we compensatephysical dot-gain problem to produce halftone imawith less gray-level distortion.

Three kinds of halftoning techniques have been pposed to compensate the dot-gain problems: (1) msurement-based calibration, (2) least-square halftonand (3) model-based halftoning.

The measurement-based calibration technique9 wasproposed to compensate for any gray-level distortIn the method, an image consisting of 256 gray levehalftoned and then printed. A tone–response curve9 isgenerated for measuring the reflectance of the prinimage. The curve is used to describe the relationbetween the perceived gray level (input reflectance)the output reflectance. The calibration technique corrthe output response of a particular halftoning algoriton a specific device using an inverse mapping functThis process works well but has three drawbacks:Each individual halftoning algorithm needs to be cabrated for each individual printer. (2) The calibrationdithered results may produce less than 256 gray lev(3) Only specific pixel patterns are measured. If the crostructure of the halftone image greatly varies frthe measured pattern, the gray levels predicted bycalibration curve may not be representative of whaprinted on paper and the tones will not be reproduaccurately.9

On the basis of the least-square-error criterionhalftoning technique can be regarded as a process oranging black dots to simulate gray levels on a bileoutput device. Hence, a halftoning technique is take

ain.

Figure 1. The physical dot gain: (a) ideal dots, (b) round dots cause dot g

(a) (b)

Chapter IV—Halftone Analysis and Modeling—363

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an optimization algorithm that searches for a suitaplacement for black dots to produce halftone images wless gray-level distortion. However, in our sense, a mary least-square halftoning is poor to compensatethe gray-level distortion; more treatments are needeget better results.

Zakhor et al.10 presented a class of dithering tecniques for black and white images. They divided an iminto small blocks and minimized the gray-level differenbetween every corresponding block pair in the origicontinuous-tone image and its low-pass filtered halftoimage. They utilized quadratic programming with lineconstraints to solve the standard optimization problHowever, they did not consider the printer charactetics needed to compensate for gray-level distortimoreover, their method needs much computation.

Saito and Kobayashi11 tried to produce a less-distortion halftone image by using evolutionary compution approaches, but they did not consider a printer moto compensate for the dot-gain problem; moreover, chastic errors are associated with the simplest selecmethod in their evolutionary algorithm.

The model-based halftoning technique relies onaccurate printer model to predicate and compensategray-level distortion to produce less-distortion halftoimages.5 Several model-based halftoning approachhave been proposed. Anastassiou12 proposed a “frequencyweighted square error criterion” to minimize the squerror between the eye-filtered binary image and the efiltered gray-level image. However, he assumed tprinters generate non-overlapped perfect dots.

Pappas and Neuhoff 6 exploited both a printer modeand a visual-perception model in a least-square mobased (LSMB) halftoning algorithm. The algorithm atempts to produce the “optimal” halftone reproductiby minimizing the square error between “the respoof the cascade of the printer and a visual model tobinary image” and “the response of the visual modethe original gray-level image.”6 However, they used aexhaustive search to find the optimal binary imageupdating the binary value of one pixel at a time and tspend a huge processing time.

In this paper, we propose a genetic algorithm cobined with a modified dot-overlap printer model for leasquare halftoning. Genetic algorithms (GAs)13–15 areprobabilistic search methods guided by the principleevolution and natural genetics. GAs are well known their ability to explore large search spaces efficienand adaptively.15 Originally, GAs were modeled and developed by Holland,16 and now have emerged as genepurpose and robust optimization techniques. The pcess of arranging the placement of black dots in a haltone image to compensate for gray-level distortiontedious. Genetic operators can dynamically arrange bdots on bilevel output devices; thus we use a GA as a sealgorithm for halftoning problems. We consider the prolem-specific knowledge in the GA and incorporateprinter model into the approach to improve the gray-lerendition.

Pappas et al.5,7–8 have proposed two printer model“circular dot-overlap” and “measurement of printer prameters” to compensate for gray-level distortion.

364—Recent Progress in Digital Halftoning II

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evaluated these two printer models and chose the clar dot-overlap printer model for further study beca(1) the circular dot-overlap printer model is simpler amore flexible than the measurement model in use;the measurement model needs to solve the constraoptimization problem, and an initial estimation of tsolution that satisfies the constraints must be provi(3) in the measurement model, several local minimaoften presented in the solution space; it is not alwpossible to determine the global minimum; and (4) parameters of the measurement model (e.g., 3 × 3 win-dow or larger) are too many to solve the whole cstrained optimization problem.

However, the circular dot-overlap printer model gerates bias error in the printed images;7 thus we proposa modified circular dot-overlap printer model to copensate for gray-level distortion and resolve the bproblem. Using the circular dot-overlap printer modthe dot radius must be known in advance. We hereuse a preproposed measurement-based method tomate the print-dot radius and make the modified doverlap printer model adaptable to a wide varietyprinters and papers.

The remaining sections of this paper are organas follows: First we present the proposed approach,the experiments and discussion, and the conclusionsummarized in the last section.

Proposed Halftoning ApproachThe most intuitive formulation of the halftonin

problem can be stated as “to find a binary image sthat the gray-level difference between an original ctinuous-tone image and its perceived bilevel imagminimized.”12 Because the human visual system acta low-pass filter, the least-square halftoning approminimizes the mean square error between the low-version of a gray-level image and that of its halftoimage.

The block diagram of the proposed least-squmodel-based GA halftoning technique is shown in F2. Initially, a least-square criterion is defined to msure the difference between the low-pass version gray-level image and that of its halftone image. A mofied circular dot-overlap printer model is taken into least-square criterion to compensate the gray-leveltortion in the halftone image. Then a genetic algoritis utilized to find the “optimal” halftone reproductiobased on the least-square criterion. Here, the “optimhalftone reproduction means that the output gray-leresponse is linear; that is, the tone-response curvestraight line of slope one through the origin.9 In process-ing, an image is divided into small blocks and the bloare processed in the raster-scan order to propa(block) gray-level errors to the right and the lower neiboring blocks.

Least-Squares Criterion. Assume that an M ¥ N gray-level image is represented by [g

i,j], i = 1, ..., M, and j = 1,

..., N. We use a 2-D Gaussian filter with impulse respo[h

i,j] to simulate the eye model to evaluate the gray-le

difference between a gray-level image and its halftimage. The original image [g

i,j] is partitioned into sev

rithm.

Figure 2. The least-square model-based GA halftoning: (a) the evaluation process; (b) the genetic algo

(b)

(a)

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eral n × n blocks, where both M/n and N/n are integersThen the evaluation function for a block is defined a

E z w

i

n

j

n

i j i j= −= =∑ ∑

1 1

2( ) ,, , (1)

wherez

i,j = g

i,j * h

i,j,

wi,j = p

i,j * h

i,j,

pi,j = P(W

i,j ),

and * indicates a convolution operator. The pi,j is a printer

parameter of the modified dot-overlap printer model lized instead of the output gray level b

i,j for compensat-

ing the gray-level distortion and described in the PrinModel Section. Next, we use a genetic algorithm asevolutionary computation to generate the halftone age [b

i,j] by minimizing the square error E.

Genetic Algorithms. A genetic algorithm (GA) is a stochastic algorithm used to solve search and optimizaproblems. The algorithm is based on the mechanicnatural selection and genetics in biological systems

In general, a GA contains a fixed-size populationpotential solutions over the search space. These stions are encoded as bit strings and called individuachromosomes. The initial population can be created

i-

rn-

nof

flu-orn-

domly or based on problem-specific knowledge. At eaiteration, called a generation, a new population is created. To generate a new population based on a preing one, the algorithm performs the following three ste(1) evaluation: each individual of the old populationevaluated by a fitness function and given a value tonote its merit, (2) selection: individuals with better fness are selected to generate the next population,(3) mating: genetic operators such as crossover andtation are applied to the selected individuals to prodnew individuals for the next generation. The above stare iterated for many generations until a satisfactorylution is found or a stopping criterion is met. A standaGA is described as the following pseudocodes:

t ← 0initialize P(t)evaluate P(t)while (the stop criterion is not met) do

begint ¨ t + 1select P(t) from P(t – 1)recombine P(t)evaluate P(t)

end,where P(t) is the population at generation t.

Typically, using a GA to solve a problem, we muprovide the following components: (1) a genetic repsentation of solutions to the problem, (2) one way to cate an initial population of solutions, (3) an evaluati

Chapter IV—Halftone Analysis and Modeling—365

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function that rates each candidate solution accordinits “fitness,” (4) genetic operators that effect genetic formation of children during reproduction, and (5) cotrol parameters (e.g., population size, crossover, mutarates, etc.)15

Solution Representation. In our utilization of a GAto find the “optimal” halftone image, a binary block encoded as a bit string. We create a population of strand evaluate each string, then select the best stringconstruct the new population. Finally, the binary blo[b

i,j] with the highest fitness value is found.

Initial Population. In general, a GA creates its starting population by filling with randomly generated bstrings; however, we can heuristically rather than rdomly generate the individuals in the initial populatito improve the search performance. Heuristic initialition may be helpful but must be done carefully to avpremature convergence. Here we use the result of sdard error diffusion as an individual and the remainindividuals are randomly generated. Note that if the tial population contains a few individuals far superiorthe rest of the population, the GA may quickly conveto a local optimum.

Fitness Function. Fitness function is the survivaarbiter for individuals. In the halftoning problem, thobjective is to find the binary halftone image block [b

i,j]

that minimizes the mean square error E. Because the fit-ness in GAs is to find the maximum profit, the fitnefunction in the halftoning problem is defined by

F C E

C z wi

n

j

n

i j i j

= −

= − −= =∑ ∑

max

max , ,( ) ,1 1

2 (2)

where Cmax

is a predefined value or the maximum of E.

Genetic Operators. Three primary genetic operators: selection, crossover, and mutation are generallyvolved in a GA.

Selection. The selection operator determines the sviving individuals. Each surviving individual is reproduced into several copies according to its relatresponsibility. Two reproductive strategies were comonly used. Generational reproduction replaces whole population in each generation, but steady-statereproduction13 only replaces the least-fitted membersa generation.

Baker17 compared various selection methods coprehensively and presented an improved version cathe stochastic universal sampling (SUS) method. A SU17

procedure is described by the following C codes:ptr = Rand( );for(sum = i = 0; i < N; i++)

for(sum += ExpVal[i]; sum > ptr; ptr++)Select_individual(i);

366—Recent Progress in Digital Halftoning II

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where Rand( ) returns a random real number betweand 1. We use steady-state reproduction and the method in the proposed halftoning approach.

Crossover. The crossover operator randomly paindividuals with a probability p

c and swaps parts of the

genetic information to produce new individuals. Sevetypes of crossover operators such as, one-point, tpoint, and uniform-type splitting have been proposHowever, DeJong and Spears18 concluded that the uniform crossover is more beneficial if the population sis small and, hence, gives a more robust performaThe uniform crossover is adopted in the proposhalftoning approach.

In the uniform crossover,13 two parents are selecteand two children are produced. One of the selected ents has the best fitness and the other is randomlylected. Each bit position in the children is createdcopying the corresponding bit from either one of tparents. The uniform crossover randomly generatecrossover mask that is a bit string with the same lengof an individual for each pair of parents. A “1” bit in thcrossover mask means that the first child inherits the gfrom the first parent and other genes from the secparent. The second child uses the opposite rule. Oneample of the uniform crossover is shown in Fig. 3.

Parent 1 1 0 0 1 1 0 1 0 0 Parent 2 0 1 1 1 0 0 0 1 1

Crossover mask 1 1 0 0 1 0 0 1 0

Child 1 1 0 1 1 1 0 0 0 1

Child 2 0 1 0 1 0 0 1 1 0

Figure 3. An example of the uniform crossover.

Mutation. The mutation operator creates a new dividual by altering one or more genes of an individwith a probability p

m to increase the variability of th

population. One or more bits in the crossover childare inverted; “1” is changed to “0” and “0” is changed“1.” In the proposed halftoning approach, the mutatoperator works as the example shown in Fig. 4. Theample shows three individuals of length 5 and a randnumber generated for each bit in each individual. Tbit changes its value when the random number tespassed. The random number that causes a bit to chis printed in bold face.

Control Parameters. The population size influencethe performance of GAs. A small-size population reduthe evaluation cost but results in premature convergebecause the population provides insufficient samplethe search space. For a large-size population, the GAgain more information to search better solutions becathe population contains more representative solutiover the search space.

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Randomnumbers

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Childindividual

1 0 0 1 0 .103 .305 .248 .851 .552 - 1 0 0 1 0

0 1 0 1 1 .203 .123 .005 .657 .341 1 0 1 1 1 1

0 0 1 1 0 .469 .778 .651 .033 .951 0 0 0 1 0 0

Figure 4. Examples of bit mutation used in the proposedproach.

Both crossover and mutation probabilities may fluence the performance of GAs. A GA may stagnatethe search for new solutions if the crossover probabis low. However, if the crossover probability is too higunstable solutions may be quickly substituted into population for individuals with better fitness. But if thmutation probability is too high, the search of the Gbecomes a random-like process. In the propohalftoning approach, extra experiments are performto find more appropriate values for the used parame

Printer Models. Most standard halftoning techniquproduce darker halftone images than the expectationto the dot-gain problem. To resolve this problem aachieve a less-distortion gray-level rendition, the princharacteristics should be considered. Model-based tniques5–8 exploited the printer characteristics to compsate for gray-level distortion.

The Pappas–Neuhoff circular dot-overlap mode5,8

assumed the print dot to be circular with a uniform dtribution of ink and the dot radius at least T / ,2 whereT is the spacing of the Cartesian grid, so that a bregion can be blackened entirely. They used r to dethe ratio of the actual dot radius to the ideal dot rad T / .2 The amount of dot-gain area at each pixel is pressed in terms of parameters α, β, and γ as shown inFig. 5(a). These parameters are the ratios of the arethe shaded regions shown in Fig. 5(a) to T 2. The param-eters a, b, and g are expressed in terms of ρ as follows:

α ρ ρρ

= − +

−−1

42 1

21

2

12

22

1sin , (3)

β πρ ρρ

ρ= −

− − +−

2 21 2

8 21

2

14

2 114

sin , (4)

γ ρ ρρ

ρ β= −

− − −−

21

2

22

21 1

21sin . (5)

The constraint of r is 1 ≤ ρ 2 . In terms of theseparameters, the circular dot-overlap model is define

p P W

f f f

bbi j i j

i j

i j, ,

,

,( )

,,

,= =+ −

==

1 101 2 3

ififα β γ

(6)

where Wi,j denotes a window consisting of b

i,j and its eight

neighbors, f1 is the number of horizontal and vertic

neighboring black dots, f2 is the number of diagona

-

-ny

e

dds.

uedrh--

-

ktes-

of

s

neighboring black dots not adjacent to any horizontavertical neighboring black dot, and f

3 is the number of

pairs of neighboring black dots in which one is a hozontal neighbor and the other is a vertical neighbor. Oexample for describing the circular dot-overlap modis given in Fig. 6(a).

Figure 5. Definition of parameters used in the circular dooverlap printer model; (a) definition of α, β, and γ; (b) defini-tion of δ and ε.

Pappas and Neuhoff 8 indicated ρ = 1.25 in the cir-cular dot-overlap printer model for HP laser printers.However, the estimated print-dot sizes are different different printers or on different papers. Here we relethe constraint of 1 ≤ ρ ≤ 2 . We use our measuremenbased method to estimate statistically the dot radiusthe printer model to adapt a wide variety of printers apapers and then we extend the circular dot-overlap pmodel to include all possible cases of print-dot radii.

Measurement of Print-Dot Radii. In general, themore print dots are considered, the more accuratearea is estimated. In order to find a reasonable dotdius, we consider a larger area of dots. At first, we fine a number of 3 × 3 print patterns with different “0 &1” permutations to analyze the physical dot gain. Th× 3 print patterns can define 29 different “0 & 1” permu-tations. We reduce the number of patterns by takingreflected or rotated patterns to be the same and then

Chapter IV—Halftone Analysis and Modeling—367

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, d,hite

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102 different “0 & 1” patterns including two special paterns: all “0” (white) and all “1” (black) patterns.

Figure 6. Two examples of the printer parameters pi,j for dif-

ferent dot radii; (a) the printer parameters pi,j when r

f ≥ T/

2 , (b) ther printer parameters pi,j when T/2 ≤ r

f < T/ 2 .

Secondly, we print out all print patterns repeatein vertical and horizontal directions as one examshown in Fig. 7 and measure the densities of theseterns using a reflection densitometer. Then the MurrDavies equation19

A

D

D

t

s= −

−

−

−1 101 10

(7)

is utilized to describe the relationship among the efftive dot area, A; the density of a solid dot, D

s; and the

resultant density of the dot area pattern, Dt . Assume that

the measured density of pattern i is Di; the measured den

sities of the all-black-dot and all-white-dot patterns D

b and D

w, respectively. Let the measured black area

pattern i be Si , then S

i can be calculated by

S S ii

D D

D D b

i w

b w= −

−=

− −

− −1 101 10

1 2 100( )

( ) , , , ..., , (8)

368—Recent Progress in Digital Halftoning II

yeat-y-

-

ef

where Sb is the 3 × 3 grid area.

Figure 7. A sample of periodic 3 × 3 print patterns. The 5 × 5dashed-line window is one example for defining the coefficia

i in calculating the black area of the internal 3 × 3 print pattern.

On the basis of the dot-overlap printer model, black area of pattern i can be calculated by

Aa a a a T T r

a a T T r Ta r r T

i =+ + − ≤

+ ≤ <<

( ) , /( ) , /

, /

1 2 3 42

1 22

12

22 2

2

α β γε δ

ρ

ififif

(9)

where r is the estimated radius of print dots; a, b, anare the ratios of the areas of the shaded regions shin Fig. 5(a) to T 2; and d and e are the ratios of the arof the shaded regions shown in Fig. 5(b) to T 2. The pa-rameters d and e are expressed in terms of r as follows:

δ ρ

ρρ=

− −−

21 2

21

2

14

2 1cos , (10)

ε πρ δ= −

2

24 . (11)

The definition of coefficients a2, a

3, and a

4 is simi-

lar to that of fi in the circular dot-overlap printer mod

Eq. 6; however, the former is more complicated. Theefficients a

2, a

3, and a

4 are defined by considering a 5×

5 window centered at a 3 × 3 print pattern in periodicprint patterns as one example shown in Fig. 7 withdashed-line block. For every white dot in the 3 × 3 pat-tern, we compute its dot-gain area in terms of a, b, gand e and then accumulate the dot-gain area for all wdots in the 3 × 3 pattern. Coefficient a

1 is the number of

black dots in the 3 × 3 print pattern. Coefficient a2 is the

accumulated number found by counting the horizonand vertical neighboring black dots for every white in the 3 × 3 print pattern noting that all neighboring bladots in the 5 × 5 window are considered. Coefficient a

3is the accumulated number found by counting theagonal neighboring black dots for every white dot in

lyacy

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3 × 3 print pattern while no black dot is simultaneousadjacent to both the white dot and the counting bldot. Coefficient a

4 is the accumulated number found b

counting the pairs of adjacent neighboring black dotwhich one is a horizontal neighbor and the other ivertical neighbor for every white dot in the 3 × 3 printpattern.

We take the measured black area and the calculblack area in a print pattern to be equal; that is, usthe equation

Si = A

i , i = 1, 2, …, 100, (12)

to estimate the radii for all print patterns. Due to tmeasured error and non-uniform roughness of prinpapers, the dot radii are different and distributed. need to determine the best-fitted dot radius for the primodel. The best-fitted dot radius means that the dodius r

f minimizes the sum of the square errors betw

the measured densities and the calculated densitieall print patterns. At first, we substitute each estimaradius r

j into every print-pattern equation (i.e., Eq. 9)

calculate the black area Airj and then substitute the are

and Eq. 8 into Eq. 12 to get the calculated density Dirj ,

D D

AS

iir

wir

b

D Dj

j

b w= − − −( )

=− −log , , ,..., .( )1 1 10 1 2 100 (13)

The sum of square errors for the estimated radiurj

is given as

SE r D D jj

ii i

rj( ) , , , ..., ,= −( ) ==∑

1

100 21 2 100 (14)

where Di is the measured density of print pattern i . The

best-fitted radius rf is the radius with the minimum SE

error.

Modified Dot-Overlap Printer Model. To includeall possible cases of print-dot radius, we extend the cular dot-overlap printer model8 as follows:

p

P W bP W bi j

i j i j

i j i j,

, ,

, ,

( ),( ),

,===

1

2

01

ifif

(15)

P W

f f f r

f Tr T

i j i j

f

, ,( )

,

, /, /

,=+ − ≤

≤ <<

1 2 3

1

0 2

α β γδ

if / 2

if /2 2if

T

T rf

f

(16)

P W

r

T

r Ti j

f

2

1

22 2

( )

,

, /

, /

,, =≤

≤ <<

if / 2

if /2 2

if

T

T rf

f

επρ

(17)

where Wi,j denotes a window consisting of b

i,j and its eight

neighbors; P1 and

P

2 are two functions for calculating

the estimated gray level pi,j for dot b

i,j; the parameters a

k

ina

edg

edeera-n ofd

ir-

b, g, d, and e are defined in Eqs. 3 through 5 and Eqand 11; and the definition of coefficients f

1, f

2, and f

3 is

the same as the definition of the Pappas–Neuhoff primodel8 given in Eq. 6. One example of p

i,j for the case of

T/2 ≤ rf < T/ 2 is shown in Fig. 6(b).

The differences between the Pappas–Neuhoff doverlap printer model and the modified dot-overlprinter model are (1) the dot radius in the PappNeuhoff model is obtained from an experience or ansumption value, but the dot radius in the modified mois measured to adapt to a wide variety of printers papers and (2) the Pappas–Neuhoff model assumedot radius is always larger than the cover square abut the modified model considers all possible casedot radius.

During the halftoning process, all image blocks aprocessed in the raster-scan order. The mutation optor incorporated with the printer model embedded in fitness function is used to test whether a binary blocgood enough. We adaptively adjust the combination0’s and 1’s through both the mutation operator and printer model to reduce gray-level distortion. After wfind a near-optimal binary block, the right-most (binarpixel values are recorded for the processing of the rblock and the lower most pixel values are also recorfor the processing of the lower block to propagate (blogray-level errors to obtain near-unbiased-error halftimages.

Experiments

We demonstrate here the experimental results of the posed approach and compare the printed results anrors with those generated by other halftoning techniquA HP LaserJet 5MP printer was used to print halftoimages, and a Macbeth RD-1200 reflection densitoeter was used to measure the densities of print-dot terns. All halftone images were printed on the sauncoated papers commonly used for copy machines

Note that the parameter values in GAs, such as mmum generation number and probability of genetic erators, always influence the performance of algorithms.20 However, GAs are always robust with respect to these parameters; thus only a few experimare needed to specify the parameter values. Of coureasonable parameters ensure good results and givto quick convergence. The parameters used in theperiments are (1) each individual represents a posspermutation of 0’s and 1’s in a 5 × 5 image block; (2)the generation number is 150; (3) the population siz30; (4) the adopted selection and crossover operatare stochastic universal sampling and uniform crossoand (5) the probabilities of crossover and mutation 0.7 and 0.1, respectively.

Nine halftoning algorithms: Floyd–Steinberg errdiffusion, Jarvis–Judice–Ninke error diffusion, Stucki eror diffusion, 4 × 4 dither-matrix ordered dither, 8 × 8dither-matrix ordered dither, dot diffusion, least-squaGA halftoning without printer model, least-square Ghalftoning with Pappas–Neuhoff circular dot-overlprinter model, and the proposed least-square halftoning with the modified circular dot-overlap print

Chapter IV—Halftone Analysis and Modeling—369

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Figure 8. Two test images were printed using the HP Lase5MP at 600 dot/inch (dpi) resolution

model, were compared. These halftoning algorithms wexamined by using two images: “Lena” and the “gralevel chart” as shown in Fig. 8. Both images have 76720 pixel resolution with 256 gray levels. Eighteen hatone images generated by the nine halftoning techniqwere printed at 300 dpi, but only four representative htone images are shown in Fig. 9.

The image obtained via the proposed least-squGA halftoning with the modified circular dot-overlaprinter model has fewer worm-like artifacts than the ror diffusion methods and does not have the false ctour as the ordered dither method produces. CompaFigs. 9(a) and 9(d), we find the proposed approach duces blocking-effect halftone images. The phenomeresulted from the fact that the proposed approach block-based halftoning method. The thresholded erin error diffusion are “diffused” pixel by pixel, but thproposed approach is based on the idea of breakinggray-level image into small blocks and solving the opmum for each block with a genetic algorithm. Zakhoral.10 also produced blocking-effect halftone images. Tblocking effect can be avoided when the image is partitioned into blocks for processing; however, su

370—Recent Progress in Digital Halftoning II

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processing expends a huge amount of time as spenthe Pappas and Neuhoff6 process. The uniformity of ahalftone image is influenced by the bandwidth of a 2Gaussian filter and the block frequency 1/n, where blocksize is n × n. When the bandwidth of a 2-D Gaussifilter is wider and the block frequency 1/n is smaller,the halftone image is very blurred.

The purpose of this study was to print halftone iages with less gray-level distortion. We compared nine halftoning algorithms not only browsing the printimages, but also inspecting the tone-response curvesprinted out the “gray-level chart” using the ninhalftoning algorithms and used a Macbeth RD-1200flection densitometer to acquire the density and thtransform to the (output) reflectance. Four represetive tone-response curves are given in Fig. 10. In thcurve charts, the abscissa denotes the perceived level from 0 (black) to 1 (white) and the ordinate dnotes the measured reflectance of a printed halftoneage. The perceived gray level is proportional to amount of ink on the printed paper.7 Thus, the tone-response curve of a tone-correct halftone image is a straline of slope one through the origin. The curves of all standard halftoning algorithms are more concave than of the proposed algorithm. This means that these sdard halftoning techniques produce more gray-level tortion than the proposed technique. There is stepoutput reflectance in the curves of the ordered dithbecause they only produce 17 and 65 gray levels,spectively.

Based on the tone-response curves, two sum-squerror criteria were used to evaluate all halftoning teniques as shown in Table I. The absolute square e(ASE) measures the difference between the tonesponse curve and the straight line of slope one throthe origin of the chart. The value of ASE describes deviation from the ideal response. The relative squerror (RSE) measures the difference between the tresponse curve and its least-square fitting straight lThe value of RSE indicates the degree of the linearitthe output reflectance. The least-squares GA halftonwith the modified circular dot-overlap printer model aways has the least error. We conclude that the propohalftoning approach produces less distorted halftimages than other commonly-used halftoning techniqu

Conclusions

On the basis of a least-squares criterion, a genetic arithm combined with the modified circular dot-overlaprinter model was proposed to produce halftone imawith less gray-level distortion. The proposed approaminimized the gray-level difference between the lopass version of a continuous-tone image and that ohalftone image. We used genetic operators, crossoand mutation in the GA to arrange the placement of blacdots in a halftone image and to find the “optimal” hatone reproduction. We proposed the modified circudot-overlap printer model to describe the printer chacteristics and then compensate for gray-level distorby means of spatial adjustment of black-dot locatioWe also quoted our measurement-based method to mate statistically the radii of print dots for the print

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Figure 9. The printed images using four representative halftoning techniques at 300-dpi resoution: (a) Jarvis–Judice–Ninke er diffu-sion; (b) least-square GA halftoning without printer model; (c) least-square GA halftoning with Pappas-Neuhoff circular dot-rlapprinter model; (d) least-square GA halftoning with the modified circular overlap printer model.

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model. The experimental results revealed the propoapproach reduces the gray-level distortion and prodmore accurate gray levels than a number of halftontechniques. From the experimental results, severapects for future work may be enumerated:

1. Incorporating problem-specific knowledge into gnetic algorithms has been acknowledged as anfective problem-solving tool in many research fielCombining more halftoning characteristics with gnetic algorithms to obtain better printed results potential research topic.

2. The human visual system plays an important roldigital halftoning. In the proposed approach, we a simple 2-D Gaussian filter as the eye model. M

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complicated or more suitable eye models shouldincorporated into the proposed approach to imprthe printed results further.

3. Recently color printers have been broadly usedthe office and home. Color printers also genercolor distortion. To apply the proposed approachcolor printing by considering the relationship amoall color attributes is worth studying.

Acknowledgment

The authors would like to thank the anonymous revieers for indicating the wrong citation of equations aproviding suggestions to improve this article’s qualand presentation.

Chapter IV—Halftone Analysis and Modeling—371

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Figure 9. The tone-response curves of four representative halftoning techniques: (a) Jarvis–Judice–Ninke error diffusion; (east-squares GA halftoning without printer model; (c) least-squares GA halftoning with Pappas–Neufhoff circular dot-overlap printermodel;(d) least-squares GA halftoning with the modified circular dot-overlap printer model.

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❈ Previously published in the Journal of Imaging Science anTechnology, 42(3), pp. 241–249, 1998.

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