1

A tone-dependent noise model for high-quality

halftones

Yik-Hing Fung and Yuk-Hee Chan†

Center of Multimedia Signal Processing

Department of Electronic and Information Engineering

The Hong Kong Polytechnic University, Hong Kong

ABSTRACT

A digital halftone of blue noise characteristics is preferred as dots in the halftone of a constant input should

be isotropically and homogeneously distributed. In practice, the placement of dots is constrained by a sampling

grid and hence aliasing happens when the input gray level is in the middle range. To solve this problem, Lau et

al. suggested replacing isolated dots by dot clusters to maintain the principal frequency of the output to be 1/2

when this happens. However, this model does not take into account that, due to the stochastic nature of the dot

distribution, there is a considerable amount of energy distributed around the principal frequency and it causes

aliasing problems even when the principal frequency of the output is 1/2. This paper presents a new noise model

which takes this factor into account. A halftoning algorithm is then proposed to generate halftones that satisfy

the new noise model. By comparing its performance with that of some other algorithms which are proposed

based on the traditional blue noise model and Lau et al.’s noise model, one can see that the proposed noise

model can be a better model to describe the noise characteristics of a high-quality halftone.

† Corresponding author (Email: enyhchan@polyu.edu.hk)

2

ITRODUCTIO

Binary digital halftoning [1] is a technique of rendering a continuous-tone image with two tone levels.

Basically, binary halftoning can be accomplished with either amplitude modulation(AM)[2] in which a halftone

is produced by varying the size of printed dots arranged along a regular grid or frequency modulation(FM) [3-

15] in which a halftone is produced by varying the relative dot density of fixed-size printed dots. It is generally

agreed that, when FM halftoning is exploited, a good quality output should bear blue noise characteristics [5].

The concept of blue noise halftoning was first introduced in [5] by Ulichney. It says that a good quality

halftone image should have a frequency spectrum that only contains high frequency random noise. In particular,

for a constant gray-level input, the dots that appear in its halftoning output should be isolated and their ideal

spatial distribution should be aperiodic, homogeneous and isotropic. Accordingly, the spectral energy of the

output should be concentrated at a particular radial frequency. This radial frequency is referred to as principal

frequency and it should be a function of input gray level g as

≤<−

≤<=

12/1 1

2/10 )(

gforg

gforggfB (1).

Little energy should be in the frequency band below the principal frequency. These spectral characteristics are

termed as blue noise characteristics.

In practice, dots are put on grid points. The grid pattern determines the sampling frequency, which in turns

confines the baseband bandwidth of the halftone output. Fig. 1 shows the spectral plane of a halftone when a

rectangular grid pattern is used. When g is less than 1/4, the principal frequency of a blue noise halftone pattern

is less than 1/2 and hence a ring pattern can be observed in the frequency spectrum as shown in Fig. 2(a).

However, when g falls in the range of 2/14/1 ≤< g , we have 2/1)( >gfB and aliasing occurs as shown in

Fig. 2(b). In the original model, Ulichney suggested packing the energy to the partial annuli regions (i.e. the

four corners) of the baseband as shown in Fig. 2(c). This adds correlation between minority pixels along the

diagonal in spatial domain and hence creates undesired visible patterns in which dots are more likely to occur

along the diagonal. In other words, the dot distribution is not isotropic and directional artifacts exist. When the

situation becomes worse, checkerboard artifacts can be observed.

In view of this, Lau and Ulichney [16] proposed a modification to the original blue noise model to prevent

this situation from happening by placing an increased emphasis on the need for maintaining the radial symmetry

of the spectrum1. Specifically, the principal frequency is bounded to be 1/2 for 4/34/1 ≤< g to maintain the

radial symmetry of the spectrum. In other words, the spectral energy of a binary dither pattern that represents

gray level g should be concentrated at a new principal frequency given as

1 In [16], based on the same philosophy, Ulichney’s noise model is modified to handle rectangular and hexagonal sampling

grids respectively. Since the focus of this paper is on the situation when a rectangular sampling grid is used, we are

referring to the modified model proposed for the rectangular sampling grid. This applies to the rest of this paper as well.

3

≤<−

≤<

≤<

=

14/3 1

4/34/1 2/1

4/10

)('

gforg

gfor

gforg

gfB (2)

in the modified blue noise model. This revised blue noise model enforces the property of radial symmetry in a

better way.

Eqn. (2) defines the desirable principal frequency of the halftone rendition of a particular input gray level

for the revised blue noise model. To produce a halftone having these desirable spectral characteristics, Lau and

Ulichney [16] suggested introducing a minimum degree of clustering. In other words, clustered dots instead of

isolated dots are distributed. It was found that, when dot clusters are distributed to generate a halftone, the

output should bear desirable green noise characteristics in which the spectral energy is concentrated at another

new principal frequency given as

≤<−

≤<=

12/1 /)1(

2/10 /)(

gforMg

gforMggfG (3),

where M is the average cluster size of the minority dots [17]. The principal frequency now depends on the

average distance between cluster centers and becomes a function of both input gray level g and average cluster

size M.

Theoretically, if a halftoning algorithm can switch from blue noise halftoning to green noise halftoning

when 4/34/1 ≤< g and adjust the average cluster size to make gM 4= for 2/14/1 ≤< g and )1(4 gM −=

for 4/32/1 ≤< g , the spectral energy will be concentrated at radial frequency

≤<−

≤<=−

≤<=

≤<

=

13/4 )1(

4/31/2 2/1/)1(

2/11/4 2/1/

4/10

)(

gforg

gforMg

gforMg

gforg

gfr (4)

and the desirable characteristics specified in formulation (2) can be achieved.

Adjusting the cluster size to modify the spectral statistics of a halftone is not a novel idea. For example, in

Levien’s EDODF [18], an output-dependent feedback path is introduced to adjust the cluster size with a

parameter called hysteresis constant. However, few of these algorithms are dedicated to produce halftones

having the spectral characteristics specified in (2) and hence whether the cluster size can be precisely and

arbitrarily adjusted with a single parameter directly is generally not their major concern. When one has to

produce clusters of precise average size M for a given gray level g to satisfy model specification (2), it becomes

a difficult task to achieve. Consequently, a tedious empirical study is required to obtain a table describing the

relationship between the average cluster size and the tuning parameter value for providing the target spectral

characteristics. Note that such a relationship may not exist or may be hard to get empirically for some cluster

sizes. In fact, a detailed study on EDODF and some of its variants was reported in [19], and it is found that this

cluster tuning approach has a performance limit for generating visually pleasing halftones inside a hysteresis

constant range. Even saying so, we cannot exclude the potential of using EDODF to solve the addressed

4

problem in the future as its possibility of varying the error weights, the hysteresis weights and the hysteresis

parameter of the diffusion filter provides it a certain extent of flexibility.

After Lau et al. introduced their revised blue noise model in [16], two iterative halftoning techniques

including Ulichney’s Void-and-Cluster initial pattern technique (VACip) [20] and Allebach’s Direct Binary

Search (DBS) [21] were tried respectively by Lau et al.[16] and González et al.[22] to produce halftones of the

spectral characteristics specified by their model. However, since VACip and DBS were not purposely

developed to manipulate the cluster size precisely and flexibly, the objective still cannot be exactly achieved to

a certain extent.

Obviously, the key to success relies on whether we can adjust the cluster size arbitrarily with a parameter

for any given input gray level. A recently proposed green noise halftoning algorithm referred to as FMEDg[23]

can help to achieve this goal. This algorithm was developed based on the multiscale error diffusion (MED)

technique proposed in [9]. FMEDg exploits a non-causal error diffusion filter which is close to isotropic to

guarantee the spatial homogeneity and, at the same time, able to produce dot clusters of any desirable average

size. By adjusting the average cluster size, one can control the average distance between clusters and hence the

principal frequency of the resultant halftone. These properties are very useful to produce halftones of any

desired spectral characteristics.

In this paper, based on FMEDg[23], we first propose a halftoning algorithm for producing halftones

bearing the noise characteristics specified by Lau et al.’s revised blue noise model [16]. This algorithm is able

to produce halftones having exactly the specific spectral characteristics, and hence its simulation results can be

used to study the performance of Lau et al.’s revised noise model. From the study, it is found both empirically

and theoretically that there is room to further improve Lau et al.’s revised noise model. Accordingly, a new

noise model for describing the noise characteristics of a high-quality halftone is suggested. Another MED-based

algorithm is then proposed to generate halftones bearing the suggested noise characteristics. By comparing its

output with those of the other relevant halftoning algorithms, one can evaluate if the new noise model is more

appropriate than the conventional noise models in describing the noise characteristics of a high-quality halftone.

The evaluation result is positive in our simulations.

The organization of this paper is as follows. As an important tool used in this paper to study the

connection between halftone quality and noise models, FMEDg[23] is briefly introduced in Section II. A

halftoning algorithm for producing halftones bearing the noise characteristics specified by Lau et al.’s revised

blue noise model [16] is also presented in this section. Then, in Section III, the weakness of Lau et al.’s noise

model is addressed and an improved noise model is suggested. A MED-based halftoning algorithm for

producing halftones bearing the noise characteristics specified by the suggested noise model is proposed in

Section IV. In Section V, a detailed analysis on the performance of the proposed halftoning algorithm in terms

of various measures is given. Simulation results on real images are provided in Section VI to evaluate the

performance of various noise models. Finally, a conclusion is given in Section VII.

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II. BASIC TOOLS FOR THE STUDY

In this section, we first provide a brief summary of FMEDg[23]. This algorithm forms the basis for solving

the problem addressed in this paper. A halftoning algorithm is then developed to produce halftones bearing the

spectral characteristics specified in (2) for studying the performance of Lau et al.’s noise model [16].

Like any other MED algorithms such as [10], [14], [24] and [25], FMEDg[23] is a two-step iterative

algorithm. At each iteration, it selects a not-yet processed pixel, quantizes its value to either 0 or 1, and diffuses

its quantization error to the pixel’s neighbors with a non-causal diffusion filter. This process repeats until all

pixels are processed.

The diffusion filter used in FMEDg, which is denoted as o

RRF ),( 21 in this paper, is an approximation of an

isotropic circular ring-shaped filter. It diffuses the error at pixel position (0,0) to a ring region defined as {(x,y)|

122

2 RyxR >+≥ } in the continuous space, where 1R and 2R are, respectively, the inner and outer radii of

the ring region. Specifically, the (m,n)th filter coefficient of

oRRF ),( 21

, ),( nmf o , is defined as

π)(

),,(),,(),(

21

22

12

RR

RnmARnmAnmf o

−

−= (5),

where ),( nm are the horizontal and vertical integer offsets from the error source, and ),,( kRnmA for k=1, 2 is

the area covered by circle kRyx ≤+ 22 in pixel (m,n). Effectively, the filter coefficient for a pixel which is

(m,n) pixels away from the point error source at the center of pixel (0,0) is proportional to the area covered by

the circular ring 122

2 RyxR >+≥ in the grid unit associated with that pixel.

Dot clusters are formed in the outputs of FMEDg. It was found that the inner radius 1R helps to determine

the average cluster size of the clusters. In particular, when 12 2RR = , we have a relationship model given as

gRM π21≈ (6),

where M is the average cluster size and g is the input gray level. The average cluster size can then be

monotonically and continuously adjusted with 1R to a certain extent. In other words, one can use diffusion filter

o

RRF

)2,( 11

to produce minority clusters of any desirable average size by simply adjusting its parameter 1R and

distribute the clusters homogeneously.

With filter o

RRF

)2,( 11

in hand, halftones bearing the spectral characteristics specified in (2) can be easily

produced by adjusting 1R to control the principal frequency of the output of a particular input gray level g. In

particular, when 0.25<g≤0.5, one can select π/21 =R to make the principal frequency be

2/1)/(1/ 1 =≈ πRMg based on spectral characteristic model (3) and relationship model (6). Green noise

halftoning is carried out in this case. When g≤0.25, the blue noise MED halftoning algorithm proposed in [14]

6

(FMED) can be used to make the principal frequency equal to g . When g>0.5, black dots become the

minority pixels. After changing the roles of black dots and white dots, the same rule applies. This specific

solution for producing halftones bearing the spectral characteristics specified in (2) is referred to as hybrid

FMED (HFMED) hereafter.

III. SUGGESTED OISE MODEL

In this section, we will show that, even when the halftone rendition of a mid-tone level bears the spectral

characteristics specified in (2), its visual quality can still be improved from spectral point of view. Directions

for improving the mid-tone rendition will be discussed and, accordingly, a suggested revision to Lau et al.’s

revised blue noise model [16] will be given.

FMEDg can also serve as a tool for us to study the impact of the principal frequency of a mid-tone level’s

halftone rendition on the rendition quality since it is able to produce halftone patterns of any arbitrary principal

frequency for any gray level by just tuning 1R . Fig. 3 shows two halftone renditions of a constant mid-tone gray

level image and their corresponding spectra. They were all generated with FMEDg and their principal

frequencies are adjusted to be 0.5 and 0.4 respectively.

Radially averaged power spectrum density (RAPSD) is a measure proposed in [5] for analyzing the spectral

characteristics of a halftone pattern and its definition is given in the appendix for reference. Fig. 4 shows the

RAPSD plots of the two halftone renditions and it confirms the locations of their principal frequencies which

are marked by the peaks.

One observation we have had is that, to achieve the ultimate goal of Lau et al.’s modification to the

traditional blue noise model, the principal frequency for 2/14/1 ≤≤ g should actually be a bit lower than ½

instead of the ½ specified in (2). In practice, dots or clusters in a halftone output are randomly distributed as

long as a stochastic halftoning algorithm is exploited, and hence its RAPSD peak will spread to a certain extent.

As shown in Fig. 4, the tail of the RAPSD peak extends to the partial annuli regions when the principal

frequency is at 1/2. When the tail is heavy, there is a considerable amount of energy accumulated in the partial

annuli regions. The correlation between diagonal pixels then becomes significant and checkerboard patterns are

still visible as shown in Fig. 3(a)(i). Besides, when the principal frequency is close to 0.5, the tail of the peak

extends out of the baseband and causes aliasing at the top, the bottom, the left and the right boundaries of the

baseband. One can see the four corresponding bright spots in the baseband spectrum shown in Fig.3(a)(ii). This

explains the appearance of the texture directionality in Fig.3(a)(i) in which there are a lot of horizontal and

vertical line segment patterns.

However, by increasing the average cluster size a bit, the principal frequency of the halftone pattern can be

shifted to the low frequency side a bit such that the checkerboard patterns can be totally eliminated as shown in

Fig. 3(b)(i). From the corresponding spectrum shown in Fig.3(b)(ii), one can see that there is negligible energy

in the partial annuli regions and there is no aliasing. The energy is isotropically distributed in the baseband and

the energy peaks form a perfect circle. This implies an isotropic distribution of dots in the halftone.

7

The spectral isotropicity is actually achieved at a cost of higher graininess as larger clusters are formed in

green noise halftoning to lower the principal frequency of the resultant halftone. Certainly there should be a

compromise between the isotropicity and the graininess, so the downshift of the principal frequency from ½ for

2/14/1 ≤≤ g should be as small as possible while optimizing the isotropicity. For reference purposes, the

compromised principal frequency for 2/14/1 ≤≤ g is referred to as fc. The details of the compromisation will

be addressed in the next section.

According to Lau et al.’s model, the desirable principal frequency for g≤1/4 should be g . When the

principal frequency for 2/14/1 ≤≤ g is lowered to fc, there is an abrupt change in the principal frequency at

g=0.25. This discontinuity may result in a visible change in the average cluster size when the input image

contains a large region in which the intensity value gradually changes across 0.25. To solve this problem, a

transition region should hence be introduced to allow the principal frequency to deviate from g gradually

when g increases from gTLB

, the lower bound of the transition region, and finally reach fc at g=0.25 as the blue

knotted curve shown in Fig. 5. To guarantee the smoothness of the transition, a twice-continuously

differentiable constraint is applied to the curve in our suggested model.

The desirable principal frequencies for the gray levels within the transition region (i.e. gTLB

≤g<0.25) can be

determined by interpolation with the samples of

≤≤

<≤=

5.025.0

0)("

gforf

ggforggf

c

TLBB (7),

where )(" gfB is the desirable principal frequency of gray level g in our suggested model. In principle, the

principal frequency for 0<g<0.5 should be monotonically increasing with g and twice continuously

differentiable to avoid any sharp change. Subject to these two constraints, the mean square difference between

the interpolated principal frequency and )(gfB should be minimized over the range of gTLB

25.0<≤ g . When a

specific curve fitting method is selected to do the interpolation, the optimal gTLB

can be determined by

gTLB

( )∑<≤

−=25.0"

2

"

)(ˆ1minarg

gg

Bog

ggf�

for 25.0"125.0 <≤ g (8),

where "g is a candidate gray level lower than 0.25, o� is the number of possible input gray levels in region

25.0" <≤ gg , and )(ˆ gfB is the interpolated principal frequency obtained when only the principal frequencies

in region 25.0" <≤ gg are interpolated with the selected curve fitting method. Once gTLB

is determined, )(" gfB

for the transition region can be determined as the )(ˆ gfB obtained when "g =gTLB

.

Function )(" gfB describes how the principal frequency should change with the gray level to improve the

mid-tone rendition in our suggested model. The curves in Fig. 5 show its difference from the traditional blue

noise model [5] and the revised blue noise model [16] graphically. To produce halftones bearing the noise

8

characteristics specified by the suggested model, green noise halftoning instead of blue noise halftoning is

performed when the gray level is a mid-tone level. Accordingly, the suggested model is a tone-dependent noise

model as it says that the noise nature of a high-quality halftone should be tone dependent.

As compared with Lau et al.’s noise model [16], the proposed noise model is better in two ways. First, the

model takes the energy around the principal frequency into account such that the energy in the partial annuli

regions can be reduced to maintain the radial symmetry of the spectrum for all gray levels. Second, a transition

region is introduced to eliminate the abrupt change in the spectral characteristics when switching between blue

and green noise halftoning.

IV. MED-BASED REALIZATIO

A model describing the desirable principal frequency for a gray level’s halftone rendition is suggested in

Section III. The issue is now how to produce halftones of the specific noise characteristics in practice.

Theoretically, as long as a halftoning algorithm can precisely adjust the principal frequency of its output for any

arbitrary input gray level as it wishes, it can be fine-tuned to produce halftones of desirable noise characteristics

according to the model and forms a solution of the addressed problem. However, few reported algorithms can

practically be tunable in this manner. In this section, we will show how a MED-based solution can produce

halftones of the desirable noise characteristics.

In our suggested model, two parameters, namely, fc and gTLB

, are intentionally open to be determined. As

mentioned earlier, theoretically one can fine-tune any appropriate halftoning algorithm to produce halftones of

the suggested noise characteristics. However, algorithms using different halftoning techniques produce outputs

of different spectral characteristics, and the width of their RAPSD peaks could be different. Accordingly, when

different halftoning techniques are used, the amount of downshift for the principal frequency for 0.5≥g≥0.25

and hence the width of the transition region should also be different. From this point of view, fc and gTLB

will be

solution dependent.

In this section, we first determine fc and gTLB

for our MED-based solution to make the model completely

well-defined for the solution. Then we will show how one can adjust parameters 1R and 2R of the ring-shaped

diffusion filter defined in eqn.(5) to, for any given constant patch, produce a halftone of desirable principal

frequency with FMEDg based on the model. Accordingly, a corresponding tone-dependent diffusion filter can

be defined. A MED-based halftoning algorithm is finally proposed to produce halftones of the suggested noise

characteristics with the tone-dependent diffusion filter.

A. Determination of the principal frequency for ¼≤g≤ ½

As mentioned earlier, the principal frequency for the gray levels in this range should be less than ½. Green

noise halftoning should hence be performed.

In the ideal case, the spatial distribution of the minority dots in a halftone rendition of a constant patch of

gray level g should be homogeneous and isotropic. In [19], Lau developed a directional distribution function

9

)(,0 α∆D to measure the directional distribution of dots in a dot pattern. Specifically, a minority dot’s circular

local region of radius ∆ is partitioned into equal sectors. Each sector is indexed by α which specifies the sector’s

directional position with respect to the minority dot. )(,0 α∆D is defined as the normalized expected number of

minority dots per unit area in a particular sector. In general, the local region is partitioned into 8 sectors and

radius ∆ is selected to be gλ , the principal wavelength of the halftone rendition of the constant patch. By

definition, gλ is the reciprocal of the principal frequency of the halftone rendition.

Based on )(,0 α∆D , a directional index function can be defined as

∑=

∆−=8

1

2,0 ))(1(

8

1

α

αDD (9)

to measure the directional characteristic of the spatial dot distribution in the halftone rendition. In the ideal case,

D should be zero for all g because an isotropic distribution of dots makes )(,0 α∆D =1 for all α[19]. The larger

the value of D, the more directional and the less isotropic the dot distribution is for the specific input gray level.

When FMEDg is used to halftone a constant patch with the diffusion filter defined in (5), the principal

frequency of the output can be adjusted directly with 1R . In fact, from spectral characteristic model (3) and

relationship model (6), one can deduce that the principal frequency is given as )/(1/ 1 πRMg ≈ when

12 2RR = . By gradually adjusting the principal frequency of the halftone rendition of a constant patch whose

gray level falls into the range from ¼ to ½, one can study how the principal frequency affects the extent of

isotropicity of the dot distribution of the halftone rendition in terms of directional index D.

Fig. 6 shows the simulation results of two constant patches whose gray levels are respectively ¼ and ½. In

both cases, the directional index D is close to zero when the principal frequency drops below 0.44. By

considering that a lower principal frequency of a halftone implies larger minority dot clusters in the halftone,

the principal frequency for ¼≤g≤ ½ is selected to be fc=0.44 in our solution for producing halftones of the

suggested noise characteristics.

As a remark, we note that fc is the minimum downshift of the principal frequency from ½ to maintain the

isotropicity of the dot distribution for 2/14/1 ≤≤ g and it changes as different screen design algorithms are

used. From Fig.1 one can easily deduce that the isotropicity can only be maintained when there is no or

comparatively negligible energy in the partial annuli regions. As discussed in Section III, the RAPSD peak

associated with the principal frequency of a halftone spreads. Its extent of spread determines how close to 0.5

the principal frequency can be under the condition that the tail of the RAPSD peak does not considerably extend

into the partial annuli regions to destroy the isotropicity. Obviously, the extent of the spread of the RAPSD peak

is algorithm dependent and so is fc. For FMEDg, the spread of the RAPSD peak is more or less the same for

2/14/1 ≤≤ g and does not extend its tail into the partial annuli regions remarkably as long as the RAPSD peak

keeps a distance of 0.06 away from the partial annuli regions, which explains the simulation result reported in

Fig.6.

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B. Determination of the principal frequency for the transition region

In our solution, the smoothing spline curve fitting is used to interpolate the desirable principal frequencies

for the gray levels in the transition region to satisfy the twice-continuously differentiable constraint. Without

loss of generality, we assume that an input image to be halftoned is of 256 gray levels. In such a case, gTLB

is a

multiple of 1/255, and only principal frequencies for gray levels g∈{gTLB

, gTLB

+1/255, …, 63/255} are required

to be interpolated. Based on formulation (7), the set of available sample points used for interpolation can be

determined as Ψ={(g, g ) | g = 0, 1/255, …, gTLB

-1/255}U {(g, 0.44) | g = 64/255, 65/255, …, 127/255}.

Any gray level in the range from 0.125 to 0.25 can be used as the gTLB

to carry out the interpolation. The

optimal gTLB

is determined as 42/255 with the criterion specified in (8) subject to the monotonic increasing

constraint and the constraint that gTLB

is a multiple of 1/255.

C. Realization for gTLB

≤g≤ ½

Green noise halftoning is carried out when gTLB

≤g≤ ½. In green noise halftoning, the principal frequency of

a halftone rendition of gray level g can be tuned by adjusting the average size of the minority dot clusters as

described in eqn. (3). When FMEDg is used to adjust the average cluster size, we have gRM π21≈ as long as

12 2RR = holds. The principal frequency is then given by )/(1/)( 1 πRMggfG ≈= . In other words,

FMEDg can produce halftones of the suggested noise characteristics for gTLB

≤g≤ ½ with diffusion filter (5) the

1R and 2R of which are given as

( )

=

⋅=

12

"1

2

)(/1

RR

gfR Bπ for g

TLB ≤ g≤0.5 (10),

where )(" gfB is the desirable principal frequency specified in the suggested model. Fig. 7 graphically shows

how average cluster size M, filter parameters 1R and 2R should change with g to produce halftones of the

suggested noise characteristics.

D. Realization for 0≤g< gTLB

As shown in Fig. 5, blue noise halftoning should be carried out when 0≤g<gTLB

. By considering that blue

noise halftoning is just a special case of green noise halftoning in which we have M=1, one can still make use of

FMEDg with diffusion filter (5) to achieve blue noise halftoning as long as appropriate 1R and 2R are selected

to maintain the average cluster size to be 1.

In our realization, we keep 1R and 2R unchanged for g≤gTLB

as

( )

=

⋅=

12

1

2

/1

RR

gRTLB

π for g≤ g

TLB (11)

11

as shown in Fig. 7. Note that the relationship model gRM π21≈ is no longer valid when g≤g

TLB even though

12 2RR = is still valid. In practice, the average cluster size cannot be smaller than 1. As the 1R at g=gTLB

already makes the average cluster size M be 1, a smaller g cannot reduce M further.

As 1R and 2R change smoothly over the range of g from 0 to 1, one can guarantee that, when the gray

levels change gradually in the input, there is no visual discontinuity in the halftone rendition. If other blue noise

halftoning algorithms such as FMED [14] are used to handle the gray levels in 0≤g< gTLB

, this continuity may

not be guaranteed as the diffusion filter used for 0≤g< gTLB

will not match with the one used for gTLB

≤g≤ ½ in

such a case.

E. MED-based halftoning Algorithm

By adjusting parameters 1R and 2R of the ring-shaped filter defined in eqn.(5) according to the input gray

level g as mentioned above, a tone-dependent diffusion filter can be defined. With this diffusion filter, a MED-

based halftoning algorithm can be easily developed based on the framework of FMEDg to produce halftones of

the suggested noise characteristics. In particular, one can just replace the default diffusion filter used in FMEDg,

which is o

RRF

)2,( 11

for all pixels, with oxRxR jiji

F ))(),(( ,2,1, where )( ,1 jixR and )( ,2 jixR are, respectively, the

desirable 1R and 2R values provided in Sections IV-C and IV-D for jix , , the gray level of pixel (i,j). Other

than this difference, the realization of the newly developed algorithm is the same as that of FMEDg. For

reference proposes, this proposed MED-based halftoning algorithm is referred to as FMEDt hereafter.

V. PERFORMACE AALYSIS

FMEDt is proposed for producing outputs bearing the spectral characteristics governed by the tone-

dependent noise model suggested in this paper. A simulation was carried out to evaluate if FMEDt can really

achieve the goal and if halftones bearing the suggested noise characteristics are of higher quality than those

bearing the noise characteristics of Lau et al.’s noise model. Accordingly, HFMED, Lau et al.’s [16] and

González et al.’s [22] were also evaluated in the simulation for comparison as they are dedicated algorithms

proposed to produce halftones according to Lau et al.’s noise model [16]. Besides, as a classical realization of

blue noise halftoning, Ulichney’s [5] algorithm was also included in the comparison as a reference. All

evaluated algorithms were applied to a set of constant gray-level images of size 256×256 and the dot

distributions of their outputs were studied.

Figs. 8 and 9, respectively, show the halftone outputs of various algorithms for images of different constant

gray levels and their corresponding frequency spectra. The selected gray levels represent different ranges of

input gray levels between 0 and 0.5. For better comparison, all spectra for the same input gray level in Fig. 9 are

normalized with respect to the maximum magnitude value of all their frequency components.

Ulichney’s algorithm [5] is basically a conventional blue noise halftoning algorithm which aims at

producing a halftone having the spectral characteristics defined in eqn. (1). Energy is packed into the partial

12

annuli regions as shown in Figs. 9(i)(b)-(d) when 5.025.0 ≤< g . As a result, the diagonal spatial correlation

among pixels is strong at the output and checkerboard patterns can be easily found in Figs. 8(i)(c) and (d).

Lau et al.’s[16], González et al.’s[22] and HFMED algorithms are proposed to produce outputs having the

spectral characteristics defined in eqn. (2). Based on the halftone outputs and their spectra shown in Figs. 8 and

9 respectively, one can see that the noise characteristics of HFMED’s output is obviously closer to the desirable

noise characteristics specified by Lau et al.’s noise model. As shown in Figs. 9(iv)(b)-(d), for each presented

input gray level g, in the spectrum of its halftone output, there is little energy in the partial annuli regions and

one can see a virtual circle formed by the energy peaks along different directions at radial frequency 0.5.

Though similar virtual circles can also be found in Figs. 9(ii) and (iii), considerable amount of energy is still

packed in their partial annuli regions. This explains why checkerboard patterns can be observed in Figs. 8(ii)

and (iii).

However, from spectral point of view, HFMED is still inferior to FMEDt. When g>0.25, HFMED

produces halftones that have their principal frequencies at 0.5. The significant amount of energy around the

principal frequency causes aliasing problems. As shown in Figs. 9(iv)(b)-(d), it contributes four bright spots at

the boundaries of the baseband. This explains why there are a lot of vertical and horizontal line segments in

Figs. 8(iv)(b)-(d). In contrast, as shown in Figs. 9(v), a prefect circle without bright spots can be observed in the

baseband spectrum of FMEDt’s output. There is no aliasing problem and the energy is distributed isotropically.

Fig. 9(a) shows the case when g=15/255<0.25. In this case, all evaluated algorithms perform blue noise

halftoning. Theoretically, the performance of FMEDt is better than that of HFMED in terms of isotropicity. It

is because HFMED exploits a 3×3 square diffusion filter as FMED [14] does while FMEDt exploits a ring-

shaped diffusion filter to produce halftones. Obviously, a ring-shaped diffusion filter diffuses error isotropically

and a better halftoning performance can be resulted.

While Fig. 9 only shows the algorithms’ spectral performance for a few input gray levels, the RAPSD plots

shown in Fig. 10(a) provide a complete picture for all input gray levels. In these plots, all RAPSD values are

clipped by 4 such that an easier comparison among the plots can be made. One can see that the proposed

FMEDt can faithfully produce the desirable characteristics specified by the suggested tone-dependent noise

model. The dot distribution of its outputs is homogeneous and isotropic as the average distance among

neighboring clusters is the same along all directions.

Fig. 10(b) shows the performance of the algorithms in terms of anisotropy. Anisotropy is a measure

proposed in [5] to measure the strength of directional artifacts, and its definition is given in the appendix for

reference. One can see that the anisotropy values of all algorithms are well below zero for 2/10 ≤< g .

Directional components are considered to be unnoticeable by human eye when this happens. It implies that the

spatial distribution of the minority dots or clusters in their outputs is radially symmetric.

As a matter of fact, since the diffusion filter used in FMEDt is an approximation of a non-causal circular

ring-shaped filter and the energy in the partial annuli regions is minimized as discussed in Sections III and IV,

the dot or dot cluster distribution in FMEDt’s output should not be only radially symmetric, but also close to

isotropic. Any imperfection that exists is mainly due to the unavoidable grid constraint.

13

VI. SIMULATIO RESULTS

A simulation was carried out to study the performance of the evaluated algorithms in handling real images.

Fig. 11 shows a set of eight 8-bit gray-level testing images used in our simulations. They are all of size 256×256

pixels.

Fig. 12 shows the performance of various algorithms in terms of vMSE . vMSE was proposed in [19] to

measure the observed distortion between an original gray-level image X and its binary halftone B . In

particular, vMSE is defined as

2),,(),,(

1MSE dpivdhvsdpivdhvs

��v BX −

×= (12),

where hvs is the HVS filter function defined in [19], vd is the viewing distance in inches and dpi is the printer

resolution. Evaluation results for different combinations of viewing distance and printer resolution were

reported in Fig. 12. While Fig. 12 shows the performance in terms of vMSE , Table I shows the performance in

terms of Universal Objective Image Quality Index (UQI) [26]. Note that the value of UQI is bounded to [-1, 1]

and a larger value indicates a better performance. One can see that, in terms of both vMSE and UQI, FMEDt

performs better than the others.

Fig. 13 shows the halftone results of various algorithms for testing image “Goldhill” for subjective

comparison. A subjective assessment study was also carried out to evaluate the performance of various

algorithms. The assessment procedure is basically the same as the one exploited by Monga et. al. in [27] to

evaluate HVS models. In each trial of assessment, an observer was forced to rank the halftoning outputs

produced with different algorithms according to their visual closeness to the original image. The proportion of

trials where one evaluated algorithm is preferred to another is recorded after 376 trials. Based on the subjective

assessment results, a preference matrix P was obtained as

=

=

5.06464.07486.08591.07652.0

3536.05.07514.08702.07486.0

2514.02486.05.08232.05746.0

1409.01298.01768.05.01685.0

2348.02514.04254.08315.05.0

5.0

5.0

5.0

5.0

5.0

EDECEBEA

DEDCDBDA

CECDCBCA

BEBDBCBA

AEADACAB

pppp

pppp

pppp

pppp

pppp

P

where XYP for X, Y∈{A, B, C, D, E} represents the proportion that algorithm X was preferred to algorithm Y,

and algorithms A, B, C, D and E correspond to Ulichney’s [5], Lau et al.’s [16], González et al.’s [22], HFMED

and FMEDt. Note that we have YXXY PP −= 1 . The preference matrix shows that the outputs of FMEDt are

visually closest to the original images.

Fig. 14 shows the halftoning results of a 1024×128 tilted gray ramp image. The gray level of the (m,n)th

pixel of the original is given by

255

1

1023

)(5.01255),( ×

+−×=

nmroundnmR for m = 0,1,…1023 and n = 0,1,…127 (13).

14

The image covers gray levels from 0.4392 to 1. One can see severe checkerboard artifacts in the mid-tone range

in the outputs of Ulichney’s[5]. This is expected as it packs the energy to the partial annuli regions in the mid-

tone range. Lau et al.’s[16], González et al.’s[22] and HFMED try to avoid packing the energy into the partial

annuli regions. Among them, HFMED is more successful in achieving this goal and it eliminates all

checkerboard artifacts. As mentioned in Section III, aliasing problems occur when the principal frequency is at

½ and it explains why in HFMED’s result there are horizontal and vertical texture patterns. By adjusting the

principal frequencies of the gray levels in the mid-tone range based on the tone-dependent noise model

suggested in this paper, FMEDt can effectively solve the aliasing problems. Besides, as a transition region is

introduced for the principal frequency to change continuously with the gray level in the suggested model, there

is no abrupt change in the halftone output of the ramp image as shown in Fig. 14(e).

Fig. 15 shows the halftoning results of a testing image in which there are mainly two mid-tone gray levels

(83/255 and 126/255). The testing image is purposely designed such that from the halftoning results one can see

how the quality of the output can be affected by the principal frequencies of these two mid-tone gray levels in

the halftoning output.

Since Lau et al.’s[16] algorithm exploits a dither array to carry out halftoning, it can be expected that its

performance in handling real images is not comparable with the other evaluated methods as shown in Fig. 12.

Accordingly, it is not included in this comparison to reduce the page length. As a replacement, the output of

direct binary searching (DBS) algorithm [21] is presented as a reference for comparison as it is generally

considered as one of the best algorithms which provide high-quality output. However, we note that DBS is not

purposely optimized according to any one of the noise model metrics concerned in this paper. It is optimized

with respect to a HVS-based error metric.

As shown in Figs. 15(b), (c) and (d), the checkerboard artifacts in the halftoning outputs contributed by

Ulichney’s[5], DBS[21] and González et al.’s[22] damage the details of the original image and make the letters

hardly recognizable. The situation is improved in HFMED’s output in which all checkerboard artifacts are

removed. However, because the principal frequencies of both major mid-tone levels in the output are 0.5, there

are horizontal/vertical texture patterns contributed by the aliasing problem. Relatively speaking, the letters are

much more recognizable in FMEDt’s output.

The principal frequencies of all mid-tone levels in FMEDt’s output are lowered to 0.44 by performing

green noise halftoning. As clusters instead of dots are introduced when handling the mid-tone gray levels in

FMEDt, worm patterns are visible in the output when the input gray level is close to 0.5. However, as

mentioned in [16], worm patterns are not necessarily bad as long as they are not directional and form twisting

and turning paths from pixel to pixel to create a smooth texture. As compared with the worm patterns appearing

in Fig. 15(d), the worm patterns in Fig. 15(e) are not mainly horizontal and vertical but are of random nature.

This makes the background region less objectionable and the details more recognizable in FMEDt’s output.

In general, people consider that blue noise halftoning is better than green noise halftoning in preserving the

feature details in the original image as green noise halftoning produces dot clusters instead of dots, increases the

graininess and reduces the spatial resolution. The example shown in Fig. 15 shows that this may not be always

15

true. In blue noise halftoning, energy is packed in the partial annuli regions and it generates checkerboard

artifacts. The checkerboard patterns are fine but come in packs as shown in Figs. 15(b)-(d). The size of a pack

of checkerboard patterns can be even larger than the size of a dot cluster produced in green noise halftoning. In

such a case, fine feature details of the original image cannot be preserved. This explains why, when a halftoning

algorithm works according to the suggested tone-dependent noise model instead of the conventional blue noise

models, it can still produce a higher quality output even though it switches from allocating dots to allocating dot

clusters for mid-tone gray levels.

The complexity of FMEDt is high when it is directly realized in the way presented in the paper as it is

basically an iterative algorithm. However, its complexity can be significantly reduced to allow real-time

processing by making use of the technique proposed in [25]. Besides, GPU technology can also be exploited to

speed up the process. Since the focus of this paper is on how to produce halftones of desirable noise

characteristics, the details of complexity reduction are not discussed in this paper.

VII. COCLUSIOS

In practice, the placement of dots in a halftone is constrained by a sampling grid and hence aliasing

happens when the input gray level is in the middle range. As suggested by Lau et al., this problem can be solved

by replacing isolated dots with dot clusters to change the noise characteristics and maintain the principal

frequency of the output to be 1/2 when this happens. However, Lau et al.’s model does not take into account the

fact that, even when the principal frequency of the output is ½, in stochastic halftoning the considerable amount

of spectral energy around the principal frequency can still cause aliasing problems. Based on this observation, a

modification to Lau et al.’s model is suggested in this paper to solve this problem.

The suggested model is a tone-dependent noise model. To produce halftones that satisfy the specification

of this model, a halftoning algorithm should control the noise characteristics according to the input gray levels.

It is not an easy task to conventional error diffusion techniques as they cannot precisely and arbitrarily tune the

principal frequencies of their halftoning outputs for each possible input gray level. In fact, it is also one of the

reasons why so far there is few dedicated solutions for producing halftones of the revised blue noise

characteristics specified by Lau et al.’s model.

FMEDg[23] is a recently proposed halftoning algorithm which allows one to flexibly tune the cluster size

and hence the principal frequency of its halftone for any given input gray level. This property makes FMEDg

capable to produce halftones of any specific noise characteristics easily. Based on FMEDg, two MED-based

halftoning algorithms, namely, HFMED and FMEDt, are separately proposed based on Lau et al.’s noise model

and the suggested tone-dependent noise model respectively in the paper.

Analysis and simulation results show that, as a dedicated solution targeted for producing halftones of the

suggested noise characteristics, FMEDt can successfully eliminate checkerboard artifacts, eliminate directional

hysteresis, preserve feature details of the original image, distribute dots or dot clusters aperiodically and

homogeneously, and provide outputs bearing the desirable noise characteristics as specified by the suggested

model. In terms of various measures, its performance is superior to HFMED and other evaluated algorithms

16

which are proposed based on Lau et al.’s model or the traditional blue noise model. Based on this observation,

we expect that a halftone can be of higher quality if it bears the suggested noise characteristics instead of the

noise characteristics specified by either of the other two models.

FMEDt is a successful example showing how to produce halftones of the suggested noise characteristics

and how the halftoning performance can be improved when the goal is achieved. With the help of this suggested

noise model, we expect that algorithms based on some other existing state-of-the-art halftoning techniques such

as adaptive threshold modulation[12], tone-dependent halftoning[13], EDODF [18], and DBS[21] can also be

developed to produce outputs of better visual quality in the future.

As a final remark, we note that this paper only presents the case when a rectangular sampling grid is used.

The same idea can be applied to the case when a hexagonal sampling grid is used. Accordingly, a corresponding

model and corresponding halftoning algorithms can be developed to handle the case.

ACKOWLEDGEMET

We would like to thank Dr. Alvaro J. González for clarifying some technical issues on his work [22] and

providing the source code of the alpha stable model described in [22] to us.

APPEDIX

Radially averaged power spectrum density (RAPSD) and anisotropy are two measures commonly used to

analyze the spectral characteristics of a halftone pattern [5]. In particular, RAPSD is defined as the average

power in an annular ring with center radius rf as follows.

∑∈

=)(

)(ˆ))((

1)(

rfRfrr fP

fR�fP (A1),

where )( rfR is an annular ring of width ∆ partitioned in the spectral domain, ))(( rfR� is the number of

frequency samples in )( rfR , and )(ˆ fP is the estimated power spectrum of the halftone pattern obtained by

averaging the periodograms of its windowed segments. Anisotropy is defined as

∑∈

−

−=

)(2

2

)(

))()(ˆ(

1))((

1)(

rfRf r

r

r

rfP

fPfP

fR�fA (A2).

It provides the noise-to-signal ratio of frequency samples of )(ˆ fP in )( rfR and is used to measure the strength

of directional artifact. Directional components are considered to be not noticeable by human eye when

dB0)( <rfA happens [5].

17

REFERECES

1. R. A. Ulichney, Digital Halftoning. Cambridge, MA:MIT Press, 1987.

2. J. C. Stoffel and J. F. Moreland, “A survey of electronic techniques for pictorial reproduction,” IEEE Trans.

Communication, 29, 1898–1925, 1981.

3. R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. S.I.D. 17(2), 75–77,

1976.

4. J. F. Jarvis, C. N. Judice, and W. H. Ninke, “A survey of techniques for the display of continuous tone

pictures on bilevel displays,” Comput. Graph. Image Processing, pp. 13-40, 1976.

5. R. A. Ulichney, “Dithering with blue noise,” Proc. IEEE, vol. 76, pp. 56–79, Jan. 1988.

6. T. N. Pappas and D. L. Neuhoff, “Printer models and error diffusion,” IEEE Trans. Image Process, vol. 4,

pp. 66–79, Jan. 1995.

7. B. Kolpatzik and C. A. Bouman, “Optimized error diffusion for image display,” Journal of Electronic

Imaging, 1(3), 277–292, 1992.

8. P. W. Wong, “Adaptive Error Diffusion and Its Application in Multiresolution Rendering,” IEEE Trans.

Image Process, vol. 5, no. 7, pp. 1184-1196, July, 1996.

9. I. Katsavounidis and C. C. J. Kuo, “A multiscale error diffusion technique for digital halftoning,” IEEE

Trans. Image Process. Vol.6, No.3, pp.483–490,1997.

10. Y.H. Chan, “A modified multiscale error diffusion technique for digital halftoning,” IEEE Signal Process.

Lett, 5(11), 277-280 (1998).

11. T. D. Kite, B. L. Evans, and A. C. Bovik, “Modeling and quality assessment of halftoning by error

diffusion,” IEEE Trans. Image Process., vol. 9, no. 5, pp. 909–921, May 2000.

12. N. Damera-Venkata and B. L. Evans, “Adaptive threshold modulation for error diffusion halftoning,” IEEE

Trans. Image Process., vol. 10, no. 1, pp. 104–116, Jan. 2001.

13. P. Li and J. P. Allebach, “Tone-Dependent Error Diffusion,” IEEE Trans. Image Process, vol. 13, no. 2, pp.

201-215, Feb. 2004.

14. Y.H. Chan and S. M. Cheung, “Feature-preserving multiscale error diffusion for digital halftoning,”

Journal of Electronic Imaging, vol.13, No.3, pp.639-645 (2004).

15. V. Monga, N, Damera-Venkata and B. L. Evans, “Design of Tone-Dependent Color-Error Diffusion

Halftoning Systems,” IEEE Trans. Image Process, vol. 16, no. 1, pp. 198-211, Jan, 2007.

16. D. L. Lau and R. A. Ulichney, “Blue-Noise Halftoning for Hexagonal Grids,” IEEE Trans. Image Process,

vol. 5, no. 5, pp. 1270-1284, May. 2006.

17. D. L. Lau, G. R. Arce, and N. C. Gallagher, “Green noise digital halftoning,” Proceedings of the IEEE 86,

pp. 2424-2442, Dec. 1998

18. R. Levien, “Output dependent feedback in error diffusion halftoning,” IS&T Imaging Science and

Technology 1, pp. 115-118, May 1993.

19. D. L. Lau and G. R. Arce, Modern Digital Halftoning, CRC Press 2nd edition 2008.

20. R.A. Ulichney, “The void-and-cluster method for dither array generation,” in Proc. SPIE, Human Vision,

Visual Processing, Digital Displays IV, 1993, Vol.1913, pp.332-343.

21. J. Allebach and Q. Lin, “FM screen design using DBS algorithm,” in Proc. IEEE Int. Conf. Image

Processing, 1996, vol.1, pp. 549-552.

22. A. J. González, J. B. Rodríguez and G. R. Arce, “Alpha stable modeling of human visual systems for digital

halftoning in rectangular and hexagonal grids,” Journal of Electronic Imaging, Vol.17, No.1, 013004, Jan-

Mar 2008.

23. Y.H. Fung and Y.H. Chan, “Green Noise Digital Halftoning with Multiscale Error Diffusion,” IEEE Trans.

Image Process, vol.19, no.7, pp. 1808-1823, Jul. 2010.

18

24. Y.H. Fung and Y.H. Chan, “Embedding halftones of different resolutions in a full-scale halftone,” IEEE

Signal Process. Lett, vol. 13, no.3, pp. 153-156, 2006.

25. Y.H. Fung, K.C. Lui and Y.H. Chan, “low-complexity high-performance multiscale error diffusion

technique for digital halftoning,” Journal of Electronic Imaging, vol. 16, No.1, pp.1-12, 2007.

26. Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett., vol. 9, no. 3,

pp.81-84, Mar. 2002.

27. V. Monga, W. S. Geisler, and B. L. Evans, “Linear, color-separable human visual system models for vector

error diffusion halftoning,” IEEE Signal Process. Lett., vol. 10, no. 4, pp. 93–97, Apr. 2003.

19

Figure caption list

Fig. 1 Spectral plane of a halftone pattern generated with a rectangular grid.

Fig. 2 Power spectra for different cases: (a) g<1/4, (b) 1/4<g<1/2, without packing the aliasing energy into the

partial annuli regions, and (c) 1/4<g<1/2 with the aliasing energy packed into the partial annuli regions.

The circles mark the location of the principal frequency.

Fig. 3 Halftone renditions of constant gray level image g=126/255 and their corresponding power spectra. The

principal frequency is (a) 0.5 and (b) 0.4.

Fig. 4 The RAPSD of Figs. 3(a)(i) and 3(b)(i).

Fig. 5 The desirable principal frequencies for a gray level when different noise halftoning models are used

Fig. 6 How the principal frequency affects the spatial directional characteristic of the halftone rendition of a

constant patch of gray level g.

Fig. 7 How M, 1R and 2R should change with g when FMEDg is used to produce halftones of the desirable

noise characteristics specified in the suggested model.

Fig. 8 Portions of the halftoning results of a 256×256 constant gray level input. (a) g=15/255, (b) g=80/255, (c)

g=100/255 and (d) g=127/255

Fig. 9 Frequency magnitude spectra of the halftoning results of a 256×256 constant gray level input. (a)

g=15/255, (b) g=80/255, (c) g=100/255 and (d) g=127/255.

Fig. 10 (a) RAPSD and (b) Anisotropy performance of various halftone algorithms: (i) Ulichney [5], (ii) Lau et

al. [16], (iii) González et al.[22], and (iv) FMEDt. In the RAPSD plots, any RAPSD value larger than 4

is clipped to be 4.

Fig. 11 Testing images

Fig. 12 Average MSEv of the halftoning results of the testing images shown in Fig. 11 at different viewing

distances for printer resolution (a) 600dpi, (b) 1200dpi and (c) 2400dpi.

Fig. 13 Halftoning results of testing image “Goldhill”: (a) Original, (b) Ulichney [5], (c) Lau et al. [16], (d)

González et al. [22], (e) HFMED and (f) FMEDt.

Fig. 14 Halftoning results of a tilted gray ramp image. (a) Ulichney [5], (b) Lau et al. [16], (c) González et al.

[22], (d) HFMED and (e) FMEDt

Fig. 15 Halftones produced with various algorithms

Table caption list

Table I. UQI performance of various algorithms

20

Fig. 1 Spectral plane of a halftone pattern generated with a rectangular grid.

(a) (b) (c)

Fig. 2 Power spectra for different cases: (a) g<1/4, (b) 1/4<g<1/2, without packing the aliasing energy into the

partial annuli regions, and (c) 1/4<g<1/2 with the aliasing energy packed into the partial annuli regions.

The circles mark the location of the principal frequency.

(i)

hal

fto

ne

(ii)

sp

ectr

um

(a) (b)

Fig. 3 Halftone renditions of constant gray level image g=126/255 and their corresponding power spectra. The

principal frequency is (a) 0.5 and (b) 0.4.

21

Fig. 4 The RAPSD of Figs. 3(a)(i) and 3(b)(i).

Fig. 5 The desirable principal frequencies for a gray level when different noise halftoning models are used

Fig. 6 How the principal frequency affects the spatial directional characteristic of the halftone rendition of a

constant patch of gray level g.

22

(a)

(b)

(c)

Fig. 7 How M, 1R and 2R should change with g when FMEDg is used to produce halftones of the desirable

noise characteristics specified in the suggested model.

23

(i)

Uli

chn

ey’s

[5

]

(ii)

Lau

et

al.’

s [1

6]

(iii

) G

on

zále

z et

al.

’s [

22

]

(iv

) H

FM

ED

(v)

FM

ED

t

(a) g=15/255 (b) g=80/255 (c) g=100/255 (d) g=127/255

Fig. 8 Portions of the halftoning results of a 256×256 constant gray level input. (a) g=15/255, (b) g=80/255, (c)

g=100/255 and (d) g=127/255

24

(i)

Uli

chn

ey’s

[5

]

(ii)

Lau

et

al.’

s [1

6]

(iii

) G

on

zále

a et

al.

’s [

22

]

(iv

) H

FM

ED

(v)

FM

ED

t

(a) g=15/255 (b) g=80/255 (c) g=100/255 (d) g=127/255

Fig. 9 Frequency magnitude spectra of the halftoning results of a 256×256 constant gray level input. (a)

g=15/255, (b) g=80/255, (c) g=100/255 and (d) g=127/255.

25

(i)

Uli

chn

ey’s

[5

]

(ii)

Lau

et

al.’

s [1

6]

(iii

) G

on

zále

z et

al.

’s [

22

]

(iv

) F

ME

Dt

(a) (b)

Fig. 10 (a) RAPSD and (b) Anisotropy performance of various halftone algorithms: (i) Ulichney’s [5], (ii) Lau

et al.’s [16], (iii) González et al.’s [22], and (iv) FMEDt. In the RAPSD plots, any RAPSD value larger

than 4 is clipped to be 4.

26

Mandrill Barbara Boat Goldhill

Lena Man Peppers Girl

Fig. 11 Testing images

(a) (b) (c)

Fig. 12 Average MSEv of the halftoning results of the testing images shown in Fig. 11 at different viewing

distances for printer resolution (a) 600dpi, (b) 1200dpi and (c) 2400dpi.

27

(a) Original (b) Ulichney’s [5]

(c) Lau et al.’s [16] (d) González et al.’s [22]

(e) HFMED (f) FMEDt

Fig. 13 Halftoning results of testing image “Goldhill”: (a) Original, (b) Ulichney’s [5], (c) Lau et al.’s [16], (d)

González et al.’s [22], (e) HFMED and (f) FMEDt

28

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

Fig. 14 Halftoning results of a tilted gray ramp image. (a) Ulichney’s [5], (b) Lau et al.’s [16], (c) González et

al.’s [22], (d) HFMED and (e) FMEDt

29

(a) Original (b) Ulichney’s [5] (c) DBS [21]

(d) González et al.’s [22] (e) HFMED (f) FMEDt

Fig. 15 Halftones produced with various algorithms

Table I. UQI PERFORMANCE OF VARIOUS ALGORITHMS

UQI

Testing Image [5] [16] [22] HFMED FMEDt

Mandrill 0.0874 0.0643 0.0719 0.1609 0.1762

Barbara 0.0914 0.0750 0.0824 0.1334 0.1423

Boat 0.0802 0.0698 0.0715 0.1179 0.1310

Golhill 0.0542 0.0463 0.0490 0.0911 0.1055

Lena 0.0675 0.0606 0.0607 0.0926 0.1018

Man 0.0703 0.0617 0.0629 0.1080 0.1208

Peppers 0.0897 0.0830 0.0862 0.1078 0.1145

Girl 0.0391 0.0359 0.0368 0.0557 0.0629

Average 0.0725 0.0621 0.0652 0.1084 0.1194