A tone-dependent noise model for high

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: [email protected])
1
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) [315] 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
 g
fB (g) = 
 1 − g
for 0 < g ≤ 1 / 2
(1).
for 1 / 2 < g ≤ 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 1 / 4 < g ≤ 1 / 2 , we have f B ( g ) > 1 / 2 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 1 / 4 < g ≤ 3 / 4 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.
2
 g

f B' ( g ) = 1 / 2
 1− g

for 0 < g ≤ 1 / 4
for 1 / 4 < g ≤ 3 / 4
for 3 / 4 < g ≤ 1
(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

g/M
fG ( g ) = 
 (1 − g ) / M
for 0 < g ≤ 1 / 2
for 1 / 2 < g ≤ 1
(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 1 / 4 < g ≤ 3 / 4 and adjust the average cluster size to make M = 4 g for 1 / 4 < g ≤ 1 / 2 and M = 4(1 − g )
for 1 / 2 < g ≤ 3 / 4 , the spectral energy will be concentrated at radial frequency



fr (g ) = 



g
g / M = 1/ 2
for 0 < g ≤ 1 / 4
for 1/4 < g ≤ 1 / 2
(1 − g ) / M = 1 / 2
for 1/2 < g ≤ 3 / 4
(1 − g )
for 3/4 < g ≤ 1
(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
3
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.
4
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 F(oR1 , R2 ) 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)|
R2 ≥
x 2 + y 2 > R1 } in the continuous space, where R1 and R2 are, respectively, the inner and outer radii of
the ring region. Specifically, the (m,n)th filter coefficient of F(oR1 , R2 ) , f o (m, n) , is defined as
f o ( m, n ) =
A(m, n, R2 ) − A(m, n, R1 )
(5),
( R22 − R12 )π
where (m, n) are the horizontal and vertical integer offsets from the error source, and A(m, n, Rk ) for k=1, 2 is
x 2 + y 2 ≤ Rk in pixel (m,n). Effectively, the filter coefficient for a pixel which is
the area covered by circle
(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 R2 ≥
x 2 + y 2 > R1 in the grid unit associated with that pixel.
Dot clusters are formed in the outputs of FMEDg. It was found that the inner radius R1 helps to determine
the average cluster size of the clusters. In particular, when R2 = 2 R1 , we have a relationship model given as
M ≈ R12πg
(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 R1 to a certain extent. In other words, one can use diffusion filter
F(oR ,
1
2 R1 )
to produce minority clusters of any desirable average size by simply adjusting its parameter R1 and
distribute the clusters homogeneously.
With filter F(oR ,
1
2 R1 )
in hand, halftones bearing the spectral characteristics specified in (2) can be easily
produced by adjusting R1 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 R1 = 2 / π
to make the principal frequency be
g / M ≈ 1 /( R1 π ) = 1 / 2 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]
5
(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 R1 . 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 1 / 4 ≤ g ≤ 1 / 2 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.
6
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
1 / 4 ≤ g ≤ 1 / 2 should be as small as possible while optimizing the isotropicity. For reference purposes, the
compromised principal frequency for 1 / 4 ≤ g ≤ 1 / 2 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 1 / 4 ≤ g ≤ 1 / 2 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 g , the lower bound of the transition region, and finally reach fc at g=0.25 as the blue
TLB
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. g ≤g<0.25) can be
TLB
determined by interpolation with the samples of
 g
f B" ( g ) = 
 f c
for 0 ≤ g < gTLB
for 0.25 ≤ g ≤ 0.5
(7),
where f B" ( g ) 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 f B (g ) should be minimized over the range of g
specific curve fitting method is selected to do the interpolation, the optimal g
g
TLB
= arg min
g"
∑
g "≤ g < 0.25
(
1 ˆ
fB (g) − g
o
TLB
TLB
≤ g < 0.25 . When a
can be determined by
)
2
for 0.125 ≤ g " < 0.25
(8),
where g " is a candidate gray level lower than 0.25, o is the number of possible input gray levels in region
g" ≤ g < 0.25 , and fˆB ( g ) is the interpolated principal frequency obtained when only the principal frequencies
in region g" ≤ g < 0.25 are interpolated with the selected curve fitting method. Once g
TLB
is determined, f B" ( g )
for the transition region can be determined as the fˆB ( g ) obtained when g " =g .
TLB
Function f B" ( g ) 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
7
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 g , are intentionally open to be determined. As
TLB
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 g
TLB
will be
solution dependent.
In this section, we first determine fc and g
TLB
for our MED-based solution to make the model completely
well-defined for the solution. Then we will show how one can adjust parameters R1 and R2 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
8
D0, ∆ (α ) 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. D0, ∆ (α ) 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 λg , the principal wavelength of the halftone rendition of the constant patch. By
definition, λg is the reciprocal of the principal frequency of the halftone rendition.
Based on D0, ∆ (α ) , a directional index function can be defined as
D=
1 8
(1 − D0, ∆ (α )) 2
8 α =1
∑
(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 D0, ∆ (α ) =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 R1 . In fact, from spectral characteristic model (3) and
relationship model (6), one can deduce that the principal frequency is given as
g / M ≈ 1 /( R1 π ) when
R2 = 2R1 . 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 1 / 4 ≤ g ≤ 1 / 2 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
1 / 4 ≤ g ≤ 1 / 2 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.
9
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, g
TLB
is a
multiple of 1/255, and only principal frequencies for gray levels g∈{g , g +1/255, …, 63/255} are required
TLB
TLB
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, …, g -1/255} U {(g, 0.44) | g = 64/255, 65/255, …, 127/255}.
TLB
Any gray level in the range from 0.125 to 0.25 can be used as the g
optimal g
TLB
TLB
to carry out the interpolation. The
is determined as 42/255 with the criterion specified in (8) subject to the monotonic increasing
constraint and the constraint that g
C. Realization for g
TLB
TLB
is a multiple of 1/255.
≤g≤ ½
Green noise halftoning is carried out when g
TLB
≤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 M ≈ R12πg as long as
R2 = 2R1 holds. The principal frequency is then given by f G ( g ) = g / M ≈ 1 /( R1 π ) . In other words,
FMEDg can produce halftones of the suggested noise characteristics for g
TLB
≤g≤ ½ with diffusion filter (5) the
R1 and R2 of which are given as
(
 R1 = 1 / π ⋅ f B" ( g )

 R2 = 2 R1
)
for g
TLB
≤ g≤0.5
(10),
where f B" ( g ) is the desirable principal frequency specified in the suggested model. Fig. 7 graphically shows
how average cluster size M, filter parameters R1 and R2 should change with g to produce halftones of the
suggested noise characteristics.
D. Realization for 0≤g< g
TLB
As shown in Fig. 5, blue noise halftoning should be carried out when 0≤g<g . By considering that blue
TLB
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 R1 and R2 are selected
to maintain the average cluster size to be 1.
In our realization, we keep R1 and R2 unchanged for g≤g
(
 R1 = 1 / π ⋅ g
TLB

 R2 = 2 R1
)
TLB
as
for g≤ g
10
TLB
(11)
as shown in Fig. 7. Note that the relationship model M ≈ R12πg is no longer valid when g≤g
TLB
even though
R2 = 2R1 is still valid. In practice, the average cluster size cannot be smaller than 1. As the R1 at g=g
TLB
already makes the average cluster size M be 1, a smaller g cannot reduce M further.
As R1 and R2 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< g , this continuity may
TLB
not be guaranteed as the diffusion filter used for 0≤g< g
TLB
will not match with the one used for g
TLB
≤g≤ ½ in
such a case.
E. MED-based halftoning Algorithm
By adjusting parameters R1 and R2 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 MEDbased 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 F(oR ,
1
2 R1 )
for all pixels, with F(oR1 ( xi , j ), R2 ( xi , j )) , where R1 ( xi , j ) and R2 ( xi , j ) are, respectively, the
desirable R1 and R2 values provided in Sections IV-C and IV-D for xi , j , 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 tonedependent 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
11
annuli regions as shown in Figs. 9(i)(b)-(d) when 0.25 < g ≤ 0.5 . 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 ringshaped 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 0 < g ≤ 1 / 2 .
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.
12
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 MSE v . MSE v was proposed in [19] to
measure the observed distortion between an original gray-level image X and its binary halftone B . In
particular, MSE v is defined as
MSE v =
1
hvs ( X, vd , dpi ) − hvs (B, vd , dpi )
×
2
(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 MSE v , 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 MSE v 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
 0 .5
p
 BA
P =  pCA

 pDA
 pEA
p AB
p AC
p AD
0 .5
pCB
pBC
0 .5
pBD
pCD
pDB
pEB
pDC
pEC
0 .5
pED
p AE   0.5
pBE  0.1685
pCE  = 0.5746
 
pDE  0.7486
0.5  0.7652
0.8315 0.4254 0.2514 0.2348
0 .5
0.1768 0.1298 0.1409
0.8232
0 .5
0.2486 0.2514

0.8702 0.7514
0 .5
0.3536
0.8591 0.7486 0.6464
0.5 
where PXY 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 PXY = 1 − PYX . 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

 0.5(m + n)   1
R (m, n) = round  255 × 1 −
 ×
1023   255


13
for m = 0,1,…1023 and n = 0,1,…127 (13).
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 midtone 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
14
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
15
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 f r as follows.
P( f r ) =
1
( R ( f r ))
∑ Pˆ ( f )
(A1),
f ∈R ( f r )
where R ( f r ) is an annular ring of width ∆ partitioned in the spectral domain, ( R ( f r )) is the number of
frequency samples in R ( f r ) , and Pˆ ( f ) is the estimated power spectrum of the halftone pattern obtained by
averaging the periodograms of its windowed segments. Anisotropy is defined as
A( f r ) =
1
( Pˆ ( f ) − P ( f r )) 2
( R ( f r )) − 1 f ∈R ( f )
P2 ( fr )
∑
(A2).
r
It provides the noise-to-signal ratio of frequency samples of Pˆ ( f ) in R ( f r ) and is used to measure the strength
of directional artifact. Directional components are considered to be not noticeable by human eye when
A( f r ) < 0dB happens [5].
16
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18
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, R1 and R2 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
19
Fig. 1 Spectral plane of a halftone pattern generated with a rectangular grid.
(a)
(b)
(c)
(ii) spectrum
(i) halftone
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.
(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.
20
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.
21
(a)
(b)
(c)
Fig. 7 How M, R1 and R2 should change with g when FMEDg is used to produce halftones of the desirable
noise characteristics specified in the suggested model.
22
(i) Ulichney’s [5]
(ii) Lau et al.’s [16]
(iii) González et al.’s [22]
(iv) HFMED
(v) FMEDt
(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
23
(i) Ulichney’s [5]
(ii) Lau et al.’s [16]
(iii) Gonzálea et al.’s [22]
(iv) HFMED
(v) FMEDt
(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.
24
(i) Ulichney’s [5]
(ii) Lau et al.’s [16]
(iii) González et al.’s [22]
(iv) FMEDt
(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.
25
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.
26
(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
27
(a)
Fig. 14
(b)
(c)
(d)
(e)
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
28
(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
Testing Image
Mandrill
Barbara
Boat
Golhill
Lena
Man
Peppers
Girl
Average
[5]
0.0874
0.0914
0.0802
0.0542
0.0675
0.0703
0.0897
0.0391
0.0725
[16]
0.0643
0.0750
0.0698
0.0463
0.0606
0.0617
0.0830
0.0359
0.0621
29
UQI
[22]
0.0719
0.0824
0.0715
0.0490
0.0607
0.0629
0.0862
0.0368
0.0652
HFMED
0.1609
0.1334
0.1179
0.0911
0.0926
0.1080
0.1078
0.0557
0.1084
FMEDt
0.1762
0.1423
0.1310
0.1055
0.1018
0.1208
0.1145
0.0629
0.1194