2718 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 12, DECEMBER 2009
Continuous Phase-Modulated Halftones
Basak Oztan, Student Member, IEEE, and Gaurav Sharma, Senior Member, IEEE
Abstract—A generalization of periodic clustered-dot halftones
is proposed, wherein the phase of the halftone spots is modulated
using a secondary signal. The process is accomplished by using an
analytic halftone threshold function that allows halftones to be gen-
erated with controlled phase variation in different regions of the
printed page. The method can also be used to modulate the screen
frequency, albeit with additional constraints. Visible artifacts are
minimized/eliminated by ensuring the continuity of the modulation
in phase. Limitations and capabilities of the method are analyzed
through a quantitative model. The technique can be exploited for
two applications that are presented in this paper: a) embedding wa-
termarks in the halftone image by encoding information in phase
or in frequency and b) modulating the screen frequency according
to the frequency content of the continuous tone image in order to
improve spatial and tonal rendering. Experimental performance is
demonstrated for both applications.
Index Terms—Clustered-dot halftones, continuous phase modu-
lation, halftone watermarking, spatial/tonal rendering.
I. INTRODUCTION
H
ALFTONING is a common method to reproduce a
continuous tone (contone) image on bi-level printers or
displays such that at normal viewing distances, the halftone
image conveys the same visual impression as the contone
image. A large number of halftoning methods have been
proposed for the rendering of halftone images on bi-level
printers [3], [4]. Because of its stability and reproducibility,
clustered-dot halftoning [5] is the primary halftoning technique
for electrophotography and lithography, which are the two
primary methods of high-volume printing.
Digital clustered-dot halftones are commonly generated by
comparing each pixel in the contone image against a corre-
sponding threshold. The thresholds for each of the pixels in the
image are obtained via periodic replication of a small set of
thresholds defined over a 2-D tiling region called a halftone cell.
Digital halftones generated by this method mimic the conven-
tional analog halftones produced by the photographic screening
Manuscript received October 09, 2008; revised June 30, 2009. First published
July 24, 2009; current version published November 13, 2009. Parts of this work
were presented in IEEE ICASSP 2006 [1] and in IEEE ICIP 2006 [2]. This
work was supported in part by a gift from the Xerox Foundation and in part by
a grant from New York State Office of Science, Technology, and Academic Re-
search (NYSTAR) through the Center for Electronic Imaging Systems (CEIS)
of University of Rochester. The associate editor coordinating the review of this
manuscript and approving it for publication was Dr. Gabriel Marcu.
B. Oztan is with the Electrical and Computer Engineering Depart-
ment, University of Rochester, Rochester, NY 14627-0126 USA (e-mail:
basak.oztan@rochester.edu).
G. Sharma is with the Electrical and Computer Engineering Depart-
ment and the Department of Biostatistics and Computational Biology,
University of Rochester, Rochester, NY 14627-0126 USA (e-mail:
gaurav.sharma@rochester.edu).
This paper has supplementary PDF material available at http://ieeexplore.
ieee.org, provided by the authors. The material is in 92 MB file size. The sup-
plementary material contains the full-scale scan of the self-modulated Library
halftone shown in Fig. 21(a) in this paper that allows the self-modulation to be
observed more clearly.
Digital Object Identifier 10.1109/TIP.2009.2028367
process [6] in the sense that the halftone spots (clusters of indi-
vidual printer dots) are centered at a periodic array of locations.
Different shades of gray are reproduced by varying the size of
these spots. The use of a single halftone cell to tile the image
plane minimizes memory requirements and simplifies compu-
tation; considerations that were important in the early days of
digital halftoning when these resources were rather expensive.
These constraints become less stringent in current digital sys-
tems and one can explore the flexibility to develop alternative
digital halftoning methods that utilize more of these resources
but offer improvements in functionality or performance.
In this paper, we describe a method to modulate the phase
of digital clustered-dot halftone screens dynamically during the
halftoning process. Instead of using a halftone threshold array,
we employ an analytic halftone threshold function, whose phase
can be modulated independently using an auxiliary signal. Suit-
able design of the auxiliary signal allows us to address two dif-
ferent applications that we describe subsequently. Since dis-
continuities in the modulated phase may produce visible arti-
facts in the halftones, we impose a requirement of continuity
on the modulated phase. Using the analogy with continuous
phase modulation in digital communications [7, pp. 598–602],
we refer to the resulting technique as continuous phase-modu-
lated (CPM) halftoning. By exploiting the relation between fre-
quency and phase, modulation of the screen frequency is also
accomplished within the CPM framework, albeit with additional
constraints. In order to gain insight into the limitations and ca-
pabilities of CPM halftones, we examine their spectral charac-
teristics by using a simplified analytic model.
We consider two practical applications of CPM halftones.
First, we demonstrate how the phase (or frequency) modula-
tion may be exploited in order to embed a watermark pattern
in the halftone image [1]. Next, we describe how the traditional
trade-off between spatial and tonal resolution for clustered-dot
halftones can be ameliorated by varying the screen frequen-
cies according to the frequency content of the contone image.
We refer to this adaptive halftoning technique as self-modulated
halftones [2].
The rest of the paper is organized as follows. Section II re-
views existing work on related applications and summarizes
our contributions. In Section III, modulating halftone phase/fre-
quency using CPM halftoning is described. Applications ex-
ploiting CPM halftones are described in Section IV along with
their limitations and the experimental results are presented in
Section V. Finally, in Section VI, we present conclusions.
II. R
ELATED WORK
Both of the applications we consider have previously been
addressed in the literature. For centuries, conventional paper
watermarks have been utilized for security and forensics ap-
plications [8, pp.64–68], [9]. In the digital domain, halftone
watermarks provide a functionality that mimics or extends the
1057-7149/$26.00 © 2009 IEEE
OZTAN AND SHARMA: CONTINUOUS PHASE-MODULATED HALFTONES 2719
capabilities of the conventional paper watermarks [8, Ch.5],
[10], [11]. Several watermarking techniques have been previ-
ously proposed for digital clustered-dot halftones [12]–[16].
Among these, methods based on modulating the phase or
screen frequency have also been investigated. Varying the
halftone screen frequency by warping the spatial domain by a
sinusoidal waveform was proposed by Ostromoukhov et al. for
protection from counterfeiting [17]. In [18], Wang proposed a
method to generate clustered-dot halftone screens with clusters
centered stochastically on the screen. The method allows phase
and frequency variations within the screen and was adopted in
[19]–[22] to embed textual watermark patterns in clustered-dot
halftones by modulating the phase of the halftone screen
according to the watermark pattern. The embedded water-
mark pattern can be visually detected by using an inexpensive
transparency decoder or by scanning the printed image and
processing it using a computer [20]. Through suitable design,
the watermark may also be decoded without requiring any addi-
tional equipment [21]. These watermarking techniques require
the generation of a (relatively) large threshold array with the
embedded information built into the phase variation over the
multiple “halftone cells” within the array. This array typically
needs to be designed in advance with the tiling requirements
in mind, which limits flexibility because only watermarks for
which tiles have been predesigned can be embedded. CPM
halftones, however, allow the information to be dynamically
embedded during the halftoning process and, thus, offer a
simpler and more flexible solution. As with other methods, the
embedded information can be retrieved using either a simple
scan-shift-overlay process or a reference transparency overlay.
The spatial and tonal rendering of clustered-dot halftones
strongly depend upon trading spatial resolution for tonal resolu-
tion [5, pp. 399–400], [23], [24]. The number of gray levels that
can be reproduced in a digital clustered-dot halftone depends
on the size of the halftone cell that is used to tile the image
plane. The number of halftone cells that fit within a linear
unit distance in the direction of halftone screen orientation is
referred to as the screen frequency, which is commonly mea-
sured in lines per inch (lpi). A low screen frequency halftone
offers more tonal but less spatial resolution due to the large and
coarsely spaced halftone spots. On the other hand, the smaller
and closely spaced halftone spots in a high screen frequency
halftone allow better rendering of spatial details despite the
limited number of gray levels that can be reproduced. A variety
of solutions have been proposed to enhance the spatial and tonal
rendering of halftones. Though conventional methodologies
usually considered these two aspects separately [25]–[27], both
of these aspects can be jointly improved. Larger size halftone
cells can be used for the rendering of smooth regions such
as backgrounds to improve the tonal resolution and smaller
size cells can be employed at spatially varying regions such as
edges to increase the spatial resolution. This image adaptive
approach is adopted in [28] and [29] by varying the size of
the dither matrices depending on the changes in image values
along the scanning path of a space filling curve. In [30], Hel-Or
et al. developed a similar iterative halftoning scheme, wherein
the halftone cells vary in size and shape based on a spatial
“busyness” metric for the image. These methods tend to be
computationally expensive. Self-modulated halftones based on
CPM offer a simpler alternative.
Our presentation in this paper also includes an analysis of
CPM halftones in the context of these two proposed applica-
tions, which has not been addressed in prior work. Both con-
siderations of visibility of the modulation and detectability of
embedded watermarks are considered in this analysis.
III. C
ONTINUOUS
PHASE-MODULATED
HALFTONES
A. Analytic Halftone Generation With Phase Modulation
A clustered-dot digital halftone image for a monochrome
contone image
is commonly generated by comparing
the image values against a periodic halftone threshold function
, where and represent the spatial coordinates along
the horizontal and vertical directions, respectively. Specifically
if
otherwise
(1)
defines the halftone image, where the values 1 and 0 correspond
to the respective decisions that ink/toner is, or is not, deposited
at the pixel position
.
As described in Section I, the halftone threshold function
is typically a predesigned periodic array of spatially varying
threshold values, whose arrangement depends on the halftone
parameters such as screen frequency, orientation, printer ad-
dressability, and spot function. Manipulation of the halftone pa-
rameters through an adjustment of the threshold array values
tends to be an unwieldy process and analytic functional descrip-
tions of the halftones are, therefore, useful when such modi-
fications are desired [31]–[33]. For images assuming intensity
values in the interval
and orthogonal screens, Pellar and
Green [32], [33] defined a useful threshold function as
(2)
where
and represent the screen frequencies along the or-
thogonal
and axes, respectively.
In order to allow controlled phase variations in the halftone
screen, we propose a modification of this function by intro-
ducing two spatially varying phase functions
and
in the respective cosine function arguments [1],
thereby obtaining a generalized form of the analytic halftone
threshold function as
(3)
Spatial discontinuities in the modulated phase can cause visible
artifacts in the printed halftones. Continuity and smoothness
of these signals are required to eliminate/reduce the visibility
of these artifacts. Consequently, we refer to the resulting
halftoning technique as continuous phase-modulated (CPM)
halftoning. This methodology can be exploited to embed sig-
nals in the halftone image. Specifically, we describe how a
watermark pattern can be embedded by controlling the halftone
phase in Section IV-A.
The halftone images in Fig. 1 illustrate the effect of different
levels of phase modulation on the halftone appearance. Fig. 1(a)
2720 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 12, DECEMBER 2009
Fig. 1. Illustration of phase modulation on the halftone appearance: (a) shows the reference halftone image without any modulation; (b) shows a phase-modulated
halftone with visible artifacts induced by the discontinuities in the phase; (c) shows a continuous phase-modulated halftone without any artifacts.
shows the halftone image generated for a constant gray level
contone image using the threshold function of (3) without any
modulation in its phase. In Fig. 1(b) and (c), the phase of the
threshold function is modulated only along the horizontal axis
with the signals
shown beneath each halftone image.
While the sudden rise and fall in the modulated phase yield vis-
ible artifacts in the halftone image of Fig. 1(b), CPM provides
seamless transitions in the halftone image of Fig. 1(c).
B. Analytic Halftone Generation With Frequency Modulation
Continuous phase modulation of clustered-dot halftones
can also be utilized to vary the screen frequencies in different
regions of the printed image. Frequency modulation is accom-
plished by controlling the rate of change in the instantaneous
phase, i.e., the instantaneous frequency, of a carrier signal.
Closed form expressions for the instantaneous frequencies of
the modified threshold function (3) are hard to obtain. If the
phase modulation terms
and do not vary
along the directions orthogonal to their subscripted indices (for
instance,
does not vary along ), they can be represented
by two 1-D functions
and , respectively, and,
consequently, (3) becomes a separable function like (2). Under
this assumption, the instantaneous frequencies along the hori-
zontal and vertical axes are simply given by
1
and (4)
(5)
respectively.
A frequency-modulated sinusoidal carrier signal
in the time domain is commonly represented by
such that the instanta-
neous frequency at time
is given by [7, pp.
328–329]. This convention has also been applied to digital
1
Through a rotation of the coordinate axes, method can also readily handle
orthogonal halftone screens that are not necessarily oriented along
0
=
90
, such
as the common 45
orientation. For notational simplicity, however, we consider
only the “vertical-horizontal” case.
modulation [34]. Mimicking conventional FM, if the phase
terms are defined as
and (6)
(7)
then using (4) and (5), the instantaneous frequencies are readily
expressed as
and (8)
(9)
where
and are the frequency deviation constants and
and are the modulating functions for the modu-
lation along the horizontal and vertical directions, respectively.
Variations in the screen frequencies are introduced by suitably
controlling these parameters. This methodology is exploited
for different applications in Sections IV-B and IV-C.
For the purpose of illustration, Fig. 2(a) and (b) show no
frequency-modulated halftone images obtained by choosing
the frequency deviation constants
and , and
using the modulating functions (
) as shown beneath each
halftone image. Neither of the frequency-modulated halftone
images exhibit any objectionable artifacts due to the inherent
continuity of the modulated phase, nonetheless, the smoother
modulating function in Fig. 2(b) yields a more appealing
halftone image.
C. Spectral Characteristics of Continuous Phase-Modulated
Halftones
The perception of halftones depends on the human visual
system (HVS), where the sensitivity of the observer to the vari-
ations in the image is affected by the spatial frequency [35]. It
is well-known from the communication theory that the modu-
lation in phase or frequency leads to a widening in the signal
bandwidth [7, pp. 333–343]. In this context, if the visual system
has substantial sensitivity in the regions where the signal spec-
trum expands, then the modulation can introduce visible distor-
tions in the halftone image. Therefore, understanding the spec-
tral characteristics of the CPM halftones helps assess visibility
of the modulated halftone images.
OZTAN AND SHARMA: CONTINUOUS PHASE-MODULATED HALFTONES 2721
Fig. 2. Illustration of frequency modulation on halftone appearance. Frequency-modulated halftones do not exhibit any discontinuity artifacts since the modulated
phase is inherently continuous. Nevertheless, the smoother modulating function shown in (b) results in a more appealing halftone image than the one shown in (a).
Spectral characterization of an arbitrarily modulated halftone
in closed form is difficult. We simplify the problem by assuming
that the halftone is formed by rectangular halftone spots
2
and the
modulations along the horizontal and vertical directions are in-
dependent. Hence, the halftone can be modeled as the product
of two 1-D modulated pulse sequences along each of the orthog-
onal axes and we focus on estimating the spectral characteristics
of a 1-D modulated pulse sequence, which provides useful in-
sight for the 2-D scenario. From the digital modulation perspec-
tive, a phase-modulated pulse sequence can be considered as a
pulse-position modulated signal, while a frequency-modulated
pulse sequence can be modeled by a pulse-frequency modulated
waveform. Under some constraints on the statistics of the modu-
lating signal, the power spectral density of the pulse-modulated
signal can be estimated [36]. The smoothness and continuity
requirement of the modulating waveform, however, motivates
consideration of a simpler deterministic signal with these prop-
erties. Therefore, we assume periodic modulation, specifically
with a sinusoid, in phase or frequency, and estimate the spectra
of the modulated halftones (pulse sequences).
We first consider a phase-modulated halftone with a normal-
ized average gray level
in the form of a pulse-position
modulated waveform as illustrated in Fig. 3. The halftone is
comprised of spots of width
each of which is located within a
halftone cell that has width
. The modulation in phase is mod-
eled as a shift of the halftone “pulse” within the cell, yielding
an overall model for the phase-modulated halftone as
(10)
where the
is a rectangular pulse of width , symmetrically
centered in the interval
, and is the shift
corresponding to the
halftone cell. Consider a sinusoidally
varying modulation signal
, where
is the initial phase, and is a positive integer
representing the period of
in number of halftone cells. It is
2
The analysis can be generalized to the 2-D case with circular halftone spots
in a conceptually straightforward fashion, albeit at the expense of significant
notational complexity.
Fig. 3. Pulse-position modulated halftone.
clear that is periodic with period and, thus, it can
be represented in terms of a Fourier series as
(11)
where (see details in Appendix A)
and the re-
maining Fourier series coefficients are given by
for .
Next, we consider a frequency-modulated halftone in the
form of a pulse-frequency modulated waveform, where the
modulation in frequency does not change the normalized
average gray level
in a given halftone cell. Motivated
by this constraint, a hybrid pulse-frequency-width modulated
sequence is illustrated in Fig. 4, which can be mathematically
expressed as
(12)
2722 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 12, DECEMBER 2009
Fig. 4. Hybrid pulse-frequency-width modulated halftone.
where is a rectangular pulse of width symmetri-
cally centered in the interval
. The sinusoidally modulated
width and period of the pulse are assumed to be
and (13)
(14)
respectively,
determines the range of variation in the
period
, and is a positive integer representing the period
of the modulating signal in number of halftone cells. It can be
shown that
is a periodic waveform with period , and
it can also be represented in terms of a Fourier series as (see
details in Appendix B)
(15)
where
and the remaining Fourier series coeffi-
cients for
are
The distortion induced by the modulations in phase/fre-
quency can be estimated, to first-order, by observing the change
in the total visible power of the halftone. Visual system models
commonly employ a linear shift-invariant filter to approximate
the contrast sensitivity function (CSF) of an average observer as
a function of the spatial frequency. Representing the magnitude
frequency response of the visual system by
and using
the Fourier series representation of the modulated halftones,
the total visible power of the modulated halftones is obtained as
(16)
where
, and correspond the Fourier series coefficients
for the phase-modulated and frequency-modulated halftones.
Even though the HVS is characterized by a band-pass re-
sponse, for halftoning, it is known that low-pass representa-
tions of the HVS are typically better than the band-pass alter-
natives [37]. We consider a low-pass approximation [38] to the
well-known Mannos-Sakrison CSF [39] as (17), shown at the
bottom of the page, where
denotes the radial frequency in
cycles per degree. The radial frequency in cycles per degree is
related to the linear frequency in cycles per inch (cpi) in terms
of the viewing distance
3
, which is assumed to be 12 inches
throughout this paper. This function is obtained by setting the
CSF to its peak value for all frequencies lower than the fre-
quency at which the peak is attained and then normalizing the
“DC-response” (i.e., response at zero-frequency) to unity.
We first investigate the effect of phase modulation on the
total visible power. For a set of halftone frequencies, by aver-
aging over several values of the initial phase of the modulating
signal
, quantitative estimates of total visible power are
obtained as a function of the period of the modulating signal
at , and halftone area coverages and plotted in
Fig. 5. It is readily seen that as the period of the modulating
signal increases, the variation in phase becomes smoother and
the amount of visible degradation it causes decreases. The total
visible power of the modulated halftone asymptotically ap-
proaches the total visible power of the nonmodulated halftone
as
is further increased. Based on this observation, for each
of the area coverage and halftone frequency, we empirically
choose the smallest period of the modulating signal, equiva-
lently the maximum rate of change in the modulating signal,
that ensures the increase in the total visible power does not
exceed
of the maximum possible increase. These values
are indicated with “cross-marks” in each of the plot in Fig. 5.
The effect of frequency modulation on the total visible power
is investigated with a similar approach. The same set of screen
frequencies are modulated such that the screen frequency sinu-
soidally deviates between
cpi with periods that
are marked with the cross-marks in Fig. 5 for different area
coverages and screen frequencies. The parameter
is deter-
mined by the maximum frequency deviation
, and total vis-
ible power corresponding to the previously used area coverages
are estimated as a function of
and shown in Fig. 6. These
plots indicate that the modulation in the screen frequency does
3
1 cycles
=
degree =
D=
180 cycles
=
inch
, where
D
is the viewing dis-
tance in inches.
for
otherwise
(17)
