Volume 62, number 5
OPTICS COMMUNICATIONS
1 June 1987
A 2-D PULSE D E N S I T Y M O D U L A T I O N BY ITERATION FOR HALFTONING
R. E S C H B A C H ~ and R. H A U C K
Physics Department. University of Essen, 4300 Essen l, Fed. Rep. Germany
Received 26 January 1987
A binarization procedure generating a 2-D pulse density modulation is presented• The concept is based on locally interacting
pulses• By iteration the pulses are shifted until a stable pulse distribution is reached.
1. Introduction
The binary representation of a continuous input
function is a necessity in a wide range o f applications. This necessity can be established by software
means, as to guarantee an intensitivity to noise in
storage and transmission, or by hardware restrictions, as i m p o s e d by binary m e d i a like metal film,
electronic displays, and matrix printers. Related to
these applications is the i n t r o d u c t i o n o f b i n a r i z a t i o n
m e t h o d s in the optical and electronical field. Examples are the implicit and explicit pulse m o d u l a t i o n
[1] ( P W M ) and m o d i f i c a t i o n s like ordered dither
[2], the pulse code m o d u l a t i o n ( P C M ) , the A-X
m o d u l a t i o n [3] and m o d i f i c a t i o n s as " m i n i m u m
average error" [4] and " e r r o r diffusion" [5], statistical methods, and the pulse density m o d u l a t i o n
( P D M ) [6]. Whereas these m o d u l a t i o n s are used in
electronical systems, the optical realization is not
necessarily straightforward. This is due to the oned i m e n s i o n a l character and the freedom in the decoding process o f electronical applications. The decoding process in optical display applications is strongly
related to a low pass filter operation, which drastically reduces the n u m b e r o f useable P C M ' s in this
field. The two d i m e n s i o n a l character o f optics leads
to problems in the two d i m e n s i o n a l generalization of
m o d u l a t i o n s like the P D M .
The two d i m e n s i o n a l generalization of the P D M
Presently at Department of Electrical Engineering and Computer Science, University of California, San Diego, La Diego,
La Jolla, CA 92093, USA.
300
is o f special interest, because m a n y hardware devices
for display situations, e.g. scanners, are able to produce pulses o f fixed size at locations continuous in
space, i.e. without the restriction o f spatial quantization. This is also true in situations, where the spatial resolution or the positioning accuracy is higher
than the m i n i m u m feature size. We like to present an
a p p r o a c h to the 2-D P D M by an iteration concept.
2. A physical model for P D M
2.1. Concept
In situations were a P D M is applied for coding a
continuous signal by binary pulses, the signal is represented by pulses of fixed size and by varying the
spacings. A possible procedure to generate such a
• pulse distribution is based on displaceable points with
forces acting between them. The points represent the
centers of the pulses and the forces are chosen
d e p e n d e n t on the spacings of the points and on the
local signal level, such that the equilibrium distribution o f the pulses represents the input signal.
The force between two points i and j is chosen as
F , =K(I,1)/D ~ ,
(1)
with D~j being the distance between the two points,
and K(I~) a p r o p o r t i o n a l i t y d e p e n d e n t on the local
signal level I , at the mean position o f the two points.
The function K(LI) will be d e t e r m i n e d in respect to
the desired signal representation.
0 030-4018/87/$03.50 © Elsevier Science Publishers B.V.
( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
Volume 62. number 5
OPTICS COMMUNICATIONS
l June 1987
2.2. Iterative solution in 1-D
(a)
The above model is first a p p l i e d to a 1-D signal
l ( x ) . In display applications, as regarded here, I ( x )
represents a spatially varying intensity. F o r a spatially constant signal I this leads to the constraint
D o = Do/l
;i ;i
Ii I
_,
(2)
in order to obtain a linear signal rendition, with Do
the m i n i m u m pulse distance and assuming an intensity range 0 < 1 4 1. We select the forces only to act
between neighboring points.
The equilibrium state, i.e. when
FI2 . . . . .
!TI ;]
F,, . . . . F x _ tj.x = F o
(3)
is valid, leads with eqs. (1) and (2) to
K( I) = D o F o l l 2 .
'",
(c)
Fig. 1. Illustration of the physical concept generating a PDM. (a)
The input signal, (b) the selected start distribution, (d) the pulse
distribution after 100 iteration cycles. In (c) the springs symbolize the forces between the pulses, and the curved lines indicate
the pulse location during iteration.
(4)
2.3. Iterative solution in 2-D
Generalizing I to I , for a varying input and inserting
eq. (4) into eq. (1) leads to
F,,=DoFo/I~,D. .
(5)
The equilibrium force Fo can be chosen arbitrarily
but unequal to zero.
The solution of eq. (3) together with eq. (5) is
achieved by means o f an iterative procedure. During
iteration, all points but point i are kept fixed. The
net force F, ~.i-F,.,+ t on that point is calculated and
the point is shifted d e p e n d e n t on the resultant net
force. This is done subsequently for all points. The
resultant pulse distribution is now regarded as the
start distribution for the next iteration cycle. This
procedure is repeated until the n u m b e r and a m o u n t
o f the shifts are smaller than defined values to stop
the iteration. This final distribution will in our case
be regarded as the equilibrium distribution. In the 1D case, any start distribution leads to the identical
equilibrium distribution.
Fig. 1 shows one example for a r a m p as input signal with l ( x ) zcx in fig. la. The start distribution, in
this case equidistant pulses, is shown in fig. l b a n d
the equilibrium distribution in fig. ld. The forces are
symbolized by springs (fig. l c ) and the curves indicate the point locations during the 100 iteration cycles
used. The resulting equilibrium distribution is in
good agreement with the r e q u i r e m e n t o f eq. (2).
The extension of the iteration concept to two
d i m e n s i o n s is straight forward. Each point is now
interacting via the local forces with a set o f points,
its socalled neighbors. In contrast to the 1-D case,
the n u m b e r of neighbors taken into account is not
obvious. In fig. 2, 6 neighboring points are selected
and the applied forces are symbolized.
In the 2-D case a linear intensity rendition requires
D E=Do~l,
(6)
because the ratio & t h e pulse area to the surrounding
area determines the average intensity level o f the
binary representation. In modification to the 1-D case
this leads to
F,/ = ev Fo D ?~/Io D ~j ,
(7)
Fig. 2. Illustration of the forces in the 2-D case.
301
Volume 62, number 5
OPTICS COMMUNICATIONS
1 June 1987
Q
O
Fig. 3. Progress of iterated pulse distribution For a constant 2-D input signal. ( a ) The selected start distribution ( random ), (b), (c) and
(d) the pulse distribution after 1, 15. and 30 iteration cycles, rcspectivcly.
w i t h e u b e i n g t h e u n i t v e c t o r in d i r e c t i o n f r o m p o i n t
i t o j . T h e i t e r a t i o n is c a r r i e d o u t in t h e s a m e m a n n e r
as in t h e I - D case, t a k i n g t h e v e c t o r c h a r a c t e r o f t h e
forces i n t o a c c o u n t . N o t e t h a t in t h e 1-D case, t h e
v e c t o r c h a r a c t e r was also i n c l u d e d b y a l l o w i n g posi-
rive a n d n e g a t i v e n e t forces a c t i n g o n p o i n t i. Fig. 3
s h o w s t h e r e s u l t for a c o n s t a n t i n t e n s i t y as i n p u t signal a n d t h e r a n d o m p u l s e d i s t r i b u t i o n o f fig. 3a as
start d i s t r i b u t i o n . Fig. 3b s h o w s t h e d i s t r i b u t i o n a f t e r
t h e first i t e r a t i o n cycle, 3c a f t e r 15 cycles, a n d 3d t h e
Fig. 4. Test for spatial resolution with a 1-D Fresnel zone plate with varying constrast as input signal. (a) The selected start distribution
based on a random process, (b) the equilibrium distribution after 43 iteration cycles.
302
Volume 62, number 5
OPTICS COMMUNICATIONS
1 June 1987
Fig. 5. Application of the algorithm 1o a portrait. (a) The selected start distribution based on random process, (b) the equilibrium
distribution after 44 iteration cycles.
equilibium distribution after 30 cycles. The itera-tion
procedure generates a good isotropy, however, in the
2-D case the resulting pulse distribution depends on
the start configuration. This might be understood
since in the 2-D case the neiborhood relation of the
points is not fixed as in the 1-D case. Due to the displacements, the points which interact by forces can
be exchanged during the iteration cycles. In the
numerical algorithm we selected for each point to be
displaced an interaction with its eight nearest
neighbors.
The next example demonstrates the spatial resolution of this PDM. A 1-D Fresnel zone plate is used
as input with increasing frequency from left to right
and increasing contrast from top to bottom. The start
distribution(random) is shown in fig. 4a and the
equilibrium pulse distribution after 43 iteration
cycles in fig. 4b. It can be seen that this concept can
generate a pulse distribution which displays a high
spatial frequency in respect to the selected pulse size.
Fig. 5 shows the application o f the algorithm to a
portrait with the start distribution ( r a n d o m ) in fig.
5a and the final distribution after 44 iteration cycles
in fig. 5b. The algorithm provides a good pulse isotropy in areas o f constant intensity and a good detail
resolution.
3. Conclusion
In case of P D M the signal is coded by pulses of
fixed size at signal dependent locations. Such a modulation is in particular suitable for scanners, it offers
a high image quality in respect to the number of
pulses used. The 1-D P D M is related to frequency
modulation and can be generated using an analytic
description [ 7]. However, the generalization to two
or more dimensions is not straightforward. The
physical concept here is based on local interactions
between the pulses, and can be formulated for any
number of dimensions. We have presented an iterative solution in 1-D and 2-D. The results of the 1D case confirm the physical concept because the generated pulse distributions agree well with those based
on frequency modulation. In 2-D, images binarized
by a random process were used as start distributions,
which demonstrates the feasibility of the physical
concept, even in case of a weak a priori information.
After iteration the presented examples show a good
pulse isotropy combined with a high detail resolution. The convergence of the algorithm and the
resulting' microstructure of the pulse distribution
depend on the start distribution and on parameters
of the algorithm as the law of the force used, the
number of neighbors which interact, and the break
criterium for the iteration process. In general, a higher
a priori information contained in the start distribution will improve the procedure. As an example
the start distribution can be generated by an error
diffusion algorithm as in ref. [5], which drastically
improves the convergence of the algorithm.
The relation of the features of this modulation to
305
Volume 62, number 5
OPTICS COMMUNICATIONS
t h e p a r a m e t e r s o f t h e a l g o r i t h m is still a s u b j e c t o f
r e s e a r c h . I n a p u r e f o r m as p r e s e n t e d h e r e t h e algor i t h m c o n s u m e s t o o m u c h C P U - t i m e to b e c o m e
p r a c t i c a b l e , h o w e v e r , t h e i d e a m a y b e i n t r o d u c e d to
e x i s t i n g g e n e r a t i o n p r o c e d u r e s f o r P D M i n o r d e r to
find a compromise between quality and calculation
effort.
Acknowledgements
W e w i s h to t h a n k P r o f e s s o r O l o f B r y n g d a h l for
m a n y v a l u a b l e d i s c u s s i o n s . R. E s c h b a c h was supported by the Deutsche Forschungsgemeinschaft.
304
I June 1987
References
[ 1 ] J.C. Stoffel and J.F. Moreland, IEEE Trans. Commun. 29
(1981) 1898.
[2] C.N. Judice, J.F. Jariv and W.H. Ninke. Proc. S.I.D. 15
(1974) p. 161.
[3] H. lnose, Y. Yasuda, J. Murakami, IRE Trans. on Space
Electronics and Telemetry SET-8 (1962 ) 204.
[4] M.R. Schroeder, IEEE Spectrum 6 (1969) 66.
[5] R.W. Floyd and L. Steinberg, Proc. SAD. 17 (1976), 75.
[6] R. Eschbach and R. Hauck, J. Opt. Soc. Am., submitted.
[ 7 ] R. Eschbach and R. Hauck, Optics Comm. 54 ( 1985 ) 71.