IEEE TRANSACTIONS ON IMAGE PROCESSING
vol. 17, No. 8, 2008
J. Bacca Rodriguez, G. R. Arce and D. L. Lau
Presented by Shu Ran
School of Electrical Engineering and Computer Science
Kyungpook National Univ.
Blue-noise multitone dithering
Abstract
Introduction of blue-noise spectra
– High-frequency white noise with minimal energy at low frequency
• Impacting on digital halftoning for binary display devices
− Inkjet printers
» Optimal distribution of black and white pixels
– Blue-noise model
• Not directly translating to printing with multiple ink intensites
− New multilevel printing and display
» Requiring development of corresponding quantization
» For multitoning
2/42
Proposed blue-noise dithering
– Developing theory and design of multitone
• For defining optimal distribution of multitone pixels
− Modeling arbitrary multitone dot patterns
» As layered superposition of stack-constrained binary patterns
– Multitone blue-noise
• Minimum energy
− At low frequencies
• Staircase-like ascending spectral pattern
− At higher frequencies
– Optimum spectral profile
• Describing by principal frequencies and amplitudes
− Requiring definition of spectral coherence structure
» Governing interaction between patterns of dots of different
intensities
3/42
Introduction
Halftoning method
– Converting continuous tone image
• into pattern of black and white dots
− Using illusion of low-pass characteristics of human eye
» Unable to discriminate printed dots
– Developing to multitoning
• Allowing reproduction of dots of different intensities
− Previous methods relying on blue-noise model
» For halftones
− Proposing comparable theory
» Explicitly for multitones
4/42
Purpose of multitoning
– Generating images visually pleasant human eye
• Using principles of blue-noise halftoning
− Desiring homogeneity and isotropy in multitone dither patterns
– Challenge of blue-noise for multitoning
• Requiring radial symmetry and low-frequency response close to zero
− Imposed by properties of human eye
• Allowing dots of intermediate intensities
− Dot patterns of different inks interfere with each other
» Creating variations in intended average value of picture
» Generating low-frequency noise
– Determining spectral profile of multitones and characteristics
• Required for its optimality threshold decomposition
− Tool for analysis of multitone patterns
» Each of patterns characterized as blue-noise pattern
5/42
– Defining spectral profile of blue-noise multitones
• Aggregation of profiles of halftones
• Cross spectra
− Generated by interaction between dots of different intensities
– Spectral correlation between halftones
• Composing blue-noise multitone
− Characterized by means of optimal spectral coherence
– Proposed method
• Given spatial and spectral characterization of optimal blue-
noise multitones
− Generating multitones arises
• Extending threshold decomposition
− Applied to continuous tone images
• Representation combined with given blue-noise halftoning
− Generating multitone dither patterns
» Showing spectral characteristics of blue-noise multitones
6/42
Spectral statistics of halftones
– Ulichney proposed dither pattern in Fourier domain
• Radially average power spectrum density
• Anisotropy measures
− Dithering of same intensity
» Bernoulli processes with probability density function
• Studying characteristics of dither pattern
− Using its power spectrum
» As average of ten periodograms
Blue-noise for binary dither pattern
, 1=1 , 0
g for H nP H n
g for H n
(1)
P f
7/42
– Calculation of RAPSD of dither pattern
• Radial average of on this annuli calculated
Fig. 1. Calculation of the RAPSD of a dither pattern.
Ten sections of 256x256 pixels are extracted from a
large dither pattern of the desired gray level, the
periodogram of each pattern is calculated and is
calculated as their average. To obtain the RAPSD, the
average of is taken over annuli of width as
indicated.
P̂ f
P̂ f
P̂ f
1 ˆ=f R f
P f P fN R f
(2)
where is central radius,
number of samples in annuli
f
N R f
8/42
Blue-noise spectra
– Ulichney stated optimal dither patterns
• Average distance between nearest-neighboring minority pixels
− Average distance between pixels in blue-noise halftone pattern
1,
2=
1, >
21
g
Sfor g
g
Sfor g
g
(3)
where is minimum distance between addressable pixels,
is referred to as principal wavelength of pattern
S
g
Fig. 2. Average distance between pixels in a
blue-noise halftone pattern. In areas of constant
intensity, minority pixels tend to spread apart an
average distance in blue-noise dithering.. g
9/42
– principal frequency
• Inverse of principal wavelength
• Ideal radial average of power spectrum
gf
1
1
1,
2=
11 ,
2
g
S g for g
f
S g for g
(4)
Fig. 3. Ideal radial average of the power spectrum of a blue-noise
halftone pattern illustrating its three main characteristics: Low-
frequency response close to zero (1), flat high-frequency region (2),
and a peak at the principal frequency of the pattern (3).
10/42
– Improving principal frequency
• Sampling grid constrained placement of dots along diagonals
1
1
1
1,
4
1 3= ,
2 4 4
31 ,
4
g
S g for g
Sf for g
S g for g
(5)
11/42
Blue-noise halftoning
– Producing patterns with blue-noise model
• Bayer’s dither array
− Screening algorithms using thresholding operation
− Resulting in periodic artifacts
– Alternative halftoning methods
• Affecting pixel being quantized and its vicinity
− Resulting in higher computational complexity
• Previous methods
− Error diffusion algorithm
− Floyd and Steinberg’s algorithm
» Geometric artifacts in gray-scale ramp
» Reflecting here as spectral peaks at principal frequency
of pattern or its multiples
12/42
• Error diffusion halftoning and halftone of gray-scale ramp
Fig. 4. Error diffusion halftoning.
Fig. 5. Halftone of a gray-scale ramp
generated with Floyd–Steinberg
error diffusion. 13/42
• RAPSD of patterns of different intensities
(a) 1/16
(c) 1/4
(b) 1/8
(d) 1/2
Fig. 6. RAPSD of halftones generated with Floyd–Steinberg error diffusion for
gray levels 1/16, 1/8 and 1/4, 1/2
14/42
– Ulichney improving error diffusion pattern
• Radial symmetry
• Cut-off frequency
− Using serpentine scan
− Introducing randomness in weights of error filter
» weights
• Halftone of gray-scale ramp
1 1 2 2, 3 1, 4 2 1 2
5 1, , 1,1 , 1,1
16 16b R b R b R b R R U R U
Fig. 7. Halftone of a gray-scale ramp
generated with Ulichney’s error diffusion.
15/42
• RAPSD of halftones by Ulichney’s error diffusion
Fig. 8. RAPSD of halftones generated with Ulichney’s error diffusion for
gray levels 1/16, 1/8 and 1/4, 1/2
16/42
– Direct binary search improving halftone under error measure
• Including models of human visual system and printing device
• Considering trial change of pixel
22
E e d g f d x x x x x (6)
where is perceived printed image and halftone,
obtained by filtering with linear filter that comprises effect
of printing process and HVS.
,f gx x
*p p h
0g m
0 1 1 0
0 1
0 0
0
0
1
,
1, 0
1, 1
,=0,
g g a p a p
g g for a swap
a for a toggleif g
for a toggleif g
a for a swapa
for a toggle
m m m m m m
m m
m
m
(7)
(8)
17/42
• Change in error measure defined in (6)
• Changing halftone every time
• DBS halftoning of grayscale ramp
2 2
0 1 0 0 1 1 0 1 0 10 2pp pe pe ppE a a c a c m a c m a a c m m (9)
where is autocorrelation function of and is the cross
correlation between linear filter and perceived error ppc p
pec
0 0 1 1pe pe pp ppc m c m a c m m a c m m (10)
Fig. 9. DBS halftoning of a grayscale
ramp.
18/42
• RAPSD using DBS
Fig. 10. RAPSD of halftones generated
with DBS for gray levels 1/16, 1/8 and 1/4,
1/2
19/42
– Differences between Ulichney’s and DBS
• DBS fewer geometric artifacts and noisy texture
Fig. 11. Section of a blue-noise halftone of a 8-bit grayscale image
generated with: Error diffusion with (left) perturbed weights and (right)
DBS. 20/42
Model for multitone dither patterns
Spectral statistics of multitones
– Multitone dither patterns
• Representing constant gray level
− Modeled as stochastic processes
– Each multitone pixel
• Considered realization of discrete random process
− Obeying probability density function
− Obtaining variance
M n
0
N
i i iP M g p
n (11)
where indicate proportion of pixels of corresponding inks included
in multitone,such that 0
N
i ip
01
N
iip
2
2 2 2 2
0
N
i i
i
Var M E M E M p g g
n n n (12)
21/42
– Analysis and synthesis of multitones
• Presenting new challenges
− Effects in spatial domain should be evaluated
» Average intensity or textures of dither pattern affected
» By superposition of dots of different intensities
» By clustering of different kinds of pixels
− Spectral domain analysis of multitones more complex
» As number of inks increasing
− Patterns formed with dots of same ink
» Having own spectral profile
» Combination generating spectral cross terms
– Correlation between multitone and halftone
• Superposition of series of halftone patterns
− Printed on top of each other with different inks
– Phenomena of overlapping halftones
• Appearance of moire
22/42
− Appearing in dispersed dot patterns
» As random fluctuations in texture as stochastic moire
– Obtaining good quality combined pattern
• Energy in cross correlation
− Compensating for energy presenting in individual pattern
» Not appearing in superposition
– Incorporating correlation between different inks
• Into analysis and synthesis of multitones
− Proposing threshold decomposition representation of signals
− Series of halftones
• Describe multitone in terms of threshold decomposition representation
1,
0,
i
i
if M gH
else
nn (13)
1
N
i i
i
M d H
n n (14)
where are the relative differences between intensities of
printable inks
1 1
N
i i i id g g
23/42
• Example of performing decomposition
Fig. 12. Decomposition of a 3-ink multitone M in a series of halftones
satisfying the stacking constraint. 3
1i iH
24/42
– Describing halftones as correlated stochastic processes
• Presenting marginal densities
• Means and variances given by
• Expressing mean of multitone
− As function of characteristics of halftones
1
0
, 1
, 0
N
j ij i
i i
j ij
p for HP H
p for H
nn
n(15)
2 1N
i j i i i
j i
p and
(16)
1 1
N N
i i i i
i i
E M E d H d g
(17)
25/42
• Variance of linear combination
• Product with is equal to
− Reducing covariance
1
2
1 1 1
2 ,
N
i i
i
N N N
i i i j i j
i i j i
Var M Var d H
d Var H d d Cov H H
(18)
where is covariance of random
processes
, ,i j i j i jCov H H E H H E H E H
,i jH H
,i jH H j i jH
, 1 ,i j j iCov H H for j i (19)
2 2
1 1 1
2 2
1 1 1
2 1
12
1
N N N
i i i j j i
i i j i
N N Nj i
i i i j i j
i i j i i j
Var M d d d
d d d
(20)
26/42
Multitone blue-noise spectra
– Defining principal frequencies
– RAPSD for two inks
1,
4
1 1 3= ,2 4 4
31 ,
4
i i
i i
i i
for
f for
for
(21)
Fig. 13. Optimal RAPSD for a 2-ink multitone dither pattern. The frequencies and
are the principal frequencies of the halftone patterns obtained by the threshold
decomposition of the multitone, is the variance of the multitone and is the
variance of the halftone pattern with the lowest principal frequency.
Af Bf
2
A2
B
27/42
– Find correlation between patterns for multitone visually pleasant
• Using cross-spectral density function
− Magnitude of CSD
» Average value of product of components of each signal for each
frequency
− Phase of CSD
» Average phase-shift between components of two signals at each fre
quency
− Fourier transform of cross correlation of two signals
» Calculated by multiplying their PSDs
» One of two signals appearing in CSD without relationship
• Frequency domain equivalent of correlation coefficient
− Measure of correlation of two signals at each frequency
2
2 xy
xy
x y
P fK
P f P f (22)
where is CSD of patterns, are PSDs of the signals xyP f ,x yP f P f ,x y
28/42
• Defining correlation coefficient
• Applying to a pair of sub-halftones
• Interesting properties of MSC
− Bounded between 0 and 1
» 0 for independent processes
» 1 is result of filtering other with linear filter
− Representing portion of power of signal at given frequency
» Accounted for by its linear regression on the other
− Coherence
» Invariant to linear filtration
» symmetric
2
2
2 2
,xy
x y
Cov x yr
(23)
2
2
2 2
, 1
1
i j j i
xy
i j i j
Cov H Hr
(24)
29/42
• Series of multitones using different mechanisms
Fig. 14. Radial MSC of multitones of gray 150 generated as the superposition of (top-left)
two independent white noise patterns, (bottom-left) two independent blue-noise patterns,
(top-right) a suboptimal multitone generated with DBS, and (bottom-right) an optimal blue-
noise multitone, with the RAPSD and the radial MSC of the patterns used for their
generation.
30/42
• Mean of pattern
• Ideal plot of this case
* *
2 2 2
0
0 0 0 0
0 0
i j j i
xy i j ijfi j
P P P PK C
P P
(25)
Fig. 15. Radial MSC of an ideal blue-noise multitone.
31/42
Blue-noise multitoning
– Representing intensity of patch
– Procedure to multitone a continuous image Y
• Following mechanism
1 1
N N
i i i i
i i
g g p g d g
(26)
where . 1,N
i j i i ij ig p g d g g
32/42
– Blue-noise multitoning process
Fig. 16. Blue-noise multitoning. A continuous tone image Y is divided in
N components that can be halftoned with any algorithm in a correlated
fashion to generate a set of halftones, the threshold decomposition
representation of the final multitone. The set of halftones is
recombined to generate the multitone using (14). 1
N
i iH
33/42
– Division of original image into subimages
• Implemented with look-up table
• Sinthesis of final multitone from subhalftones is linear combination
− Blue-noise multitoning with error diffusion
− Blue-noise multitoning with DBS
» Quality metric determining change to including all sub-halftones
1
11, 1
2
0,
e
i i i
i
if Y H and HH
else
n n nn (27)
where is error diffused to pixel and . e
iH n iH n 1, ,i N
1
2
,
ˆ ˆ
N
i
i
i i i
E E
E H Y dx
x x
(28)
34/42
Simulations
Test effectiveness of algorithms
– Two kinds of inks
Fig. 17. Two different concentrations of (solid) black and (dashed) gray
inks to use with blue-noise multitoning.
(a) (b)
35/42
– Results obtained with error diffusion
Fig. 18. Multitones of a gray-scale ramp generated with blue-noise
multitoning error diffusion using the gray level concentrations in Fig.
17(a) and (b), respectively.
36/42
– Result of blue-noise multitoning with DBS
Fig. 19. Multitones of a gray-scale ramp generated with blue-noise
multitoning DBS using the gray level concentrations in Fig. 17(a) and
(b), respectively.
37/42
– Results of RAPSD of multitones in figure 17
Fig. 20. RAPSD of multitones generated for
both gray level distributions in Fig. 17. Top to
bottom: ED for gray level 1/16, DBS for gray
level 1/16,ED for gray level 1/8, DBS for gray
level 1/8.
Fig. 21. RAPSD of multitones for gray level 1/4. Top
to bottom: ED with the gray level distribution in Fig.
17(a), DBS with the same gray level distribution,ED
with the gray level distribution in Fig. 17(b), DBS with
the same gray level distribution. 38/42
– Corresponding MSC for these patterns
Fig. 22. MSC of the multitones generated with (top) blue-noise error diffusion
and (bottom) DBS for gray level 1/4 and the gray level distribution in Fig. 17(a).
39/42
– RAPSD and MSC of patterns
• Applying both methods with both gray level
− A pattern of intensity 1/2
Fig. 23. RAPSD of multitones generated with
blue-noise error diffusion and DBS for different
gray level concentrations for gray level 1/2.
Fig. 24. MSC of multitones generated with
blue-noise error diffusion and DBS for different
gray level concentrations for gray level 1/2. 40/42
– Results obtained applying blue noise multitoning
• Error diffusion and DBS with different gray level
Fig. 25. Multitones of a natural image generated
with: (top) blue-noise multitoning error diffusion
and (bottom) DBS using the gray level
concentrations in Fig. 17(a) (right) and Fig. 17(b)
(left).
41/42
Conclusion and future work
Proposed method
– Designed through extensions of halftoning
– Introduction of model characterizing ideal spectral statistics
• Aperiodic
• Dispersed-dot
• Multilevel dither patterns
• Binary counterparts
• Minimize low-frequency graininess
− Illustrating these results
42/42
