Transcript Digital Halftoning

Digital Halftoning
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What is Digital halftoning
History
Goals of halftoning
Methods of halftoning
1.1 What is digital halftoning?
• Digital halftoning is the process of rendering a continuoustone image with a device that is capable of generating only
two or a few levels of gray at each point on the device
output surface.
• The perception of additional levels of gray depends on a
local average of the binary or multilevel texture.
•Detail is rendered by local modulation of the texture. Different dot patterns for
different levels of gray.
• Halftone is the reprographic technique that simulates continuous tone
imagery through the use of dots, varying either in size or in spacing.
'Halftone' can also be used to refer specifically to the image that is
produced by this process.
• Where continuous tone imagery (film photography, for example)
contains an infinite range of colors or greys, the halftone process
reduces visual reproductions to a binary image that is printed with only
one color of ink. This binary reproduction relies on a basic optical
illusion—that these tiny halftone dots are blended into smooth tones
by the human eye.
• (At a microscopic level, developed black and white photographic film
also consists of only two colors, and not an infinite range of
continuous tones. For details, see film grain . Next slide)
• Just as color photography evolved with the addition of filters and film
layers, color printing is made possible by repeating the halftone
process for each subtractive color—most commonly using what is
called the 'CMYK color model.' The semi-opaque property of ink
allows halftone dots of different colors to create another optical
effect—full-color imagery.
Photomicrograph of grain of different photographic plates
Left: Halftone dots. Right:
How the human eye would
see this sort of arrangement
from a sufficient distance.
History
• Fundamental concepts have been used for centuries in
weaving (by varying the no. of black threads) and
engraving (thickness/frequency of the engraved lines)
History (cont.)
• Photomechanical halftoning process was introduced by
Talbot in 1852.
The first printed photo using a halftone, December 2, 1873.
History (cont.)
• In 1882 the German patented a halftone process in
England. His invention was based on the previous ideas of
Berchtold and Swan. He used single lined screens which
were turned during exposure to produce cross-lined effects.
He was the first to achieve any commercial success with
relief halftones.
• Shortly afterwards, Ives, this time in collaboration with
Louis and Max Levy, improved the process further with the
invention and commercial production of quality cross-lined
screens.
• The relief halftone process proved almost immediately to
be a success. The use of halftone blocks in popular
journals became regular during the early 1890s
http://en.wikipedia.org/wiki/Halftone
Three examples of color halftoning with CMYK separations. From left to
right: The cyan separation, the magenta separation, the yellow separation,
the black separation, the combined halftone pattern and finally how the
human eye would observe the combined halftone pattern from a sufficient
distance.
History (cont.)
• Digital halftoning algorithms first appeared in the early 70s
as computer graphics displays and hardcopy devices
became more widely available.
Bayer - 1972
Floyd-Steinberg - 1976
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IBM J. RES. & DEV. VOL. 47 NO. 1 JANUARY 2003
The mathematics of Halftoning - paper
The problem of halftoning in digital printing has provided a splendid
new example of a mathematical opportunity in modern technology.
Digital halftoning is the technique used to display an image with a
few immiscible colors discretely applied to paper. At any point on the
paper there is only one color dot ( no mixing/superimposition of
colors).
This problem involves various fields of science such as the physics of
light, the biology of the human visual system, and the mathematics
of analog to digital conversion.
Digital halftoning can be considered as a huge optimization problem,
but we shall see that by sacrificing optimality one can construct
efficient algorithms which achieve very satisfactory results.
Halftoning is above all an art, yet mathematics, particularly the theory of
dynamical systems, helps to improve this art, while the art suggests
a rich collection of mathematical problems and insights.
• Specifically, we address the question of how full-color images should
be imitated by printers which print at positions on a lattice with colors
limited to a few available choices.
Purists use the term pixels for screens and instead
use pels for printing, but we shall stick to pixels.
Pixel colors are for the colors of inks or toners. In
some printers the choice of pixel colors can be
augmented by superposition of these substances,
but this is not allowed in highlight printers.
The two images of Brian Wu (see Figure 1) illustrate
halftoning for gray scale: Figure 1(b) is a coarsegrain halftone image of the fine-grain image in
Figure 1(a), which is approximately a full gray-scale
image.
Fig 1
(a) Gray-scale image. (b) halftone image (using clustered dither mask).
Color Half toning
• In color half toning one uses R,G,B pens which
print /produce dots of full color (255) of each or
leave blank ( i.e. r =g=b=0). These dots are
equally spaced with varying sizes.
• Halftoning is for approximate reproducing a
given color patch/area and not for a color point.
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In color half toning, one uses R,G,B
pens which print /produce dots of full
color (255) of each or leave blank ( i.e.
r =g=b=0). These dots are equally
spaced with varying sizes.
Halftoning is for approximate
reproducing a given color patch/area
and not for a color point.
The human eye perceives diffusion as
a mixture of the colors within it.
Reducing the color depth of an image
can often have significant visual sideeffects.
If the original image is a photograph, it
is likely to have thousands, or even
millions of distinct colors. The process
of constraining the available colors to a
specific color palette effectively throws
away a certain amount of color
information.
An illustration of dithering. Red and blue are the only colors
used, but as the pixels become smaller, the patch appears violet
(R+B).
• Colors and gamuts
• Perception of color is modelled as a vector addition in color space.
Standard ink and toner colors are cyan (C), magenta (M), yellow (Y), and black
(K). The letters in parentheses represent color vectors. Added to this set is
white (W), taken for the color of paper. The primary colors for light are red
(R), green (G), and blue (B). In color space we have the following relations:
W =R+G+B
M=W-G=R+B
C =W-R=G+B
Y =W-B =R+G
Since inks and toners act as light absorbers, only blue light
is reflected from a combination of magenta and cyan on
white paper.
B=M+C
Similarly, only red light is reflected from magenta and yellow, and green from
cyan and yellow.
The eye averages received light over areas consisting of many pixels.
When all eight colors C, M, Y, R, G, B, K,
and W are available to a printer and
arranged in the cube, as in Figure 2,
the Cartesian product structure of the
cube allows for significant mathematical
simplification in printing, as follows:
1. Each component of a color vector along
the C, M, Y axes is treated separately
with W in the same way one would deal
with a black and white image. The page
is overprinted three times, one for each
pixel color, and possibly a fourth time in
order to print as much K as possible
(black ink or toner is cheaper, and K, W
render better grays than C, M, Y, W).
2. When overprinting the sheet multiple
times, one must take care of
interference (Moire´) patterns due to
misalignment in the superposition of
colors, a phenomenon which is now well
understood.
Actually, the eight color vectors form
the vertices of a six-sided figure which
is far from cubic in any perceptual
color space such as the one given by
the CIE colorimetric system.
Furthermore, constraints such as the
impossibility of superimposing inks to
get R, G, or B, or the addition of other
shades to augment printer gamuts,
such as “light magenta” and “light
cyan,” also lead to significant
departures from the cube.
• For example, an original image
(Figure 1) may be reduced to the 216color color palette.
• If the original pixel colors are simply
translated into the closest available
color from the palette, no halftoning
occurs (Figure 2).
• Typically, this approach results in flat
areas and a loss of detail, and may
produce patches of color that are
significantly different from the original.
Shaded or gradient areas may appear
as color bands, which may be
distracting.
Figure 1
Figure 2
•
The application of halftoning can help to minimize such visual artifacts, and
usually results in a better representation of the original (Figure 3). Halftoning
helps to reduce color banding and flatness.
Figure 3
Need for Digital Image Halftoning
• Examples of reduced grayscale/color resolution
– Laser and inkjet printers
– Facsimile machines
– Low-cost liquid crystal displays
• Halftoning is wordlength reduction for images
– Grayscale: 8-bit to 1-bit (binary)
– Color displays: 24-bit RGB to 8-bit RGB
– Color printers: 24-bit RGB to CMY (each color binarized)
• Halftoning tries to reproduce full range of gray/ color while
preserving quality & spatial resolution
– Screening methods are pixel-parallel, fast, and simple
– Error diffusion gives better results on some media
Applications
• Display hardware, including early computer video adapters
and many modern LCDs used in mobile phones and
inexpensive digital cameras, are only capable of showing a
smaller color range than more advanced displays. One
common application of dithering is to more accurately display
graphics containing a greater range of colors than the
hardware is capable of showing.
• For example, dithering might be used in order to display a
photographic image containing millions of colors on video
hardware that is only capable of showing 256 colors at a time.
The 256 available colors would be used to generate a
dithered approximation of the original image. Without
dithering, the colors in the original image might be simply be
"rounded off" to the closest available color, resulting in a new
image that is a poor representation of the original. Dithering
takes advantage of the human eye's tendency to "mix"
two colors in close proximity to one another. ex: light
colors can be created by adding white dots in between the
colors.
• But even when the total number of available colors in the display
hardware is high enough when rendering full color digital
photographs, as those 15- and 16-bit RGB Hicolor 32,768/65,536
color modes, banding can be evident to the eye, especially in large
areas of smooth shade transitions (although the original image file
has no banding at all). Dithering the 32 or 64 RGB levels will result
in a pretty good "pseudo truecolor" display approximation, which the
eye cannot resolve as grainy.
• Another useful application of dithering is for situations in which the
graphic file format is the limiting factor. In particular, the commonlyused GIF format is restricted to the use of 256 or fewer colors in
many graphics editing software. Images in other file formats, such
as PNG, may also have such a restriction imposed on them for the
sake of a reduction in file size. Images such as these have a fixed
color palette defining all the colors that the image may use. For such
situations, graphical editing software may be responsible for
dithering images prior to saving them in such restrictive formats.
Traditional screening
Modulation Strategies
• The most common method of creating screens—amplitude
modulation—produces a regular grid of dots that vary in
size.
• The other method of creating screens—frequency
modulation—is used in a process named “Stochastic
screening”.
• Amplitude modulation - dot size varies, dot spacing is fixed.
• Frequency modulation - dot spacing varies, dot size is
fixed.
Conventional AM halftoning
• Indigo’s “Sequin” halftoning
– Dot frequency is fixed
– Dot size varies to represent tone
• Disadvantages
– Rosette patterns : different magnitudes in
different directions
– Tone jumps
– Detail rendition suffers
– Scan quality suffers
Rosette pattern When preparing color separations for printing, the screen
angles are rotated for each process color ink. A symmetrical (but nonobjectionable) "rosette" dot pattern can result, which the eye can merge into
smooth color gradations.
However, incorrect screening angles or the shifting of the paper during
printing can result in objectionable patterns; see "moiré' patterns".
FM halftoning ( “blue-noise”)
• Photo printers
– Dot size is fixed
– Dot frequency varies
• Advantages
– Does not cause Moiré
– Rosette patterns eliminated
– Tone jumps not abrupt
– Better detail rendition
– Better quality at consumer grade
scan resolutions
• Disadvantages
– Depends heavily on fidelity of
isolated dot reproduction
dot dropout and
banding in
highlights
clumping in the
midtones results
in grain
AM-FM halftoning ( “green-noise”)
• AM-FM
– Dot size and dot frequency varies
• Advantages
– Promise of the “best of both worlds”
• Disadvantages
– Design depends on particular dot
formation characteristics
– Difficult design problem
highlights
midtones
AM
FM
FM
AM
AM-FM
AM & FM Halftone
AM
FM
(looks better)
Multiple screens and color halftoning
• When different screens are combined, a number of distracting
visual effects can occur, including the edges being overly
emphasized, as well as a moiré pattern. This problem can be
reduced by rotating the screens in relation to each other. This
screen angle is another common measurement used in printing,
measured in degrees clockwise from a line running to the left (9
o'clock is zero degrees).
• Halftoning is also commonly used for printing color pictures. The
general idea is the same, by varying the density of the four primary
printing colors, cyan, magenta, yellow and black (abbreviation
CMYK), any particular shade can be reproduced. In this case there
is an additional problem that can occur. In the simple case, one
could create a halftone using the same techniques used for printing
shades of grey, but in this case the different printing colors have to
remain physically close to each other to fool the eye into thinking
they are a single color. To do this the industry has standardized on
a set of known angles, which result in the dots forming into small
circles or rosettes.
• The dots cannot easily be seen by the naked eye, but can be
discerned through a microscope or a magnifying glass
COLOR PRINT
•
Three secondary colors
•
And Black
• RED (R, MY)
• GREEN (G, CY)
• BLUE (B, CM)
• BLACK (K, CMY)
Resolution of halftone screens
• The resolution of a halftone screen is measured in lines per inch (lpi).
This is the number of lines of dots in one inch, measured parallel with the
screen's angle. Known as the screen ruling, the resolution of a screen is
written either with the suffix lpi or a hash mark. E.g. 150lpi or 150#.
• The higher the pixel resolution of a source file, the greater the detail that
can be reproduced. However, such increase also requires a
corresponding increase in screen ruling or the output will suffer from
posterization. Therefore file resolution is matched to the output
resolution.
Typical Halftone Resolutions
Screen Printing
Laser Printer (300dpi)
Laser Printer (600dpi)
45-65 lpi
65 lpi
85-105 lpi
Offset Press (newsprint paper)
85 lpi
Offset Press (coated paper)
85-185 lpi
Posterization
• Posterization of an image
occurs when a region of
an image with a
continuous gradation of
tone is replaced with
several regions of fewer
tones, resulting in an
abrupt change from one
tone to another. This
creates an effect
somewhat similar to that
of a simple usual graphic
poster.
AM HALFTONE
same angle for C, M, Y & K
Conventional Color Halftoning
Same raster angle
Error in position can cause color shift
(translation error)
Conventional Color Halftoning
Same raster angle
Error in raster angle can cause Moiré
( rotation error)
AM HALFTONE
different angles for C, M, Y and K
15, 75, 0 and 45 degrees
Conventional Color Halftoning
Different raster angle, 0, 15, 75 and 45 degrees
AM different angles
Rosette patterns
FM
ROSETTE PATTERN
ROSETTE PATTERN
FM (Stochastic) Halftone, 4-color print
“Moiré pattern” and “Rosette pattern” disappear
COLOR PRINT
Original
COLOR PRINT
AM
COLOR PRINT
FM
Architectures for halftoning algorithms
• There are three basic architectures for halftoning
algorithms.
– Dithering – point-to-point operation.
– Error diffusion – neighborhood processing.
– Search-based methods – usually are iterative.
» Ex: Direct Binary Search (DBS)
The Two Fundamental Goals
of Digital Halftoning
• Representation of Tone
– smooth, homogeneous texture.
– free from visible structure or contouring.
Diamond dot
screen
Bayer
screen
Error
diffusion
Direct
Binary
Search
(DBS)
The Two Fundamental Goals
of Digital Halftoning (cont.)
• Representation of Detail
– sharp, distinct, and good contrast in rendering of fine
structure in image.
– good rendering of lines, edges, and type characters.
– freedom from Moire due to interference between
halftone algorithm and image content
Diamond dot
screen
Error diffusion
DBS
Basic Structure of Dithering Algorithm
1
1
1
1
1 1 1
3 3 1
3 3 1
1 1
Compare
1
Halftone Image
ContinuousTone Image
0.5 3.5
2.5 1.5
Threshold
Matrix
The threshold matrix is periodically tiled over the entire continuous-tone
image. If threshold is satisfied, put a dot (black) . 2x2 threshold matrix (i.e.
four thresholds) renders five levels of gray, which is shown below. The
locations of the dots correspond to the respective locations of the threshold
matrix values.
HALFTONE CELL
Pixel (/a number of pixels)
Halftone cell
The fractional area covered
by the ink corresponds to
the value of the pixel (or the area)
Dithering algorithms
Dithering methods include:
• Thresholding (also average dithering): each pixel value is
compared against a fixed threshold. This may be the simplest
dithering algorithm there is, but it results in immense loss of detail
and contouring.
• Random dithering was the first attempt (at least as early as 1951)
to remedy the drawbacks of thresholding. Each pixel value is
compared against a random threshold, resulting in a noisy image.
Although this method doesn't generate patterned artifacts, the noise
tends to swamp the detail of the image
• Ordered dithering: dithers using a fixed pattern. For every pixel in
the image the value of the pattern at the corresponding location is
used as a threshold. Different patterns can generate completely
different dithering effects.
Thresholding (average dithering)
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The Average Dithering is a basic two-level algorithm for halftone
image. It consists in choosing a certain constant gray level,
in particular the average value of image pixels, and using it as a
global threshold in deciding whether a pixel should be quantized
to 0 or to 1. All pixels whose intensity level lies above the average
value (the threshold) are quantized to 1; all others get a value of
0.
This method is simple to implement but it has a disadvantage:
quantization contouring is quite perceptible.
Original
Threshold Dithering
Random dithering
• It is not really acceptable as a production method, but it is
very simple to describe and implement.
• For each value in the image, simply generate a random
number 1..256; if it is greater than the image value at that
point, plot the point white, otherwise plot it black. That's it.
• This generates a picture with a lot of "white noise", which
looks like TV picture "snow". Though the image produced
is very inaccurate and noisy, it is free from "artifacts" which
are phenomena produced by digital signal processing.
• Many techniques exist for the reduction of digital artifacts
like these, most of which involve using a little randomness
to 'perturb' a regular algorithm a little. Random dither
obviously takes this to extreme.
• While random dither adds a lot of high-frequency noise to
a picture, it is useful in reproducing very low-frequency
images where the absence of artifacts is more important
than noise.
• For example, a whole screen containing a gradient of all
levels from black to white would actually look best with a
random dither. In this case, ordered dithering would
produce diagonal patterns, and error dispersion would
produce clustering.
Original
Random dithering
Ordered dithering
• Halftone dithering looks similar to halftone screening in
newspapers. This is a form of clustered dithering, in that
dots tend to cluster together. This can help hide the
adverse effects of blurry pixels found on some older output
devices.
– Two approaches
» Dispersed: turn on the pixel individually
» Clustered: group pixels to clusters
• Bayer dithering produces a cross-hatch pattern. This is a
form of dispersed dithering. Because the dots don't cluster,
the result looks much less grainy
Dithering is based on a completely local determination
which is both simple and fast.
We first consider the easier case of black and white (BW) printing, for which the
color dimension is 1.
Furthermore, we assume that the index dimension is 2.
A dithering mask (mask for short) is specified by an n x m matrix M of threshold
coefficients M(i, j).
The numbers M(i, j) range over the unit interval in a uniform fashion.
The image to be halftoned is a h x v matrix T of input gray levels T(i, j), also
numbers in the unit interval. Typically, n and m are much smaller than either
v or h. i.e. the dither mask is much smaller than the given image.
The output image ( halftone image) is given by an h x v zero-one matrix φ [(i,
j)], which controls the printing of black as follows: At pixel location (i, j), black
is printed if and only if φ(i, j) = 1. The matrix φ(i, j) is defined by
φ(i, j) = 1 if T(i,j) > M ( i mod n, j mod m ) --- print black
0 therwise
See fig 3
M( i, j )
T( i, j )
Fig 3 Dither mask
if threshold exceeds, print
black
φ( i, j )
Note: The -45 degrees white
diagonal in T(i,j) is approximately
retained in φ( i, j )
• The problem of dithering is to arrange the numbers n x m in matrix
M(i, j) in the mask, so that good image quality results.
• This can be quite printer-dependent, at least when it comes to fine
tuning.
• The more thresholds in the mask, the more continuity in the gray
spectrum and the smoother the picture.
• A commonly used upper limit to the number of gray levels is 256.
Masks are constructed so that constant gray images look good, the
idea being that natural images, which are nearly constant small
areas, will then also look good.
• There is a price to be paid for the instant decision-making ability of a
mask: Namely, it is always possible to construct improbable images
on which it will fail badly.
• However, this does not happen for the usual pictures for which
masks are designed.
Selection of Threshold Values
• For an MxN halftone cell,can print 0, 1, 2, …, MN dots,
yielding average absorptances 0, 1/MN, 2/MN, …, 1,
respectively.
• As the input gray level increases, each time a threshold is
exceeded, we add a new dot, thereby increasing the
rendered absorptance by 1/MN.
• It follows that the threshold levels should be uniformly
spaced over the range of gray values of the input image.
How Tone is Rendered
• If we threshold the screen against a constant gray value,
we obtain the binary texture used to represent that constant
level of absorptance.
This is the halftone
image for the
original continuous
tone image.
Is this a good mask?
No, because we see diagonal lines in the uniform region (original).
Dot Profile Function
• The family of binary textures used to render each level of
constant tone is called the dot profile function.
• There is a one-to-one relationship between the dot profile
and the screen.
Dot Profile patterns correspond to the respective locations of the
values in the given threshold matrix .
Threshold Matrix
6
13
14
7
12
2
3
15
11
1
4
16
5
10
9
8
Selecting Patterns
• Avoid Symmetry
• At each level we select all
the pixels on the previous
level and turn on one pixel
more
• Black circles should be as
close as possible to the
center of the pattern.
Half Tone and Dithering
Half tone Pattern is a rectangular pixel regions used to approximate the
halftone production
2x2 Halftone
Pattern
Arrangement
possibilities
for one dot
assignment
HALFTONING
If pattern is like this
Clustered
If pattern is like this
Dispersed
Rendering of Detail - Partial Dotting
Values represent
Average gray levels
Cluster dither mask
Partial Dotting - Example
Spatial Arrangement of Thresholds (cont.)
• For dispersed dot textures, thresholds that are close in
value are located far apart in the threshold matrix.
Dispersed dither mask
Rule: If image gray value is greater than the
corresponding mask value, put/print a dot (black).
Detail Rendition with Dispersed Dot Screens
• Compute the halftone image in the example given below to
show how detail is rendered with a dispersed dot screen.
Dispersed dither mask
THRESHOLD MATRIX
Example: Line
1 2
5 6
9 10
13 14
3
7
11
15
4
8
12
16
Thresholds are arranged in
lines.
THRESHOLD MATRIX
Example: Spiral
1
12
11
10
2
13
16
9
3
14
15
8
4
5
6
7
Thresholds are arranged in
spiral manner.
THRESHOLD MATRIX
Clustered & Dispersed, 45 degrees
14
5
6
9
19
28
27
24
12 13
4 3
1 2
7 8
21 20
29 30
32 31
26 25
16
10
11
15
17
23
22
18
19
28
27
24
14
5
6
9
21 20
29 30
32 31
26 25
12 13
4 3
1 2
7 8
Clustered
17
23
22
18
16
10
11
15
1 30 8 28
17 9 24 16
5 25 3 32
21 13 19 11
2 29 7 27
18 10 23 15
6 26 4 31
22 14 20 12
2 29 7 27
18 10 23 15
6 26 4 31
22 14 20 12
1 30 8 28
17 9 24 16
5 25 3 32
21 13 19 11
Dispersed
The 4x4 submatrices are repeated diagonally (45 degrees).
Clustered vs. Dispersed Dots
Attribute
Texture Visibility
Detail Rendering
Stability
Clustered Dot
High
Fair
High
Dispersed Dot
Low
Good
Low
• Note that these assessments are relative.
• For example, at sufficiently high resolution, clustered dot
textures will also have low visibility and good detail
rendition.
One might think that numbers M(i, j) should be placed in the
mask as randomly as possible.
However, this produces undesirable irregular clusters and
other unpleasant defects.
To avoid such shortcomings, one tries to place numbers in the
mask so that φ(i, j) = 1 will be well dispersed for each gray
level.
Masks by their nature (dispersed or clustered) impose an
increasing gray-level constraint (or stacking constraint);
i.e., if some pixel is printed for a gray level, it is also
printed for any darker one (see fig. below). This
constraint presents a serious difficulty in designing masks.
•
•
•
•
•
•
Another disturbing feature of an improperly constructed dispersed
mask is that periods n and m of the mask dimensions can
sometimes be detected.
To avoid this, masks are often constructed so that the product n x m
is much larger than the number of distinct threshold levels.
Particularly popular in this respect are blue noise masks, so called
because low frequencies in the power spectrum of the patterns turn
out to be attenuated.
The most pleasing pattern for gray level 0.5 is the maximally
dispersed checkerboard pattern, but too much regularity in the
presence of noise is undesirable, so checkerboard patterns are
avoided in most blue noise masks (high frequency).
Adler, Thompson, Tresser, and Wu based an invention the
aperiodic Thue–Morse sequence1 in order to tile large mask by two
much smaller ones.
The memory required for this is considerably less than that for one
large mask alone. This gives an alternative (or a complement) to
blue noise masks, and can be used in the clustered mask case also.
Thue-Morse sequence
• In mathematics and its applications, the Thue-Morse sequence,
or Prouhet-Thue-Morse sequence, is a certain binary sequence
whose initial segments alternate (in a certain sense).
• Bitwise negation
• T0 =0.
• T1 = 01.
• T2 = 0110.
• T3 = 01101001.
• T4 = 0110100110010110.
• T5 = 01101001100101101001011001101001.
• T6 = 01101001100101101001011001101001
10010110011010010110100110010110.
• And so on.
Construction of a 2-D Thue-Morse pattern
• A 2-D Thue-Morse pattern,
where each row and column
is a Thue-Morse sequence,
can also be formed by a
recursive concatenation as
described in previous slide.
The complement of an array
of symbols is concatenated
to itself, first in one direction
and then the other:
8 th generation
1st generation
(corresponding to
:
a b ...
ba
3rd generation
2nd generation
(corresponding to :
a b b a ...
baab
baab
abba
... )
4th generation
Clustered masks
Size of clustered mask should larger than number of gray
levels in the given image.
Furthermore, there is a tradeoff between the number of
gray levels to be rendered and the size of clusters: The
bigger the cluster, the more levels the mask can render,
but the more noticeable the cluster.
Thompson, Tresser, and Wu, using ideas inspired by
celestial and statistical mechanics, have addressed this
difficulty in patents.
The main idea behind the new masks is to start with a large
mask consisting of several copies of a small mask with
256 levels.
• There are certain gray levels for which there is an ideal
pattern: for example, the checkerboard for the 0.5 gray
level. One attempts to keep these patterns and obtain
the intermediate threshold values by interpolation.
• This involves a trial-and-error procedure in which one
repetitively resets thresholds and judges print quality.
• This method is very versatile, and it can be used for a
variety of situations: for example, to create several
masks for machines whose behavior degrades between
maintenances, to adapt a mask to a new printer that has
been tuned for another, etc. Other inventions mentioned
above concern variations on this theme.
Such ideas can also be used for the simpler problem of
building dispersed masks.
For standard color dithering, four different masks are used, one for
each nonwhite color (including K, which is applied first, since it is
usually the cheapest ink).
Using the same mask four times is usually avoided because of the
possibility of misregistration that causes disturbing Moire´ patterns.
In the past, color printing was done with screens that performed like
masks (indeed, the word screen is still used in lieu of dithering
mask).
To minimize Moire´ patterns, the screens for separate colors were set at
different angles. For digital printing, there is a technique designing
the different masks that is equivalent to the effect of rotating a
screen.
Humans vision is least sensitive to 45 degrees.
For a description of this method. A pending patent by Rao, Thompson,
Tresser, and Wu proposes to build what they call semidigital
printers: printers intermediate to digital and analog ones, in which
each color plane is treated as in the usual digital printing, but the
screens/dither masks are rotated as in traditional analog printing.