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  1. 134
      src/ImageSharp/Dithering/DHALF.TXT
  2. 530
      src/ImageSharp/Dithering/DITHER.TXT

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src/ImageSharp/Dithering/DHALF.TXT
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@ -12,10 +12,10 @@ What follows is everything you ever wanted to know (for the time being)
about digital halftoning, or dithering. I'm sure it will be out of date as about digital halftoning, or dithering. I'm sure it will be out of date as
soon as it is released, but it does serve to collect data from a wide soon as it is released, but it does serve to collect data from a wide
variety of sources into a single document, and should save you considerable variety of sources into a single document, and should save you considerable
searching time. searching time.
Numbers in brackets (e.g. [4] or [12]) are references. A list of these Numbers in brackets (e.g. [4] or [12]) are references. A list of these
works appears at the end of this document. works appears at the end of this document.
Because this document describes ideas and algorithms which are constantly Because this document describes ideas and algorithms which are constantly
changing, I expect that it may have many editions, additions, and changing, I expect that it may have many editions, additions, and
@ -23,7 +23,7 @@ corrections before it gets to you. I will list my name below as original
author, but I do not wish to deter others from adding their own thoughts and author, but I do not wish to deter others from adding their own thoughts and
discoveries. This is not copyrighted in any way, and was created solely discoveries. This is not copyrighted in any way, and was created solely
for the purpose of organizing my own knowledge on the subject, and sharing for the purpose of organizing my own knowledge on the subject, and sharing
this with others. Please distribute it to anyone who might be interested. this with others. Please distribute it to anyone who might be interested.
If you add anything to this document, please feel free to include your name If you add anything to this document, please feel free to include your name
below as a contributor or as a reference. I would particularly like to see below as a contributor or as a reference. I would particularly like to see
@ -31,7 +31,7 @@ additions to the "Other books of interest" section. Please keep the text in
this simple format: no margins, no pagination, no lines longer than 79 this simple format: no margins, no pagination, no lines longer than 79
characters, and no non-ASCII or non-printing characters other than a CR/LF characters, and no non-ASCII or non-printing characters other than a CR/LF
pair at the end of each line. It is intended that this be read on as many pair at the end of each line. It is intended that this be read on as many
different machines as possible. different machines as possible.
Original Author: Original Author:
@ -50,7 +50,7 @@ Contributors:
COMMENTS BY MIKE MORRA COMMENTS BY MIKE MORRA
I first entered the world of imaging in the fall of 1990 when my employer, I first entered the world of imaging in the fall of 1990 when my employer,
Epson America Inc., began shipping the ES-300C color flatbed scanner. Epson America Inc., began shipping the ES-300C color flatbed scanner.
Suddenly, here I was, a field systems analyst who had worked almost Suddenly, here I was, a field systems analyst who had worked almost
exclusively with printers and PCs, thrust into a new and arcane world of exclusively with printers and PCs, thrust into a new and arcane world of
look-up tables and dithering and color reduction and .GIF files! I realized look-up tables and dithering and color reduction and .GIF files! I realized
@ -60,10 +60,10 @@ Graphics Support Forum on a very regular basis.
Lee Crocker's excellent paper called DITHER.TXT was one of the first pieces Lee Crocker's excellent paper called DITHER.TXT was one of the first pieces
of information that I came across, and it went a very long way toward of information that I came across, and it went a very long way toward
answering a lot of questions that I'd had about the subject of dithering. answering a lot of questions that I'd had about the subject of dithering.
It also provided me with the names of other essential reference works upon It also provided me with the names of other essential reference works upon
which Lee had based his paper, and I immediately began an eager search for which Lee had based his paper, and I immediately began an eager search for
these other references. these other references.
In the course of my self-study, however, I found that DITHER.TXT does In the course of my self-study, however, I found that DITHER.TXT does
presume the reader's familiarity with some fundamental imaging concepts, presume the reader's familiarity with some fundamental imaging concepts,
@ -90,7 +90,7 @@ second, distinct document. Too, I may very well have misconstrued or
misinterpreted some factual information in my revision. As such, I welcome misinterpreted some factual information in my revision. As such, I welcome
criticism and comment from all the original authors and contributors, and criticism and comment from all the original authors and contributors, and
any readers, with the hope that their feedback will help me to address these any readers, with the hope that their feedback will help me to address these
issues. issues.
If this revision it is received favorably, I will submit it to the public If this revision it is received favorably, I will submit it to the public
domain; if it is met with brickbats (for whatever reason), I will withdraw domain; if it is met with brickbats (for whatever reason), I will withdraw
@ -103,7 +103,7 @@ my questions that I needed. I'd like to publicly thank the whole Forum
community in general for putting up with my unending barrage of questions community in general for putting up with my unending barrage of questions
and inquiries over the past few months <g>. In particular, I would thank and inquiries over the past few months <g>. In particular, I would thank
John Swenson, Chris Young, and (of course) Lee Crocker for their invaluable John Swenson, Chris Young, and (of course) Lee Crocker for their invaluable
assistance. assistance.
Mike Morra [76703,4051] Mike Morra [76703,4051]
June 20, 1991 June 20, 1991
@ -118,7 +118,7 @@ digitized images on display devices which were incapable of reproducing the
full spectrum of intensities or colors present in the source image. The full spectrum of intensities or colors present in the source image. The
challenge is even more pronounced in today's world of personal computing challenge is even more pronounced in today's world of personal computing
because of the technology gap between image generation and image rendering because of the technology gap between image generation and image rendering
equipment. equipment.
Today, we now have affordable 24-bit image scanners which can generate Today, we now have affordable 24-bit image scanners which can generate
nearly true-to-life scans having as many as 256 shades of gray, or in excess nearly true-to-life scans having as many as 256 shades of gray, or in excess
@ -127,7 +127,7 @@ behind with 16- and 256-color VGA/SVGA video monitors and printers with
binary (black/white) "marking engines" as the norm. Without specialized binary (black/white) "marking engines" as the norm. Without specialized
techniques for color reduction -- the process of finding the "best fit" of techniques for color reduction -- the process of finding the "best fit" of
the display device's available gray shades and/or colors -- the imaging the display device's available gray shades and/or colors -- the imaging
experimenter would be plagued with blotchy, noisy, off-color images. experimenter would be plagued with blotchy, noisy, off-color images.
(As of this writing, "true color" 24-bit video display devices, capable of (As of this writing, "true color" 24-bit video display devices, capable of
reproducing all of the color/intensity information in the source image, are reproducing all of the color/intensity information in the source image, are
@ -135,7 +135,7 @@ now beginning to migrate downward into the PC environment, but they exact a
premium in cost and processor power which many users are loathe to pay. So- premium in cost and processor power which many users are loathe to pay. So-
called "high-color" video displays -- typically 16-bit, with 32,768-color called "high-color" video displays -- typically 16-bit, with 32,768-color
capability -- are moving into the mainstream, but color reduction techniques capability -- are moving into the mainstream, but color reduction techniques
would still be required with these devices.) would still be required with these devices.)
The science of digital halftoning (more commonly referred to as dithering, The science of digital halftoning (more commonly referred to as dithering,
or spatial dithering) is one of the techniques used to achieve satisfactory or spatial dithering) is one of the techniques used to achieve satisfactory
@ -146,7 +146,7 @@ full white pixels, or on printers which could produce only full black spots
on a printed page. Indeed, Ulichney [3] gives a definition of digital on a printed page. Indeed, Ulichney [3] gives a definition of digital
halftoning as "... any algorithmic process which creates the illusion of halftoning as "... any algorithmic process which creates the illusion of
continuous-tone images from the judicious arrangement of binary picture continuous-tone images from the judicious arrangement of binary picture
elements." elements."
Ulichney's study, as well as the earlier literature on the subject (and this Ulichney's study, as well as the earlier literature on the subject (and this
paper itself), discusses the process mostly in this context. Since we in paper itself), discusses the process mostly in this context. Since we in
@ -164,8 +164,8 @@ range of colors or gray shades that are contained in the source image.
Intensity/Color Resolution Intensity/Color Resolution
The concept of resolution is essential to the understanding of digital The concept of resolution is essential to the understanding of digital
halftoning. Resolution can be defined as "fineness" and is used to halftoning. Resolution can be defined as "fineness" and is used to
describe the level of detail in a digitally sampled signal. describe the level of detail in a digitally sampled signal.
Typically, when we hear the term "resolution" applied to images, we think of Typically, when we hear the term "resolution" applied to images, we think of
what's known as "spatial resolution," which is the basic sampling rate for what's known as "spatial resolution," which is the basic sampling rate for
@ -194,7 +194,7 @@ display device has a higher spatial resolution than the image you are trying
to reproduce, it can show a very good image even if its color resolution is to reproduce, it can show a very good image even if its color resolution is
less. This is what most of us know as "dithering" and is the subject of less. This is what most of us know as "dithering" and is the subject of
this paper. (The other tradeoff, i.e., trading color resolution for spatial this paper. (The other tradeoff, i.e., trading color resolution for spatial
resolution, is called "anti-aliasing," and is not discussed here.) resolution, is called "anti-aliasing," and is not discussed here.)
For the following discussions I will assume that we are given a grayscale For the following discussions I will assume that we are given a grayscale
@ -205,7 +205,7 @@ printer, or an HP LaserJet laser printer. Most of these methods can be
extended in obvious ways to deal with displays that have more than two extended in obvious ways to deal with displays that have more than two
levels (but still fewer than the source image), or to color images. Where levels (but still fewer than the source image), or to color images. Where
such extension is not obvious, or where better results can be obtained, I such extension is not obvious, or where better results can be obtained, I
will go into more detail. will go into more detail.
===================================== =====================================
@ -217,7 +217,7 @@ black and white device. This is accomplished by establishing a demarcation
point, or threshold, at the 50% gray level. Each dot of the source image is point, or threshold, at the 50% gray level. Each dot of the source image is
compared against this threshold value: if it is darker than the value, the compared against this threshold value: if it is darker than the value, the
device plots it black, and if it's lighter, the device plots it white. device plots it black, and if it's lighter, the device plots it white.
What happens to the image during this operation? Well, some detail What happens to the image during this operation? Well, some detail
survives, but our perception of gray levels is completely gone. This means survives, but our perception of gray levels is completely gone. This means
that a lot of the image content is obliterated. Take an area of the image that a lot of the image content is obliterated. Take an area of the image
@ -281,16 +281,16 @@ categories:
3. Ordered dither 3. Ordered dither
4. Error-diffusion halftoning 4. Error-diffusion halftoning
Each of these methods is generally better than those listed before it, but Each of these methods is generally better than those listed before it, but
other considerations such as processing time, memory constraints, etc. may other considerations such as processing time, memory constraints, etc. may
weigh in favor of one of the simpler methods. weigh in favor of one of the simpler methods.
To convert any of the first three methods into color, simply apply the To convert any of the first three methods into color, simply apply the
algorithm separately for each primary color and mix the resulting values. algorithm separately for each primary color and mix the resulting values.
This assumes that you have at least eight output colors: black, red, green, This assumes that you have at least eight output colors: black, red, green,
blue, cyan, magenta, yellow, and white. Though this will work for error blue, cyan, magenta, yellow, and white. Though this will work for error
diffusion as well, there are better methods which will be discussed in more diffusion as well, there are better methods which will be discussed in more
detail later. detail later.
===================================== =====================================
@ -307,7 +307,7 @@ While it is not really acceptable as a production method, it is very simple
to describe and implement. For each dot in our grayscale image, we generate to describe and implement. For each dot in our grayscale image, we generate
a random number in the range 0 - 255: if the random number is greater than a random number in the range 0 - 255: if the random number is greater than
the image value at that dot, the display device plots the dot white; the image value at that dot, the display device plots the dot white;
otherwise, it plots it black. That's it. otherwise, it plots it black. That's it.
This generates a picture with a lot of "white noise", which looks like TV This generates a picture with a lot of "white noise", which looks like TV
picture "snow". Although inaccurate and grainy, the image is free from picture "snow". Although inaccurate and grainy, the image is free from
@ -317,7 +317,7 @@ 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 gradient of all levels from black to white would actually look best with a
random dither. With this image, other digital halftoning algorithms would random dither. With this image, other digital halftoning algorithms would
produce significant artifacts like diagonal patterns (in ordered dithering) produce significant artifacts like diagonal patterns (in ordered dithering)
and clustering (in error diffusion halftones). and clustering (in error diffusion halftones).
I should mention, of course, that unless your computer has a hardware-based I should mention, of course, that unless your computer has a hardware-based
random number generator (and most don't), there may be some artifacts from random number generator (and most don't), there may be some artifacts from
@ -399,7 +399,7 @@ like:
because if they were repeated over a large area (a common occurrence in many because if they were repeated over a large area (a common occurrence in many
images [1]) they would create vertical, horizontal, or diagonal lines. images [1]) they would create vertical, horizontal, or diagonal lines.
Also, studies [1] have shown that the patterns should form a "growth Also, studies [1] have shown that the patterns should form a "growth
sequence:" once a pixel is intensified for a particular value, it should sequence:" once a pixel is intensified for a particular value, it should
remain intensified for all subsequent values. In this fashion, each pattern remain intensified for all subsequent values. In this fashion, each pattern
@ -407,7 +407,7 @@ is a superset of the previous one; this similarity between adjacent
intensity patterns minimizes any contouring artifacts. intensity patterns minimizes any contouring artifacts.
Here is a good pattern for a 3-by-3 matrix which subscribes to the rules set Here is a good pattern for a 3-by-3 matrix which subscribes to the rules set
forth above: forth above:
--- --- --- -X- -XX -XX -XX -XX XXX XXX --- --- --- -X- -XX -XX -XX -XX XXX XXX
@ -427,7 +427,7 @@ greater than that of the image.
Another limitation of patterning is that the effective spatial resolution is Another limitation of patterning is that the effective spatial resolution is
decreased, since a multiple-pixel "cell" is used to simulate the single, decreased, since a multiple-pixel "cell" is used to simulate the single,
larger halftone dot. The more intensity resolution we want, the larger the larger halftone dot. The more intensity resolution we want, the larger the
halftone cell used and, by extension, the lower the spatial resolution. halftone cell used and, by extension, the lower the spatial resolution.
In the above example, using 3 x 3 patterning, we are able to simulate 10 In the above example, using 3 x 3 patterning, we are able to simulate 10
intensity levels (not a very good rendering) but we must reduce the spatial intensity levels (not a very good rendering) but we must reduce the spatial
@ -437,22 +437,22 @@ eight-fold decrease in spatial resolution. And to get the full 256 levels
of intensity in our source image, we would need a 16 x 16 pattern and would of intensity in our source image, we would need a 16 x 16 pattern and would
incur a 16-fold reduction in spatial resolution. Because of this size incur a 16-fold reduction in spatial resolution. Because of this size
distortion of the image, and with the development of more effective digital distortion of the image, and with the development of more effective digital
halftoning methods, patterning is only infrequently used today. halftoning methods, patterning is only infrequently used today.
To extend this method to color images, we would use patterns of colored To extend this method to color images, we would use patterns of colored
pixels to represent shades not directly printable by the hardware. For pixels to represent shades not directly printable by the hardware. For
example, if your hardware is capable of printing only red, green, blue, and example, if your hardware is capable of printing only red, green, blue, and
black (the minimal case for color dithering), other colors can be black (the minimal case for color dithering), other colors can be
represented with 2 x 2 patterns of these four: represented with 2 x 2 patterns of these four:
Yellow = R G Cyan = G B Magenta = R B Gray = R G Yellow = R G Cyan = G B Magenta = R B Gray = R G
G R B G B R B K G R B G B R B K
(B here represents blue, K is black). In this particular example, there are (B here represents blue, K is black). In this particular example, there are
a total of 31 such distinct patterns which can be used; their enumeration is a total of 31 such distinct patterns which can be used; their enumeration is
left "as an exercise for the reader" (don't you hate books that do that?). left "as an exercise for the reader" (don't you hate books that do that?).
===================================== =====================================
@ -503,13 +503,13 @@ PATTERN. Returning to our example of a 3 x 3 pattern, this means that we
would be mapping NINE image dots into this pattern. would be mapping NINE image dots into this pattern.
The simplest way to do this in programming is to map the X and Y coordinates The simplest way to do this in programming is to map the X and Y coordinates
of each image dot into the pixel (X mod 3, Y mod 3) in the pattern. of each image dot into the pixel (X mod 3, Y mod 3) in the pattern.
Returning to our two patterns (clustered and dispersed) as defined earlier, Returning to our two patterns (clustered and dispersed) as defined earlier,
we can derive an effective mathematical algorithm that can be used to plot we can derive an effective mathematical algorithm that can be used to plot
the correct pixel patterns. Because each of the patterns above is a the correct pixel patterns. Because each of the patterns above is a
superset of the previous, we can express the patterns in a compact array superset of the previous, we can express the patterns in a compact array
form as the order of pixels added: form as the order of pixels added:
8 3 4 1 7 4 8 3 4 1 7 4
@ -534,7 +534,7 @@ allows) is preferred in order to decrease the graininess of the displayed
images. Bayer [2] has shown that for matrices of orders which are powers of images. Bayer [2] has shown that for matrices of orders which are powers of
two there is an optimal pattern of dispersed dots which results in the two there is an optimal pattern of dispersed dots which results in the
pattern noise being as high-frequency as possible. The pattern for a 2x2 pattern noise being as high-frequency as possible. The pattern for a 2x2
and 4x4 matrices are as follows: and 4x4 matrices are as follows:
1 3 1 9 3 11 These patterns (and their rotations 1 3 1 9 3 11 These patterns (and their rotations
@ -548,7 +548,7 @@ patterns. (To fully reproduce our 256-level image, we would need to use an
8x8 pattern.) 8x8 pattern.)
The Bayer ordered dither is in very common use and is easily identified by The Bayer ordered dither is in very common use and is easily identified by
the cross-hatch pattern artifacts it produces in the resulting display. the cross-hatch pattern artifacts it produces in the resulting display.
This artifacting is the major drawback of an otherwise powerful and very This artifacting is the major drawback of an otherwise powerful and very
fast technique. fast technique.
@ -616,7 +616,7 @@ we use for processing this point.
If we are dithering our sample grayscale image for output to a black-and- If we are dithering our sample grayscale image for output to a black-and-
white device, the "find closest intensity/color" operation is just a simple white device, the "find closest intensity/color" operation is just a simple
thresholding (the closest intensity is going to be either black or white). thresholding (the closest intensity is going to be either black or white).
In color imaging -- for instance, color-reducing a 24-bit true color Targa In color imaging -- for instance, color-reducing a 24-bit true color Targa
file to an 8-bit, mapped GIF file -- this involves matching the input color file to an 8-bit, mapped GIF file -- this involves matching the input color
to the closest available hardware color. Depending on how the display to the closest available hardware color. Depending on how the display
@ -653,7 +653,7 @@ position. The expression in parentheses is the divisor used to break up the
error weights. In the Floyd-Steinberg filter, each pixel "communicates" error weights. In the Floyd-Steinberg filter, each pixel "communicates"
with 4 "neighbors." The pixel immediately to the right gets 7/16 of the with 4 "neighbors." The pixel immediately to the right gets 7/16 of the
error value, the pixel directly below gets 5/16 of the error, and the error value, the pixel directly below gets 5/16 of the error, and the
diagonally adjacent pixels get 3/16 and 1/16. diagonally adjacent pixels get 3/16 and 1/16.
The weighting shown is for the traditional left-to-right scanning of the The weighting shown is for the traditional left-to-right scanning of the
image. If the line were scanned right-to-left (more about this later), this image. If the line were scanned right-to-left (more about this later), this
@ -683,11 +683,11 @@ The output from this filter is nowhere near as good as that from the real
Floyd-Steinberg filter. There aren't enough weights to the dispersion, Floyd-Steinberg filter. There aren't enough weights to the dispersion,
which means that the error value isn't distributed finely enough. With the which means that the error value isn't distributed finely enough. With the
entire image scanned left-to-right, the artifacting produced would be entire image scanned left-to-right, the artifacting produced would be
totally unacceptable. totally unacceptable.
Much better results would be obtained by using an alternating, or Much better results would be obtained by using an alternating, or
serpentine, raster scan: processing the first line left-to-right, the next serpentine, raster scan: processing the first line left-to-right, the next
line right-to-left, and so on (reversing the filter pattern appropriately). line right-to-left, and so on (reversing the filter pattern appropriately).
Serpentine scanning -- which can be used with any of the error-diffusion Serpentine scanning -- which can be used with any of the error-diffusion
filters detailed here -- introduces an additional perturbation which filters detailed here -- introduces an additional perturbation which
contributes more randomness to the resultant halftone. Even with serpentine contributes more randomness to the resultant halftone. Even with serpentine
@ -702,9 +702,9 @@ If the false Floyd-Steinberg filter fails because the error isn't
distributed well enough, then it follows that a filter with a wider distributed well enough, then it follows that a filter with a wider
distribution would be better. This is exactly what Jarvis, Judice, and distribution would be better. This is exactly what Jarvis, Judice, and
Ninke [6] did in 1976 with their filter: Ninke [6] did in 1976 with their filter:
* 7 5
* 7 5
3 5 7 5 3 3 5 7 5 3
1 3 5 3 1 (1/48) 1 3 5 3 1 (1/48)
@ -723,7 +723,7 @@ requires extra memory and time for processing.
The Stucki filter The Stucki filter
P. Stucki [7] offered a rework of the Jarvis, Judice, and Ninke filter in P. Stucki [7] offered a rework of the Jarvis, Judice, and Ninke filter in
1981: 1981:
* 8 4 * 8 4
@ -731,12 +731,12 @@ P. Stucki [7] offered a rework of the Jarvis, Judice, and Ninke filter in
1 2 4 2 1 (1/42) 1 2 4 2 1 (1/42)
Once again, division by 42 is quite slow to calculate (requiring DIVs). Once again, division by 42 is quite slow to calculate (requiring DIVs).
However, after the initial 8/42 is calculated, some time can be saved by However, after the initial 8/42 is calculated, some time can be saved by
producing the remaining fractions by shifts. The Stucki filter has been producing the remaining fractions by shifts. The Stucki filter has been
observed to give very clean, sharp output, which helps to offset the slow observed to give very clean, sharp output, which helps to offset the slow
processing time. processing time.
===================================== =====================================
The Burkes filter The Burkes filter
@ -752,7 +752,7 @@ in 1988:
Notice that this is just a simplification of the Stucki filter with the Notice that this is just a simplification of the Stucki filter with the
bottom row removed. The main improvement is that the divisor is now 32, bottom row removed. The main improvement is that the divisor is now 32,
which allows the error values to be calculated using shifts once more, and which allows the error values to be calculated using shifts once more, and
the number of neighbors communicated with has been reduced to seven. the number of neighbors communicated with has been reduced to seven.
Furthermore, the removal of one row reduces the memory requirements of the Furthermore, the removal of one row reduces the memory requirements of the
filter by eliminating the second forward array which would otherwise be filter by eliminating the second forward array which would otherwise be
needed. needed.
@ -807,9 +807,9 @@ would also include HP-compatible and PostScript desktop laser printers using
Some displays may use "rectangular pixels," where the horizontal and Some displays may use "rectangular pixels," where the horizontal and
vertical spacings are unequal. This would include various EGA and CGA video vertical spacings are unequal. This would include various EGA and CGA video
modes and other specialized video displays, and most dot-matrix printers. modes and other specialized video displays, and most dot-matrix printers.
In many cases, the filters described earlier will do a decent job on In many cases, the filters described earlier will do a decent job on
rectangular pixel grids, but an optimized filter would be preferred. rectangular pixel grids, but an optimized filter would be preferred.
Slinkman [10] describes one such filter for his 640 x 240 monochrome display Slinkman [10] describes one such filter for his 640 x 240 monochrome display
with a 1:2 aspect ratio. with a 1:2 aspect ratio.
@ -832,7 +832,7 @@ be interested in further information.
While technically not an error-diffusion filter, a method proposed by Gozum While technically not an error-diffusion filter, a method proposed by Gozum
[11] offers color resolutions in excess of 256 colors by plotting red, [11] offers color resolutions in excess of 256 colors by plotting red,
green, and blue pixel "triplets" or triads to simulate an "interlaced" green, and blue pixel "triplets" or triads to simulate an "interlaced"
television display (sacrificing some horizontal resolution in the process). television display (sacrificing some horizontal resolution in the process).
Again, I would refer interested readers to his document for more Again, I would refer interested readers to his document for more
information. information.
@ -848,7 +848,7 @@ various filters in the same program, but the speed benefits are enormous.
It is critical with all of these algorithms that when error values are added It is critical with all of these algorithms that when error values are added
to neighboring pixels, the resultant summed values must be truncated to fit to neighboring pixels, the resultant summed values must be truncated to fit
within the limits of hardware. Otherwise, an area of very intense color may within the limits of hardware. Otherwise, an area of very intense color may
cause streaks into an adjacent area of less intense color. cause streaks into an adjacent area of less intense color.
This truncation is known as "clipping," and is analogous to the audio This truncation is known as "clipping," and is analogous to the audio
world's concept of the same name. As in the case of an audio amplifier, world's concept of the same name. As in the case of an audio amplifier,
@ -856,13 +856,13 @@ clipping adds undesired noise to the data. Unlike the audio world, however,
the visual clipping performed in error-diffusion halftoning is acceptable the visual clipping performed in error-diffusion halftoning is acceptable
since it is not nearly so offensive as the color streaking that would occur since it is not nearly so offensive as the color streaking that would occur
otherwise. It is mainly for this reason that the larger filters work better otherwise. It is mainly for this reason that the larger filters work better
-- they split the errors up more finely and produce less clipping noise. -- they split the errors up more finely and produce less clipping noise.
With all of these filters, it is also important to ensure that the sum of With all of these filters, it is also important to ensure that the sum of
the distributed error values is equal to the original error value. This is the distributed error values is equal to the original error value. This is
most easily accomplished by subtracting each fraction, as it is calculated, most easily accomplished by subtracting each fraction, as it is calculated,
from the whole error value, and using the final remainder as the last from the whole error value, and using the final remainder as the last
fraction. fraction.
===================================== =====================================
@ -880,7 +880,7 @@ constantly-varying pattern). As you might imagine, any of these methods
incur a penalty in processing time. incur a penalty in processing time.
Indeed, some of the above filters (particularly the simpler ones) can be Indeed, some of the above filters (particularly the simpler ones) can be
greatly improved by skewing the weights with a little randomness [3]. greatly improved by skewing the weights with a little randomness [3].
===================================== =====================================
@ -888,7 +888,7 @@ Nearest available color
Calculating the nearest available intensity is trivial with a monochrome Calculating the nearest available intensity is trivial with a monochrome
image; calculating the nearest available color in a color image requires image; calculating the nearest available color in a color image requires
more work. more work.
A table of RGB values of all available colors must be scanned sequentially A table of RGB values of all available colors must be scanned sequentially
for each input pixel to find the closest. The "distance" formula most often for each input pixel to find the closest. The "distance" formula most often
@ -896,7 +896,7 @@ used is a simple pythagorean "least squares". The difference for each color
is squared, and the three squares added to produce the distance value. This is squared, and the three squares added to produce the distance value. This
value is equivalent to the square of the distance between the points in RGB- value is equivalent to the square of the distance between the points in RGB-
space. It is not necessary to compute the square root of this value because space. It is not necessary to compute the square root of this value because
we are not interested in the actual distance, only in which is smallest. we are not interested in the actual distance, only in which is smallest.
The square root function is a monotonic increasing function and does not The square root function is a monotonic increasing function and does not
affect the order of its operands. If the total number of colors with which affect the order of its operands. If the total number of colors with which
you are dealing is small, this part of the algorithm can be replaced by a you are dealing is small, this part of the algorithm can be replaced by a
@ -907,7 +907,7 @@ results can be achieved by selecting colors from the image itself. You must
reserve at least 8 colors for the primaries, secondaries, black, and white reserve at least 8 colors for the primaries, secondaries, black, and white
for best results. If you do not know the colors in your image ahead of for best results. If you do not know the colors in your image ahead of
time, or if you are going to use the same map to dither several different time, or if you are going to use the same map to dither several different
images, you will have to fill your color map with a good range of colors. images, you will have to fill your color map with a good range of colors.
This can be done either by assigning a certain number of bits to each This can be done either by assigning a certain number of bits to each
primary and computing all combinations, or by a smoother distribution as primary and computing all combinations, or by a smoother distribution as
suggested by Heckbert [8]. suggested by Heckbert [8].
@ -927,7 +927,7 @@ the "raw" scan is then already in a 1- or 2-bit/pixel format. While this
feature would probably be unsuitable for cases where the image would need feature would probably be unsuitable for cases where the image would need
further processing (see the "Loss of image information" section below), it further processing (see the "Loss of image information" section below), it
is very useful where the operator wants to generate a final image, ready for is very useful where the operator wants to generate a final image, ready for
printing or displaying, with little or no subsequent processing. printing or displaying, with little or no subsequent processing.
As an example, the Epson ES-300C color scanner (and its European equivalent, As an example, the Epson ES-300C color scanner (and its European equivalent,
the Epson GT-6000) offers three internal halftone modes. One is a standard the Epson GT-6000) offers three internal halftone modes. One is a standard
@ -953,12 +953,12 @@ needs to be rendered on a bilevel display device. In this situation, one
would almost never want to store the dithered image. would almost never want to store the dithered image.
On the other hand, when color images are dithered for display on color On the other hand, when color images are dithered for display on color
displays with a lower color resolution, the dithered images are more useful. displays with a lower color resolution, the dithered images are more useful.
In fact, the bulk of today's scanned-image GIF files which abound on In fact, the bulk of today's scanned-image GIF files which abound on
electronic BBSs and information services are 8-bit (256 color), colormapped electronic BBSs and information services are 8-bit (256 color), colormapped
and dithered files created from 24-bit true-color scans. Only rarely are and dithered files created from 24-bit true-color scans. Only rarely are
the 24-bit files exchanged, because of the huge amount of data contained in the 24-bit files exchanged, because of the huge amount of data contained in
them. them.
In some cases, these mapped GIF files may be further processed with special In some cases, these mapped GIF files may be further processed with special
paint/processing utilities, with very respectable results. However, the paint/processing utilities, with very respectable results. However, the
@ -1214,10 +1214,10 @@ Bibliography
[2] Bayer, B.E., "An Optimum Method for Two-Level Rendition of Continuous [2] Bayer, B.E., "An Optimum Method for Two-Level Rendition of Continuous
Tone Pictures," IEEE International Conference on Communications, Tone Pictures," IEEE International Conference on Communications,
Conference Records, 1973, pp. 26-11 to 26-15. Conference Records, 1973, pp. 26-11 to 26-15.
A short article proving the optimality of Bayer's pattern in the A short article proving the optimality of Bayer's pattern in the
dispersed-dot ordered dither. dispersed-dot ordered dither.
[3] Ulichney, R., Digital Halftoning, The MIT Press, Cambridge, MA, 1987. [3] Ulichney, R., Digital Halftoning, The MIT Press, Cambridge, MA, 1987.
@ -1226,7 +1226,7 @@ Bibliography
higher math may come in handy) and wonderful illustrations. It higher math may come in handy) and wonderful illustrations. It
does not contain any code, but don't let that keep you from does not contain any code, but don't let that keep you from
getting this book. Computer Literacy normally carries it but the getting this book. Computer Literacy normally carries it but the
title is often sold out. title is often sold out.
[MFM note: I can't describe how much information I got from this [MFM note: I can't describe how much information I got from this
book! Several different writers have praised this reference to book! Several different writers have praised this reference to
@ -1242,8 +1242,8 @@ Bibliography
Scale." SID 1975, International Symposium Digest of Technical Papers, Scale." SID 1975, International Symposium Digest of Technical Papers,
vol 1975m, pp. 36-37. vol 1975m, pp. 36-37.
Short article in which Floyd and Steinberg introduce their filter. Short article in which Floyd and Steinberg introduce their filter.
[5] Daniel Burkes is unpublished, but can be reached at this address: [5] Daniel Burkes is unpublished, but can be reached at this address:
Daniel Burkes Daniel Burkes
@ -1266,7 +1266,7 @@ Bibliography
for bilevel image hardcopy reproduction." Research Report RZ1060, IBM for bilevel image hardcopy reproduction." Research Report RZ1060, IBM
Research Laboratory, Zurich, Switzerland, 1981. Research Laboratory, Zurich, Switzerland, 1981.
[8] Heckbert, P. "Color Image Quantization for Frame Buffer Display." [8] Heckbert, P. "Color Image Quantization for Frame Buffer Display."
Computer Graphics (SIGGRAPH 82), vol. 16, pp. 297-307, 1982. Computer Graphics (SIGGRAPH 82), vol. 16, pp. 297-307, 1982.
[9] Frankie Sierra is unpublished, but can be reached via CIS at UID# [9] Frankie Sierra is unpublished, but can be reached via CIS at UID#
@ -1317,13 +1317,13 @@ York, 1985.
Rogers, D.F. and J. A. Adams, Mathematical Elements for Computer Graphics, Rogers, D.F. and J. A. Adams, Mathematical Elements for Computer Graphics,
McGraw-Hill, New York, 1976. McGraw-Hill, New York, 1976.
A good detailed discussion of producing graphic images on a computer. A good detailed discussion of producing graphic images on a computer.
Plenty of sample code. Plenty of sample code.
Kuto, S., "Continuous Color Presentation Using a Low-Cost Ink Jet Printer," Kuto, S., "Continuous Color Presentation Using a Low-Cost Ink Jet Printer,"
Proc. Computer Graphics Tokyo 84, 24-27 April, 1984, Tokyo, Japan. Proc. Computer Graphics Tokyo 84, 24-27 April, 1984, Tokyo, Japan.
Mitchell, W.J., R.S. Liggett, and T. Kvan, The Art of Computer Graphics Mitchell, W.J., R.S. Liggett, and T. Kvan, The Art of Computer Graphics
Programming, Van Nostrand Reinhold Co., New York, 1987. Programming, Van Nostrand Reinhold Co., New York, 1987.
Pavlidis, T., Algorithms for Graphics and Image Processing, Computer Science Pavlidis, T., Algorithms for Graphics and Image Processing, Computer Science

530
src/ImageSharp/Dithering/DITHER.TXT
View File

@ -1,28 +1,28 @@
DITHER.TXT DITHER.TXT
What follows is everything you ever wanted to know (for the time being) about What follows is everything you ever wanted to know (for the time being) about
dithering. I'm sure it will be out of date as soon as it is released, but it dithering. I'm sure it will be out of date as soon as it is released, but it
does serve to collect data from a wide variety of sources into a single does serve to collect data from a wide variety of sources into a single
document, and should save you considerable searching time. document, and should save you considerable searching time.
Numbers in brackets (like this [0]) are references. A list of these works Numbers in brackets (like this [0]) are references. A list of these works
appears at the end of this document. appears at the end of this document.
Because this document describes ideas and algorithms which are constantly Because this document describes ideas and algorithms which are constantly
changing, I expect that it may have many editions, additions, and corrections changing, I expect that it may have many editions, additions, and corrections
before it gets to you. I will list my name below as original author, but I before it gets to you. I will list my name below as original author, but I
do not wish to deter others from adding their own thoughts and discoveries. do not wish to deter others from adding their own thoughts and discoveries.
This is not copyrighted in any way, and was created solely for the purpose of This is not copyrighted in any way, and was created solely for the purpose of
organizing my own knowledge on the subject, and sharing this with others. organizing my own knowledge on the subject, and sharing this with others.
Please distribute it to anyonw who might be interested. Please distribute it to anyonw who might be interested.
If you add anything to this document, please feel free to include your name If you add anything to this document, please feel free to include your name
below as a contributor or as a reference. I would particularly like to see below as a contributor or as a reference. I would particularly like to see
additions to the "Other books of interest" section. Please keep the text in additions to the "Other books of interest" section. Please keep the text in
this simple format: no margins, no pagination, no lines longer that 79 this simple format: no margins, no pagination, no lines longer that 79
characters, and no non-ASCII or non-printing characters other than a CR/LF characters, and no non-ASCII or non-printing characters other than a CR/LF
pair at the end of each line. It is intended that this be read on as many pair at the end of each line. It is intended that this be read on as many
different machines as possible. different machines as possible.
Original Author: Original Author:
@ -35,37 +35,37 @@ Contributors:
======================================================================== ========================================================================
What is Dithering? What is Dithering?
Dithering, also called Halftoning or Color Reduction, is the process of Dithering, also called Halftoning or Color Reduction, is the process of
rendering an image on a display device with fewer colors than are in the rendering an image on a display device with fewer colors than are in the
image. The number of different colors in an image or on a device I will call image. The number of different colors in an image or on a device I will call
its Color Resolution. The term "resolution" means "fineness" and is used to its Color Resolution. The term "resolution" means "fineness" and is used to
describe the level of detail in a digitally sampled signal. It is used most describe the level of detail in a digitally sampled signal. It is used most
often in referring to the Spatial Resolution, which is the basic sampling often in referring to the Spatial Resolution, which is the basic sampling
rate for a digitized image. rate for a digitized image.
Spatial resolution describes the fineness of the "dots" used in an image. Spatial resolution describes the fineness of the "dots" used in an image.
Color resolution describes the fineness of detail available at each dot. The Color resolution describes the fineness of detail available at each dot. The
higher the resolution of a digital sample, the better it can reproduce high higher the resolution of a digital sample, the better it can reproduce high
frequency detail. A compact disc, for example, has a temporal (time) frequency detail. A compact disc, for example, has a temporal (time)
resolution of 44,000 samples per second, and a dynamic (volume) resolution of resolution of 44,000 samples per second, and a dynamic (volume) resolution of
16 bits (0..65535). It can therefore reproduce sounds with a vast dynamic 16 bits (0..65535). It can therefore reproduce sounds with a vast dynamic
range (from barely audible to ear-splitting) with great detail, but it has range (from barely audible to ear-splitting) with great detail, but it has
problems with very high-frequency sounds, like violins and piccolos. problems with very high-frequency sounds, like violins and piccolos.
It is often possible to "trade" one kind of resolution for another. If your It is often possible to "trade" one kind of resolution for another. If your
display device has a higher spatial resolution than the image you are trying display device has a higher spatial resolution than the image you are trying
to reproduce, it can show a very good image even if its color resolution is to reproduce, it can show a very good image even if its color resolution is
less. This is what we will call "dithering" and is the subject of this less. This is what we will call "dithering" and is the subject of this
paper. The other tradeoff, i.e., trading color resolution for spatial paper. The other tradeoff, i.e., trading color resolution for spatial
resolution, is called "anti-aliasing" and is not discussed here. resolution, is called "anti-aliasing" and is not discussed here.
It is important to emphasize here that dithering is a one-way operation. It is important to emphasize here that dithering is a one-way operation.
Once an image has been dithered, although it may look like a good Once an image has been dithered, although it may look like a good
reproduction of the original, information is permanently lost. Many image reproduction of the original, information is permanently lost. Many image
processing functions fail on dithered images. For these reasons, dithering processing functions fail on dithered images. For these reasons, dithering
must be considered only as a way to produce an image on hardware that would must be considered only as a way to produce an image on hardware that would
otherwise be incapable of displaying it. The data representing an image otherwise be incapable of displaying it. The data representing an image
should always be kept in full detail. should always be kept in full detail.
======================================================================== ========================================================================
@ -78,84 +78,84 @@ The classes of dithering algorithms we will discuss here are these:
3. Ordered 3. Ordered
4. Error dispersion 4. Error dispersion
Each of these methods is generally better than those listed before it, but Each of these methods is generally better than those listed before it, but
other considerations such as processing time, memory constraints, etc. may other considerations such as processing time, memory constraints, etc. may
weigh in favor of one of the simpler methods. weigh in favor of one of the simpler methods.
For the following discussions I will assume that we are given an image with For the following discussions I will assume that we are given an image with
256 shades of gray (0=black..255=white) that we are trying to reproduce on a 256 shades of gray (0=black..255=white) that we are trying to reproduce on a
black and white ouput device. Most of these methods can be extended in black and white ouput device. Most of these methods can be extended in
obvious ways to deal with displays that have more than two levels but fewer obvious ways to deal with displays that have more than two levels but fewer
than the image, or to color images. Where such extension is not obvious, or than the image, or to color images. Where such extension is not obvious, or
where better results can be obtained, I will go into more detail. where better results can be obtained, I will go into more detail.
To convert any of the first three methods into color, simply apply the To convert any of the first three methods into color, simply apply the
algorithm separately for each primary and mix the resulting values. This algorithm separately for each primary and mix the resulting values. This
assumes that you have at least eight output colors: black, red, green, blue, assumes that you have at least eight output colors: black, red, green, blue,
cyan, magenta, yellow, and white. Though this will work for error dispersion cyan, magenta, yellow, and white. Though this will work for error dispersion
as well, there are better methods in this case. as well, there are better methods in this case.
======================================================================== ========================================================================
Random dither Random dither
This is the bubblesort of dithering algorithms. It is not really acceptable This is the bubblesort of dithering algorithms. It is not really acceptable
as a production method, but it is very simple to describe and implement. For 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 each value in the image, simply generate a random number 1..256; if it is
geater than the image value at that point, plot the point white, otherwise geater 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 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 noise", which looks like TV picture "snow". Though the image produced is
very inaccurate and noisy, it is free from "artifacts" which are phenomena very inaccurate and noisy, it is free from "artifacts" which are phenomena
produced by digital signal processing. produced by digital signal processing.
The most common type of artifact is the Moire pattern (Contributors: please The most common type of artifact is the Moire pattern (Contributors: please
resist the urge to put an accent on the "e", as no portable character set resist the urge to put an accent on the "e", as no portable character set
exists for this). If you draw several lines close together radiating from a exists for this). If you draw several lines close together radiating from a
single point on a computer display, you will see what appear to be flower- single point on a computer display, you will see what appear to be flower-
like patterns. These patterns are not part of the original idea of lines, like patterns. These patterns are not part of the original idea of lines,
but are an illusion produced by the jaggedness of the display. but are an illusion produced by the jaggedness of the display.
Many techniques exist for the reduction of digital artifacts like these, most 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 of which involve using a little randomness to "perturb" a regular algorithm a
little. Random dither obviously takes this to extreme. little. Random dither obviously takes this to extreme.
I should mention, of course, that unless your computer has a hardware-based I should mention, of course, that unless your computer has a hardware-based
random number generator (and most don't) there may be some artifacts from the random number generator (and most don't) there may be some artifacts from the
random number generation algorithm itself. random number generation algorithm itself.
While random dither adds a lot of high-frequency noise to a picture, it is 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 useful in reproducing very low-frequency images where the absence of
artifacts is more important than noise. For example, a whole screen artifacts is more important than noise. For example, a whole screen
containing a gradient of all levels from black to white would actually look 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 best with a random dither. In this case, ordered dithering would produce
diagonal patterns, and error dispersion would produce clustering. diagonal patterns, and error dispersion would produce clustering.
For efficiency, you can take the random number generator "out of the loop" by For efficiency, you can take the random number generator "out of the loop" by
generating a list of random numbers beforehand for use in the dither. Make generating a list of random numbers beforehand for use in the dither. Make
sure that the list is larger than the number of pixels in the image or you sure that the list is larger than the number of pixels in the image or you
may get artifacts from the reuse of numbers. The worst case would be if the may get artifacts from the reuse of numbers. The worst case would be if the
size of your list of random numbers is a multiple or near-multiple of the size of your list of random numbers is a multiple or near-multiple of the
horizontal size of the image, in which case unwanted vertical or diagonal horizontal size of the image, in which case unwanted vertical or diagonal
lines will appear. lines will appear.
======================================================================== ========================================================================
Pattern dither Pattern dither
This is also a simple concept, but much more effective than random dither. This is also a simple concept, but much more effective than random dither.
For each possible value in the image, create a pattern of dots that For each possible value in the image, create a pattern of dots that
approximates that value. For instance, a 3-by-3 block of dots can have one approximates that value. For instance, a 3-by-3 block of dots can have one
of 512 patterns, but for our purposes, there are only 10; the number of black of 512 patterns, but for our purposes, there are only 10; the number of black
dots in the pattern determines the darkness of the pattern. dots in the pattern determines the darkness of the pattern.
Which 10 patterns do we choose? Obviously, we need the all-white and all- Which 10 patterns do we choose? Obviously, we need the all-white and all-
black patterns. We can eliminate those patterns which would create vertical black patterns. We can eliminate those patterns which would create vertical
or horizontal lines if repeated over a large area because many images have or horizontal lines if repeated over a large area because many images have
such regions of similar value [1]. It has been shown [1] that patterns for such regions of similar value [1]. It has been shown [1] that patterns for
adjacent colors should be similar to reduce an artifact called "contouring", adjacent colors should be similar to reduce an artifact called "contouring",
or visible edges between regions of adjacent values. One easy way to assure or visible edges between regions of adjacent values. One easy way to assure
this is to make each pattern a superset of the previous. Here are two good this is to make each pattern a superset of the previous. Here are two good
sets of patterns for a 3-by-3 matrix: sets of patterns for a 3-by-3 matrix:
--- --- --- -X- -XX -XX -XX -XX XXX XXX --- --- --- -X- -XX -XX -XX -XX XXX XXX
--- -X- -XX -XX -XX -XX XXX XXX XXX XXX --- -X- -XX -XX -XX -XX XXX XXX XXX XXX
@ -165,102 +165,102 @@ or
--- --- --- --X --X X-X X-X X-X XXX XXX --- --- --- --X --X X-X X-X X-X XXX XXX
--- --- -X- -X- -X- -X- XX- XX- XX- XXX --- --- -X- -X- -X- -X- XX- XX- XX- XXX
The first set of patterns above are "clustered" in that as new dots are added The first set of patterns above are "clustered" in that as new dots are added
to each pattern, they are added next to dots already there. The second set to each pattern, they are added next to dots already there. The second set
is "dispersed" as the dots are spread out more. This distinction is more is "dispersed" as the dots are spread out more. This distinction is more
important on larger patterns. Dispersed-dot patterns produce less grainy important on larger patterns. Dispersed-dot patterns produce less grainy
images, but require that the output device render each dot distinctly. When images, but require that the output device render each dot distinctly. When
this is not the case, as with a printing press which smears the dots a this is not the case, as with a printing press which smears the dots a
little, clustered patterns are better. little, clustered patterns are better.
For each pixel in the image we now print the pattern which is closest to its For each pixel in the image we now print the pattern which is closest to its
value. This will triple the size of the image in each direction, so this value. This will triple the size of the image in each direction, so this
method can only be used where the display spatial resolution is much greater method can only be used where the display spatial resolution is much greater
than that of the image. than that of the image.
We can exploit the fact that most images have large areas of similar value to We can exploit the fact that most images have large areas of similar value to
reduce our need for extra spatial resolution. Instead of plotting a whole reduce our need for extra spatial resolution. Instead of plotting a whole
pattern for each pixel, map each pixel in the image to a dot in the pattern pattern for each pixel, map each pixel in the image to a dot in the pattern
an only plot the corresponding dot for each pixel. an only plot the corresponding dot for each pixel.
The simplest way to do this is to map the X and Y coordinates of each pixel The simplest way to do this is to map the X and Y coordinates of each pixel
into the dot (X mod 3, Y mod 3) in the pattern. Large areas of constant into the dot (X mod 3, Y mod 3) in the pattern. Large areas of constant
value will come out as repetitions of the pattern as before. value will come out as repetitions of the pattern as before.
To extend this method to color images, we must use patterns of colored dots To extend this method to color images, we must use patterns of colored dots
to represent shades not directly printable by the hardware. For example, if to represent shades not directly printable by the hardware. For example, if
your hardware is capable of printing only red, green, blue, and black (the your hardware is capable of printing only red, green, blue, and black (the
minimal case for color dithering), other colors can be represented with minimal case for color dithering), other colors can be represented with
patterns of these four: patterns of these four:
Yellow = R G Cyan = G B Magenta = R B Gray = R G Yellow = R G Cyan = G B Magenta = R B Gray = R G
G R B G B R B K G R B G B R B K
(B here represents blue, K is black). There are a total of 31 such distinct (B here represents blue, K is black). There are a total of 31 such distinct
patterns which can be used; I will leave their enumeration "as an exercise patterns which can be used; I will leave their enumeration "as an exercise
for the reader" (don't you hate books that do that?). for the reader" (don't you hate books that do that?).
======================================================================== ========================================================================
Ordered dither Ordered dither
Because each of the patterns above is a superset of the previous, we can Because each of the patterns above is a superset of the previous, we can
express the patterns in compact form as the order of dots added: express the patterns in compact form as the order of dots added:
8 3 4 and 1 7 4 8 3 4 and 1 7 4
6 1 2 5 8 3 6 1 2 5 8 3
7 5 9 6 2 9 7 5 9 6 2 9
Then we can simply use the value in the array as a threshhold. If the value Then we can simply use the value in the array as a threshhold. If the value
of the pixel (scaled into the 0-9 range) is less than the number in the of the pixel (scaled into the 0-9 range) is less than the number in the
corresponding cell of the matrix, plot that pixel black, otherwise, plot it corresponding cell of the matrix, plot that pixel black, otherwise, plot it
white. This process is called ordered dither. As before, clustered patterns white. This process is called ordered dither. As before, clustered patterns
should be used for devices which blur dots. In fact, the clustered pattern should be used for devices which blur dots. In fact, the clustered pattern
ordered dither is the process used by most newspapers, and the term ordered dither is the process used by most newspapers, and the term
halftoning refers to this method if not otherwise qualified. halftoning refers to this method if not otherwise qualified.
Bayer [2] has shown that for matrices of orders which are powers of two there Bayer [2] has shown that for matrices of orders which are powers of two there
is an optimal pattern of dispersed dots which results in the pattern noise is an optimal pattern of dispersed dots which results in the pattern noise
being as high-frequency as possible. The pattern for a 2x2 and 4x4 matrices being as high-frequency as possible. The pattern for a 2x2 and 4x4 matrices
are as follows: are as follows:
1 3 1 9 3 11 These patterns (and their rotations 1 3 1 9 3 11 These patterns (and their rotations
4 2 13 5 15 7 and reflections) are optimal for a 4 2 13 5 15 7 and reflections) are optimal for a
4 12 2 10 dispersed-pattern ordered dither. 4 12 2 10 dispersed-pattern ordered dither.
16 8 14 6 16 8 14 6
Ulichney [3] shows a recursive technique can be used to generate the larger Ulichney [3] shows a recursive technique can be used to generate the larger
patterns. To fully reproduce our 256-level image, we would need to use the patterns. To fully reproduce our 256-level image, we would need to use the
8x8 pattern. 8x8 pattern.
Bayer's method is in very common use and is easily identified by the cross- Bayer's method is in very common use and is easily identified by the cross-
hatch pattern artifacts it produces in the resulting display. This hatch pattern artifacts it produces in the resulting display. This
artifacting is the major drawback of the technique wich is otherwise very artifacting is the major drawback of the technique wich is otherwise very
fast and powerful. Ordered dithering also performs very badly on images fast and powerful. Ordered dithering also performs very badly on images
which have already been dithered to some extent. As stated earlier, which have already been dithered to some extent. As stated earlier,
dithering should be the last stage in producing a physical display from a dithering should be the last stage in producing a physical display from a
digitally stored image. The dithered image should never be stored itself. digitally stored image. The dithered image should never be stored itself.
======================================================================== ========================================================================
Error dispersion Error dispersion
This technique generates the best results of any method here, and is This technique generates the best results of any method here, and is
naturally the slowest. In fact, there are many variants of this technique as naturally the slowest. In fact, there are many variants of this technique as
well, and the better they get, the slower they are. well, and the better they get, the slower they are.
Error dispersion is very simple to describe: for each point in the image, Error dispersion is very simple to describe: for each point in the image,
first find the closest color available. Calculate the difference between the first find the closest color available. Calculate the difference between the
value in the image and the color you have. Now divide up these error values value in the image and the color you have. Now divide up these error values
and distribute them over the neighboring pixels which you have not visited and distribute them over the neighboring pixels which you have not visited
yet. When you get to these later pixels, just add the errors distributed yet. When you get to these later pixels, just add the errors distributed
from the earlier ones, clip the values to the allowed range if needed, then from the earlier ones, clip the values to the allowed range if needed, then
continue as above. continue as above.
If you are dithering a grayscale image for output to a black-and-white If you are dithering a grayscale image for output to a black-and-white
device, the "find closest color" is just a simle threshholding operation. In device, the "find closest color" is just a simle threshholding operation. In
color, it involves matching the input color to the closest available hardware color, it involves matching the input color to the closest available hardware
color, which can be difficult depending on the hardware palette. color, which can be difficult depending on the hardware palette.
There are many ways to distribute the errors and many ways to scan the There are many ways to distribute the errors and many ways to scan the
image, but I will deal here with only a few. The two basic ways to scan the image, but I will deal here with only a few. The two basic ways to scan the
@ -269,22 +269,22 @@ alternating left-to-right then right-to-left raster. The latter method
generally produces fewer artifacts and can be used with all the error generally produces fewer artifacts and can be used with all the error
diffusion patterns discussed below. diffusion patterns discussed below.
The different ways of dividing up the error can be expressed as patterns The different ways of dividing up the error can be expressed as patterns
(called filters, for reasons too boring to go into here). (called filters, for reasons too boring to go into here).
X 7 This is the Floyd and Steinberg [4] X 7 This is the Floyd and Steinberg [4]
3 5 1 error diffusion filter. 3 5 1 error diffusion filter.
In this filter, the X represents the pixel you are currently scanning, and In this filter, the X represents the pixel you are currently scanning, and
the numbers (called weights, for equally boring reasons) represent the the numbers (called weights, for equally boring reasons) represent the
proportion of the error distributed to the pixel in that position. Here, the proportion of the error distributed to the pixel in that position. Here, the
pixel immediately to the right gets 7/16 of the error (the divisor is 16 pixel immediately to the right gets 7/16 of the error (the divisor is 16
because the weights add to 16), the pixel directly below gets 5/16 of the because the weights add to 16), the pixel directly below gets 5/16 of the
error, and the diagonally adjacent pixels get 3/16 and 1/16. When scanning a error, and the diagonally adjacent pixels get 3/16 and 1/16. When scanning a
line right-to-left, this pattern is reversed. This pattern was chosen line right-to-left, this pattern is reversed. This pattern was chosen
carefully so that it would produce a checkerboard pattern in areas with carefully so that it would produce a checkerboard pattern in areas with
intensity of 1/2 (or 128 in our image). It is also fairly easy to calculate intensity of 1/2 (or 128 in our image). It is also fairly easy to calculate
when the division by 16 is replaced by shifts. when the division by 16 is replaced by shifts.
Another filter in common use, but not recommended: Another filter in common use, but not recommended:
@ -306,24 +306,24 @@ the bottom row removed. The main improvement is that the divisor is now 32,
which makes calculating the errors faster, and the removal of one row which makes calculating the errors faster, and the removal of one row
reduces the memory requirements of the method. reduces the memory requirements of the method.
This is also fairly easy to calculate and produces better results than Floyd This is also fairly easy to calculate and produces better results than Floyd
and Steinberg. Jarvis, Judice, and Ninke [6] use the following: and Steinberg. Jarvis, Judice, and Ninke [6] use the following:
X 7 5 The Jarvis, et al. pattern. X 7 5 The Jarvis, et al. pattern.
3 5 7 5 3 3 5 7 5 3
1 3 5 3 1 1 3 5 3 1
The divisor here is 48, which is a little more expensive to calculate, and The divisor here is 48, which is a little more expensive to calculate, and
the errors are distributed over three lines, requiring extra memory and time the errors are distributed over three lines, requiring extra memory and time
for processing. Probably the best filter is from Stucki [7]: for processing. Probably the best filter is from Stucki [7]:
X 8 4 The Stucki pattern. X 8 4 The Stucki pattern.
2 4 8 4 2 2 4 8 4 2
1 2 4 2 1 1 2 4 2 1
This one takes a division by 42 for each pixel and is therefore slow if math This one takes a division by 42 for each pixel and is therefore slow if math
is done inside the loop. After the initial 8/42 is calculated, some time can is done inside the loop. After the initial 8/42 is calculated, some time can
be saved by producing the remaining fractions by shifts. be saved by producing the remaining fractions by shifts.
The speed advantages of the simpler filters can be eliminated somewhat by The speed advantages of the simpler filters can be eliminated somewhat by
performing the divisions beforehand and using lookup tables instead of per- performing the divisions beforehand and using lookup tables instead of per-
@ -334,52 +334,52 @@ It is critical with all of these algorithms that when error values are added
to neighboring pixels, the values must be truncated to fit within the limits to neighboring pixels, the values must be truncated to fit within the limits
of hardware, otherwise and area of very intense color may cause streaks into of hardware, otherwise and area of very intense color may cause streaks into
an adjacent area of less intense color. This truncation adds noise to the an adjacent area of less intense color. This truncation adds noise to the
image anagous to clipping in an audio amplifier, but it is not nearly so image anagous to clipping in an audio amplifier, but it is not nearly so
offensive as the streaking. It is mainly for this reason that the larger offensive as the streaking. It is mainly for this reason that the larger
filters work better--they split the errors up more finely and produce less of filters work better--they split the errors up more finely and produce less of
this clipping noise. this clipping noise.
With all of these filters, it is also important to ensure that the errors With all of these filters, it is also important to ensure that the errors
you distribute properly add to the original error value. This is easiest to you distribute properly add to the original error value. This is easiest to
accomplish by subtracting each fraction from the whole error as it is accomplish by subtracting each fraction from the whole error as it is
calculated, and using the final remainder as the last fraction. calculated, and using the final remainder as the last fraction.
Some of these methods (particularly the simpler ones) can be greatly improved Some of these methods (particularly the simpler ones) can be greatly improved
by skewing the weights with a little randomness [3]. by skewing the weights with a little randomness [3].
Calculating the "nearest available color" is trivial with a monochrome image; Calculating the "nearest available color" is trivial with a monochrome image;
with color images it requires more work. A table of RGB values of all with color images it requires more work. A table of RGB values of all
available colors must be scanned sequentially for each input pixel to find available colors must be scanned sequentially for each input pixel to find
the closest. The "distance" formula most often used is a simple pythagorean the closest. The "distance" formula most often used is a simple pythagorean
"least squares". The difference for each color is squared, and the three "least squares". The difference for each color is squared, and the three
squares added to produce the distance value. This value is equivalent to the squares added to produce the distance value. This value is equivalent to the
square of the distance between the points in RGB-space. It is not necessary square of the distance between the points in RGB-space. It is not necessary
to compute the square root of this value because we are not interested in the to compute the square root of this value because we are not interested in the
actual distance, only in which is smallest. The square root function is a actual distance, only in which is smallest. The square root function is a
monotonic increasing function and does not affect the order of its operands. monotonic increasing function and does not affect the order of its operands.
If the total number of colors with which you are dealing is small, this part If the total number of colors with which you are dealing is small, this part
of the algorithm can be replaced by a lookup table as well. of the algorithm can be replaced by a lookup table as well.
When your hardware allows you to select the available colors, very good When your hardware allows you to select the available colors, very good
results can be achieved by selecting colors from the image itself. You must results can be achieved by selecting colors from the image itself. You must
reserve at least 8 colors for the primaries, secondaries, black, and white reserve at least 8 colors for the primaries, secondaries, black, and white
for best results. If you do not know the colors in your image ahead of time, for best results. If you do not know the colors in your image ahead of time,
or if you are going to use the same map to dither several different images, or if you are going to use the same map to dither several different images,
you will have to fill your color map with a good range of colors. This can you will have to fill your color map with a good range of colors. This can
be done either by assigning a certain number of bits to each primary and be done either by assigning a certain number of bits to each primary and
computing all combinations, or by a smoother distribution as suggested by computing all combinations, or by a smoother distribution as suggested by
Heckbert [8]. Heckbert [8].
======================================================================== ========================================================================
Sample code Sample code
Despite my best efforts in expository writing, nothing explains an algorithm Despite my best efforts in expository writing, nothing explains an algorithm
better than real code. With that in mind, presented here below is an better than real code. With that in mind, presented here below is an
algorithm (in somewhat incomplete, very inefficient pseudo-C) which algorithm (in somewhat incomplete, very inefficient pseudo-C) which
implements error diffusion dithering with the Floyd and Steinberg filter. It implements error diffusion dithering with the Floyd and Steinberg filter. It
is not efficiently coded, but its purpose is to show the method, which I is not efficiently coded, but its purpose is to show the method, which I
believe it does. believe it does.
/* Floyd/Steinberg error diffusion dithering algorithm in color. The array /* Floyd/Steinberg error diffusion dithering algorithm in color. The array
** line[][] contains the RGB values for the current line being processed; ** line[][] contains the RGB values for the current line being processed;
@ -466,34 +466,34 @@ dither()
Bibliography Bibliography
[1] Foley, J. D. and Andries Van Dam (1982) [1] Foley, J. D. and Andries Van Dam (1982)
Fundamentals of Interactive Computer Graphics. Reading, MA: Addisson Fundamentals of Interactive Computer Graphics. Reading, MA: Addisson
Wesley. Wesley.
This is a standard reference for many graphic techniques which has not This is a standard reference for many graphic techniques which has not
declined with age. Highly recommended. declined with age. Highly recommended.
[2] Bayer, B. E. (1973) [2] Bayer, B. E. (1973)
"An Optimum Method for Two-Level Rendition of Continuous Tone Pictures," "An Optimum Method for Two-Level Rendition of Continuous Tone Pictures,"
IEEE International Conference on Communications, Conference Records, pp. IEEE International Conference on Communications, Conference Records, pp.
26-11 to 26-15. 26-11 to 26-15.
A short article proving the optimality of Bayer's pattern in the A short article proving the optimality of Bayer's pattern in the
dispersed-dot ordered dither. dispersed-dot ordered dither.
[3] Ulichney, R. (1987) [3] Ulichney, R. (1987)
Digital Halftoning. Cambridge, MA: The MIT Press. Digital Halftoning. Cambridge, MA: The MIT Press.
This is the best book I know of for describing the various black and This is the best book I know of for describing the various black and
white dithering methods. It has clear explanations (a little higher math white dithering methods. It has clear explanations (a little higher math
may come in handy) and wonderful illustrations. It does not contain any may come in handy) and wonderful illustrations. It does not contain any
code, but don't let that keep you from getting this book. Computer code, but don't let that keep you from getting this book. Computer
Literacy carries it but is often sold out. Literacy carries it but is often sold out.
[4] Floyd, R.W. and L. Steinberg (1975) [4] Floyd, R.W. and L. Steinberg (1975)
"An Adaptive Algorithm for Spatial Gray Scale." SID International "An Adaptive Algorithm for Spatial Gray Scale." SID International
Symposium Digest of Technical Papers, vol 1975m, pp. 36-37. Symposium Digest of Technical Papers, vol 1975m, pp. 36-37.
Short article in which Floyd and Steinberg introduce their filter. Short article in which Floyd and Steinberg introduce their filter.
[5] Daniel Burkes is unpublished, but can be reached at this address: [5] Daniel Burkes is unpublished, but can be reached at this address:
@ -502,21 +502,21 @@ Bibliography
2351 College Station Road Suite 563 2351 College Station Road Suite 563
Athens, GA 30305 Athens, GA 30305
or via CompuServe's Graphics Support Forum, ID # 72077,356. or via CompuServe's Graphics Support Forum, ID # 72077,356.
[6] Jarvis, J. F., C. N. Judice, and W. H. Ninke (1976) [6] Jarvis, J. F., C. N. Judice, and W. H. Ninke (1976)
"A Survey of Techniques for the Display of Continuous Tone Pictures on "A Survey of Techniques for the Display of Continuous Tone Pictures on
Bi-Level Displays." Computer Graphics and Image Processing, vol. 5, pp. Bi-Level Displays." Computer Graphics and Image Processing, vol. 5, pp.
13-40. 13-40.
[7] Stucki, P. (1981) [7] Stucki, P. (1981)
"MECCA - a multiple-error correcting computation algorithm for bilevel "MECCA - a multiple-error correcting computation algorithm for bilevel
image hardcopy reproduction." Research Report RZ1060, IBM Research image hardcopy reproduction." Research Report RZ1060, IBM Research
Laboratory, Zurich, Switzerland. Laboratory, Zurich, Switzerland.
[8] Heckbert, Paul (9182) [8] Heckbert, Paul (9182)
"Color Image Quantization for Frame Buffer Display." Computer Graphics "Color Image Quantization for Frame Buffer Display." Computer Graphics
(SIGGRAPH 82), vol. 16, pp. 297-307. (SIGGRAPH 82), vol. 16, pp. 297-307.
======================================================================== ========================================================================
@ -536,12 +536,12 @@ Mathematical Elements for Computer Graphics. New York: McGraw-Hill.
======================================================================== ========================================================================
About CompuServe Graphics Support Forum: About CompuServe Graphics Support Forum:
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