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@ -1,28 +1,28 @@ |
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DITHER.TXT |
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What follows is everything you ever wanted to know (for the time being) about |
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dithering. I'm sure it will be out of date as soon as it is released, but it |
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|
does serve to collect data from a wide variety of sources into a single |
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|
document, and should save you considerable searching time. |
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Numbers in brackets (like this [0]) are references. A list of these works |
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|
appears at the end of this document. |
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|
Because this document describes ideas and algorithms which are constantly |
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changing, I expect that it may have many editions, additions, and corrections |
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|
before it gets to you. I will list my name below as original author, but I |
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|
do not wish to deter others from adding their own thoughts and discoveries. |
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|
This is not copyrighted in any way, and was created solely for the purpose of |
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organizing my own knowledge on the subject, and sharing this with others. |
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|
Please distribute it to anyonw who might be interested. |
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If you add anything to this document, please feel free to include your name |
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|
below as a contributor or as a reference. I would particularly like to see |
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|
additions to the "Other books of interest" section. Please keep the text in |
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|
this simple format: no margins, no pagination, no lines longer that 79 |
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|
characters, and no non-ASCII or non-printing characters other than a CR/LF |
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|
pair at the end of each line. It is intended that this be read on as many |
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|
|
different machines as possible. |
|
|
|
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 |
|
|
|
does serve to collect data from a wide variety of sources into a single |
|
|
|
document, and should save you considerable searching time. |
|
|
|
|
|
|
|
Numbers in brackets (like this [0]) are references. A list of these works |
|
|
|
appears at the end of this document. |
|
|
|
|
|
|
|
Because this document describes ideas and algorithms which are constantly |
|
|
|
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 |
|
|
|
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 |
|
|
|
organizing my own knowledge on the subject, and sharing this with others. |
|
|
|
Please distribute it to anyonw who might be interested. |
|
|
|
|
|
|
|
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 |
|
|
|
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 |
|
|
|
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 |
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|
|
different machines as possible. |
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Original Author: |
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@ -35,37 +35,37 @@ Contributors: |
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======================================================================== |
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What is Dithering? |
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Dithering, also called Halftoning or Color Reduction, is the process of |
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rendering an image on a display device with fewer colors than are in the |
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image. The number of different colors in an image or on a device I will call |
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its Color Resolution. The term "resolution" means "fineness" and is used to |
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describe the level of detail in a digitally sampled signal. It is used most |
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often in referring to the Spatial Resolution, which is the basic sampling |
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rate for a digitized image. |
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Spatial resolution describes the fineness of the "dots" used in an image. |
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Color resolution describes the fineness of detail available at each dot. The |
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higher the resolution of a digital sample, the better it can reproduce high |
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frequency detail. A compact disc, for example, has a temporal (time) |
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resolution of 44,000 samples per second, and a dynamic (volume) resolution of |
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16 bits (0..65535). It can therefore reproduce sounds with a vast dynamic |
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range (from barely audible to ear-splitting) with great detail, but it has |
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problems with very high-frequency sounds, like violins and piccolos. |
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|
It is often possible to "trade" one kind of resolution for another. If your |
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display device has a higher spatial resolution than the image you are trying |
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|
to reproduce, it can show a very good image even if its color resolution is |
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|
less. This is what we will call "dithering" and is the subject of this |
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|
paper. The other tradeoff, i.e., trading color resolution for spatial |
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resolution, is called "anti-aliasing" and is not discussed here. |
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|
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It is important to emphasize here that dithering is a one-way operation. |
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Once an image has been dithered, although it may look like a good |
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reproduction of the original, information is permanently lost. Many image |
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|
processing functions fail on dithered images. For these reasons, dithering |
|
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|
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 |
|
|
|
should always be kept in full detail. |
|
|
|
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 |
|
|
|
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 |
|
|
|
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 |
|
|
|
rate for a digitized 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 |
|
|
|
higher the resolution of a digital sample, the better it can reproduce high |
|
|
|
frequency detail. A compact disc, for example, has a temporal (time) |
|
|
|
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 |
|
|
|
range (from barely audible to ear-splitting) with great detail, but it has |
|
|
|
problems with very high-frequency sounds, like violins and piccolos. |
|
|
|
|
|
|
|
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 |
|
|
|
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 |
|
|
|
paper. The other tradeoff, i.e., trading color resolution for spatial |
|
|
|
resolution, is called "anti-aliasing" and is not discussed here. |
|
|
|
|
|
|
|
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 |
|
|
|
reproduction of the original, information is permanently lost. Many image |
|
|
|
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 |
|
|
|
otherwise be incapable of displaying it. The data representing an image |
|
|
|
should always be kept in full detail. |
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======================================================================== |
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@ -78,84 +78,84 @@ The classes of dithering algorithms we will discuss here are these: |
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3. Ordered |
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4. Error dispersion |
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Each of these methods is generally better than those listed before it, but |
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other considerations such as processing time, memory constraints, etc. may |
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weigh in favor of one of the simpler methods. |
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Each of these methods is generally better than those listed before it, but |
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other considerations such as processing time, memory constraints, etc. may |
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weigh in favor of one of the simpler methods. |
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For the following discussions I will assume that we are given an image with |
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256 shades of gray (0=black..255=white) that we are trying to reproduce on a |
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black and white ouput device. Most of these methods can be extended in |
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obvious ways to deal with displays that have more than two levels but fewer |
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than the image, or to color images. Where such extension is not obvious, or |
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where better results can be obtained, I will go into more detail. |
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|
For the following discussions I will assume that we are given an image with |
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|
256 shades of gray (0=black..255=white) that we are trying to reproduce on a |
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|
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 |
|
|
|
than the image, or to color images. Where such extension is not obvious, or |
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|
where better results can be obtained, I will go into more detail. |
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To convert any of the first three methods into color, simply apply the |
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algorithm separately for each primary and mix the resulting values. This |
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assumes that you have at least eight output colors: black, red, green, blue, |
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cyan, magenta, yellow, and white. Though this will work for error dispersion |
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as well, there are better methods in this case. |
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To convert any of the first three methods into color, simply apply the |
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algorithm separately for each primary and mix the resulting values. This |
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assumes that you have at least eight output colors: black, red, green, blue, |
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cyan, magenta, yellow, and white. Though this will work for error dispersion |
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as well, there are better methods in this case. |
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======================================================================== |
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Random dither |
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This is the bubblesort of dithering algorithms. It is not really acceptable |
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as a production method, but it is very simple to describe and implement. For |
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each value in the image, simply generate a random number 1..256; if it is |
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geater than the image value at that point, plot the point white, otherwise |
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plot it black. That's it. This generates a picture with a lot of "white |
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noise", which looks like TV picture "snow". Though the image produced is |
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very inaccurate and noisy, it is free from "artifacts" which are phenomena |
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produced by digital signal processing. |
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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 |
|
|
|
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 |
|
|
|
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 |
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|
|
produced by digital signal processing. |
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The most common type of artifact is the Moire pattern (Contributors: please |
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resist the urge to put an accent on the "e", as no portable character set |
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exists for this). If you draw several lines close together radiating from a |
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exists for this). If you draw several lines close together radiating from a |
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single point on a computer display, you will see what appear to be flower- |
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like patterns. These patterns are not part of the original idea of lines, |
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but are an illusion produced by the jaggedness of the display. |
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like patterns. These patterns are not part of the original idea of lines, |
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but are an illusion produced by the jaggedness of the display. |
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Many techniques exist for the reduction of digital artifacts like these, most |
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of which involve using a little randomness to "perturb" a regular algorithm a |
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little. Random dither obviously takes this to extreme. |
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Many techniques exist for the reduction of digital artifacts like these, most |
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of which involve using a little randomness to "perturb" a regular algorithm a |
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little. Random dither obviously takes this to extreme. |
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I should mention, of course, that unless your computer has a hardware-based |
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random number generator (and most don't) there may be some artifacts from the |
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random number generation algorithm itself. |
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I should mention, of course, that unless your computer has a hardware-based |
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random number generator (and most don't) there may be some artifacts from the |
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random number generation algorithm itself. |
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While random dither adds a lot of high-frequency noise to a picture, it is |
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useful in reproducing very low-frequency images where the absence of |
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artifacts is more important than noise. For example, a whole screen |
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containing a gradient of all levels from black to white would actually look |
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best with a random dither. In this case, ordered dithering would produce |
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diagonal patterns, and error dispersion would produce clustering. |
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|
While random dither adds a lot of high-frequency noise to a picture, it is |
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|
useful in reproducing very low-frequency images where the absence of |
|
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|
artifacts is more important than noise. For example, a whole screen |
|
|
|
containing a gradient of all levels from black to white would actually look |
|
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|
best with a random dither. In this case, ordered dithering would produce |
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diagonal patterns, and error dispersion would produce clustering. |
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For efficiency, you can take the random number generator "out of the loop" by |
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generating a list of random numbers beforehand for use in the dither. Make |
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sure that the list is larger than the number of pixels in the image or you |
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may get artifacts from the reuse of numbers. The worst case would be if the |
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size of your list of random numbers is a multiple or near-multiple of the |
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horizontal size of the image, in which case unwanted vertical or diagonal |
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lines will appear. |
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For efficiency, you can take the random number generator "out of the loop" by |
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|
generating a list of random numbers beforehand for use in the dither. Make |
|
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|
sure that the list is larger than the number of pixels in the image or you |
|
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|
may get artifacts from the reuse of numbers. The worst case would be if the |
|
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|
size of your list of random numbers is a multiple or near-multiple of the |
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horizontal size of the image, in which case unwanted vertical or diagonal |
|
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|
lines will appear. |
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======================================================================== |
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Pattern dither |
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This is also a simple concept, but much more effective than random dither. |
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For each possible value in the image, create a pattern of dots that |
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approximates that value. For instance, a 3-by-3 block of dots can have one |
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of 512 patterns, but for our purposes, there are only 10; the number of black |
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dots in the pattern determines the darkness of the pattern. |
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This is also a simple concept, but much more effective than random dither. |
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|
For each possible value in the image, create a pattern of dots that |
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|
approximates that value. For instance, a 3-by-3 block of dots can have one |
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of 512 patterns, but for our purposes, there are only 10; the number of black |
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dots in the pattern determines the darkness of the pattern. |
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Which 10 patterns do we choose? Obviously, we need the all-white and all- |
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black patterns. We can eliminate those patterns which would create vertical |
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or horizontal lines if repeated over a large area because many images have |
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such regions of similar value [1]. It has been shown [1] that patterns for |
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adjacent colors should be similar to reduce an artifact called "contouring", |
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|
or visible edges between regions of adjacent values. One easy way to assure |
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|
this is to make each pattern a superset of the previous. Here are two good |
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sets of patterns for a 3-by-3 matrix: |
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|
black patterns. We can eliminate those patterns which would create vertical |
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|
or horizontal lines if repeated over a large area because many images have |
|
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|
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", |
|
|
|
or visible edges between regions of adjacent values. One easy way to assure |
|
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|
this is to make each pattern a superset of the previous. Here are two good |
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sets of patterns for a 3-by-3 matrix: |
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--- --- --- -X- -XX -XX -XX -XX XXX XXX |
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--- -X- -XX -XX -XX -XX XXX XXX XXX XXX |
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@ -165,102 +165,102 @@ or |
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--- --- --- --X --X X-X X-X X-X XXX XXX |
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--- --- -X- -X- -X- -X- XX- XX- XX- XXX |
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The first set of patterns above are "clustered" in that as new dots are added |
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to each pattern, they are added next to dots already there. The second set |
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is "dispersed" as the dots are spread out more. This distinction is more |
|
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|
important on larger patterns. Dispersed-dot patterns produce less grainy |
|
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|
images, but require that the output device render each dot distinctly. When |
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|
this is not the case, as with a printing press which smears the dots a |
|
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|
little, clustered patterns are better. |
|
|
|
|
|
|
|
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 |
|
|
|
method can only be used where the display spatial resolution is much greater |
|
|
|
than that of the image. |
|
|
|
|
|
|
|
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 |
|
|
|
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. |
|
|
|
|
|
|
|
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 |
|
|
|
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 represent shades not directly printable by the hardware. For example, if |
|
|
|
your hardware is capable of printing only red, green, blue, and black (the |
|
|
|
minimal case for color dithering), other colors can be represented with |
|
|
|
patterns of these four: |
|
|
|
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 |
|
|
|
is "dispersed" as the dots are spread out more. This distinction is more |
|
|
|
important on larger patterns. Dispersed-dot patterns produce less grainy |
|
|
|
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 |
|
|
|
little, clustered patterns are better. |
|
|
|
|
|
|
|
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 |
|
|
|
method can only be used where the display spatial resolution is much greater |
|
|
|
than that of the image. |
|
|
|
|
|
|
|
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 |
|
|
|
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. |
|
|
|
|
|
|
|
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 |
|
|
|
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 represent shades not directly printable by the hardware. For example, if |
|
|
|
your hardware is capable of printing only red, green, blue, and black (the |
|
|
|
minimal case for color dithering), other colors can be represented with |
|
|
|
patterns of these four: |
|
|
|
|
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|
|
Yellow = R G Cyan = G B Magenta = R B Gray = R G |
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|
G R B G B R B K |
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|
|
|
|
(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 |
|
|
|
for the reader" (don't you hate books that do that?). |
|
|
|
(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 |
|
|
|
for the reader" (don't you hate books that do that?). |
|
|
|
|
|
|
|
|
|
|
|
======================================================================== |
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|
|
Ordered dither |
|
|
|
|
|
|
|
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: |
|
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|
Because each of the patterns above is a superset of the previous, we can |
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|
express the patterns in compact form as the order of dots added: |
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8 3 4 and 1 7 4 |
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6 1 2 5 8 3 |
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7 5 9 6 2 9 |
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Then we can simply use the value in the array as a threshhold. If the value |
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of the pixel (scaled into the 0-9 range) is less than the number in the |
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|
corresponding cell of the matrix, plot that pixel black, otherwise, plot it |
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|
white. This process is called ordered dither. As before, clustered patterns |
|
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|
should be used for devices which blur dots. In fact, the clustered pattern |
|
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|
ordered dither is the process used by most newspapers, and the term |
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|
halftoning refers to this method if not otherwise qualified. |
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|
Then we can simply use the value in the array as a threshhold. If the value |
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|
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 |
|
|
|
white. This process is called ordered dither. As before, clustered patterns |
|
|
|
should be used for devices which blur dots. In fact, the clustered pattern |
|
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|
ordered dither is the process used by most newspapers, and the term |
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|
halftoning refers to this method if not otherwise qualified. |
|
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|
Bayer [2] has shown that for matrices of orders which are powers of two there |
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|
is an optimal pattern of dispersed dots which results in the pattern noise |
|
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|
being as high-frequency as possible. The pattern for a 2x2 and 4x4 matrices |
|
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|
are as follows: |
|
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|
Bayer [2] has shown that for matrices of orders which are powers of two there |
|
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|
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 |
|
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|
are as follows: |
|
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1 3 1 9 3 11 These patterns (and their rotations |
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4 2 13 5 15 7 and reflections) are optimal for a |
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4 12 2 10 dispersed-pattern ordered dither. |
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16 8 14 6 |
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Ulichney [3] shows a recursive technique can be used to generate the larger |
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patterns. To fully reproduce our 256-level image, we would need to use the |
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|
8x8 pattern. |
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|
Ulichney [3] shows a recursive technique can be used to generate the larger |
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patterns. To fully reproduce our 256-level image, we would need to use the |
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8x8 pattern. |
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Bayer's method is in very common use and is easily identified by the cross- |
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|
hatch pattern artifacts it produces in the resulting display. This |
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|
artifacting is the major drawback of the technique wich is otherwise very |
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|
fast and powerful. Ordered dithering also performs very badly on images |
|
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|
which have already been dithered to some extent. As stated earlier, |
|
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|
dithering should be the last stage in producing a physical display from a |
|
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|
digitally stored image. The dithered image should never be stored itself. |
|
|
|
hatch pattern artifacts it produces in the resulting display. This |
|
|
|
artifacting is the major drawback of the technique wich is otherwise very |
|
|
|
fast and powerful. Ordered dithering also performs very badly on images |
|
|
|
which have already been dithered to some extent. As stated earlier, |
|
|
|
dithering should be the last stage in producing a physical display from a |
|
|
|
digitally stored image. The dithered image should never be stored itself. |
|
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|
|
|
|
|
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|
======================================================================== |
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|
Error dispersion |
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This technique generates the best results of any method here, and is |
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naturally the slowest. In fact, there are many variants of this technique as |
|
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|
well, and the better they get, the slower they are. |
|
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|
This technique generates the best results of any method here, and is |
|
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|
naturally the slowest. In fact, there are many variants of this technique as |
|
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|
well, and the better they get, the slower they are. |
|
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|
Error dispersion is very simple to describe: for each point in the image, |
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|
first find the closest color available. Calculate the difference between the |
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|
value in the image and the color you have. Now divide up these error values |
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|
and distribute them over the neighboring pixels which you have not visited |
|
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|
yet. When you get to these later pixels, just add the errors distributed |
|
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|
from the earlier ones, clip the values to the allowed range if needed, then |
|
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|
continue as above. |
|
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|
Error dispersion is very simple to describe: for each point in the image, |
|
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|
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 |
|
|
|
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 |
|
|
|
from the earlier ones, clip the values to the allowed range if needed, then |
|
|
|
continue as above. |
|
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|
|
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|
If you are dithering a grayscale image for output to a black-and-white |
|
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|
device, the "find closest color" is just a simle threshholding operation. In |
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|
color, it involves matching the input color to the closest available hardware |
|
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|
color, which can be difficult depending on the hardware palette. |
|
|
|
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 |
|
|
|
color, it involves matching the input color to the closest available hardware |
|
|
|
color, which can be difficult depending on the hardware palette. |
|
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|
There are many ways to distribute the errors and many ways to scan the |
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|
image, but I will deal here with only a few. The two basic ways to scan the |
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|
@ -269,22 +269,22 @@ alternating left-to-right then right-to-left raster. The latter method |
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|
generally produces fewer artifacts and can be used with all the error |
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|
diffusion patterns discussed below. |
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|
The different ways of dividing up the error can be expressed as patterns |
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|
The different ways of dividing up the error can be expressed as patterns |
|
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|
(called filters, for reasons too boring to go into here). |
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|
X 7 This is the Floyd and Steinberg [4] |
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|
3 5 1 error diffusion filter. |
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|
In this filter, the X represents the pixel you are currently scanning, and |
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|
the numbers (called weights, for equally boring reasons) represent the |
|
|
|
proportion of the error distributed to the pixel in that position. Here, the |
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|
|
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 |
|
|
|
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 |
|
|
|
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 |
|
|
|
when the division by 16 is replaced by shifts. |
|
|
|
the numbers (called weights, for equally boring reasons) represent 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 |
|
|
|
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 |
|
|
|
line right-to-left, this pattern is reversed. This pattern was chosen |
|
|
|
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 |
|
|
|
when the division by 16 is replaced by shifts. |
|
|
|
|
|
|
|
Another filter in common use, but not recommended: |
|
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|
|
|
|
@ -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 |
|
|
|
reduces the memory requirements of the method. |
|
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|
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|
|
This is also fairly easy to calculate and produces better results than Floyd |
|
|
|
and Steinberg. Jarvis, Judice, and Ninke [6] use the following: |
|
|
|
This is also fairly easy to calculate and produces better results than Floyd |
|
|
|
and Steinberg. Jarvis, Judice, and Ninke [6] use the following: |
|
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|
|
|
|
X 7 5 The Jarvis, et al. pattern. |
|
|
|
3 5 7 5 3 |
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|
|
1 3 5 3 1 |
|
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|
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 |
|
|
|
for processing. Probably the best filter is from Stucki [7]: |
|
|
|
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 |
|
|
|
for processing. Probably the best filter is from Stucki [7]: |
|
|
|
|
|
|
|
X 8 4 The Stucki pattern. |
|
|
|
2 4 8 4 2 |
|
|
|
1 2 4 2 1 |
|
|
|
|
|
|
|
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 |
|
|
|
be saved by producing the remaining fractions by shifts. |
|
|
|
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 |
|
|
|
be saved by producing the remaining fractions by shifts. |
|
|
|
|
|
|
|
The speed advantages of the simpler filters can be eliminated somewhat by |
|
|
|
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 |
|
|
|
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 |
|
|
|
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 |
|
|
|
filters work better--they split the errors up more finely and produce less of |
|
|
|
this clipping noise. |
|
|
|
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 |
|
|
|
filters work better--they split the errors up more finely and produce less of |
|
|
|
this clipping noise. |
|
|
|
|
|
|
|
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 |
|
|
|
accomplish by subtracting each fraction from the whole error as it is |
|
|
|
calculated, and using the final remainder as the last fraction. |
|
|
|
|
|
|
|
Some of these methods (particularly the simpler ones) can be greatly improved |
|
|
|
by skewing the weights with a little randomness [3]. |
|
|
|
|
|
|
|
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 |
|
|
|
available colors must be scanned sequentially for each input pixel to find |
|
|
|
the closest. The "distance" formula most often 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 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 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 affect the order of its operands. |
|
|
|
If the total number of colors with which you are dealing is small, this part |
|
|
|
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 |
|
|
|
calculated, and using the final remainder as the last fraction. |
|
|
|
|
|
|
|
Some of these methods (particularly the simpler ones) can be greatly improved |
|
|
|
by skewing the weights with a little randomness [3]. |
|
|
|
|
|
|
|
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 |
|
|
|
available colors must be scanned sequentially for each input pixel to find |
|
|
|
the closest. The "distance" formula most often 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 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 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 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 lookup table as well. |
|
|
|
|
|
|
|
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 |
|
|
|
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, |
|
|
|
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 |
|
|
|
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 |
|
|
|
Heckbert [8]. |
|
|
|
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 |
|
|
|
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, |
|
|
|
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 |
|
|
|
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 |
|
|
|
Heckbert [8]. |
|
|
|
|
|
|
|
|
|
|
|
======================================================================== |
|
|
|
Sample code |
|
|
|
|
|
|
|
Despite my best efforts in expository writing, nothing explains an algorithm |
|
|
|
better than real code. With that in mind, presented here below is an |
|
|
|
algorithm (in somewhat incomplete, very inefficient pseudo-C) which |
|
|
|
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 |
|
|
|
believe it does. |
|
|
|
Despite my best efforts in expository writing, nothing explains an algorithm |
|
|
|
better than real code. With that in mind, presented here below is an |
|
|
|
algorithm (in somewhat incomplete, very inefficient pseudo-C) which |
|
|
|
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 |
|
|
|
believe it does. |
|
|
|
|
|
|
|
/* Floyd/Steinberg error diffusion dithering algorithm in color. The array |
|
|
|
** line[][] contains the RGB values for the current line being processed; |
|
|
|
@ -466,34 +466,34 @@ dither() |
|
|
|
Bibliography |
|
|
|
|
|
|
|
[1] Foley, J. D. and Andries Van Dam (1982) |
|
|
|
Fundamentals of Interactive Computer Graphics. Reading, MA: Addisson |
|
|
|
Wesley. |
|
|
|
Fundamentals of Interactive Computer Graphics. Reading, MA: Addisson |
|
|
|
Wesley. |
|
|
|
|
|
|
|
This is a standard reference for many graphic techniques which has not |
|
|
|
declined with age. Highly recommended. |
|
|
|
This is a standard reference for many graphic techniques which has not |
|
|
|
declined with age. Highly recommended. |
|
|
|
|
|
|
|
[2] Bayer, B. E. (1973) |
|
|
|
"An Optimum Method for Two-Level Rendition of Continuous Tone Pictures," |
|
|
|
IEEE International Conference on Communications, Conference Records, pp. |
|
|
|
26-11 to 26-15. |
|
|
|
"An Optimum Method for Two-Level Rendition of Continuous Tone Pictures," |
|
|
|
IEEE International Conference on Communications, Conference Records, pp. |
|
|
|
26-11 to 26-15. |
|
|
|
|
|
|
|
A short article proving the optimality of Bayer's pattern in the |
|
|
|
dispersed-dot ordered dither. |
|
|
|
A short article proving the optimality of Bayer's pattern in the |
|
|
|
dispersed-dot ordered dither. |
|
|
|
|
|
|
|
[3] Ulichney, R. (1987) |
|
|
|
Digital Halftoning. Cambridge, MA: The MIT Press. |
|
|
|
|
|
|
|
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 |
|
|
|
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 |
|
|
|
Literacy carries it but is often sold out. |
|
|
|
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 |
|
|
|
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 |
|
|
|
Literacy carries it but is often sold out. |
|
|
|
|
|
|
|
[4] Floyd, R.W. and L. Steinberg (1975) |
|
|
|
"An Adaptive Algorithm for Spatial Gray Scale." SID International |
|
|
|
Symposium Digest of Technical Papers, vol 1975m, pp. 36-37. |
|
|
|
"An Adaptive Algorithm for Spatial Gray Scale." SID International |
|
|
|
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: |
|
|
|
|
|
|
|
@ -502,21 +502,21 @@ Bibliography |
|
|
|
2351 College Station Road Suite 563 |
|
|
|
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) |
|
|
|
"A Survey of Techniques for the Display of Continuous Tone Pictures on |
|
|
|
Bi-Level Displays." Computer Graphics and Image Processing, vol. 5, pp. |
|
|
|
13-40. |
|
|
|
"A Survey of Techniques for the Display of Continuous Tone Pictures on |
|
|
|
Bi-Level Displays." Computer Graphics and Image Processing, vol. 5, pp. |
|
|
|
13-40. |
|
|
|
|
|
|
|
[7] Stucki, P. (1981) |
|
|
|
"MECCA - a multiple-error correcting computation algorithm for bilevel |
|
|
|
image hardcopy reproduction." Research Report RZ1060, IBM Research |
|
|
|
Laboratory, Zurich, Switzerland. |
|
|
|
"MECCA - a multiple-error correcting computation algorithm for bilevel |
|
|
|
image hardcopy reproduction." Research Report RZ1060, IBM Research |
|
|
|
Laboratory, Zurich, Switzerland. |
|
|
|
|
|
|
|
[8] Heckbert, Paul (9182) |
|
|
|
"Color Image Quantization for Frame Buffer Display." Computer Graphics |
|
|
|
(SIGGRAPH 82), vol. 16, pp. 297-307. |
|
|
|
"Color Image Quantization for Frame Buffer Display." Computer Graphics |
|
|
|
(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: |
|
|
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|
|
|
|
CompuServe Information Service is a service of the H&R Block companies |
|
|
|
providing computer users with electronic mail, teleconferencing, and many |
|
|
|
other telecommunications services. Call 800-848-8199 for more information. |
|
|
|
CompuServe Information Service is a service of the H&R Block companies |
|
|
|
providing computer users with electronic mail, teleconferencing, and many |
|
|
|
other telecommunications services. Call 800-848-8199 for more information. |
|
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|
|
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|
The Graphics Support Forum is dedicated to helping its users get the most out |
|
|
|
of their computers' graphics capabilities. It has a small staff and a large |
|
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number of "Developers" who create images and software on all types of |
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machines from Apple IIs to Sun workstations. While on CompuServe, type GO |
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PICS from any "!" prompt to gain access to the forum. |
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The Graphics Support Forum is dedicated to helping its users get the most out |
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of their computers' graphics capabilities. It has a small staff and a large |
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number of "Developers" who create images and software on all types of |
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machines from Apple IIs to Sun workstations. While on CompuServe, type GO |
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PICS from any "!" prompt to gain access to the forum. |
James Jackson-South
9 years ago