diff --git a/src/ImageSharp/Dithering/DHALF.TXT b/src/ImageSharp/Dithering/DHALF.TXT index dce9ef924..ec6ba3d6e 100644 --- a/src/ImageSharp/Dithering/DHALF.TXT +++ b/src/ImageSharp/Dithering/DHALF.TXT @@ -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 I'm sure it will be out of date as soon as it is released, but it does serve to collect data from a wide variety of sources into a single document, and should save you considerable -searching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. +searching time. 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 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 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 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 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 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 -different machines as possible. +different machines as possible. Original Author: @@ -50,7 +50,7 @@ Contributors: COMMENTS BY MIKE MORRA 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 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 @@ -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 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 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 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 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 -issues. +issues. 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 @@ -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 and inquiries over the past few months . In particular, I would thank John Swenson, Chris Young, and (of course) Lee Crocker for their invaluable -assistance. +assistance. Mike Morra [76703,4051] 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 challenge is even more pronounced in today's world of personal computing 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 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 techniques for color reduction -- the process of finding the "best fit" of 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 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- called "high-color" video displays -- typically 16-bit, with 32,768-color 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, 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 halftoning as "... any algorithmic process which creates the illusion of 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 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 The concept of resolution is essential to the understanding of digital -halftoning. Resolution can be defined as "fineness" and is used to -describe the level of detail in a digitally sampled signal. +halftoning. Resolution can be defined as "fineness" and is used to +describe the level of detail in a digitally sampled signal. 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 @@ -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 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 -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 @@ -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 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 -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 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. - + What happens to the image during this operation? Well, some detail 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 @@ -281,16 +281,16 @@ categories: 3. Ordered dither 4. Error-diffusion halftoning -Each of these methods is generally better than those listed before it, but -other considerations such as processing time, memory constraints, etc. may -weigh in favor of one of the simpler methods. +Each of these methods is generally better than those listed before it, but +other considerations such as processing time, memory constraints, etc. may +weigh in favor of one of the simpler methods. 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, 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 -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 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; -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 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 random dither. With this image, other digital halftoning algorithms would 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 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 -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 sequence:" once a pixel is intensified for a particular value, it should 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. 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 @@ -427,7 +427,7 @@ greater than that of the image. Another limitation of patterning is that the effective spatial resolution is decreased, since a multiple-pixel "cell" is used to simulate the single, 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 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 incur a 16-fold reduction in spatial resolution. Because of this size 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 pixels 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 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 (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 -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. 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, 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 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 @@ -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 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 -and 4x4 matrices are as follows: +and 4x4 matrices are as follows: 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.) 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 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- 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 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 @@ -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" 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 -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 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, which means that the error value isn't distributed finely enough. With the 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 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 filters detailed here -- introduces an additional perturbation which 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 distribution would be better. This is exactly what Jarvis, Judice, and Ninke [6] did in 1976 with their filter: - - * 7 5 + + * 7 5 3 5 7 5 3 1 3 5 3 1 (1/48) @@ -723,7 +723,7 @@ requires extra memory and time for processing. The Stucki filter P. Stucki [7] offered a rework of the Jarvis, Judice, and Ninke filter in -1981: +1981: * 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) -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 producing the remaining fractions by shifts. The Stucki filter has been observed to give very clean, sharp output, which helps to offset the slow processing time. - + ===================================== The Burkes filter @@ -752,7 +752,7 @@ in 1988: 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, 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 filter by eliminating the second forward array which would otherwise be 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 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 -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 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 [11] offers color resolutions in excess of 256 colors by plotting red, 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 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 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 -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 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 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 --- 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 the distributed error values is equal to the original error value. This is most easily accomplished by subtracting each fraction, as it is calculated, 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. 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 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 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 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. +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 @@ -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 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. +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]. @@ -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 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 -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, 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. 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 electronic BBSs and information services are 8-bit (256 color), colormapped 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 -them. +them. In some cases, these mapped GIF files may be further processed with special 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 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 - dispersed-dot ordered dither. + dispersed-dot ordered dither. [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 does not contain any code, but don't let that keep you from 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 book! Several different writers have praised this reference to @@ -1242,8 +1242,8 @@ Bibliography Scale." SID 1975, 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: Daniel Burkes @@ -1266,7 +1266,7 @@ Bibliography for bilevel image hardcopy reproduction." Research Report RZ1060, IBM 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. [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, 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. Kuto, S., "Continuous Color Presentation Using a Low-Cost Ink Jet Printer," 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. Pavlidis, T., Algorithms for Graphics and Image Processing, Computer Science diff --git a/src/ImageSharp/Dithering/DITHER.TXT b/src/ImageSharp/Dithering/DITHER.TXT index 4d29a533e..1f49fd6eb 100644 --- a/src/ImageSharp/Dithering/DITHER.TXT +++ b/src/ImageSharp/Dithering/DITHER.TXT @@ -1,28 +1,28 @@ DITHER.TXT -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 -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 +different machines as possible. Original Author: @@ -35,37 +35,37 @@ Contributors: ======================================================================== What is Dithering? -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. +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. ======================================================================== @@ -78,84 +78,84 @@ The classes of dithering algorithms we will discuss here are these: 3. Ordered 4. Error dispersion -Each of these methods is generally better than those listed before it, but -other considerations such as processing time, memory constraints, etc. may -weigh in favor of one of the simpler methods. +Each of these methods is generally better than those listed before it, but +other considerations such as processing time, memory constraints, etc. may +weigh in favor of one of the simpler methods. -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 -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 -where better results can be obtained, I will go into more detail. +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 +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 +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 -algorithm separately for each primary and mix the resulting values. 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 dispersion -as well, there are better methods in this case. +To convert any of the first three methods into color, simply apply the +algorithm separately for each primary and mix the resulting values. 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 dispersion +as well, there are better methods in this case. ======================================================================== Random dither -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 -produced by digital signal processing. +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 +produced by digital signal processing. 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 -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- -like patterns. These patterns are not part of the original idea of lines, -but are an illusion produced by the jaggedness of the display. +like patterns. These patterns are not part of the original idea of lines, +but are an illusion produced by the jaggedness of the display. -Many techniques exist for the reduction of digital artifacts like these, most -of which involve using a little randomness to "perturb" a regular algorithm a -little. Random dither obviously takes this to extreme. +Many techniques exist for the reduction of digital artifacts like these, most +of which involve using a little randomness to "perturb" a regular algorithm a +little. Random dither obviously takes this to extreme. -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 generation algorithm itself. +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 generation algorithm itself. -While random dither adds a lot of high-frequency noise to a picture, it is -useful in reproducing very low-frequency images where the absence of -artifacts is more important than noise. For example, a whole screen -containing a gradient of all levels from black to white would actually look -best with a random dither. In this case, ordered dithering would produce -diagonal patterns, and error dispersion would produce clustering. +While random dither adds a lot of high-frequency noise to a picture, it is +useful in reproducing very low-frequency images where the absence of +artifacts is more important than noise. For example, a whole screen +containing a gradient of all levels from black to white would actually look +best with a random dither. In this case, ordered dithering would produce +diagonal patterns, and error dispersion would produce clustering. -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 -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 -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 -lines will appear. +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 +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 +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 +lines will appear. ======================================================================== Pattern 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 -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 -dots in the pattern determines the darkness of the pattern. +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 +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 +dots in the pattern determines the darkness of the pattern. 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 -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 -adjacent colors should be similar to reduce an artifact called "contouring", -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 -sets of patterns for a 3-by-3 matrix: +black patterns. We can eliminate those patterns which would create vertical +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 +adjacent colors should be similar to reduce an artifact called "contouring", +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 +sets of patterns for a 3-by-3 matrix: --- --- --- -X- -XX -XX -XX -XX 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- XX- XX- XX- XXX -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: +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: Yellow = R G Cyan = G B Magenta = R B Gray = R G G R B G B R B K -(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?). ======================================================================== 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: +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: 8 3 4 and 1 7 4 6 1 2 5 8 3 7 5 9 6 2 9 -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 -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 -ordered dither is the process used by most newspapers, and the term -halftoning refers to this method if not otherwise qualified. +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 +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 +ordered dither is the process used by most newspapers, and the term +halftoning refers to this method if not otherwise qualified. -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 -being as high-frequency as possible. The pattern for a 2x2 and 4x4 matrices -are as follows: +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 +being as high-frequency as possible. The pattern for a 2x2 and 4x4 matrices +are as follows: 1 3 1 9 3 11 These patterns (and their rotations 4 2 13 5 15 7 and reflections) are optimal for a 4 12 2 10 dispersed-pattern ordered dither. 16 8 14 6 -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 -8x8 pattern. +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 +8x8 pattern. 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 -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. +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. ======================================================================== Error dispersion -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 -well, and the better they get, the slower they are. +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 +well, and the better they get, the slower they are. -Error dispersion is very simple to describe: for each point in the image, -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. +Error dispersion is very simple to describe: for each point in the image, +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. -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. +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. 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 @@ -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 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). X 7 This is the Floyd and Steinberg [4] 3 5 1 error diffusion filter. In this filter, the X represents the pixel you are currently scanning, and -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. +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: @@ -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. -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: X 7 5 The Jarvis, et al. pattern. 3 5 7 5 3 1 3 5 3 1 -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: -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. -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 -number of "Developers" who create images and software on all types of -machines from Apple IIs to Sun workstations. While on CompuServe, type GO -PICS from any "!" prompt to gain access to the forum. \ No newline at end of file +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 +number of "Developers" who create images and software on all types of +machines from Apple IIs to Sun workstations. While on CompuServe, type GO +PICS from any "!" prompt to gain access to the forum. \ No newline at end of file