FAQ about Color and Gamma

In video, computer graphics and image processing the "gamma" symbolrepresents a numerical parameter that describes the nonlinearity ofintensity reproduction. The Gamma FAQ section of this documentclarifies aspects of nonlinear image coding.The Color FAQ section of this document clarifies aspects of colorspecification and image coding that are important to computer graphics,image processing, video, and the transfer of digital images to print. Adrian Ford and Alan Roberts have written "Colour Space Conversions"that details transforms among color spaces such as RGB, HSI, CMY andvideo. Find it at <http://www.wmin.ac.uk/ITRG/docs/coloureq/COL_.htm>.Steve Westland has written "Frequently asked questions about ColourPhysics", available at <http://www.keele.ac.uk/depts/co/cpfaq.html>.I retain copyright to this note. You have permission to use it, but youmay not publish it.CONTENTS    G-0   Where do these documents live?Frequently Asked Questions about Gamma     G-1   What is intensity?    G-2   What is luminance?    G-3   What is lightness?    G-4   What is gamma?    G-5   What is gamma correction?    G-6   Does NTSC use a gamma of 2.2?    G-7   Does PAL use a gamma of 2.8?    G-8   I pulled an image off the net and it looks murky.    G-9   I pulled an image off the net and it looks a little too contrasty.    G-10  What is luma?    G-11  What is contrast ratio?    G-12  How many bits do I need to smoothly shade from black to white?    G-13  How is gamma handled in video, computer graphics and desktop            computing?    G-14  What is the gamma of a Macintosh?    G-15  Does the gamma of CRTs vary wildly?    G-16  How should I adjust my monitor's brightness and contrast controls?    G-17  Should I do image processing operations on linear or nonlinear             image data?    G-18  What's the transfer function of offset printing?    G-19  ReferencesFrequently Asked Questions about Color     C-1   What is color?    C-2   What is intensity?    C-3   What is luminance?    C-4   What is lightness?    C-5   What is hue?    C-6   What is saturation?    C-7   How is color specified?    C-8   Should I use a color specification system for image data?    C-9   What weighting of red, green and blue corresponds to brightness?    C-10  Can blue be assigned fewer bits than red or green?    C-11  What is "luma"?    C-12  What are CIE XYZ components?    C-13  Does my scanner use the CIE spectral curves?    C-14  What are CIE x and y chromaticity coordinates?    C-15  What is white?    C-16  What is color temperature?    C-17  How can I characterize red, green and blue?    C-18  How do I transform between CIE XYZ and a particular set of RGB            primaries?    C-19  Is RGB always device-dependent?    C-20  How do I transform data from one set of RGB primaries to another?    C-21  Should I use RGB or XYZ for image synthesis?    C-22  What is subtractive color?    C-23  Why did my grade three teacher tell me that the primaries are red,            yellow and blue?    C-24  Is CMY just one-minus-RGB?    C-25  Why does offset printing use black ink in addition to CMY?    C-26  What are color differences?    C-27  How do I obtain color difference components from tristimulus values?    C-28  How do I encode Y'PBPR components?    C-29  How do I encode Y'CBCR components from R'G'B' in [0, +1]?    C-30  How do I encode Y'CBCR components from computer R'G'B' ?    C-31  How do I encode Y'CBCR components from studio video?    C-32  How do I decode R'G'B' from PhotoYCC?    C-33  Will you tell me how to decode Y'UV and Y'IQ?    C-34  How should I test my encoders and decoders?    C-35  What is perceptual uniformity?    C-36  What are HSB and HLS?    C-37  What is true color?    C-38  What is indexed color?    C-39  I want to visualize a scalar function of two variables. Should I use            RGB values corresponding to the colors of the rainbow?    C-40  What is dithering?    C-41  How does halftoning relate to color?    C-42  What's a color management system?    C-43  How does a CMS know about particular devices?    C-44  Is a color management system useful for color specification?    C-45  I'm not a color expert. What parameters should I use to             code my images?    C-46  References    C-47  ContributorsG-0   WHERE DO THESE DOCUMENTS LIVE?Each document GammaFAQ and ColorFAQ is available in four formats --Adobe Acrobat (PDF), hypertext (HTML), PostScript, and plain 7-bitASCII text-only. You are reading the concatenation of the text versionsof GammaFAQ and ColorFAQ. The text formats are devoid of graphs andillustrations, of course; I strongly recommend the PDF versions.The hypertext version is linked from my color page,    <http://www.inforamp.net/~poynton/Poynton-color.html>The PDF, PostScript and text formats are available by ftp:    <ftp://ftp.inforamp.net/pub/users/poynton/doc/color/>If you have access to Internet e-mail but not to ftp, use a mailer thatis properly configured with your return address to send mail to<[e-mail]> with en empty subject and the single word helpin the body.PDF NotesAdobe's Acrobat Reader is freely available for Windows, Mac, MS-DOS andSPARC. If you don't already have a reader, you can obtain one from    <ftp://ftp.adobe.com/pub/adobe/acrobatreader/>    in a subdirectory and file appropriate for your platform. On CompuServe, GO Acrobat.On America Online, for Mac, use Keyword Adobe -> Adobe Software Library-> New! Adobe Acrobat Reader 3.0, then choose a platform.Transfer PDF files in binary mode, particularly to Windows or MS-DOSmachines. PDF files contain "bookmarks" corresponding to the table ofcontents. Clicking a bookmark takes you to  the topic. Also,cross-references in the PDF files are links.PostScript NotesAcrobat Reader allows viewing on-screen on multiple platforms, printingto both PostScript and non-PostScript printers, and permits viewing andprinting independent of the fonts that you have installed. But forthose people who cannot or do not wish to run Acrobat Reader, I providePostScript versions of the notes.The documents use only Times, Helvetica, Palatino and Symbol fonts andare laid out with generous margins for US Letter size paper. I confessI don't know how well they print to A4 but it should fit. If anyoneusing A4 has suggestions to improve the PostScript please let me know.The PostScript files are compressed with Gnu zip compression, and aregiven the file suffix ".gz". Gzip for UNIX (and maybe other platformsas well) is available from the usual gnu sites. If you use a Macintosh,the freeware StuffIt Expander 4.0 will decode gnu zip files.------------------------------FREQUENTLY ASKED QUESTIONS ABOUT GAMMA G-1   WHAT IS INTENSITY?Intensity is a measure over some interval of the electromagnetic spectrum ofthe flow of power that is radiated from, or incident on, a surface.Intensity is what I call a "linear-light measure", expressed in units such aswatts per square meter.The voltages presented to a CRT monitor control the intensities of thecolor components, but in a nonlinear manner. CRT voltages are notproportional to intensity.Image data stored in a file (TIFF, JFIF, PPM, etc.) may or may notrepresent intensity, even if it is so described. The I component of acolor described as HSI (hue, saturation, intensity) does not accuratelyrepresent intensity if HSI is computed according to any of the usualformulae.G-2   WHAT IS LUMINANCE?Brightness is defined by the Commission Internationale de L'Eclairage (CIE)as the attribute of a visual sensation according to which an area appearsto emit more or less light. Because brightness perception is very complex,the CIE defined a more tractable quantity luminance, denoted Y, which isradiant power weighted by a spectral sensitivity function that ischaracteristic of vision. To learn about the relationship between physicalspectra and perceived brightness, and other color issues, refer to thecompanion Frequently Asked Questions about Color.The magnitude of luminance is proportional to physical power. In that senseit is like intensity. But the spectral composition of luminance is relatedto the brightness sensitivity of human vision.G-3   WHAT IS LIGHTNESS?Human vision has a nonlinear perceptual response to brightness: a sourcehaving a luminance only 18% of a reference luminance appears about half asbright. The perceptual response to luminance is called Lightness and isdefined by the CIE [1] as a modified cube root of luminance:  Lstar = -16 + 116 * pow(Y / Yn, 1. / 3.)Yn is the luminance of the white reference. If you normalize luminance toreference white then you need not compute the fraction. The CIE definitionapplies a linear segment with a slope of 903.3 near black, for (Y/Yn) <0.008856. The linear segment is unimportant for practical purposes but ifyou don't use it, make sure that you limit L* at zero. L* has a range of 0to 100, and a "delta L-star" of unity is taken to be roughly the thresholdof visibility.Stated differently, lightness perception is roughly logarithmic. You candetect an intensity difference between two patches when the ratio of theirintensities differs by more than about one percent.Video systems approximate the lightness response of vision using RGBsignals that are each subject to a 0.45 power function. This is comparableto the 1/3 power function defined by L*.The L component of a color described as HLS (hue, lightness, saturation)does not accurately represent lightness if HLS is computed according toany of the usual formulae. See Frequently Asked Questions about Color.G-4   WHAT IS GAMMA?The intensity of light generated by a physical device is not usually alinear function of the applied signal. A conventional CRT has a power-lawresponse to voltage: intensity produced at the face of the display isapproximately the applied voltage, raised to the 2.5 power. The numericalvalue of the exponent of this power function is colloquially known asgamma. This nonlinearity must be compensated in order to achieve correctreproduction of intensity.As mentioned above (What is lightness?), human vision has a nonuniformperceptual response to intensity. If intensity is to be coded into a smallnumber of steps, say 256, then in order for the most effective perceptualuse to be made of the available codes, the codes must be assigned tointensities according to the properties of perception.Here is a graph of an actual CRT's transfer function, at three differentcontrast settings:<< A nice graph is found in the .PDF and .PS versions. >>This graph indicates a video signal having a voltage from zero to 700 mV.In a typical eight-bit digital-to-analog converter on a framebuffer card,black is at code zero and white is at code 255.Through an amazing coincidence, vision's response to intensity iseffectively the inverse of a CRT's nonlinearity. If you apply a transferfunction to code a signal to take advantage of the properties of lightnessperception - a function similar to the L* function - the coding will beinverted by a CRT.G-5   WHAT IS GAMMA CORRECTION?In a video system, linear-light intensity is transformed to a nonlinearvideo signa by gamma correction, which is universally done at the camera.The Rec. 709 transfer function [2] takes linear-light intensity (here R) toa nonlinear component (here Rprime), for example, voltage in a videosystem:  Rprime = ( R <= 0.018 ?              4.5 * R :              -0.099 + 1.099 * pow(R, 0.45)            );The linear segment near black minimizes the effect of sensor noise inpractical cameras and scanners. Here is a graph of the Rec. 709 transferfunction, for a signal range from zero to unity:<< An attractive graph is presented in the .PDF and .PS versions. >>An idealized monitor inverts the transform:  R = ( Rprime <= 0.081 ?         Rprime / 4.5 :         pow((Rprime + 0.099) / 1.099, 1. / 0.45)       );Real monitors are not as exact as this equation suggests, and have no linearsegment, but the precise definition is necessary for accurate intermediateprocessing in the linear-light domain. In a color system, an identicaltransfer function is applied to each of the three tristimulus(linear-light) RGB components. See Frequently Asked Questions aboutColor.By the way, the nonlinearity of a CRT is a function of the electrostaticsof the cathode and the grid of an electron gun; it has nothing to do withthe phosphor. Also, the nonlinearity is a power function (which has theform f(x) = x^a), not an exponential function (which has the form f(x) =a^x). For more detail, read Poynton's article [3].G-6   DOES NTSC USE A GAMMA OF 2.2?Television is usually viewed in a dim environment. If an images's correctphysical intensity is reproduced in a dim surround, a subjective effectcalled simultaneous contrast causes the reproduced image to appear lackingin contrast. The effect can be overcome by applying an end-to-end powerfunction whose exponent is about 1.1 or 1.2. Rather than having eachreceiver provide this correction, the assumed 2.5-power at the CRT isunder-corrected at the camera by using an exponent of about 1/2.2 insteadof 1/2.5. The assumption of a dim viewing environment is built into videocoding.G-7   DOES PAL USE A GAMMA OF 2.8?Standards for 625/50 systems mention an exponent of 2.8 at the decoder,however this value is unrealistically high to be used in practice. If anexponent different from 0.45 is chosen for a power function with a linearsegment near black like Rec. 709, the other parameters need to be changedto maintain function and tangent continuity.G-8   I PULLED AN IMAGE OFF THE NET AND IT LOOKS MURKY.If an image originates in linear-light form, gamma correction needs to beapplied exactly once. If gamma correction is not applied and linear-lightimage data is applied to a CRT, the midtones will be reproduced too dark.If gamma correction is applied twice, the midtones will be too light.G-9   I PULLED AN IMAGE OFF THE NET AND IT LOOKS A LITTLE TOO CONTRASTY.Viewing environments typical of computing are quite bright. When an imageis coded according to video standards it implicitly carries the assumptionof a dim surround. If it is displayed without correction in a brightambient, it will appear contrasty. In this circumstance you should apply apower function with an exponent of about 1/1.1 or 1/1.2 to correct for yourbright surround.Ambient lighting is rarely taken into account in the exchange of computerimages. If an image is created in a dark environment and transmitted to aviewer in a bright environment, the recipient will find it to haveexcessive contrast.If an image originated in a bright environment and viewed in a brightenvironment, it will need no modification no matter what coding is applied.But then it will carry an assumption of a bright surround. Video standardsare widespread and well optimized for vision, so it makes sense to codewith a power function of 0.45 and retain a single standard for the assumedviewing environment.In the long term, for everyone to get the best results in image interchangeamong applications, an image originator should remove the effect of hisambient environment when he transmits an image. The recipient of an imageshould insert a transfer function appropriate for his viewing environment.In the short term, you should include with your image data tags thatspecify the parameters that you used to encode. TIFF 6.0 has provisions forthis data. You can correct for your own viewing environment as appropriate,but until image interchange standards incorporate viewing conditions, youwill also have to compensate for the originator's viewing conditions.G-10  WHAT IS LUMA?In video it is standard to represent brightness information not as anonlinear function of true CIE luminance, but as a weighted sum ofnonlinear R'G'B' components called luma. For more information, consult thecompanion document Frequently Asked Questions about Color.G-11  WHAT IS CONTRAST RATIO?Contrast ratio is the ratio of intensity between the brightest white andthe darkest black of a particular device or a particular environment.Projected cinema film - or a photographic reflection print - has a contrastratio of about 80:1. Television assumes a contrast ratio - in your livingroom - of about 30:1. Typical office viewing conditions restrict contrastratio of CRT display to about 5:1.G-12  HOW MANY BITS DO I NEED TO SMOOTHLY SHADE FROM BLACK TO WHITE?At a particular level of adaptation, human vision responds to about ahundred-to-one contrast ratio of intensity from white to black. Call theseintensities 100 and 1. Within this range, vision can detect that twointensities are different if the ratio between them exceeds about 1.01,corresponding to a contrast sensitivity of one percent.To shade smoothly over this range, so as to produce no perceptible steps,at the black end of the scale it is necessary to have coding thatrepresents different intensity levels 1.00, 1.01, 1.02 and so on. If linearlight coding is used, the "delta" of 0.01 must be maintained all the way upthe scale to white. This requires about 9,900 codes, or about fourteen bitsper component.If you use nonlinear coding, then the 1.01 "delta" required at the blackend of the scale applies as a ratio, not an absolute increment, andprogresses like compound interest up to white. This results in about 460codes, or about nine bits per component. Eight bits, nonlinearly codedaccording to Rec. 709, is sufficient for broadcast-quality digitaltelevision at a contrast ratio of about 50:1.If poor viewing conditions or poor display quality restrict the contrastratio of the display, then fewer bits can be employed.If a linear light system is quantized to a small number of bits, with blackat code zero, then the ability of human vision to discern a 1.01 ratiobetween adjacent intensity levels takes effect below code 100. If a linearlight system has only eight bits, then the top end of the scale is only255, and contouring in dark areas will be perceptible even in very poorviewing conditions.G-13  HOW IS GAMMA HANDLED IN VIDEO, COMPUTER GRAPHICS AND DESKTOP      COMPUTING?As outlined above, gamma correction in video effectively codes into aperceptually uniform domain. In video, a 0.45-power function is applied atthe camera, as shown in the top row of this diagram:<< A nice diagram is presented in the .PDF and .PS versions. >>Synthetic computer graphics calculates the interaction of light andobjects. These interactions are in the physical domain, and must becalculated in linear-light values. It is conventional in computer graphicsto store linear-light values in the framebuffer, and introduce gammacorrection at the lookup table at the output of the framebuffer. This isillustrated in the middle row above.If linear-light is represented in just eight bits, near black the stepsbetween codes will be perceptible as banding in smoothly-shaded images.This is the eight-bit bottleneck in the sketch.Desktop computers are optimized neither for image synthesis nor for video.They have programmable "gamma" and either poor standards or no standards.Consequently, image interchange among desktop computers is fraught withdifficulty.G-14  WHAT IS THE GAMMA OF A MACINTOSH?Apple offers no definition of the nonlinearity - or loosely speaking, gamma- that is intrinsic in QuickDraw. But the combination of a defaultQuickDraw lookup table and a standard monitor causes intensity to representthe 1.8-power of the R, G and B values presented to QuickDraw. It iswrongly believed that Macintosh computers use monitors whose transferfunction is different from the rest of the industry. The unconventionalQuickDraw handling of nonlinearity is the root of this misconception.Macintosh coding is shown in the bottom row of the diagram<< provided in the PDF and PS versions >>.The transfer of image data in computing involves various transferfunctions: at coding, in the framebuffer, at the lookup table, and at themonitor. Strictly speaking the term gamma applies to the exponent of thepower function at the monitor. If you use the term loosely, in the case ofa Mac you could call the gamma 1.4, 1.8 or 2.5 depending which part of thesystem you were discussing. More detail is available [4].I recommend using the Rec. 709 transfer function, with its 0.45-power law,for best perceptual performance and maximum ease of interchange withdigital video. If you need Mac compatibility you will have to codeintensity with a 1/1.8-power law, anticipating QuickDraw's 1/1.4-power inthe lookup table. This coding has adequate performance in the brightviewing environments typical of desktop applications, but suffers in darkerviewing conditions that have high contrast ratio.G-15  DOES THE GAMMA OF CRTS VARY WILDLY?Gamma of a properly adjusted conventional CRT varies anywhere between about2.35 and 2.55.CRTs have acquired a reputation for wild variation for two reasons. First,if the model intensity=voltage^gamma is naively fitted to a display withblack-level error, the exponent deduced will be as much a function of theblack error as the true exponent. Second, input devices, graphics librariesand application programs all have the potential to introduce their owntransfer functions. Nonlinearities from these sources are often categorizedas gamma and attributed to the display.G-16  HOW SHOULD I ADJUST MY MONITOR'S BRIGHTNESS AND CONTRAST CONTROLS?On a CRT monitor, the control labelled contrast controls overall intensity,and the control labelled brightness controls offset (black level). Displaya picture that is predominantly black. Adjust brightness so that themonitor reproduces true black on the screen, just at the threshold where itis not so far down as to "swallow" codes greater than the black code, butnot so high that the picture sits on a "pedestal" of dark grey. When thecritical point is reached, put a piece of tape over the brightness control.Then set contrast to suit your preference for display intensity.For more information, consult "Black Level" and "Picture", <ftp://ftp.inforamp.net/pub/users/poynton/doc/color/>.G-17  SHOULD I DO IMAGE PROCESSING OPERATIONS ON LINEAR OR NONLINEAR IMAGE      DATA?If you wish to simulate the physical world, linear-light coding isnecessary. For example, if you want to produce a numerical simulation of alens performing a Fourier transform, you should use linear coding. If youwant to compare your model with the transformed image captured from a reallens by a video camera, you will have to "remove" the nonlinear gammacorrection that was imposed by the camera, to convert the image data backinto its linear-light representation.On the other hand, if your computation involves human perception, anonlinear representation may be required. For example, if you perform adiscrete cosine transform on image data as the first step in imagecompression, as in JPEG, then you ought to use nonlinear coding thatexhibits perceptual uniformity, because you wish to minimize theperceptibility of the errors that will be introduced during quantization.The image processing literature rarely discriminates between linear andnonlinear coding. In the JPEG and MPEG standards there is no mention oftransfer function, but nonlinear (video-like) coding is implicit:unacceptable results are obtained when JPEG or MPEG are applied tolinear-light data. In computer graphic standards such as PHIGS and CGMthere is no mention of transfer function, but linear-light coding isimplicit. These discrepancies make it very difficult to exchange image databetween systems.When you ask a video engineer if his system is linear, he will say "Ofcourse!" referring to linear voltage. If you ask an optical engineer if hersystem is linear, she will say "Of course!" referring to linear intensity.But when a nonlinear transform lies between the two systems, as in video, alinear transformation performed in one domain is not linear in the other.G-18  WHAT'S THE TRANSFER FUNCTION OF OFFSET PRINTING?A image destined for halftone printing conventionally specifies each pixelin terms of dot percentage in film. An imagesetter's halftoning machinerygenerates dots whose areas are proportional to the requested coverage. Inprinciple, dot percentage in film is inversely proportional to linear-lightreflectance.Two phenomena distort the requested dot coverage values. First, printinginvolves a mechanical smearing of the ink that causes dots to enlarge.Second, optical effects within the bulk of the paper cause more light to beabsorbed than would be expected from the surface coverage of the dot alone.These phenomena are collected under the term dot gain, which is thepercentage by which the light absorption of the printed dots exceeds therequested dot coverage.Standard offset printing involves a dot gain at 50% of about 24%: when 50%absorption is requested, 74% absorption is obtained. The midtones printdarker than requested. This results in a transfer function from code toreflectance that closely resembles the voltage-to-light curve of a CRT.Correction of dot gain is conceptually similar to gamma correction invideo: physical correction of the "defect" in the reproduction process isvery well matched to the lightness perception of human vision. Coding animage in terms of dot percentage in film involves coding into a roughlyperceptually uniform space. The standard dot gain functions employed inNorth America and Europe correspond to intensity being reproduced as apower function of the digital code, where the numerical value of theexponent is about 1.75, compared to about 2.2 for video. This is lower thanthe optimum for perception, but works well for the low contrast ratio ofoffset printing.The Macintosh has a power function that is close enough to printing practicethat raw QuickDraw codes sent to an imagesetter produce acceptable results.High-end publishing software allows the user to specify the parameters ofdot gain compensation.I have described the linearity of conventional offset printing. Otherhalftoned devices have different characteristics, and require differentcorrections.G-19  REFERENCES[1] Publication CIE No 15.2, Colorimetry, Second Edition (1986), CentralBureau of the Commission Internationale de L'Eclairage, Vienna, Austria.[2] ITU-R Recommendation BT.709, Basic Parameter Values for the HDTVStandard for the Studio and for International Programme Exchange (1990),[formerly CCIR Rec. 709], ITU, 1211 Geneva 20, Switzerland.[3] Charles A. Poynton, "Gamma and Its Disguises" in Journal of the Societyof Motion Picture and Television Engineers, Vol. 102, No. 12 (December1993), 1099-1108.[4] Charles A. Poynton, "Gamma on the Apple Macintosh", <ftp://ftp.inforamp.net/pub/users/poynton/doc/Mac/>.------------------------------FREQUENTLY ASKED QUESTIONS ABOUT COLORC-1   WHAT IS COLOR?Color is the perceptual result of light in the visible region of thespectrum, having wavelengths in the region of 400 nm to 700 nm, incidentupon the retina. Physical power (or radiance) is expressed in a spectralpower distribution (SPD), often in 31 components each representing a 10 nmband.The human retina has three types of color photoreceptor cone cells, whichrespond to incident radiation with somewhat different spectral responsecurves. A fourth type of photoreceptor cell, the rod, is also present inthe retina. Rods are effective only at extremely low light levels(colloquially, night vision), and although important for vision play norole in image reproduction.Because there are exactly three types of color photoreceptor, threenumerical components are necessary and sufficient to describe a color,providing that appropriate spectral weighting functions are used. This isthe concern of the science of colorimetry. In 1931, the CommissionInternationale de L'Eclairage (CIE) adopted standard curves for ahypothetical Standard Observer. These curves specify how an SPD can betransformed into a set of three numbers that specifies a color.The CIE system is immediately and almost universally applicable toself-luminous sources and displays. However the colors produced byreflective systems such as photography, printing or paint are a functionnot only of the colorants but also of the SPD of the ambient illumination.If your application has a strong dependence upon the spectrum of theilluminant, you may have to resort to spectral matching.Sir Isaac Newton said, "Indeed rays, properly expressed, are not coloured."SPDs exist in the physical world, but color exists only in the eye and thebrain.C-2   WHAT IS INTENSITY?Intensity is a measure over some interval of the electromagnetic spectrumof the flow of power that is radiated from, or incident on, a surface.Intensity is what I call a linear-light measure, expressed in units such aswatts per square meter.The voltages presented to a CRT monitor control the intensities of thecolor components, but in a nonlinear manner. CRT voltages are notproportional to intensity.C-3   WHAT IS LUMINANCE?Brightness is defined by the CIE as the attribute of a visual sensationaccording to which an area appears to emit more or less light. Becausebrightness perception is very complex, the CIE defined a more tractablequantity luminance which is radiant power weighted by a spectralsensitivity function that is characteristic of vision. The luminousefficiency of the Standard Observer is defined numerically, is everywherepositive, and peaks at about 555 nm. When an SPD is integrated using thiscurve as a weighting function, the result is CIE luminance, denoted Y.The magnitude of luminance is proportional to physical power. In that senseit is like intensity. But the spectral composition of luminance is relatedto the brightness sensitivity of human vision.Strictly speaking, luminance should be expressed in a unit such as candelasper meter squared, but in practice it is often normalized to 1 or 100 unitswith respect to the luminance of a specified or implied white reference.For example, a studio broadcast monitor has a white reference whoseluminance is about 80 cd*m -2, and Y = 1 refers to this value.C-4   WHAT IS LIGHTNESS?Human vision has a nonlinear perceptual response to brightness: a sourcehaving a luminance only 18% of a reference luminance appears about half asbright. The perceptual response to luminance is called Lightness. It isdenoted L* and is defined by the CIE as a modified cube root of luminance:  Lstar = -16 + 116 * (pow(Y / Yn), 1. / 3.)Yn is the luminance of the white reference. If you normalize luminance toreference white then you need not compute the fraction. The CIE definitionapplies a linear segment with a slope of 903.3 near black, for (Y/Yn) <=0.008856. The linear segment is unimportant for practical purposes but ifyou don't use it, make sure that you limit L* at zero. L* has a range of 0to 100, and a "delta L-star" of unity is taken to be roughly the thresholdof visibility.Stated differently, lightness perception is roughly logarithmic. Anobserver can detect an intensity difference between two patches when theirintensities differ by more than one about percent.Video systems approximate the lightness response of vision using R'G'B'signals that are each subject to a 0.45 power function. This is comparableto the 1/3 power function defined by L*.C-5   WHAT IS HUE?According to the CIE [1], hue is the attribute of a visual sensationaccording to which an area appears to be similar to one of the perceivedcolors, red, yellow, green and bue, or a combination of two of them.Roughly speaking, if the dominant wavelength of an SPD shifts, the hue ofthe associated color will shift.C-6   WHAT IS SATURATION?Again from the CIE, saturation is the colorfulness of an area judged inproportion to its brightness. Saturation runs from neutral gray throughpastel to saturated colors. Roughly speaking, the more an SPD isconcentrated at one wavelength, the more saturated will be the associatedcolor. You can desaturate a color by adding light that contains power atall wavelengths.C-7   HOW IS COLOR SPECIFIED?The CIE system defines how to map an SPD to a triple of numericalcomponents that are the mathematical coordinates of color space. Theirfunction is analagous to coordinates on a map. Cartographers have differentmap projections for different functions: some map projections preserveareas, others show latitudes and longitudes as straight lines. No singlemap projection fills all the needs of map users. Similarly, no singlecolor system fills all of the needs of color users.The systems useful today for color specification include CIE XYZ, CIE xyY,CIE L*u*v* and CIE L*a*b*. Numerical values of hue and saturation are notvery useful for color specification, for reasons to be discussed insection 36.A color specification system needs to be able to represent any color withhigh precision. Since few colors are handled at a time, a specificationsystem can be computationally complex. Any system for color specificationmust be intimately related to the CIE specifications.You can specify a single "spot" color using a color order system such asMunsell. Systems like Munsell come with swatch books to enable visualcolor matches, and have documented methods of transforming betweencoordinates in the system and CIE values. Systems like Munsell are notuseful for image data. You can specify an ink color by specifying theproportions of standard (or secret) inks that can be mixed to make thecolor. That's how pantone(tm) works. Although widespread, it'sproprietary. No translation to CIE is publicly available.C-8   SHOULD I USE A COLOR SPECIFICATION SYSTEM FOR IMAGE DATA?A digitized color image is represented as an array of pixels, where eachpixel contains numerical components that define a color. Three componentsare necessary and sufficient for this purpose, although in printing it isconvenient to use a fourth (black) component.In theory, the three numerical values for image coding could be provided bya color specification system. But a practical image coding system needs tobe computationally efficient, cannot afford unlimited precision, need notbe intimately related to the CIE system and generally needs to cover only areasonably wide range of colors and not all of the colors. So imagecoding uses different systems than color specification.The systems useful for image coding are linear RGB, nonlinear R'G'B',nonlinear CMY, nonlinear CMYK, and derivatives of nonlinear R'G'B' suchas Y'CBCR. Numerical values of hue and saturation are not useful in colorimage coding.If you manufacture cars, you have to match the color of paint on the doorwith the color of paint on the fender. A color specification system willbe necessary. But to convey a picture of the car, you need image coding.You can afford to do quite a bit of computation in the first case becauseyou have only two colored elements, the door and the fender. In the secondcase, the color coding must be quite efficient because you may have amillion colored elements or more.For a highly readable short introduction to color image coding, seeDeMarsh and Giorgianni [2]. For a terse, complete technical treatment, readSchreiber [3].C-9   WHAT WEIGHTING OF RED, GREEN AND BLUE CORRESPONDS TO BRIGHTNESS?Direct acquisition of luminance requires use of a very specific spectralweighting. However, luminance can also be computed as a weighted sum ofred, green and blue components.If three sources appear red, green and blue, and have the same radiance inthe visible spectrum, then the green will appear the brightest of the threebecause the luminous efficiency function peaks in the green region of thespectrum. The red will appear less bright, and the blue will be the darkestof the three. As a consequence of the luminous efficiency function, allsaturated blue colors are quite dark and all saturated yellows are quitelight. If luminance is computed from red, green and blue, the coefficientswill be a function of the particular red, green and blue spectral weightingfunctions employed, but the green coefficient will be quite large, the redwill have an intermediate value, and the blue coefficient will be thesmallest of the three.Contemporary CRT phosphors are standardized in Rec. 709 [8], to bedescribed in section 17. The weights to compute true CIE luminance fromlinear red, green and blue (indicated without prime symbols), for the Rec.709, are these:  Y = 0.212671 * R + 0.715160 * G + 0.072169 * B;This computation assumes that the luminance spectral weighting can beformed as a linear combination of the scanner curves, and assumes that thecomponent signals represent linear-light. Either or both of theseconditions can be relaxed to some extent depending on the application.Some computer systems have computed brightness using (R+G+B)/3. This is atodds with the properties of human vision, as will be discussed under Whatare HSB and HLS? in section 36.The coefficients 0.299, 0.587 and 0.114 properly computed luminance formonitors having phosphors that were contemporary at the introduction ofNTSC television in 1953. They are still appropriate for computing videoluma to be discussed below in section 11. However, these coefficients donot accurately compute luminance for contemporary monitors.C-10  CAN BLUE BE ASSIGNED FEWER BITS THAN RED OR GREEN?Blue has a small contribution to the brightness sensation. However, humanvision has extraordinarily good color discrimination capability in bluecolors. So if you give blue fewer bits than red or green, you willintroduce noticeable contouring in blue areas of your pictures.C-11  WHAT IS "LUMA"?It is useful in a video system to convey a component representative ofluminance and two other components representative of color. It isimportant to convey the component representative of luminance in such a waythat noise (or quantization) introduced in transmission, processing andstorage has a perceptually similar effect across the entire tone scale fromblack to white. The ideal way to accomplish these goals would be to form aluminance signal by matrixing RGB, then subjecting luminance to a nonlineartransfer function similar to the L* function.There are practical reasons in video to perform these operations in theopposite order. First a nonlinear transfer function - gamma correction - isapplied to each of the linear R, G and B. Then a weighted sum of thenonlinear components is computed to form a signal representative ofluminance. The resulting component is related to brightness but is not CIEluminance. Many video engineers call it luma and give it the symbol Y'. Itis often carelessly called luminance and given the symbol Y. You must becareful to determine whether a particular author assigns a linear ornonlinear interpretation to the term luminance and the symbol Y.The coefficients that correspond to the "NTSC" red, green and blue CRTphosphors of 1953 are standardized in ITU-R Recommendation BT. 601-2(formerly CCIR Rec. 601-2). I call it Rec. 601. To compute nonlinear videoluma from nonlinear red, green and blue:    Yprime = 0.299 * Rprime + 0.587 * Gprime + 0.114 * Bprime;The prime symbols in this equation, and in those to follow, denotenonlinear components.C-12  WHAT ARE CIE XYZ COMPONENTS?The CIE system is based on the description of color as a luminancecomponent Y, as described above, and two additional components X and Z. Thespectral weighting curves of X and Z have been standardized by the CIEbased on statistics from experiments involving human observers. XYZtristimulus values can describe any color. (RGB tristimulus values will bedescribed later.)The magnitudes of the XYZ components are proportional to physical energy,but their spectral composition corresponds to the color matchingcharacteristics of human vision.The CIE system is defined in Publication CIE No 15.2, Colorimetry, SecondEdition (1986) [4].C-13  DOES MY SCANNER USE THE CIE SPECTRAL CURVES?Probably not. Scanners are most often used to scan images such as colorphotographs and color offset prints that are already "records" of threecomponents of color information. The usual task of a scanner is notspectral analysis but extraction of the values of the three components thathave already been recorded. Narrowband filters are more suited to this taskthan filters that adhere to the principles of colorimetry.If you place on your scanner an original colored object that has"original" SPDs that are not already a record of three components, chancesare your scanner will not very report accurate RGB values. This is becausemost scanners do not conform very closely to CIE standards.C-14  WHAT ARE CIE x AND y CHROMATICITY COORDINATES?It is often convenient to discuss "pure" color in the absence ofbrightness. The CIE defines a normalization process to compute "little" xand y chromaticity coordinates:  x = X / (X + Y + Z);      y = Y / (X + Y + Z);A color plots as a point in an (x, y) chromaticity diagram. When anarrowband SPD comprising power at just one wavelength is swept across therange 400 nm to 700 nm, it traces a shark-fin shaped spectral locus in (x,y) coordinates. The sensation of purple cannot be produced by a singlewavelength: to produce purple requires a mixture of shortwave and longwavelight. The line of purples on a chromaticity diagram joins extreme blue toextreme red. All colors are contained in the area in (x, y) bounded by theline of purples and the spectral locus.A color can be specified by its chromaticity and luminance, in the form ofan xyY triple. To recover X and Z from chromaticities and luminance, usethese relations:  X = (x / y) * Y;    Z = (1 - x - y) / y * Y;The bible of color science is Wyszecki and Styles, Color Science [5]. Butit's daunting. For Wyszecki's own condensed version, see Color in Business,Science and Industry, Third Edition [6]. It is directed to the colorindustry: ink, paint and the like. For an approachable introduction to thesame theory, accompanied by descriptions of image reproduction, try to finda copy of R.W.G. Hunt, The Reproduction of Colour [7]. But sorry to report,as I write this, it's out of print.C-15  WHAT IS WHITE?In additive image reproduction, the white point is the chromaticity of thecolor reproduced by equal red, green and blue components. White point is afunction of the ratio (or balance) of power among the primaries. Insubtractive reproduction, white is the SPD of the illumination, multipliedby the SPD of the media. There is no unique physical or perceptualdefinition of white, so to achieve accurate color interchange you mustspecify the characteristics of your white.It is often convenient for purposes of calculation to define white as auniform SPD. This white reference is known as the equal-energy illuminant,or CIE Illuminant E.A more realistic reference that approximates daylight has been specifiednumerically by the CIE as Illuminant D65. You should use this unless youhave a good reason to use something else. The print industry commonly usesD50 and photography commonly uses D55. These represent compromises betweenthe conditions of indoor (tungsten) and daylight viewing.C-16  WHAT IS COLOR TEMPERATURE?Planck determined that the SPD radiated from a hot object - a black bodyradiator - is a function of the temperature to which the object is heated.Many sources of illumination have, at their core, a heated object, so it isoften useful to characterize an illuminant by specifying the temperature(in units of kelvin, K) of a black body radiator that appears to have thesame hue.Although an illuminant can be specified informally by its colortemperature, a more complete specification is provided by the chromaticitycoordinates of the SPD of the source.Modern blue CRT phosphors are more efficient with respect to human visionthan red or green. In a quest for brightness at the expense of coloraccuracy, it is common for a computer display to have excessive bluecontent, about twice as blue as daylight, with white at about 9300 K.Human vision adapts to white in the viewing environment. An image viewed inisolation - such as a slide projected in a dark room - creates its ownwhite reference, and a viewer will be quite tolerant of errors in the whitepoint. But if the same image is viewed in the presence of an external whitereference or a second image, then differences in white point can beobjectionable.Complete adaptation seems to be confined to the range 5000 K to 5500 K. Formost people, D65 has a little hint of blue. Tungsten illumination, at about3200 K, always appears somewhat yellow.C-17  HOW CAN I CHARACTERIZE RED, GREEN AND BLUE?Additive reproduction is based on physical devices that produceall-positive SPDs for each primary. Physically and mathematically, thespectra add. The largest range of colors will be produced with primariesthat appear red, green and blue. Human color vision obeys the principle ofsuperposition, so the color produced by any additive mixture of threeprimary spectra can be predicted by adding the corresponding fractions ofthe XYZ components of the primaries: the colors that can be mixed from aparticular set of RGB primaries are completely determined by the colors ofthe primaries by themselves. Subtractive reproduction is much morecomplicated: the colors of mixtures are determined by the primaries and bythe colors of their combinations.An additive RGB system is specified by the chromaticities of its primariesand its white point. The extent (gamut) of the colors that can be mixedfrom a given set of RGB primaries is given in the (x, y) chromaticitydiagram by a triangle whose vertices are the chromaticities of theprimaries.In computing there are no standard primaries or white point. If you have anRGB image but have no information about its chromaticities, you cannotaccurately reproduce the image.The NTSC in 1953 specified a set of primaries that were representative ofphosphors used in color CRTs of that era. But phosphors changed over theyears, primarily in response to market pressures for brighter receivers,and by the time of the first the videotape recorder the primaries in usewere quite different than those "on the books". So although you may see theNTSC primary chromaticities documented, they are of no use today.Contemporary studio monitors have slightly different standards in NorthAmerica, Europe and Japan. But international agreement has been obtained onprimaries for high definition television (HDTV), and these primaries areclosely representative of contemporary monitors in studio video, computingand computer graphics. The primaries and the D65 white point of Rec. 709[8] are:         x       y       zR        0.6400  0.3300  0.0300G        0.3000  0.6000  0.1000B        0.1500  0.0600  0.7900 white    0.3127  0.3290  0.3582For a discussion of nonlinear RGB in computer graphics, see Lindbloom [9]. For technical details on monitor calibration, consult Cowan [10].C-18  HOW DO I TRANSFORM BETWEEN CIE XYZ AND A PARTICULAR SET OF RGB      PRIMARIES?RGB values in a particular set of primaries can be transformed to and fromCIE XYZ by a three-by-three matrix transform. These transforms involvetristimulus values, that is, sets of three linear-light components thatconform to the CIE color matching functions. CIE XYZ is a special case oftristimulus values. In XYZ, any color is represented by a positive set ofvalues.Details can be found in SMPTE RP 177-1993 [11].To transform from CIE XYZ into Rec. 709 RGB (with its D65 white point), putan XYZ column vector to the right of this matrix, and multiply: [ R709 ] [ 3.240479 -1.53715  -0.498535 ] [ X ]  [ G709 ]=[-0.969256  1.875991  0.041556 ]*[ Y ]  [ B709 ] [ 0.055648 -0.204043  1.057311 ] [ Z ] As a convenience to C programmers, here are the coefficients as a C array:{{ 3.240479,-1.53715 ,-0.498535}, {-0.969256, 1.875991, 0.041556}, { 0.055648,-0.204043, 1.057311}}This matrix has some negative coefficients: XYZ colors that are out ofgamut for a particular RGB transform to RGB where one or more RGBcomponents is negative or greater than unity.Here's the inverse matrix. Because white is normalized to unity, themiddle row sums to unity: [ X ] [ 0.412453  0.35758   0.180423 ] [ R709 ]  [ Y ]=[ 0.212671  0.71516   0.072169 ]*[ G709 ]  [ Z ] [ 0.019334  0.119193  0.950227 ] [ B709 ]  {{ 0.412453, 0.35758 , 0.180423}, { 0.212671, 0.71516 , 0.072169}, { 0.019334, 0.119193, 0.950227}}To recover primary chromaticities from such a matrix, compute little x andy for each RGB column vector. To recover the white point, transform RGB=[1,1, 1] to XYZ, then compute x and y.C-19  IS RGB ALWAYS DEVICE-DEPENDENT?Video standards specify abstract R'G'B' systems that are closelymatched to the characteristics of real monitors. Physical devices thatproduce additive color involve tolerances and uncertainties, but if youhave a monitor that conforms to Rec. 709 within some tolerance, you canconsider the monitor to be device-independent.The importance of Rec. 709 as an interchange standard in studio video,broadcast television and high definition television, and the perceptualbasis of the standard, assures that its parameters will be used even bydevices such as flat-panel displays that do not have the same physics asCRTs.C-20  HOW DO I TRANSFORM DATA FROM ONE SET OF RGB PRIMARIES TO ANOTHER?RGB values in a system employing one set of primaries can be transformedinto another set by a three-by-three linear-light matrix transform.Generally these matrices are normalized for a white point luminance ofunity. For details, see Television Engineering Handbook [12].As an example, here is the transform from SMPTE 240M (or SMPTE RP 145) RGBto Rec. 709: [ R709 ] [ 0.939555  0.050173  0.010272 ] [ R240M ]  [ G709 ]=[ 0.017775  0.965795  0.01643  ]*[ G240M ]  [ B709 ] [-0.001622 -0.004371  1.005993 ] [ B240M ] {{ 0.939555, 0.050173, 0.010272}, { 0.017775, 0.965795, 0.01643 }, {-0.001622,-0.004371, 1.005993}}All of these terms are close to either zero or one. In a case like this, ifthe transform is computed in the nonlinear (gamma-corrected) R'G'B'domain the resulting errors will be insignificant.Here's another example. To transform EBU 3213 RGB to Rec. 709: [ R709 ] [ 1.044036 -0.044036  0.       ] [ R240M ]  [ G709 ]=[ 0.        1.        0.       ]*[ G240M ]  [ B709 ] [ 0.        0.011797  0.988203 ] [ B240M ] {{ 1.044036,-0.044036, 0.      }, { 0.      , 1.      , 0.      }, { 0.      , 0.011797, 0.988203}}Transforming among RGB systems may lead to an out of gamut RGB result whereone or more RGB components is negative or greater than unity.C-21  SHOULD I USE RGB OR XYZ FOR IMAGE SYNTHESIS?Once light is on its way to the eye, any tristimulus-based system willwork. But the interaction of light and objects involves spectra, nottristimulus values. In synthetic computer graphics, the calculations areactually simulating sampled SPDs, even if only three components are used.Details concerning the resultant errors are found in Hall [13].C-22  WHAT IS SUBTRACTIVE COLOR?Subtractive systems involve colored dyes or filters that absorb power fromselected regions of the spectrum. The three filters are placed in tandem. Adye that appears cyan absobs longwave (red) light. By controlling theamount of cyan dye (or ink), you modulate the amount of red in the image.In physical terms the spectral transmission curves of the colorantsmultiply, so this method of color reproduction should really be called"multiplicative". Photographers and printers have for decades measuredtransmission in base-10 logarithmic density units, where transmission ofunity corresponds to a density of 0, transmission of 0.1 corresponds to adensity of 1, transmission of 0.01 corresponds to a density of 2 and so on.When a printer or photographer computes the effect of filters in tandem, hesubtracts density values instead of multiplying transmission values, so hecalls the system subtractive.To achieve a wide range of colors in a subtractive system requires filtersthat appear colored cyan, yellow and magenta (CMY). Cyan in tandem withmagenta produces blue, cyan with yellow produces green, and magenta withyellow produces red. Smadar Nehab suggests this memory aid:  ----+             ----------+   R  | G    B        R    G  | B      |                       |   Cy | Mg   Yl       Cy   Mg | Yl      +----------             +-----Additive primaries are at the top, subtractive at the bottom. On the left,magenta and yellow filters combine to produce red. On the right, red andgreen sources add to produce yellow.C-23  WHY DID MY GRADE THREE TEACHER TELL ME THAT THE PRIMARIES ARE RED,YELLOW AND BLUE?To get a wide range of colors in an additive system, the primaries mustappear red, green and blue (RGB). In a subtractive system the primariesmust appear yellow, cyan and magenta (CMY). It is complicated to predictthe colors produced when mixing paints, but roughly speaking, paints mixadditively to the extent that they are opaque (like oil paints), andsubtractively to the extent that they are transparent (like watercolors).This question also relates to color names: your grade three "red" wasprobably a little on the magenta side, and "blue" was probably quite cyan.For a discussion of paint mixing from a computer graphics perspective,consult Haase [14].C-24  IS CMY JUST ONE-MINUS-RGB?In a theoretical subtractive system, CMY filters could have spectralabsorption curves with no overlap. The color reproduction of the systemwould correspond exactly to additive color reproduction using the red,green and blue primaries that resulted from pairs of filters incombination.Practical photographic dyes and offset printing inks have spectralabsorption curves that overlap significantly. Most magenta dyes absorbmediumwave (green) light as expected, but incidentally absorb about halfthat amount of shortwave (blue) light. If reproduction of a color, saybrown, requires absorption of all shortwave light then the incidentalabsorption from the magenta dye is not noticed. But for other colors, the"one minus RGB" formula produces mixtures with much less blue thanexpected, and therefore produce pictures that have a yellow cast in the midtones. Similar but less severe interactions are evident for the other pairsof practical inks and dyes.Due to the spectral overlap among the colorants, converting CMY using the"one-minus-RGB" method works for applications such as business graphicswhere accurate color need not be preserved, but the method fails toproduce acceptable color images.Multiplicative mixture in a CMY system is mathematically nonlinear, and theeffect of the unwanted absorptions cannot be easily analyzed orcompensated. The colors that can be mixed from a particular set of CMYprimaries cannot be determined from the colors of the primariesthemselves, but are also a function of the colors of the sets ofcombinations of the primaries.Print and photographic reproduction is also complicated by nonlinearitiesin the response of the three (or four) channels. In offset printing, thephysical and optical processes of dot gain introduce nonlinearity that isroughly comparable to gamma correction in video. In a typical system usedfor print, a black code of 128 (on a scale of 0 to 255) produces areflectance of about 0.26, not the 0.5 that you would expect from a linearsystem. Computations cannot be meaningfully performed on CMY componentswithout taking nonlinearity into account.For a detailed discussion of transferring colorimetric image data to printmedia, see Stone [15].C-25  WHY DOES OFFSET PRINTING USE BLACK INK IN ADDITION TO CMY?Printing black by overlaying cyan, yellow and magenta ink in offsetprinting has three major problems. First, colored ink is expensive.Replacing colored ink by black ink - which is primarily carbon - makeseconomic sense. Second, printing three ink layers causes the printed paperto become quite wet. If three inks can be replaced by one, the ink will drymore quickly, the press can be run faster, and the job will be lessexpensive. Third, if black is printed by combining three inks, andmechanical tolerances cause the three inks to be printed slightly out ofregister, then black edges will suffer colored tinges. Vision is mostdemanding of spatial detail in black and white areas. Printing black with asingle ink minimizes the visibility of registration errors.Other printing processes may or may not be subject to similar constraints.C-26  WHAT ARE COLOR DIFFERENCES?This term is ambiguous. In its first sense, color difference refers tonumerical differences between color specifications. The perception ofcolor differences in XYZ or RGB is highly nonuniform. The study ofperceptual uniformity concerns numerical differences that correspond tocolor differences at the threshold of perceptibility (just noticeabledifferences, or JNDs).In its second sense, color difference refers to color components wherebrightness is "removed". Vision has poor response to spatial detail incolored areas of the same luminance, compared to its response to luminancespatial detail. If data capacity is at a premium it is advantageous totransmit luminance with full detail and to form two color differencecomponents each having no contribution from luminance. The two colorcomponents can then have spatial detail removed by filtering, and can betransmitted with substantially less information capacity than luminance.Instead of using a true luminance component to represent brightness, it isubiquitous for practical reasons to use a luma signal that is computednonlinearly as outlined above ( What is luma?  ).The easiest way to "remove" brightness information to form two colorchannels is to subtract it. The luma component already contains a largefraction of the green information from the image, so it is standard to formthe other two components by subtracting luma from nonlinear blue (to formB'-Y') and by subtracting luma from nonlinear red (to form R'-Y').These are called chroma.Various scale factors are applied to (B'-Y') and (R'-Y') for differentapplications. The Y  'PBPR scale factors are optimized for component analogvideo. The Y  'CBCR scaling is appropriate for component digital video suchas studio video, JPEG and MPEG. Kodak's PhotoYCC(tm) uses scale factorsoptimized for the gamut of film colors. Y'UV scaling is appropriate as anintermediate step in the formation of composite NTSC or PAL video signals,but is not appropriate when the components are kept separate. The Y'UVnomenclature is now used rather loosely, and it sometimes denotes anyscaling of (B'-Y') and (R'-Y'). Y  'IQ coding is obsolete.The subscripts in CBCR and PBPR are often written in lower case. I findthis to compromise readability, so without introducing any ambiguity Iwrite them in uppercase. Authors with great attention to detail sometimes"prime" these quantities to indicate their nonlinear nature, but because nopractical image coding system employs linear color differences I considerit safe to omit the primes.C-27  HOW DO I OBTAIN COLOR DIFFERENCE COMPONENTS FROM TRISTIMULUS      VALUES?Here is the block diagram for luma/color difference encoding anddecoding:<< A nice diagram is included in the .PDF and .PS versions. >>From linear XYZ - or linear R1 G1 B1 whose chromaticity coordinates aredifferent from the interchange standard - apply a 3x3 matrix transformto obtain linear RGB according to the interchange primaries. Apply a anonlinear transfer function ("gamma correction") to each of the componentsto get nonlinear R'G'B'. Apply a 3x3 matrix to obtain colordifference components such as Y'PBPR , Y'CBCR or PhotoYCC. If necessary,apply a color subsampling filter to obtain subsampled color differencecomponents. To decode, invert the above procedure: run through the blockdiagram right-to-left using the inverse operations. If your monitorconforms to the interchange primaries, decoding need not explicitly use atransfer function or the tristimulus 3x3.The block diagram emphasizes that 3x3 matrix transforms are used for twodistinctly different tasks. When someone hands you a 3x3, you have to askfor which task it is intended.C-28  HOW DO I ENCODE Y'PBPR COMPONENTS?Although the following matrices could in theory be used for tristimulussignals, it is ubiquitous to use them with gamma-corrected signals.To encode Y'PBPR , start with the basic Y', (B'-Y') and (R'-Y')relationships:Eq 1 [  Y'   601 ] [ 0.299  0.587  0.114 ] [ R' ]  [ B'-Y' 601 ]=[-0.299 -0.587  0.886 ]*[ G' ]  [ R'-Y' 601 ] [ 0.701 -0.587 -0.114 ] [ B' ] {{ 0.299, 0.587, 0.114}, {-0.299,-0.587, 0.886}, { 0.701,-0.587,-0.114}}Y'PBPR components have unity excursion, where Y' ranges [0..+1] and eachof PB and PR ranges [-0.5..+0.5]. The (B'-Y') and (R'-Y') rows need tobe scaled. To encode from R'G'B' where reference black is 0and reference white is +1:Eq 2 [  Y'  601 ] [ 0.299     0.587     0.114    ] [ R' ]  [  PB  601 ]=[-0.168736 -0.331264  0.5      ]*[ G' ]  [  PR  601 ] [ 0.5      -0.418688 -0.081312 ] [ B' ] {{ 0.299   , 0.587   , 0.114   }, {-0.168736,-0.331264, 0.5     }, { 0.5     ,-0.418688,-0.081312}}The first row comprises the luma coefficients; these sum to unity. Thesecond and third rows each sum to zero, a necessity for color differencecomponents. The +0.5 entries reflect the maximum excursion of PB and PR of+0.5, for the blue and red primaries [0, 0, 1] and [1, 0, 0].The inverse, decoding matrix is this: [ R' ] [ 1.        0.        1.402    ] [  Y'  601 ]  [ G' ]=[ 1.       -0.344136 -0.714136 ]*[  PB  601 ]  [ B' ] [ 1.        1.772     0.       ] [  PR  601 ] {{ 1.      , 0.      , 1.402   }, { 1.      ,-0.344136,-0.714136}, { 1.      , 1.772   , 0.      }}C-29  HOW DO I ENCODE Y'CBCR COMPONENTS FROM R'G'B' IN [0, +1]?Rec. 601 specifies eight-bit coding where Y' has an excursion of 219 andan offset of +16. This coding places black at code 16 and white at code235, reserving the extremes of the range for signal processing headroom andfootroom. CB and CR have excursions of +/-112 and offset of +128, for arange of 16 through 240 inclusive.To compute Y'CBCR from R'G'B' in the range [0..+1], scale the rows ofthe matrix of Eq 2 by the factors 219, 224 and 224, corresponding to theexcursions of each of the components:Eq 3{{    65.481,   128.553,    24.966}, {   -37.797,   -74.203,   112.   }, {   112.   ,   -93.786,   -18.214}}Add [16, 128, 128] to the product to get Y'CBCR. Summing the first row of the matrix yields 219, the luma excursion fromblack to white. The two entries of 112 reflect the positive CBCR extrema ofthe blue and red primaries.Clamp all three components to the range 1 through 254 inclusive, since Rec.601 reserves codes 0 and 255 for synchronization signals.To recover R'G'B' in the range [0..+1] from Y'CBCR, subtract [16, 128, 128]from Y'CBCR, then multiply by the inverse of the matrix in Eq 3 above:{{ 0.00456621, 0.        , 0.00625893}, { 0.00456621,-0.00153632,-0.00318811}, { 0.00456621, 0.00791071, 0.        }}This looks scary, but the Y'CBCR components are integers in eightbits and the reconstructed R'G'B' are scaled down to the range[0..+1].C-30  HOW DO I ENCODE Y'CBCR COMPONENTS FROM COMPUTER R'G'B' ?In computing it is conventional to use eight-bit coding with black at code 0and white at 255. To encode Y'CBCR from R'G'B' in the range [0..255], usingeight-bit binary arithmetic, scale the Y'CBCR matrix of Eq 3 by 256/255:{{    65.738,   129.057,    25.064}, {   -37.945,   -74.494,   112.439}, {   112.439,   -94.154,   -18.285}}The entries in this matrix have been scaled up by 256, assuming that you willimplement the equation in fixed-point binary arithmetic, using a shift by eightbits. Add [16, 128, 128] to the product to get Y'CBCR. To decode R'G'B' in the range [0..255] from Rec. 601 Y'CBCR, usingeight-bit binary arithmetic , subtract [16, 128, 128] from Y'CBCR, then multiply by the inverse of the matrix above, scaled by 256:Eq 4{{   298.082,     0.   ,   408.583}, {   298.082,  -100.291,  -208.12 }, {   298.082,   516.411,     0.   }}You can remove a factor of 1/256 from these coefficients, then accomplish themultiplication by shifting. Some of the coefficients, when scaled by 256, arelarger than unity. These coefficients will need more than eight multiplierbits.For implementation in binary arithmetic the matrix coefficients have to berounded. When you round, take care to preserve the row sums of [1, 0, 0].The matrix of Eq 4 will decode standard Y'CBCR components to RGBcomponents in the range [0..255], subject to roundoff error. You must takecare to avoid overflow due to roundoff error. But you must protect againstoverflow in any case, because studio video signals use the extremes of thecoding range to handle signal overshoot and undershoot, and these willrequire clipping when decoded to an RGB range that has no headroom orfootroom.C-31  HOW DO I ENCODE Y'CBCR COMPONENTS FROM STUDIO VIDEO?Studio R'G'B' signals use the same 219 excursion as the luma componentof Y'CBCR. To encode Y'CBCR from R'G'B' in the range [0..219], usingeight-bit binary arithmetic, scale the Y'CBCR encoding matrix of Eq 3above by 256/219. Here is the encoding matrix for studio video:{{    65.481,   128.553,    24.966}, {   -37.797,   -74.203,   112.   }, {   112.   ,   -93.786,   -18.214}}To decode R'G'B' in the range [0..219] from Y'CBCR, using eight-bitbinary arithmetic, use this matrix:{{   256.   ,     0.   ,   350.901}, {   256.   ,   -86.132,  -178.738}, {   256.   ,   443.506,     0.   }} When scaled by 256, the first column in this matrix is unity, indicatingthat the corresponding component can simply be added: there is no need fora multiplication operation. This matrix contains entries larger than 256;the corresponding multipliers will need capability for nine bits.The matrices in this section conform to Rec. 601 and apply directly toconventional 525/59.94 and 625/50 video. It is not yet decided whetheremerging HDTV standards will use the same matrices, or adopt a new set ofmatrices having different luma coefficients. In my view it would beunfortunate if different matrices were adopted, because then image codingand decoding would depend on whether the picture was small (conventionalvideo) or large (HDTV).In digital video, Rec. 601 standardizes subsampling denoted 4:2:2, where CBand CR components are subsampled horizontally by a factor of two withrespect to luma. JPEG and MPEG conventionally subsample by a factor of twoin the vertical dimension as well, denoted 4:2:0.Color difference coding is standardized in Rec. 601. For details on colordifference coding as used in video, consult Poynton [16].C-32  HOW DO I DECODE R'G'B' FROM PHOTOYCC?Kodak's PhotoYCC uses the Rec. 709 primaries, white point and transferfunction. Reference white codes to luma 189; this preserves filmhighlights. The color difference coding is asymmetrical, to encompass filmgamut. You are unlikely to encounter any raw image data in PhotoYCC formbecause YCC is closely associated with the PhotoCD(tm) system whosecompression methods are proprietary. But just in case, the followingequation is comparable to  in that it produces R'G'B' in the range[0..+1] from integer YCC. If you want to return R'G'B' in a differentrange, or implement the equation in eight-bit integer arithmetic, use thetechniques in the section above.[ R'709 ] [ 0.0054980  0.0000000  0.0051681 ]    [ Y'601,189 ]   [   0 ][ G'709 ]=[ 0.0054980 -0.0015446 -0.0026325 ]* ( [    C1     ] - [ 156 ] )[ B'709 ] [ 0.0054980  0.0079533  0.0000000 ]    [    C2     ]   [ 137 ]{{ 0.0054980,  0.0000000,  0.0051681}, { 0.0054980, -0.0015446, -0.0026325}, { 0.0054980,  0.0079533,  0.0000000}}Decoded R'G'B' components from PhotoYCC can exceed unity or go belowzero. PhotoYCC extends the Rec. 709 transfer function above unity, andreflects it around zero, to accommodate wide excursions of R'G'B'. Todecode to CRT primaries, clip R'G'B' to the range zero to one.C-33  WILL YOU TELL ME HOW TO DECODE Y'UV AND Y'IQ?No, I won't! Y'UV and Y'IQ have scale factors appropriate to compositeNTSC and PAL. They have no place in component digital video! You shouldn'tcode into these systems, and if someone hands you an image claiming it'sY'UV, chances are it's actually Y'CBCR, it's got the wrong scale factors,or it's linear-light.Well OK, just this once. To transform Y', (B'-Y') and (R'-Y')components from Eq 1 to Y'UV, scale (B'-Y') by 0.492111 to get U andscale R'-Y' by 0.877283 to get V. The factors are chosen to limitcomposite NTSC or PAL amplitude for all legal R'G'B' values:  << Equation omitted -- see PostScript or PDF version. >>To transform to Y'IQ to Y'UV, perform a 33 degree rotation and an exchangeof color difference axes:  << Equation omitted -- see PostScript or PDF version. >>C-34  HOW SHOULD I TEST MY ENCODERS AND DECODERS?To test your encoding and decoding, ensure that colorbars are handledcorrectly. A colorbar signal comprises a binary RGB sequence ordered fordecreasing luma: white, yellow, cyan, green, magenta, red, blue and black.  [ 1 1 0 0 1 1 0 0 ]  [ 1 1 1 1 0 0 0 0 ]  [ 1 0 1 0 1 0 1 0 ]To ensure that your scale factors are correct and that clipping is notbeing invoked, test 75% bars, a colorbar sequence having 75%-amplitudebars instead of 100%.C-35  WHAT IS PERCEPTUAL UNIFORMITY?A system is perceptually uniform if a small perturbation to a componentvalue is approximately equally perceptible across the range of that value.The volume control on your radio is designed to be perceptually uniform:rotating the knob ten degrees produces approximately the same perceptualincrement in volume anywhere across the range of the control. If thecontrol were physically linear, the logarithmic nature of human loudnessperception would place all of the perceptual "action" of the control at thebottom of its range.The XYZ and RGB systems are far from exhibiting perceptual uniformity.Finding a transformation of XYZ into a reasonably perceptually-uniformspace consumed a decade or more at the CIE and in the end no single systemcould be agreed. So the CIE standardized two systems, L*u*v* and L*a*b*,sometimes written CIELUV and CIELAB. (The u and v are unrelated to video Uand V.) Both L*u*v* and L*a*b* improve the 80:1 or so perceptualnonuniformity of XYZ to about 6:1. Both demand too much computation toaccommodate real-time display, although both have been successfully appliedto image coding for printing.Computation of CIE L*u*v* involves intermediate u' and v ' quantities,where the prime denotes the successor to the obsolete 1960 CIE u and vsystem:  uprime = 4 * X / (X + 15 * Y + 3 * Z);   vprime = 9 * Y / (X + 15 * Y + 3 * Z); First compute un' and vn' for your reference white Xn , Yn  and Zn. Thencompute u' and v ' - and L* as discussed earlier - for your colors.Finally, compute:  ustar = 13 * Lstar * (uprime - unprime);  vstar = 13 * Lstar * (vprime - vnprime);L*a*b* is computed as follows, for (X/Xn, Y/Yn, Z/Zn) > 0.01:  astar = 500 * (pow(X / Xn, 1./3.) - pow(Y / Yn, 1./3.));  bstar = 200 * (pow(Y / Yn, 1./3.) - pow(Z / Zn, 1./3.));These equations are great for a few spot colors, but no fun for a millionpixels. Although it was not specifically optimized for this purpose, thenonlinear R'G'B' coding used in video is quite perceptually uniform,and has the advantage of being fast enough for interactive applications.C-36  WHAT ARE HSB AND HLS?HSB and HLS were developed to specify numerical Hue, Saturation andBrightness (or Hue, Lightness and Saturation) in an age when users had tospecify colors numerically. The usual formulations of HSB and HLS areflawed with respect to the properties of color vision. Now that users canchoose colors visually, or choose colors related to other media (such asPANTONE), or use perceptually-based systems like L*u*v* and L*a*b*, HSB andHLS should be abandoned.Here are some of problems of HSB and HLS. In color selection where"lightness" runs from zero to 100, a lightness of 50 should appear to behalf as bright as a lightness of 100. But the usual formulations of HSB andHLS make no reference to the linearity or nonlinearity of the underlyingRGB, and make no reference to the lightness perception of human vision.The usual formulation of HSB and HLS compute so-called "lightness" or"brightness" as (R+G+B)/3. This computation conflicts badly with theproperties of color vision, as it computes yellow to be about six timesmore intense than blue with the same "lightness" value (say L=50).HSB and HSL are not useful for image computation because of thediscontinuity of hue at 360 degrees. You cannot perform arithmetic mixturesof colors expressed in polar coordinates.Nearly all formulations of HSB and HLS involve different computationsaround 60 degree segments of the hue circle. These calculations introducevisible discontinuities in color space.Although the claim is made that HSB and HLS are "device independent", theubiquitous formulations are based on RGB components whose chromaticitiesand white point are unspecified. Consequently, HSB and HLS are useless forconveyance of accurate color information.If you really need to specify hue and saturation by numerical values,rather than HSB and HSL you should use polar coordinate version of u* andv*: h*uv for hue angle and c*uv  for chroma.C-37  WHAT IS TRUE COLOR?True color is the provision of three separate components for additive red,green and blue reproduction. True color systems often provide eight bitsfor each of the three components, so true color is sometimes referred toas 24-bit color.A true color system usually interposes a lookup table between eachcomponent of the framestore and each channel to the display. This makes itpossible to use a true color system with either linear or nonlinearcoding. In the X Window System, direct color refers to fixed lookup tables,and truecolor refers to lookup tables that are under the control ofapplication software.C-38  WHAT IS INDEXED COLOR?Indexed color (or pseudocolor), is the provision of a relatively smallnumber, say 256, of discrete colors in a colormap or palette. Theframebuffer stores, at each pixel, the index number of a color. At theoutput of the framebuffer, a lookup table uses the index to retrieve red,green and blue components that are then sent to the display.The colors in the map may be fixed systematically at the design of asystem. As an example, 216 index entries an eight-bit indexed color systemcan be partitioned systematically into a 6x6x6 "cube" to implement whatamounts to a direct color system where each of red, green and blue has avalue that is an integer in the range zero to five.An RGB image can be converted to a predetermined colormap by choosing, foreach pixel in the image, the colormap index corresponding to the "closest"RGB triple. With a systematic colormap such as a 6x6x6 colorcube thisis straightforward. For an arbitrary colormap, the colormap has to besearched looking for entries that are "close" to the requested color."Closeness" should be determined according to the perceptibility of colordifferences. Using color systems such as CIE L*u*v* or L*a*b* iscomputationally prohibitive, but in practice it is adequate to use aEuclidean distance metric in R'G'B' components coded nonlinearlyaccording to video practice.A direct color image can be converted to indexed color with animage-dependent colormap by a process of color quantization that searchesthrough all of the triples used in the image, and chooses the palette forthe image based on the colors that are in some sense most "important".Again, the decisions should be made according to the perceptibility ofcolor differences. Adobe Photoshop(tm) can perform this conversion.UNIX(tm) users can employ the pbm package.If your system accommodates arbitrary colormaps, when the map associatedwith the image in a particular window is loaded into the hardware colormap,the maps associated with other windows may be disturbed. In window systemsuch as the X Window System(tm) running on a multitasking operating systemsuch as UNIX, even moving the cursor between two windows with differentmaps can cause annoying colormap flashing.An eight-bit indexed color system requires less data to represent apicture than a twenty-four bit truecolor system. But this data reductioncomes at a high price. The truecolor system can represent each of itsthree components according to the principles of sampled continuous signals.This makes it possible to accomplish, with good quality, operations such asresizing the image. In indexed color these operations introduce severeartifacts because the underlying representation lacks the properties of acontinuous representation, even if converted back to RGB.In graphic file formats such as GIF of TIFF, an indexed color image isaccompanied by its colormap. Generally such a colormap has RGB entries thatare gamma corrected: the colormap's RGB codes are intended to be presenteddirectly to a CRT, without further gamma correction.C-39  I WANT TO VISUALIZE A SCALAR FUNCTION OF TWO VARIABLES. SHOULD I USE RGB      VALUES CORRESPONDING TO THE COLORS OF THE RAINBOW?When you look at a rainbow you do not see a smooth gradation of colors.Instead, some bands appear quite narrow, and others are quite broad.Perceptibility of hue variation near 540 nm is half that of either 500 nmor 600 nm. If you use the rainbow's colors to represent data, thevisibility of differences among your data values will depend on where theylie in the spectrum.If you are using color to aid in the visual detection of patterns, youshould use colors chosen according to the principles of perceptualuniformity. This an open research problem, but basing your system on CIEL*a*b* or L*u*v*, or on nonlinear video-like RGB, would be a good start.C-40  WHAT IS DITHERING?A display device may have only a small number of choices of greyscalevalues or color values at each device pixel. However if the viewer issufficiently distant from the display, the value of neighboring pixels canbe set so that the viewer's eye integrates several pixels to achieve anapparent improvement in the number of levels or colors that can bereproduced.Computer displays are generally viewed from distances where the devicepixels subtend a rather large angle at the viewer's eye, relative to hisvisual acuity. Applying dither to a conventional computer display oftenintroduces objectionable artifacts. However, careful application of dithercan be effective. For example, human vision has poor acuity for bluespatial detail but good color discrimination capability in blue. Blue canbe dithered across two-by-two pixel arrays to produce four times the numberof blue levels, with no perceptible penalty at normal viewing distances.C-41  HOW DOES HALFTONING RELATE TO COLOR?The processes of offset printing and conventional laser printing areintrinsically bilevel: a particular location on the page is either coveredwith ink or not. However, each of these devices can reproduceclosely-spaced dots of variable size. An array of small dots produces theperception of light gray, and an array of large dots produces dark gray.This process is called halftoning or screening. In a sense this isdithering, but with device dots so small that acceptable pictures can beproduced at reasonable viewing distances.Halftone dots are usually placed in a regular grid, although stochasticscreening has recently been introduced that modulates the spacing of thedots rather than their size.In color printing it is conventional to use cyan, magenta, yellow andblack grids that have exactly the same dot pitch but differentcarefully-chosen screen angles. The recently introduced technique ofFlamenco screening uses the same screen angles for all screens, but itsregistration requirements are more stringent than conventional offsetprinting.Agfa's booklet [17] is an excellent introduction to practical concerns ofprinting. And it's in color! The standard reference to halftoningalgorithms is Ulichney [18], but that work does not detail thenonlinearities found in practical printing systems. For details aboutscreening for color reproduction, consult Fink [19]. Consult FrequentlyAsked Questions about Gamma for an introduction to the transfer function ofoffset printing.C-42  WHAT'S A COLOR MANAGEMENT SYSTEM?Software and hardware for scanner, monitor and printer calibration have hadlimited success in dealing with the inaccuracies of color handling indesktop computing. These solutions deal with specific pairs of devices butcannot address the end-to-end system. Certain application developers haveadded color transformation capability to their applications, but themajority of application developers have insufficient expertise andinsufficient resources to invest in accurate color.A color management system (CMS) is a layer of software resident on acomputer that negotiates color reproduction between the application andcolor devices. It cooperates with the operating system and the graphicslibrary components of the platform software. Color management systemsperform the color transformations necessary to exchange accurate colorbetween diverse devices, in various color coding systems including RGB,CMYK and CIE L*a*b*.The CMS makes available to the application a set of facilities whereby theapplication can determine what color devices and what color spaces areavailable. When the application wishes to access a particular device, itrequests that the color manager perform a mathematical transform from onespace to another. The color spaces involved can be device-independentabstract color spaces such as CIE XYZ, CIE L*a*b* or calibrated RGB.Alternatively a color space can be associated with a particular device. Inthe second case the Color manager needs access to characterization datafor the device, and perhaps also to calibration data that reflects thestate of the particular instance of the device.Sophisticated color management systems are commercially available fromKodak, Electronics for Imaging (EFI) and Agfa. Apple's ColorSync(tm)provides an interface between a Mac application program and colormanagement capabilities either built-in to ColorSync or provided by aplug-in. Sun has announced that Kodak's CMS will be shipped with the nextversion of Solaris.The basic CMS services provided with desktop operating systems are likelyto be adequate for office users, but are unlikely to satisfy high-end userssuch as in prepress. All of the announced systems have provisions forplug-in color management modules (CMMs) that can provide sophisticatedtransform machinery. Advanced color management modules will becommercially available from third parties. C-43  HOW DOES A CMS KNOW ABOUT PARTICULAR DEVICES?A CMS needs access to information that characterizes the colorreproduction capabilities of particular devices. The set ofcharacterization data for a device is called a device profile. Industryagreement has been reached on the format of device profiles, althoughdetails have not yet been publicly disseminated. Apple has announced thatthe forthcoming ColorSync version 2.0 will adhere to this agreement.Vendors of color peripherals will soon provide industry-standard profileswith their devices, and they will have to make, buy or rentcharacterization services.If you have a device that has not been characterized by its manufacturer,Agfa's FotoTune(tm) software - part of Agfa's FotoFlow(tm) color manager -can create device profiles.C-44  IS A COLOR MANAGEMENT SYSTEM USEFUL FOR COLOR SPECIFICATION?Not yet. But color management system interfaces in the future are likelyto include the ability to accommodate commercial proprietary colorspecification systems such as pantone(tm) and colorcurve(tm). These vendorsare likely to provide their color specification systems in shrink-wrappedform to plug into color managers. In this way, users will have guaranteedcolor accuracy among applications and peripherals, and application vendorswill no longer need to pay to license these systems individually.C-45  I'M NOT A COLOR EXPERT. WHAT PARAMETERS SHOULD I USE TO CODE MY      IMAGES?Use the CIE D65 white point (6504 K) if you can.Use the Rec. 709 primary chromaticities. Your monitor is probably alreadyquite close to this. Rec. 709 has international agreement, offers excellentperformance, and is the basis for HDTV development so it's future-proof.If you need to operate in linear light, so be it. Otherwise, for bestperceptual performance and maximum ease of interchange with digital video,use the Rec. 709 transfer function, with its 0.45-power law. If you needMac compatibility you will have to suffer a penalty in perceptualperformance. Raise tristimulus values to the 1/1.8-power before presentingthem to QuickDraw.To code luma, use the Rec. 601 luma coefficients 0.299, 0.587 and 0.114.Use Rec. 601 digital video coding with black at 16 and white at 235.Use prime symbols (') to denote all of your nonlinear components!PhotoCD uses all of the preceding measures. PhotoCD codes colordifferences asymmetrically, according to film gamut. Unless you have arequirement for film gamut, you should code into color differences usingY'CBCR coding with Rec. 601 studio video (16..235/128+/-112) excursion.Tag your image data with the primary and white chromaticity, transferfunction and luma coefficients that you are using. TIFF 6.0 tags have beendefined for these parameters. This will enable intelligent readers, todayor in the future, to determine the parameters of your coded image and giveyou the best possible results.

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