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Stock Answer: Gamma and Color FAQ

In video, computer graphics and image processing the "gamma" symbol represents a numerical parameter that describes the nonlinearity of intensity reproduction. The Gamma FAQ section of this document clarifies aspects of nonlinear image coding.

The Colour FAQ section of this document clarifies aspects of colour specification and image coding that are important to computer graphics, image processing, video, and the transfer of digital images to print.

Colorspace-faq was originated by David Bourgin <dbourgin@turing.imag.fr>, who has ceded responsibility for maintenance at RTFM and periodic posting. This version of colorspace-faq comprises concatenated GammaFAQ and ColorFAQ documents which were previously available -- and remain available -- by ftp; see G-0 below.

Adrian Ford and Alan Roberts have written a Colour Equations FAQ that details transforms among colour spaces such as RGB, HSI, HSL, CMY and video. Find it at ftp.wmin.ac.uk/pub/itrg/coloureq.txt(48 KB).

Frequently Asked Questions about Gamma

  1. G-1 What is intensity?
  2. G-2 What is luminance?
  3. G-3 What is lightness?
  4. G-4 What is gamma?
  5. G-5 What is gamma correction?
  6. G-6 Does NTSC use a gamma of 2.2?
  7. G-7 Does PAL use a gamma of 2.8?
  8. G-8 I pulled an image off the net and it looks murky.
  9. G-9 I pulled an image off the net and it looks a little too contrasty.
  10. G-10 What is luma?
  11. G-11 What is contrast ratio?
  12. G-12 How many bits do I need to smoothly shade from black to white?
  13. G-13 How is gamma handled in video, computer graphics and desktop computing?
  14. G-14 What is the gamma of a Macintosh?
  15. G-15 Does the gamma of CRTs vary wildly?
  16. G-16 How should I adjust my monitor's brightness and contrast controls?
  17. G-17 Should I do image processing operations on linear or nonlinear image data?
  18. G-18 What's the transfer function of offset printing?
  19. G-19 References

Frequently Asked Questions about Color

  1. C-1 What is colour?
  2. C-2 What is intensity?
  3. C-3 What is luminance?
  4. C-4 What is lightness?
  5. C-5 What is hue?
  6. C-6 What is saturation?
  7. C-7 How is colour specified?
  8. C-8 Should I use a colour specification system for image data?
  9. C-9 What weighting of red, green and blue corresponds to brightness?
  10. C-10 Can blue be assigned fewer bits than red or green?
  11. C-11 What is "luma"?
  12. C-12 What are CIE XYZ components?
  13. C-13 Does my scanner use the CIE spectral curves?
  14. C-14 What are CIE x and y chromaticity coordinates?
  15. C-15 What is white?
  16. C-16 What is colour temperature?
  17. C-17 How can I characterize red, green and blue?
  18. C-18 How do I transform between CIE XYZ and a particular set of RGB primaries?
  19. C-19 Is RGB always device-dependent?
  20. C-20 How do I transform data from one set of RGB primaries to another?
  21. C-21 Should I use RGB or XYZ for image synthesis?
  22. C-22 What is subtractive colour?
  23. C-23 Why did my grade three teacher tell me that the primaries are red, yellow and blue?
  24. C-24 Is CMY just one-minus-RGB?
  25. C-25 Why does offset printing use black ink in addition to CMY?
  26. C-26 What are colour differences?
  27. C-27 How do I obtain colour difference components from tristimulus values?
  28. C-28 How do I encode Y'PBPR components?
  29. C-29 How do I encode Y'CBCR components from R'G'B' in [0, +1]?
  30. C-30 How do I encode Y'CBCR components from computer R'G'B' ?
  31. C-31 How do I encode Y'CBCR components from studio video?
  32. C-32 How do I decode R'G'B' from PhotoYCC?
  33. C-33 Will you tell me how to decode Y'UV and Y'IQ?
  34. C-34 How should I test my encoders and decoders?
  35. C-35 What is perceptual uniformity?
  36. C-36 What are HSB and HLS?
  37. C-37 What is true colour?
  38. C-38 What is indexed colour?
  39. C-39 I want to visualize a scalar function of two variables. Should I use RGB values corresponding to the colours of the rainbow?
  40. C-40 What is dithering?
  41. C-41 How does halftoning relate to colour?
  42. C-42 What's a colour management system?
  43. C-43 How does a CMS know about particular devices?
  44. C-44 Is a colour management system useful for colour specification?
  45. C-45 I'm not a colour expert. What parameters should I use to code my images?
  46. C-46 References

FREQUENTLY ASKED QUESTIONS ABOUT GAMMA


G-1 WHAT IS INTENSITY?


Intensity is a measure over some interval of the electromagnetic spectrum of 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 as watts per square meter.
The voltages presented to a CRT monitor control the intensities of the colour components, but in a nonlinear manner. CRT voltages are not proportional to intensity.
Image data stored in a file (TIFF, JFIF, PPM, etc.) may or may not represent intensity, even if it is so described. The I component of a color described as HSI (hue, saturation, intensity) does not accurately represent intensity if HSI is computed according to any of the usual formulae.

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 appears to emit more or less light. Because brightness perception is very complex, the CIE defined a more tractable quantity luminance, denoted Y, which is radiant power weighted by a spectral sensitivity function that is characteristic of vision. To learn about the relationship between physical spectra and perceived brightness, and other color issues, refer to the companion Frequently Asked Questions about Colour.
The magnitude of luminance is proportional to physical power. In that sense it is like intensity. But the spectral composition of luminance is related to the brightness sensitivity of human vision.

G-3 WHAT IS LIGHTNESS?


Human vision has a nonlinear perceptual response to brightness: a source having a luminance only 18% of a reference luminance appears about half as bright. The perceptual response to luminance is called Lightness and is defined 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 to reference white then you need not compute the fraction. The CIE definition applies 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 if you don't use it, make sure that you limit L* at zero. L* has a range of 0 to 100, and a "delta L-star" of unity is taken to be roughly the threshold of visibility.
Stated differently, lightness perception is roughly logarithmic. You can detect an intensity difference between two patches when the ratio of their intensities differs by more than about one percent.
Video systems approximate the lightness response of vision using RGB signals that are each subject to a 0.45 power function. This is comparable to 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 to any of the usual formulae. See Frequently Asked Questions about Colour.

G-4 WHAT IS GAMMA?


The intensity of light generated by a physical device is not usually a linear function of the applied signal. A conventional CRT has a power-law response to voltage: intensity produced at the face of the display is approximately the applied voltage, raised to the 2.5 power. The numerical value of the exponent of this power function is colloquially known as gamma. This nonlinearity must be compensated in order to achieve correct reproduction of intensity.
As mentioned above (What is lightness?), human vision has a nonuniform perceptual response to intensity. If intensity is to be coded into a small number of steps, say 256, then in order for the most effective perceptual use to be made of the available codes, the codes must be assigned to intensities according to the properties of perception.
Here is a graph of an actual CRT's transfer function, at three different contrast 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 is effectively the inverse of a CRT's nonlinearity. If you apply a transfer function to code a signal to take advantage of the properties of lightness perception - a function similar to the L* function - the coding will be inverted by a CRT.

G-5 WHAT IS GAMMA CORRECTION?


In a video system, linear-light intensity is transformed to a nonlinear video signa by gamma correction, which is universally done at the camera. The Rec. 709 transfer function [2] takes linear-light intensity (here R) to a nonlinear component (here Rprime), for example, voltage in a video system: 
  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 in practical cameras and scanners. Here is a graph of the Rec. 709 transfer function, 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 linear segment, but the precise definition is necessary for accurate intermediate processing in the linear-light domain. In a colour system, an identical transfer function is applied to each of the three tristimulus (linear-light) RGB components. See Frequently Asked Questions about Colour.
By the way, the nonlinearity of a CRT is a function of the electrostatics of the cathode and the grid of an electron gun; it has nothing to do with the phosphor. Also, the nonlinearity is a power function (which has the form 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 correct physical intensity is reproduced in a dim surround, a subjective effect called simultaneous contrast causes the reproduced image to appear lacking in contrast. The effect can be overcome by applying an end-to-end power function whose exponent is about 1.1 or 1.2. Rather than having each receiver provide this correction, the assumed 2.5-power at the CRT is under-corrected at the camera by using an exponent of about 1/2.2 instead of 1/2.5. The assumption of a dim viewing environment is built into video coding.

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 an exponent different from 0.45 is chosen for a power function with a linear segment near black like Rec. 709, the other parameters need to be changed to 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 be applied exactly once. If gamma correction is not applied and linear-light image 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 image is coded according to video standards it implicitly carries the assumption of a dim surround. If it is displayed without correction in a bright ambient, it will appear contrasty. In this circumstance you should apply a power function with an exponent of about 1/1.1 or 1/1.2 to correct for your bright surround.
Ambient lighting is rarely taken into account in the exchange of computer images. If an image is created in a dark environment and transmitted to a viewer in a bright environment, the recipient will find it to have excessive contrast.
If an image originated in a bright environment and viewed in a bright environment, it will need no modification no matter what coding is applied. But then it will carry an assumption of a bright surround. Video standards are widespread and well optimized for vision, so it makes sense to code with a power function of 0.45 and retain a single standard for the assumed viewing environment.
In the long term, for everyone to get the best results in image interchange among applications, an image originator should remove the effect of his ambient environment when he transmits an image. The recipient of an image should insert a transfer function appropriate for his viewing environment. In the short term, you should include with your image data tags that specify the parameters that you used to encode. TIFF 6.0 has provisions for this data. You can correct for your own viewing environment as appropriate, but until image interchange standards incorporate viewing conditions, you will 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 a nonlinear function of true CIE luminance, but as a weighted sum of nonlinear R'G'B' components called luma. For more information, consult the companion document Frequently Asked Questions about Colour.

G-11 WHAT IS CONTRAST RATIO?


Contrast ratio is the ratio of intensity between the brightest white and the darkest black of a particular device or a particular environment. Projected cinema film - or a photographic reflection print - has a contrast ratio of about 80:1. Television assumes a contrast ratio - in your living room - of about 30:1. Typical office viewing conditions restrict contrast ratio 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 a hundred-to-one contrast ratio of intensity from white to black. Call these intensities 100 and 1. Within this range, vision can detect that two intensities 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 that represents different intensity levels 1.00, 1.01, 1.02 and so on. If linear light coding is used, the "delta" of 0.01 must be maintained all the way up the scale to white. This requires about 9,900 codes, or about fourteen bits per component.
If you use nonlinear coding, then the 1.01 "delta" required at the black end of the scale applies as a ratio, not an absolute increment, and progresses like compound interest up to white. This results in about 460 codes, or about nine bits per component. Eight bits, nonlinearly coded according to Rec. 709, is sufficient for broadcast-quality digital television at a contrast ratio of about 50:1.
If poor viewing conditions or poor display quality restrict the contrast ratio of the display, then fewer bits can be employed.
If a linear light system is quantized to a small number of bits, with black at code zero, then the ability of human vision to discern a 1.01 ratio between adjacent intensity levels takes effect below code 100. If a linear light system has only eight bits, then the top end of the scale is only 255, and contouring in dark areas will be perceptible even in very poor viewing conditions.

G-13 HOW IS GAMMA HANDLED IN VIDEO, COMPUTER GRAPHICS AND DESKTOP COMPUTING?


As outlined above, gamma correction in video effectively codes into a perceptually uniform domain. In video, a 0.45-power function is applied at the 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 and objects. These interactions are in the physical domain, and must be calculated in linear-light values. It is conventional in computer graphics to store linear-light values in the framebuffer, and introduce gamma correction at the lookup table at the output of the framebuffer. This is illustrated in the middle row above.
If linear-light is represented in just eight bits, near black the steps between 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 with difficulty.

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 default QuickDraw lookup table and a standard monitor causes intensity to represent the 1.8-power of the R, G and B values presented to QuickDraw. It is wrongly believed that Macintosh computers use monitors whose transfer function is different from the rest of the industry. The unconventional QuickDraw 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 transfer functions: at coding, in the framebuffer, at the lookup table, and at the monitor. Strictly speaking the term gamma applies to the exponent of the power function at the monitor. If you use the term loosely, in the case of a Mac you could call the gamma 1.4, 1.8 or 2.5 depending which part of the system 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 with digital video. If you need Mac compatibility you will have to code intensity with a 1/1.8-power law, anticipating QuickDraw's 1/1.4-power in the lookup table. This coding has adequate performance in the bright viewing environments typical of desktop applications, but suffers in darker viewing 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 about 2.35 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 with black-level error, the exponent deduced will be as much a function of the black error as the true exponent. Second, input devices, graphics libraries and application programs all have the potential to introduce their own transfer functions. Nonlinearities from these sources are often categorized as 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). Display a picture that is predominantly black. Adjust brightness so that the monitor reproduces true black on the screen, just at the threshold where it is not so far down as to "swallow" codes greater than the black code, but not so high that the picture sits on a "pedestal" of dark grey. When the critical 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/colour/Black_and_Picture.pdf>.

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 is necessary. For example, if you want to produce a numerical simulation of a lens performing a Fourier transform, you should use linear coding. If you want to compare your model with the transformed image captured from a real lens by a video camera, you will have to "remove" the nonlinear gamma correction that was imposed by the camera, to convert the image data back into its linear-light representation.
On the other hand, if your computation involves human perception, a nonlinear representation may be required. For example, if you perform a discrete cosine transform on image data as the first step in image compression, as in JPEG, then you ought to use nonlinear coding that exhibits perceptual uniformity, because you wish to minimize the perceptibility of the errors that will be introduced during quantization.
The image processing literature rarely discriminates between linear and nonlinear coding. In the JPEG and MPEG standards there is no mention of transfer function, but nonlinear (video-like) coding is implicit: unacceptable results are obtained when JPEG or MPEG are applied to linear-light data. In computer graphic standards such as PHIGS and CGM there is no mention of transfer function, but linear-light coding is implicit. These discrepancies make it very difficult to exchange image data between systems.
When you ask a video engineer if his system is linear, he will say "Of course!" referring to linear voltage. If you ask an optical engineer if her system is linear, she will say "Of course!" referring to linear intensity. But when a nonlinear transform lies between the two systems, as in video, a linear 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 pixel in terms of dot percentage in film. An imagesetter's halftoning machinery generates dots whose areas are proportional to the requested coverage. In principle, dot percentage in film is inversely proportional to linear-light reflectance.
Two phenomena distort the requested dot coverage values. First, printing involves a mechanical smearing of the ink that causes dots to enlarge. Second, optical effects within the bulk of the paper cause more light to be absorbed than would be expected from the surface coverage of the dot alone. These phenomena are collected under the term dot gain, which is the percentage by which the light absorption of the printed dots exceeds the requested 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 print darker than requested. This results in a transfer function from code to reflectance that closely resembles the voltage-to-light curve of a CRT. Correction of dot gain is conceptually similar to gamma correction in video: physical correction of the "defect" in the reproduction process is very well matched to the lightness perception of human vision. Coding an image in terms of dot percentage in film involves coding into a roughly perceptually uniform space. The standard dot gain functions employed in North America and Europe correspond to intensity being reproduced as a power function of the digital code, where the numerical value of the exponent is about 1.75, compared to about 2.2 for video. This is lower than the optimum for perception, but works well for the low contrast ratio of offset printing.
The Macintosh has a power function that is close enough to printing practice that raw QuickDraw codes sent to an imagesetter produce acceptable results. High-end publishing software allows the user to specify the parameters of dot gain compensation.
I have described the linearity of conventional offset printing. Other halftoned devices have different characteristics, and require different corrections.

G-19 REFERENCES


[1] Publication CIE No 15.2, Colorimetry, Second Edition (1986), Central Bureau of the Commission Internationale de L'Eclairage, Vienna, Austria.
[2] ITU-R Recommendation BT.709, Basic Parameter Values for the HDTV Standard 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 Society of Motion Picture and Television Engineers, Vol. 102, No. 12 (December 1993), 1099-1108.
[4] Charles A. Poynton, "Gamma on the Apple Macintosh", <ftp://ftp.inforamp.net/pub/users/poynton/doc/Mac/>.

FREQUENTLY ASKED QUESTIONS ABOUT COLOR


C-1 WHAT IS COLOUR?


Colour is the perceptual result of light in the visible region of the spectrum, having wavelengths in the region of 400 nm to 700 nm, incident upon the retina. Physical power (or radiance) is expressed in a spectral power distribution (SPD), often in 31 components each representing a 10 nm band.
The human retina has three types of colour photoreceptor cone cells, which respond to incident radiation with somewhat different spectral response curves. A fourth type of photoreceptor cell, the rod, is also present in the retina. Rods are effective only at extremely low light levels (colloquially, night vision), and although important for vision play no role in image reproduction.
Because there are exactly three types of colour photoreceptor, three numerical components are necessary and sufficient to describe a colour, providing that appropriate spectral weighting functions are used. This is the concern of the science of colorimetry. In 1931, the Commission Internationale de L'Eclairage (CIE) adopted standard curves for a hypothetical Standard Observer. These curves specify how an SPD can be transformed into a set of three numbers that specifies a colour.
The CIE system is immediately and almost universally applicable to self-luminous sources and displays. However the colours produced by reflective systems such as photography, printing or paint are a function not only of the colourants but also of the SPD of the ambient illumination. If your application has a strong dependence upon the spectrum of the illuminant, 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 colour exists only in the eye and the brain.

C-2 WHAT IS INTENSITY?


Intensity is a measure over some interval of the electromagnetic spectrum of 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 as watts per square meter.
The voltages presented to a CRT monitor control the intensities of the colour components, but in a nonlinear manner. CRT voltages are not proportional to intensity.

C-3 WHAT IS LUMINANCE?


Brightness is defined by the CIE as the attribute of a visual sensation according to which an area appears to emit more or less light. Because brightness perception is very complex, the CIE defined a more tractable quantity luminance which is radiant power weighted by a spectral sensitivity function that is characteristic of vision. The luminous efficiency of the Standard Observer is defined numerically, is everywhere positive, and peaks at about 555 nm. When an SPD is integrated using this curve as a weighting function, the result is CIE luminance, denoted Y.
The magnitude of luminance is proportional to physical power. In that sense it is like intensity. But the spectral composition of luminance is related to the brightness sensitivity of human vision.
Strictly speaking, luminance should be expressed in a unit such as candelas per meter squared, but in practice it is often normalized to 1 or 100 units with respect to the luminance of a specified or implied white reference. For example, a studio broadcast monitor has a white reference whose luminance 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 source having a luminance only 18% of a reference luminance appears about half as bright. The perceptual response to luminance is called Lightness. It is denoted 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 to reference white then you need not compute the fraction. The CIE definition applies 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 if you don't use it, make sure that you limit L* at zero. L* has a range of 0 to 100, and a "delta L-star" of unity is taken to be roughly the threshold of visibility.
Stated differently, lightness perception is roughly logarithmic. An observer can detect an intensity difference between two patches when their intensities 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 comparable to 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 sensation according to which an area appears to be similar to one of the perceived colours, red, yellow, green and bue, or a combination of two of them. Roughly speaking, if the dominant wavelength of an SPD shifts, the hue of the associated colour will shift.

C-6 WHAT IS SATURATION?


Again from the CIE, saturation is the colourfulness of an area judged in proportion to its brightness. Saturation runs from neutral gray through pastel to saturated colours. Roughly speaking, the more an SPD is concentrated at one wavelength, the more saturated will be the associated colour. You can desaturate a colour by adding light that contains power at all wavelengths.

C-7 HOW IS COLOUR SPECIFIED?


The CIE system defines how to map an SPD to a triple of numerical components that are the mathematical coordinates of colour space. Their function is analagous to coordinates on a map. Cartographers have different map projections for different functions: some map projections preserve areas, others show latitudes and longitudes as straight lines. No single map projection fills all the needs of map users. Similarly, no single colour system fills all of the needs of colour users.
The systems useful today for colour specification include CIE XYZ, CIE xyY, CIE L*u*v* and CIE L*a*b*. Numerical values of hue and saturation are not very useful for colour specification, for reasons to be discussed in section 36.
A colour specification system needs to be able to represent any colour with high precision. Since few colours are handled at a time, a specification system can be computationally complex. Any system for colour specification must be intimately related to the CIE specifications.
You can specify a single "spot" colour using a colour order system such as Munsell. Systems like Munsell come with swatch books to enable visual colour matches, and have documented methods of transforming between coordinates in the system and CIE values. Systems like Munsell are not useful for image data. You can specify an ink colour by specifying the proportions of standard (or secret) inks that can be mixed to make the colour. That's how pantone(tm) works. Although widespread, it's proprietary. No translation to CIE is publicly available.

C-8 SHOULD I USE A COLOUR SPECIFICATION SYSTEM FOR IMAGE DATA?


A digitized colour image is represented as an array of pixels, where each pixel contains numerical components that define a colour. Three components are necessary and sufficient for this purpose, although in printing it is convenient to use a fourth (black) component.
In theory, the three numerical values for image coding could be provided by a colour specification system. But a practical image coding system needs to be computationally efficient, cannot afford unlimited precision, need not be intimately related to the CIE system and generally needs to cover only a reasonably wide range of colours and not all of the colours. So image coding uses different systems than colour 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' such as Y'CBCR. Numerical values of hue and saturation are not useful in colour image coding.
If you manufacture cars, you have to match the colour of paint on the door with the colour of paint on the fender. A colour specification system will be 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 because you have only two coloured elements, the door and the fender. In the second case, the colour coding must be quite efficient because you may have a million coloured elements or more.
For a highly readable short introduction to colour image coding, see DeMarsh and Giorgianni [2]. For a terse, complete technical treatment, read Schreiber [3].

C-9 WHAT WEIGHTING OF RED, GREEN AND BLUE CORRESPONDS TO BRIGHTNESS?


Direct acquisition of luminance requires use of a very specific spectral weighting. However, luminance can also be computed as a weighted sum of red, green and blue components.
If three sources appear red, green and blue, and have the same radiance in the visible spectrum, then the green will appear the brightest of the three because the luminous efficiency function peaks in the green region of the spectrum. The red will appear less bright, and the blue will be the darkest of the three. As a consequence of the luminous efficiency function, all saturated blue colours are quite dark and all saturated yellows are quite light. If luminance is computed from red, green and blue, the coefficients will be a function of the particular red, green and blue spectral weighting functions employed, but the green coefficient will be quite large, the red will have an intermediate value, and the blue coefficient will be the smallest of the three.
Contemporary CRT phosphors are standardized in Rec. 709 [8], to be described in section 17. The weights to compute true CIE luminance from linear 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 be formed as a linear combination of the scanner curves, and assumes that the component signals represent linear-light. Either or both of these conditions can be relaxed to some extent depending on the application.
Some computer systems have computed brightness using (R+G+B)/3. This is at odds with the properties of human vision, as will be discussed under What are HSB and HLS? in section 36.
The coefficients 0.299, 0.587 and 0.114 properly computed luminance for monitors having phosphors that were contemporary at the introduction of NTSC television in 1953. They are still appropriate for computing video luma to be discussed below in section 11. However, these coefficients do not 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, human vision has extraordinarily good colour discrimination capability in blue colours. So if you give blue fewer bits than red or green, you will introduce 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 of luminance and two other components representative of colour. It is important to convey the component representative of luminance in such a way that noise (or quantization) introduced in transmission, processing and storage has a perceptually similar effect across the entire tone scale from black to white. The ideal way to accomplish these goals would be to form a luminance signal by matrixing RGB, then subjecting luminance to a nonlinear transfer function similar to the L* function.
There are practical reasons in video to perform these operations in the opposite order. First a nonlinear transfer function - gamma correction - is applied to each of the linear R, G and B. Then a weighted sum of the nonlinear components is computed to form a signal representative of luminance. The resulting component is related to brightness but is not CIE luminance. Many video engineers call it luma and give it the symbol Y'. It is often carelessly called luminance and given the symbol Y. You must be careful to determine whether a particular author assigns a linear or nonlinear interpretation to the term luminance and the symbol Y.
The coefficients that correspond to the "NTSC" red, green and blue CRT phosphors of 1953 are standardized in ITU-R Recommendation BT. 601-2 (formerly CCIR Rec. 601-2). I call it Rec. 601. To compute nonlinear video luma 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, denote nonlinear components.

C-12 WHAT ARE CIE XYZ COMPONENTS?


The CIE system is based on the description of colour as a luminance component Y, as described above, and two additional components X and Z. The spectral weighting curves of X and Z have been standardized by the CIE based on statistics from experiments involving human observers. XYZ tristimulus values can describe any colour. (RGB tristimulus values will be described later.)
The magnitudes of the XYZ components are proportional to physical energy, but their spectral composition corresponds to the colour matching characteristics of human vision.
The CIE system is defined in Publication CIE No 15.2, Colorimetry, Second Edition (1986) [4].

C-13 DOES MY SCANNER USE THE CIE SPECTRAL CURVES?


Probably not. Scanners are most often used to scan images such as colour photographs and colour offset prints that are already "records" of three components of colour information. The usual task of a scanner is not spectral analysis but extraction of the values of the three components that have already been recorded. Narrowband filters are more suited to this task than filters that adhere to the principles of colorimetry.
If you place on your scanner an original coloured object that has "original" SPDs that are not already a record of three components, chances are your scanner will not very report accurate RGB values. This is because most 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" colour in the absence of brightness. The CIE defines a normalization process to compute "little" x and y chromaticity coordinates: 
  x = X / (X + Y + Z);  
  
  y = Y / (X + Y + Z);

A colour plots as a point in an (x, y) chromaticity diagram. When a narrowband SPD comprising power at just one wavelength is swept across the range 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 single wavelength: to produce purple requires a mixture of shortwave and longwave light. The line of purples on a chromaticity diagram joins extreme blue to extreme red. All colours are contained in the area in (x, y) bounded by the line of purples and the spectral locus.
A colour can be specified by its chromaticity and luminance, in the form of an xyY triple. To recover X and Z from chromaticities and luminance, use these relations: 
  X = (x / y) * Y;
  
  Z = (1 - x - y) / y * Y;

The bible of colour science is Wyszecki and Styles, Color Science [5]. But it's daunting. For Wyszecki's own condensed version, see Color in Business, Science and Industry, Third Edition [6]. It is directed to the colour industry: ink, paint and the like. For an approachable introduction to the same theory, accompanied by descriptions of image reproduction, try to find a 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 the colour reproduced by equal red, green and blue components. White point is a function of the ratio (or balance) of power among the primaries. In subtractive reproduction, white is the SPD of the illumination, multiplied by the SPD of the media. There is no unique physical or perceptual definition of white, so to achieve accurate colour interchange you must specify the characteristics of your white.
It is often convenient for purposes of calculation to define white as a uniform SPD. This white reference is known as the equal-energy illuminant, or CIE Illuminant E.
A more realistic reference that approximates daylight has been specified numerically by the CIE as Illuminant D65. You should use this unless you have a good reason to use something else. The print industry commonly uses D50 and photography commonly uses D55. These represent compromises between the conditions of indoor (tungsten) and daylight viewing.

C-16 WHAT IS COLOUR TEMPERATURE?


Planck determined that the SPD radiated from a hot object - a black body radiator - 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 is often useful to characterize an illuminant by specifying the temperature (in units of kelvin, K) of a black body radiator that appears to have the same hue.
Although an illuminant can be specified informally by its colour temperature, a more complete specification is provided by the chromaticity coordinates of the SPD of the source.
Modern blue CRT phosphors are more efficient with respect to human vision than red or green. In a quest for brightness at the expense of colour accuracy, it is common for a computer display to have excessive blue content, about twice as blue as daylight, with white at about 9300 K.
Human vision adapts to white in the viewing environment. An image viewed in isolation - such as a slide projected in a dark room - creates its own white reference, and a viewer will be quite tolerant of errors in the white point. But if the same image is viewed in the presence of an external white reference or a second image, then differences in white point can be objectionable.
Complete adaptation seems to be confined to the range 5000 K to 5500 K. For most people, D65 has a little hint of blue. Tungsten illumination, at about 3200 K, always appears somewhat yellow.
C-17 HOW CAN I CHARACTERIZE RED, GREEN AND BLUE?
Additive reproduction is based on physical devices that produce all-positive SPDs for each primary. Physically and mathematically, the spectra add. The largest range of colours will be produced with primaries that appear red, green and blue. Human colour vision obeys the principle of superposition, so the colour produced by any additive mixture of three primary spectra can be predicted by adding the corresponding fractions of the XYZ components of the primaries: the colours that can be mixed from a particular set of RGB primaries are completely determined by the colours of the primaries by themselves. Subtractive reproduction is much more complicated: the colours of mixtures are determined by the primaries and by the colours of their combinations.
An additive RGB system is specified by the chromaticities of its primaries and its white point. The extent (gamut) of the colours that can be mixed from a given set of RGB primaries is given in the (x, y) chromaticity diagram by a triangle whose vertices are the chromaticities of the primaries.
In computing there are no standard primaries or white point. If you have an RGB image but have no information about its chromaticities, you cannot accurately reproduce the image.
The NTSC in 1953 specified a set of primaries that were representative of phosphors used in colour CRTs of that era. But phosphors changed over the years, primarily in response to market pressures for brighter receivers, and by the time of the first the videotape recorder the primaries in use were quite different than those "on the books". So although you may see the NTSC primary chromaticities documented, they are of no use today.
Contemporary studio monitors have slightly different standards in North America, Europe and Japan. But international agreement has been obtained on primaries for high definition television (HDTV), and these primaries are closely representative of contemporary monitors in studio video, computing and computer graphics. The primaries and the D65 white point of Rec. 709 [8] are: 
         x       y       z
R        0.6400  0.3300  0.0300
G        0.3000  0.6000  0.1000
B        0.1500  0.0600  0.7900
 
white    0.3127  0.3290  0.3582

For 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 from CIE XYZ by a three-by-three matrix transform. These transforms involve tristimulus values, that is, sets of three linear-light components that conform to the CIE colour matching functions. CIE XYZ is a special case of tristimulus values. In XYZ, any colour is represented by a positive set of values.
Details can be found in SMPTE RP 177-1993 [11].
To transform from CIE XYZ into Rec. 709 RGB (with its D65 white point), put an 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 colours that are out of gamut for a particular RGB transform to RGB where one or more RGB components is negative or greater than unity.
Here's the inverse matrix. Because white is normalized to unity, the middle 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 and y 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 closely matched to the characteristics of real monitors. Physical devices that produce additive colour involve tolerances and uncertainties, but if you have a monitor that conforms to Rec. 709 within some tolerance, you can consider 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 perceptual basis of the standard, assures that its parameters will be used even by devices such as flat-panel displays that do not have the same physics as CRTs.

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 transformed into another set by a three-by-three linear-light matrix transform. Generally these matrices are normalized for a white point luminance of unity. For details, see Television Engineering Handbook [12].
As an example, here is the transform from SMPTE 240M (or SMPTE RP 145) RGB to 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, if the 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 where one 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 will work. But the interaction of light and objects involves spectra, not tristimulus values. In synthetic computer graphics, the calculations are actually 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 COLOUR?


Subtractive systems involve coloured dyes or filters that absorb power from selected regions of the spectrum. The three filters are placed in tandem. A dye that appears cyan absobs longwave (red) light. By controlling the amount of cyan dye (or ink), you modulate the amount of red in the image.
In physical terms the spectral transmission curves of the colourants multiply, so this method of colour reproduction should really be called "multiplicative". Photographers and printers have for decades measured transmission in base-10 logarithmic density units, where transmission of unity corresponds to a density of 0, transmission of 0.1 corresponds to a density 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, he subtracts density values instead of multiplying transmission values, so he calls the system subtractive.
To achieve a wide range of colours in a subtractive system requires filters that appear coloured cyan, yellow and magenta (CMY). Cyan in tandem with magenta produces blue, cyan with yellow produces green, and magenta with yellow 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 and green 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 colours in an additive system, the primaries must appear red, green and blue (RGB). In a subtractive system the primaries must appear yellow, cyan and magenta (CMY). It is complicated to predict the colours produced when mixing paints, but roughly speaking, paints mix additively to the extent that they are opaque (like oil paints), and subtractively to the extent that they are transparent (like watercolours). This question also relates to colour names: your grade three "red" was probably 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 spectral absorption curves with no overlap. The colour reproduction of the system would correspond exactly to additive colour reproduction using the red, green and blue primaries that resulted from pairs of filters in combination.
Practical photographic dyes and offset printing inks have spectral absorption curves that overlap significantly. Most magenta dyes absorb mediumwave (green) light as expected, but incidentally absorb about half that amount of shortwave (blue) light. If reproduction of a colour, say brown, requires absorption of all shortwave light then the incidental absorption from the magenta dye is not noticed. But for other colours, the "one minus RGB" formula produces mixtures with much less blue than expected, and therefore produce pictures that have a yellow cast in the mid tones. Similar but less severe interactions are evident for the other pairs of practical inks and dyes.
Due to the spectral overlap among the colourants, converting CMY using the "one-minus-RGB" method works for applications such as business graphics where accurate colour need not be preserved, but the method fails to produce acceptable colour images.
Multiplicative mixture in a CMY system is mathematically nonlinear, and the effect of the unwanted absorptions cannot be easily analyzed or compensated. The colours that can be mixed from a particular set of CMY primaries cannot be determined from the colours of the primaries themselves, but are also a function of the colours of the sets of combinations of the primaries.
Print and photographic reproduction is also complicated by nonlinearities in the response of the three (or four) channels. In offset printing, the physical and optical processes of dot gain introduce nonlinearity that is roughly comparable to gamma correction in video. In a typical system used for print, a black code of 128 (on a scale of 0 to 255) produces a reflectance of about 0.26, not the 0.5 that you would expect from a linear system. Computations cannot be meaningfully performed on CMY components without taking nonlinearity into account.
For a detailed discussion of transferring colorimetric image data to print media, 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 offset printing has three major problems. First, coloured ink is expensive. Replacing coloured ink by black ink - which is primarily carbon - makes economic sense. Second, printing three ink layers causes the printed paper to become quite wet. If three inks can be replaced by one, the ink will dry more quickly, the press can be run faster, and the job will be less expensive. Third, if black is printed by combining three inks, and mechanical tolerances cause the three inks to be printed slightly out of register, then black edges will suffer coloured tinges. Vision is most demanding of spatial detail in black and white areas. Printing black with a single ink minimizes the visibility of registration errors.
Other printing processes may or may not be subject to similar constraints.

C-26 WHAT ARE COLOUR DIFFERENCES?


This term is ambiguous. In its first sense, colour difference refers to numerical differences between colour specifications. The perception of colour differences in XYZ or RGB is highly nonuniform. The study of perceptual uniformity concerns numerical differences that correspond to colour differences at the threshold of perceptibility (just noticeable differences, or JNDs).
In its second sense, colour difference refers to colour components where brightness is "removed". Vision has poor response to spatial detail in coloured areas of the same luminance, compared to its response to luminance spatial detail. If data capacity is at a premium it is advantageous to transmit luminance with full detail and to form two colour difference components each having no contribution from luminance. The two colour components can then have spatial detail removed by filtering, and can be transmitted with substantially less information capacity than luminance.
Instead of using a true luminance component to represent brightness, it is ubiquitous for practical reasons to use a luma signal that is computed nonlinearly as outlined above ( What is luma? ).
The easiest way to "remove" brightness information to form two colour channels is to subtract it. The luma component already contains a large fraction of the green information from the image, so it is standard to form the other two components by subtracting luma from nonlinear blue (to form B'-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 different applications. The Y 'PBPR scale factors are optimized for component analog video. The Y 'CBCR scaling is appropriate for component digital video such as studio video, JPEG and MPEG. Kodak's PhotoYCC(tm) uses scale factors optimized for the gamut of film colours. Y'UV scaling is appropriate as an intermediate step in the formation of composite NTSC or PAL video signals, but is not appropriate when the components are kept separate. The Y'UV nomenclature is now used rather loosely, and it sometimes denotes any scaling of (B'-Y') and (R'-Y'). Y 'IQ coding is obsolete.
The subscripts in CBCR and PBPR are often written in lower case. I find this to compromise readability, so without introducing any ambiguity I write them in uppercase. Authors with great attention to detail sometimes "prime" these quantities to indicate their nonlinear nature, but because no practical image coding system employs linear colour differences I consider it safe to omit the primes.

C-27 HOW DO I OBTAIN COLOUR DIFFERENCE COMPONENTS FROM TRISTIMULUS VALUES?


Here is the block diagram for luma/colour difference encoding and decoding:
<< A nice diagram is included in the .PDF and .PS versions. >>
From linear XYZ - or linear R1 G1 B1 whose chromaticity coordinates are different from the interchange standard - apply a 3x3 matrix transform to obtain linear RGB according to the interchange primaries. Apply a a nonlinear transfer function ("gamma correction") to each of the components to get nonlinear R'G'B'. Apply a 3x3 matrix to obtain colour difference components such as Y'PBPR , Y'CBCR or PhotoYCC. If necessary, apply a colour subsampling filter to obtain subsampled colour difference components. To decode, invert the above procedure: run through the block diagram right-to-left using the inverse operations. If your monitor conforms to the interchange primaries, decoding need not explicitly use a transfer function or the tristimulus 3x3.
The block diagram emphasizes that 3x3 matrix transforms are used for two distinctly different tasks. When someone hands you a 3x3, you have to ask for which task it is intended.

C-28 HOW DO I ENCODE Y'PBPR COMPONENTS?


Although the following matrices could in theory be used for tristimulus signals, 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 each of PB and PR ranges [-0.5..+0.5]. The (B'-Y') and (R'-Y') rows need to be scaled. To encode from R'G'B' where reference black is 0 and 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. The second and third rows each sum to zero, a necessity for colour difference components. 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 and an offset of +16. This coding places black at code 16 and white at code 235, reserving the extremes of the range for signal processing headroom and footroom. CB and CR have excursions of +/-112 and offset of +128, for a range of 16 through 240 inclusive.
To compute Y'CBCR from R'G'B' in the range [0..+1], scale the rows of the matrix of Eq 2 by the factors 219, 224 and 224, corresponding to the excursions 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 from black to white. The two entries of 112 reflect the positive CBCR extrema of the 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 eight bits 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 0 and white at 255. To encode Y'CBCR from R'G'B' in the range [0..255], using eight-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 will implement the equation in fixed-point binary arithmetic, using a shift by eight bits. 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, using eight-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 the multiplication by shifting. Some of the coefficients, when scaled by 256, are larger than unity. These coefficients will need more than eight multiplier bits.
For implementation in binary arithmetic the matrix coefficients have to be rounded. 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 RGB components in the range [0..255], subject to roundoff error. You must take care to avoid overflow due to roundoff error. But you must protect against overflow in any case, because studio video signals use the extremes of the coding range to handle signal overshoot and undershoot, and these will require clipping when decoded to an RGB range that has no headroom or footroom.

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 component of Y'CBCR. To encode Y'CBCR from R'G'B' in the range [0..219], using eight-bit binary arithmetic, scale the Y'CBCR encoding matrix of Eq 3 above 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-bit binary 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, indicating that the corresponding component can simply be added: there is no need for a 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 to conventional 525/59.94 and 625/50 video. It is not yet decided whether emerging HDTV standards will use the same matrices, or adopt a new set of matrices having different luma coefficients. In my view it would be unfortunate if different matrices were adopted, because then image coding and decoding would depend on whether the picture was small (conventional video) or large (HDTV).
In digital video, Rec. 601 standardizes subsampling denoted 4:2:2, where CB and CR components are subsampled horizontally by a factor of two with respect to luma. JPEG and MPEG conventionally subsample by a factor of two in the vertical dimension as well, denoted 4:2:0.
Colour difference coding is standardized in Rec. 601. For details on colour difference coding as used in video, consult Watkinson [16].

C-32 HOW DO I DECODE R'G'B' FROM PHOTOYCC?


Kodak's PhotoYCC uses the Rec. 709 primaries, white point and transfer function. Reference white codes to luma 189; this preserves film highlights. The colour difference coding is asymmetrical, to encompass film gamut. You are unlikely to encounter any raw image data in PhotoYCC form because YCC is closely associated with the PhotoCD(tm) system whose compression methods are proprietary. But just in case, the following equation 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 different range, or implement the equation in eight-bit integer arithmetic, use the techniques 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 below zero. PhotoYCC extends the Rec. 709 transfer function above unity, and reflects it around zero, to accommodate wide excursions of R'G'B'. To decode 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 composite NTSC and PAL. They have no place in component digital video! You shouldn't code into these systems, and if someone hands you an image claiming it's Y'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 and scale R'-Y' by 0.877283 to get V. The factors are chosen to limit composite 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 exchange of colour 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 colourbars are handled correctly. A colourbar signal comprises a binary RGB sequence ordered for decreasing 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 not being invoked, test 75% bars, a colourbar sequence having 75%-amplitude bars instead of 100%.

C-35 WHAT IS PERCEPTUAL UNIFORMITY?


A system is perceptually uniform if a small perturbation to a component value 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 perceptual increment in volume anywhere across the range of the control. If the control were physically linear, the logarithmic nature of human loudness perception would place all of the perceptual "action" of the control at the bottom of its range.
The XYZ and RGB systems are far from exhibiting perceptual uniformity. Finding a transformation of XYZ into a reasonably perceptually-uniform space consumed a decade or more at the CIE and in the end no single system could 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 U and V.) Both L*u*v* and L*a*b* improve the 80:1 or so perceptual nonuniformity of XYZ to about 6:1. Both demand too much computation to accommodate real-time display, although both have been successfully applied to 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 v system: 
        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. Then compute u' and v ' - and L* as discussed earlier - for your colours. 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 colours, but no fun for a million pixels. Although it was not specifically optimized for this purpose, the nonlinear 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 and Brightness (or Hue, Lightness and Saturation) in an age when users had to specify colours numerically. The usual formulations of HSB and HLS are flawed with respect to the properties of colour vision. Now that users can choose colours visually, or choose colours related to other media (such as PANTONE), or use perceptually-based systems like L*u*v* and L*a*b*, HSB and HLS should be abandoned.
Here are some of problems of HSB and HLS. In colour selection where "lightness" runs from zero to 100, a lightness of 50 should appear to be half as bright as a lightness of 100. But the usual formulations of HSB and HLS make no reference to the linearity or nonlinearity of the underlying RGB, 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 the properties of colour vision, as it computes yellow to be about six times more intense than blue with the same "lightness" value (say L=50).
HSB and HSL are not useful for image computation because of the discontinuity of hue at 360 degrees. You cannot perform arithmetic mixtures of colours expressed in polar coordinates.
Nearly all formulations of HSB and HLS involve different computations around 60 degree segments of the hue circle. These calculations introduce visible discontinuities in colour space.
Although the claim is made that HSB and HLS are "device independent", the ubiquitous formulations are based on RGB components whose chromaticities and white point are unspecified. Consequently, HSB and HLS are useless for conveyance of accurate colour 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* and v*: h*uv for hue angle and c*uv for chroma.

C-37 WHAT IS TRUE COLOUR?


True colour is the provision of three separate components for additive red, green and blue reproduction. True colour systems often provide eight bits for each of the three components, so true colour is sometimes referred to as 24-bit colour.
A true colour system usually interposes a lookup table between each component of the framestore and each channel to the display. This makes it possible to use a true colour system with either linear or nonlinear coding. In the X Window System, true colour refers to fixed lookup tables, and direct colour refers to lookup tables that are under the control of application software.

C-38 WHAT IS INDEXED COLOUR?


Indexed colour (or pseudocolour), is the provision of a relatively small number, say 256, of discrete colours in a colormap or palette. The framebuffer stores, at each pixel, the index number of a colour. At the output of the framebuffer, a lookup table uses the index to retrieve red, green and blue components that are then sent to the display.
The colours in the map may be fixed systematically at the design of a system. As an example, 216 index entries an eight-bit indexed colour system can be partitioned systematically into a 6x6x6 "cube" to implement what amounts to a direct colour system where each of red, green and blue has a value that is an integer in the range zero to five.
An RGB image can be converted to a predetermined colormap by choosing, for each pixel in the image, the colormap index corresponding to the "closest" RGB triple. With a systematic colormap such as a 6x6x6 colourcube this is straightforward. For an arbitrary colormap, the colormap has to be searched looking for entries that are "close" to the requested colour. "Closeness" should be determined according to the perceptibility of colour differences. Using colour systems such as CIE L*u*v* or L*a*b* is computationally prohibitive, but in practice it is adequate to use a Euclidean distance metric in R'G'B' components coded nonlinearly according to video practice.
A direct colour image can be converted to indexed colour with an image-dependent colormap by a process of colour quantization that searches through all of the triples used in the image, and chooses the palette for the image based on the colours that are in some sense most "important". Again, the decisions should be made according to the perceptibility of colour 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 associated with the image in a particular window is loaded into the hardware colormap, the maps associated with other windows may be disturbed. In window system such as the X Window System(tm) running on a multitasking operating system such as UNIX, even moving the cursor between two windows with different maps can cause annoying colormap flashing.
An eight-bit indexed colour system requires less data to represent a picture than a twenty-four bit truecolour system. But this data reduction comes at a high price. The truecolour system can represent each of its three components according to the principles of sampled continuous signals. This makes it possible to accomplish, with good quality, operations such as resizing the image. In indexed colour these operations introduce severe artifacts because the underlying representation lacks the properties of a continuous representation, even if converted back to RGB.
In graphic file formats such as GIF of TIFF, an indexed colour image is accompanied by its colormap. Generally such a colormap has RGB entries that are gamma corrected: the colormap's RGB codes are intended to be presented directly 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 COLOURS OF THE RAINBOW?
When you look at a rainbow you do not see a smooth gradation of colours. Instead, some bands appear quite narrow, and others are quite broad. Perceptibility of hue variation near 540 nm is half that of either 500 nm or 600 nm. If you use the rainbow's colours to represent data, the visibility of differences among your data values will depend on where they lie in the spectrum.
If you are using colour to aid in the visual detection of patterns, you should use colours chosen according to the principles of perceptual uniformity. This an open research problem, but basing your system on CIE L*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 greyscale values or colour values at each device pixel. However if the viewer is sufficiently distant from the display, the value of neighboring pixels can be set so that the viewer's eye integrates several pixels to achieve an apparent improvement in the number of levels or colours that can be reproduced.
Computer displays are generally viewed from distances where the device pixels subtend a rather large angle at the viewer's eye, relative to his visual acuity. Applying dither to a conventional computer display often introduces objectionable artifacts. However, careful application of dither can be effective. For example, human vision has poor acuity for blue spatial detail but good colour discrimination capability in blue. Blue can be dithered across two-by-two pixel arrays to produce four times the number of blue levels, with no perceptible penalty at normal viewing distances.

C-41 HOW DOES HALFTONING RELATE TO COLOUR?


The processes of offset printing and conventional laser printing are intrinsically bilevel: a particular location on the page is either covered with ink or not. However, each of these devices can reproduce closely-spaced dots of variable size. An array of small dots produces the perception of light gray, and an array of large dots produces dark gray. This process is called halftoning or screening. In a sense this is dithering, but with device dots so small that acceptable pictures can be produced at reasonable viewing distances.
Halftone dots are usually placed in a regular grid, although stochastic screening has recently been introduced that modulates the spacing of the dots rather than their size.
In colour printing it is conventional to use cyan, magenta, yellow and black grids that have exactly the same dot pitch but different carefully-chosen screen angles. The recently introduced technique of Flamenco screening uses the same screen angles for all screens, but its registration requirements are more stringent than conventional offset printing.
Agfa's booklet [17] is an excellent introduction to practical concerns of printing. And it's in colour! The standard reference to halftoning algorithms is Ulichney [18], but that work does not detail the nonlinearities found in practical printing systems. For details about screening for colour reproduction, consult Fink [19]. Consult Frequently Asked Questions about Gamma for an introduction to the transfer function of offset printing.

C-42 WHAT'S A COLOUR MANAGEMENT SYSTEM?


Software and hardware for scanner, monitor and printer calibration have had limited success in dealing with the inaccuracies of colour handling in desktop computing. These solutions deal with specific pairs of devices but cannot address the end-to-end system. Certain application developers have added colour transformation capability to their applications, but the majority of application developers have insufficient expertise and insufficient resources to invest in accurate colour.
A colour management system (CMS) is a layer of software resident on a computer that negotiates colour reproduction between the application and colour devices. It cooperates with the operating system and the graphics library components of the platform software. Colour management systems perform the colour transformations necessary to exchange accurate colour between diverse devices, in various colour coding systems including RGB, CMYK and CIE L*a*b*.
The CMS makes available to the application a set of facilities whereby the application can determine what colour devices and what colour spaces are available. When the application wishes to access a particular device, it requests that the colour manager perform a mathematical transform from one space to another. The colour spaces involved can be device-independent abstract colour spaces such as CIE XYZ, CIE L*a*b* or calibrated RGB. Alternatively a colour space can be associated with a particular device. In the second case the Colour manager needs access to characterization data for the device, and perhaps also to calibration data that reflects the state of the particular instance of the device.
Sophisticated colour management systems are commercially available from Kodak, Electronics for Imaging (EFI) and Agfa. Apple's ColorSync(tm) provides an interface between a Mac application program and colour management capabilities either built-in to ColorSync or provided by a plug-in. Sun has announced that Kodak's CMS will be shipped with the next version of Solaris.
The basic CMS services provided with desktop operating systems are likely to be adequate for office users, but are unlikely to satisfy high-end users such as in prepress. All of the announced systems have provisions for plug-in colour management modules (CMMs) that can provide sophisticated transform machinery. Advanced colour management modules will be commercially available from third parties. For an application developer's prespective on colour management, see Aldus [20].

C-43 HOW DOES A CMS KNOW ABOUT PARTICULAR DEVICES?


A CMS needs access to information that characterizes the colour reproduction capabilities of particular devices. The set of characterization data for a device is called a device profile. Industry agreement has been reached on the format of device profiles, although details have not yet been publicly disseminated. Apple has announced that the forthcoming ColorSync version 2.0 will adhere to this agreement. Vendors of colour peripherals will soon provide industry-standard profiles with their devices, and they will have to make, buy or rent characterization 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) colour manager - can create device profiles.

C-44 IS A COLOUR MANAGEMENT SYSTEM USEFUL FOR COLOUR SPECIFICATION?


Not yet. But colour management system interfaces in the future are likely to include the ability to accommodate commercial proprietary colour specification systems such as pantone(tm) and colorcurve(tm). These vendors are likely to provide their colour specification systems in shrink-wrapped form to plug into colour managers. In this way, users will have guaranteed colour accuracy among applications and peripherals, and application vendors will no longer need to pay to license these systems individually.

C-45 I'M NOT A COLOUR 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 already quite close to this. Rec. 709 has international agreement, offers excellent performance, 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 best perceptual performance and maximum ease of interchange with digital video, use the Rec. 709 transfer function, with its 0.45-power law. If you need Mac compatibility you will have to suffer a penalty in perceptual performance. Raise tristimulus values to the 1/1.8-power before presenting them 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 colour differences asymmetrically, according to film gamut. Unless you have a requirement for film gamut, you should code into colour differences using Y'CBCR coding with Rec. 601 studio video (16..235/128+/-112) excursion.
Tag your image data with the primary and white chromaticity, transfer function and luma coefficients that you are using. TIFF 6.0 tags have been defined for these parameters. This will enable intelligent readers, today or in the future, to determine the parameters of your coded image and give you the best possible results.

C-46 REFERENCES


[1] Publication CIE No 17.4, International Lighting Vocabulary. Central Bureau of the Commission Internationale de L'Eclairage, Vienna, Austria.
[2] LeRoy E. DeMarsh and Edward J. Giorgianni, "Color Science for Imaging Systems", Physics Today, September 1989, 44-52.
[3] W.F. Schreiber, Fundamentals of Electronic Imaging Systems, Second Edition (Springer-Verlag, 1991), ISBN 0-387-53272-2.
[4] Publication CIE No 15.2, Colorimetry, Second Edition (1986), Central Bureau of the Commission Internationale de L'Eclairage, Vienna, Austria.
[5] Guenter Wyszecki and W.S. Styles, Color Science: Concepts and Methods, Quantitative Data and Formulae, Second Edition (John Wiley & Sons, New York, 1982), ISBN 0-471-02106-7.
[6] Guenter Wyszecki and D.B. Judd, Color in Business, Science and Industry, Third Edition (John Wiley, New York, 1975), ISBN 0-471-45212-2.
[7] R.W.G. Hunt, The Reproduction of Colour in Photography, Printing and Television, Fourth Edition (Fountain Press, Tolworth, England, 1987), ISBN 0-86343-088-0.
[8] ITU-R Recommendation BT.709, Basic Parameter Values for the HDTV Standard for the Studio and for International Programme Exchange (1990), [formerly CCIR Rec. 709], ITU, 1211 Geneva 20, Switzerland.
[9] Bruce J. Lindbloom, "Accurate Color Reproduction for Computer Graphics Applications", Computer Graphics, Vol. 23, No. 3 (July 1989), 117-126 (proceedings of SIGGRAPH '89).
[10] William B. Cowan, "An Inexpensive Scheme for Calibration of a Colour Monitor in terms of CIE Standard Coordinates", Computer Graphics, Vol. 17, No. 3 (July 1983), 315-321.
[11] SMPTE RP 177-1993, Derivation of Basic Television Color Equations.
[12] Television Engineering Handbook, Featuring HDTV Systems, Revised Edition by K. Blair Benson, revised by Jerry C. Whitaker (McGraw-Hill, 1992), ISBN 0-07-004788-X. This supersedes the Second Edition.
[13] Roy Hall, Illumination and Color in Computer Generated Imagery (Springer-Verlag, 1989), ISBN 0-387-96774-5.
[14] Chet S. Haase and Gary W. Meyer, "Modelling Pigmented Materials for Realistic Image Synthesis", ACM Transactions on Graphics, Vol. 11, No. 4, 1992, p. 305.
[15] Maureen C. Stone, William B. Cowan and John C. Beatty, "Color Gamut Mapping and the Printing of Digital Color Images", ACM Transactions on Graphics, Vol. 7, No. 3, October 1988.
[16] John Watkinson, An Introduction to Digital Video (Focal Press, Sevenoaks, Kent, England, 1994), ISBN 0-240-51380-0.
[17] Agfa Corporation, An introduction to Digital Color Prepress, Volumes 1 and 2 (1990), Prepress Education Resources, P.O. Box 7917 Mt.Prospect, IL 60056-7917. 800-395-7007.
[18] Robert Ulichney, Digital Halftoning (MIT Press, Cambridge, MA, 1988), ISBN 0-262-21009-6.
[19] Peter Fink, PostScript Screening: Adobe Accurate Screens (Adobe Press, 1992), ISBN 0-672-48544-3.
[20] Color management systems: Getting reliable color from start to finish, Aldus Corporation, <ftp://www.adobe.com/PDFs/FaxYI/500301.pdf>.
[21] Overview of color publishing, Aldus Corporation, <ftp://www.adobe.com/PDFs/FaxYI/500302.pdf>. Despite appearances and title, this document is in greyscale, not colour.
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