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From: dbourgin@turing.imag.fr (David Bourgin (The best player).)
Newsgroups: comp.graphics,sci.image.processing,comp.answers,sci.answers,news.answers
Subject: Color space FAQ
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Date: 8 Apr 1995 00:07:05 GMT
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Summary: This posting contains a list of Frequently Asked
         Questions (and their answers) about colors and color spaces.
         It provides an extension to the short 4 and 5 items
         of comp.graphics FAQ. Read item 1 for more details.
         A copy of this document is available by anonymous ftp
         in rtfm.mit.edu: /pub/usenet/news.answers/graphics/colorspace-faq
         or turing.imag.fr: /pub/compression/colorspace-faq
Keywords: Color space FAQ
X-Last-Updated: 1995/03/06
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Archive-name: graphics/colorspace-faq
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Last-modified: 28/2/95

###########################################################
Color spaces FAQ - David Bourgin
Date: 28/1/95 (Major modifications)
Last update: 6/1/95 (23/1/95: minor modifications)

---------------------------
     Table of contents
---------------------------
1 - Purpose of this FAQ
2 - Some definitions
3 - What is an image based on a color look-up table (LUT)?
4 - What is this gamma component?
5 - Color space conversions
5.1 - RGB, CMY, and CMYK
5.2 - HSI and related color spaces
5.3 - CIE XYZ and gray level (monochrome included) pictures
5.4 - CIE Luv and CIE Lab
5.5 - LCH and CIE LSH
5.6 - The associated standards: YUV, YIQ, and YCbCr
5.7 - SMPTE-C RGB
5.8 - SMPTE-240M YPbPr (HD televisions)
5.9 - Xerox Corporation YES
5.10- Kodak Photo CD YCC
6 - References
7 - Comments and thanks

---------------------------
          Contents
---------------------------
1 - Purpose of this FAQ (D. Bourgin)

    I did a (too) long period of research in the video domain (video cards,
    image file formats, and so on) and I've decided to provide to all people
    who need some information about that.
    I aim to cover a part of the Frequently Asked Questions (FAQ) in the video
    works, it means to provide some (useful?) information about the colors,
    and more especially about color spaces. If you have some information
    to ask/add to this document, please read item 7.

2 - Some definitions (A. Ford, D.Bourgin)

    Color is defined as an experience in human perception. In physics terms, a
    color is the result of an observed light on the retina of the eye. The
    light must have a wavelength in the range of 400 to 700 nm. The radient
    flux of observed light at each wavelength in the visible spectrum is
    associated to a Spectral Power Distribution (SPD).
    An SPD is created by cascading the SPD of the light source with the
    Spectral Reflectance of the object in the scene. In addition the optics of
    any imaging device will have an effect.
    Strictly though, color is a visual sensation, so a `color' is created
    when we observe a specific SPD.
    We see color by means of cones in the retina. There are three types of
    cones sensitive to wavelengths that approximately correspond to red, green
    and blue light. Together with information from rod cells (which are not
    sensitive to color) the cone information is encoded and sent to higher
    brain centres along the optic nerve. The encoding, known as opponent
    process theory, consists of three opponent channels, these are:

	Red     -       Green
	Blue    -       Yellow
	Black   -       White

    Note: Actually, recent studies show that eyes use addtionnal cone types.
    (cf. "La Recherche", n.272, january, 1995)

    This is different to tri-chromatic theory (e.g. Red, Green, Blue additive
    color) which you may be used to, but when we describe colors we do not say
    "it is a reddy green" or "that's a bluey yellow".
    Perceptually we require three attributes to describe a color. Generally
    any three will do, but there need's to be three.

    For human purpose color descriptions these attributes have been by the CIE
    recommendations, as follows:

    Brightness. The attribute of a visual sensation according to which an area
    appears to exhibit more or less light. You can blurry or enhance an image
    by modifying this attribute.

    Hue. The attribute of a visual sensation according to which an area appears
    to be similar to one, or to proportions of two, of the perceived colors
    red, yellow, green and blue.

    Colorfulness. The attribute of a visual sensation according to which an
    area appears to exhibit more or less of its hue. You can go from a sky blue
    to a deep blue by changing this attribute.

    So, a color is a visual sensation produced by a stimulus which is a
    specific SPD. It should be noted however, that two different SPD's may
    produce the same visual sensation - an effect known as metarmerism.

    What is a color space?

    A color space is a method by which we can specify, create and visualise
    color. As human's, we may define a color by its attributes of brightness,
    hue and colorfulness. A computer will define a color in terms of the
    excitations of red, green and blue phosphors on the CRT faceplate. A
    printing press defines color in terms of the reflectance and absorbance of
    cyan, magenta, yellow and black inks on the paper.  

    If we imagine that each of the three attributes used to describe a color
    are axes in a three dimensional space then this defines a color space.
    The colors that we can percieve can be represented by the CIE system, other
    color spaces are subsets of this perceptual space. For instance RGB color
    space, as used by television displays, can be visualised as a cube with
    red, green and blue axes. This cube lies within our perceptual space, since
    the RGB space is smaller and represents less colors than we can see. CMY
    space would be represented by a second cube, with a different orientation
    and a different position within the perceptual space.

    So, a color space is a mathematical representation of our perceptions. E.g.
    RGB is a tri-dimensional space with red, green and blue axes. It's useful
    to think so because computers are in fond of numbers and equations...

    Why is there more than one color space?

    Different color spaces are better for different applications, some
    equipment has limiting factors that dictate the size and type of color
    space that can be used.  Some color spaces are perceptually linear, i.e. a
    10 unit change in stimulus will produce the same change in perception
    wherever it is applied. Many color spaces, particularly in computer
    graphics are not linear in this way. Some color spaces are intuitive to
    use, i.e. it is easy for the user to navigate within them and creating
    desired colors is relativly easy. Finally, some color spaces are device
    dependent while others are not (so called device independent).

    What's the difference between device dependent and device independent?

    A device dependent color space is a color space where the color produced
    depends on the equipment and the set-up used to produce it. For example the
    color produced using pixel values of [rgb = 250,134,67] will alter as you
    change the brightness and contrast on your display. In the same way if you
    change your monitor the red, green and blue phosphors will have slightly
    different SPD's and the color produced will change. Thus RGB is a color
    space that is dependent on the system being used, it is device dependent.
    A device indepent color space is one where the coordinates used to specify
    the color will produce the same color wherever they are applied.
    An example of a device independent color space (if it has been implemented
    properly) is the CIE L*a*b* color space (known as CIELab). This is based on
    the HVS as described by the CIE system (see below to know what CIE stands
    for).

    Another way of looking a device dependancy is to imagine our RGB cube
    within our perceptual color space. We define a color by the values on the
    three axes. However the exact color will depend on the position of the cube
    within the perceptual color space, move the cube (by changing the set-up)
    and the color will change even if the RGB values remain the same.
    Some device dependent color spaces have their position within CIE space
    defined, these are known as device callibrated color spaces and are a kind
    of half way house between dependent and independent color spaces. For
    example, a graphics file that contains colorimetric information, i.e. the
    white point, transfer functions, and phosphor chromaticities, would enable
    device dependent RGB data to be modified for whatever device was being used
    - i.e. callibrated to specific devices. In other words, if you have a
    device independent color space, you must adapt your device as defined in
    the color space and not the color space to the device.

    What is a color gamut?

    A color gamut is the boundary of the color space. Gamut's are best
    shown and evaluated using the CIE system.

    What is the CIE System?

    The CIE has defined a system that classifies color according to the HVS (it
    started producing specifications in 1931). Using this system we can specify
    any color in terms of its CIE coordinates. 

    The CIE system works by weighting the SPD of an object in terms of three
    color matching functions. These functions are the sensitivities of a
    standard observer to light of different wavelengths. The weighting is 
    performed over the visual spectrum, from around 360nm to 830nm in 1nm 
    intervals. However, the illuminant, and lighting and viewing geometry are 
    carefully defined. This process produces three CIE tri-stimulus values, 
    XYZ, which describe the color. 

    There are many measures that can be derived from the tri-stimulus values,
    these include chromaticity coordinates and color spaces.

    What color space should I use?

    That depends on what you want to do, but heres a list of the pros and cons
    of some of the more common, computer related, color spaces - we will see
    in item 5 how to convert the (most common) color spaces between
    themselves - :

    RGB (Red Green Blue)

    Additive color system based on trichromatic theory, commonly used by CRT
    displays where proportions of excitation of red, green and blue emmiting
    phosphors produce colors when visually fused. Easy to implement, non
    linear, device dependent, unintuitive, common (used in television cameras,
    computer graphics etc).

    CMY(K) (Cyan Magenta Yellow (Black))

    Subtractive color. Used in printing and photography. Printers often include
    the fourth component, black ink, to improve the color gamut (by increasing
    the density range), improving blacks, saving money and speeding drying (less
    ink to dry). Fairly easy to implement, difficult to transfer *properly* from
    RGB (simple transforms are, well, simple), device dependent, non-linear,
    unintuitive.

    HSL (Hue Saturation and Lightness)

    This represents a wealth of similar color spaces, alternatives include HSI
    (intensity), HSV (value), HCI (chroma / colorfulness),  HVC, TSD (hue
    saturation and darkness) etc etc. All these color spaces are non-linear
    transforms from RGB they are thus, device dependent, non-linear but very
    intuitive. In addition the seperation of the luminance component has
    advantages in image processing and other applications. (But take care, the
    complete isolation of the separate components will require a space
    optimised for your device. See later notes on CIE color spaces)

    YIQ, YUV, YCbCr, YCC  (Luminance - Chrominance)

    These are the television transmission color spaces (YIQ and YUV analogue
    (NTSC and PAL) and YCbCr digital). They separate luminance from chrominance
    (lightness from color) and are useful in compression and image processing
    applications. YIQ and YUV are, if used according to their relative
    specifications, linear. They are all device dependent and, unless you are a
    TV engineer, unintuitive. Kodaks PhotoCD system uses a type of YCC color
    space, PhotoYCC, which is a device calibrated color space.

    CIE 

    The HVS based color specification system. There are two CIE based color
    spaces, CIELuv and CIELab. They are near linear (as close as any color
    space is expected to sensibly get), device independent (unless your in the
    habit of swapping your eye balls with aliens), but not very intuitive to
    use. 

    From CIELuv you can derive CIELhs or CIELhc where h is the hue (an angle),
    s the saturation and c the chroma. CIELuv has an associated chromaticity
    diagram, a two dimensional chart which makes additive color mixing very
    easy to visualise, hence CIELuv is widely used in additive color
    applications, like television.
    CIELab has no associated two dimensional chromaticity diagram and no 
    correlate of saturation so only Lhc can be used.

    Since there is such a wide variet of color spaces, its useful to understand
    a bit more about them and how to convert between them.

    The color space conversions are essentially provided for programmers. If
    you are a specialist then skip to the references in item 6. Many of the
    conversions are based on linear matrix transforms. (Was it Jim Blinn who
    said that any problem in computer graphics can be solved by a matrix 
    transform ?). As an example:

    E.g. RGB -> CIE XYZccir601-1 (D65) provides the following matrix of numbers
    (see item 5.3):
    |  0.607   0.174   0.200 |
    |  0.299   0.587   0.114 |
    |  0.000   0.066   1.111 |
    and CIE XYZccir601-1 (D65) -> RGB provides the following matrix:
    |  1.910  -0.5338 -0.2891  |
    | -0.9844  1.9990 -0.0279 |
    | 0.0585  -0.1187 -0.9017 |
    
    These two matrices are the (approximate) inversion of each other. If you
    are a beginner in this mathematical stuff, skip the previous explainations,
    and just use the results...

    Other definitions.

    Photometric terms:  illuminance   - luminous flux per unit area incident on
                                        a surface
                        luminance     - luminous flux per unit solid angle and
                                        per unit projected area, in a given
                                        direction, at a point on a surface.
                        luminous flux - radient flux weighted by the V(landa)
                                        function.
	                                I.e. weighted by the eye's sensitivity.
                        radient flux  - total power / energy of the incident
                                        radiation.

    Other terms:        brightness    - the human sensation by which an area
                                        exhibits more or less light.
                        lightness     - the sensation of an area's brightness
                                        relative to  a reference white in the
                                        scene.
                        chroma        - the colorfulness of an area relative to
                                        the brightness of a reference white.
                        saturation    - the colorfulness of an area relative to
                                        its  brightness. 

    Note: This list is not exhaustive, some terms have alternative meanings
    but we assume these to be the fundamentals.

3 - What is an image based on a color look-up table (LUT)?

    All of the pictures don't use the full color space. That's why we often
    use another scheme to improve the encoding of the picture (especially
    to get a file which takes less space). To do so, you have two
    possibilities:
    - You reduce the bits/sample. It means you use less bits for each component
    that describe the color. The colors are described as direct colors, it
    means that all the pixels (or vectors, for vectorial descriptions) are
    directly stored with the full components. For example, with a RGB (see
    item 5.1 to know what RGB is) bitmapped image with a width of 5 pixels and
    a height of 8 pixels, you have:
    (R11,G11,B11) (R12,G12,B12) (R13,G13,B13) (R14,G14,B14) (R15,G15,B15)
    (R21,G21,B21) (R22,G22,B22) (R23,G23,B23) (R24,G24,B24) (R25,G25,B25)
    (R31,G31,B31) (R32,G32,B32) (R33,G33,B33) (R34,G34,B34) (R35,G35,B35)
    (R41,G41,B41) (R42,G42,B42) (R43,G43,B43) (R44,G44,B44) (R45,G45,B45)
    (R51,G51,B51) (R52,G52,B52) (R53,G53,B53) (R54,G54,B54) (R55,G55,B55)
    (R61,G61,B61) (R62,G62,B62) (R63,G63,B63) (R64,G64,B64) (R65,G65,B65)
    (R71,G71,B71) (R72,G72,B72) (R73,G73,B73) (R74,G74,B74) (R75,G75,B75)
    (R81,G81,B81) (R82,G82,B82) (R83,G83,B83) (R84,G84,B84) (R85,G85,B85)
    where Ryx, Gyx, Byx are respectively the Red, Green, and Blue components
    you need to render a color for the pixel located at (x;y).
    - You use a palette. In this case, all the colors are stored in a table
    called a palette. This table is usually small. And the components of all
    the colors for each pixel (or the vector data) are removed to be replaced
    with a number. This number is an index in the palette. It explains why we
    call the palette, a color look-up table (LUT or CLUT).

4 - What is this gamma component?

    Many image processing operations, and also color space transforms that
    involve device independent color spaces, like the CIE system based ones,
    must be performed in a linear luminance domain.
    By this we really mean that the relationship between pixel values specified
    in software and the luminance of a specific area on the CRT display must be
    known. CRTa will have a non-linear response.
    The luminance of a CRT is generally modelled using a power function with an
    exponent, gamma, somewhere between 2.2 [NTSC and SMPTE specifications] and
    2.8 [as given by Hunt and Sproson]. Recent measurements performed at the
    BBC in the UK (Richard Salmon and Alan Roberts) have shown that the actual
    value of gamma is very dependent upon the accurate setting of the CRTs
    black level. For correctly set-up CRTs gamma is 2.35 +/- 0.1. This
    relationship is given as follows:

    Luminance = voltage ^ gamma

    Where luminance and voltage are normalised. For Liquid Crystal Displays the
    response is more closely followed by an "S" shaped curve with a vicious
    hook near black and a slow roll-off near white.

    In order to display image information as linear luminance we need to modify
    the voltages sent to the CRT. This process stems from television systems
    where the camera and reciever had different transfer functions (which,
    unless corrected, would cause problems with tone reproduction). The
    modification applied is known as gamma correction and is given below:

    New_Voltage = Old_Voltage ^ (1/gamma)

    (Both voltages are normalised and gamma is the value of the exponent of the
    power function that most closely models the luminance-voltage relationship
    of the display being used.)

    For a color computer system we can replace the voltages by the pixel
    values selected, this of course assumes that your graphics card converts
    digital values to analogue voltages in a linear way. (For precision work
    you should check this). The color relationships are:

    Red  = a* (Red' ^gamma) +b
    Green= a* (Green' ^gamma) +b
    Blue = a* (Blue' ^gamma) +b

    where Red', Green', and Blue' are the normalised input RGB pixel values and
    Red, Green, and Blue are the normalised gamma corrected signals sent to the
    graphics card. The values of the constants a and b componsate for the
    overall system gain and system offset respectively. (Essentially gain is
    contrast and offset is intensity.) For basic applications the value of a,
    b and gamma can be assumed to be consistent between color channels,
    however for precise applications they must be measured for each channel
    separatley.
    	   
    It is common to perform gamma correction by calculating the corrected value
    for each possible input value and storing this in an array as a Look Up
    Table (see item 3). In some cases the LUT is part of the graphics card, in
    others it needs to be implemented in software.

    Research by Cowan and Post (see references in item 6) has shown that not
    all CRT displays can be accurately modelled by a simple power relationship.
    Cowan found errors of up to 100% between measured and calculated values. To
    prevent this a simple LUT generating gamma correction function cannot be
    used. The best method is to measure the absolute luminance of the display
    at various pixel values and to linearily interpolate between them to
    produce values for the LUT.

    It should be noted at this point that correct gamma correction depends on a
    number of factors, in addition to the characteristics of the CRT display.
    These include the gamma of the input device (scanner, video grabber etc),
    the viewing conditions and target tone reproduction characteristics. In
    addition linear luminance is not always desirable, for computer generated
    images a system which is linear with our visual perception may be
    preferable. If luminance is represented by L then lightness, our visual
    sensation of an objects brightness relative to a similarly illuminated
    area that appears white is given by L:

    { L=116*((Y/Yn)^(1/3))-16 if Y/Yn>0.008856
    { L=903.3*Y/Yn            if Y/Yn<=0.008856

    This relationship is in fact the CIE 1976 Lightness (L*) equation which will
    be discussed in section 5.4. 

    Most un-gamma corrected displays have a near linear L* response, thus for
    computer generated applications, or applications where pixel values in an
    image need to be intuitivly changed, this set up may be better.

    Note: Gamma correction performed in integer maths is prone to large
    quantisation errors. For example, applying a gamma correction of 1/2.2
    to an image with an original gamma of one (linear luminance) produced a
    drop in the number of grey levels from 245 to 196. Take care not to alter
    the transfer characteristics more than is necessary, if you need to gamma
    correct images try to keep the originals so that you can pass them on to
    others without passing on the degradations that you've produced ;-).

    In some image file formats or in graphics applications in general, you
    need sometimes some other kinds of correction. These corrections provide
    some specific processings rather than true gamma correction curves.
    This is often the case, for examples, with the printing devices or
    in animation. In the first case, it is interesting to specify that a color
    must be re-affected in order you get a better rendering, as we see it later
    in CMYK item. In the second case, some animations can need an extra
    component associated to each pixel. This component can be, for example,
    used as a transparency mask. We *improperly* call this extra component
    gamma correction.
    -> not to to be confused with gamma.

5 - Color space conversions

    Except an historical point of view, most of you are - I hope - interested
    in color spaces to make renderings and, if possible, on your favorite
    computer. Most of computers display in the RGB color space but you may need
    sometimes the CMYK color space for printing, the YCbCr or CIE Lab to
    compress with JPEG scheme, and so on. That is why we are going to see,
    from here, what are all these color spaces and how to convert them from one
    to another (and primary from one to RGB and vice-versa, this was my purpose
    when I started this FAQ).

    I provide the color space conversions for programmers. The specialists
    don't need most of these infos or they can give a glance to all the stuff
    and read carefully the item 6. Many of the conversions are based on linear
    functions. The best example is given in item 5.3. These conversions can
    be seen in matrices. A matrix is in mathematics an array of values. And to
    go from one to another space color, you just make a matrix inversion.
    E.g. RGB -> CIE XYZrec601-1 (C illuminant) provides the following matrix
    of numbers (see item 5.3):
    |  0.607   0.174   0.200 |
    |  0.299   0.587   0.114 |
    |  0.000   0.066   1.116 |
    and CIE XYZrec601-1 (C illuminant) -> RGB provides the following matrix:
    |  1.910  -0.532  -0.288 |
    | -0.985   1.999  -0.028 |
    |  0.058  -0.118   0.898 |
    These two matrices are the (approximative) inversion of each other.
    If you are a beginner in this mathematical stuff, skip the previous
    explainations, and just use the result...

5.1 - RGB, CMY, and CMYK

    The most popular color spaces are RGB and CMY. These two acronyms stand
    for Red-Green-Blue and Cyan-Magenta-Yellow. They're device-dependent.
    The first is normally used on monitors, the second on printers.

    RGB are known as additive primary colors, because a color is produced by
    adding different qunatities of the three components, red, green, and blue.

    CMY are known as subtractive (or secondary) colors, because the color is
    generated by subtracting different quantities of cyan, magenta and yellow
    from white light.

    The primaries used by artists, red, yellow and blue are different because
    they are concerned with mixing pigments rather than lights or dyes.

    RGB -> CMY				| CMY -> RGB
    Red   = 1-Cyan	(0<=Cyan<=1)	| Cyan    = 1-Red (0<=Red<=1)
    Green = 1-Magenta	(0<=Magenta<=1)	| Magenta = 1-Green (0<=Green<=1)
    Blue  = 1-Yellow	(0<=Yellow<=1)	| Yellow  = 1-Blue (0<=Blue<=1)

    On printer devices, a component of black is added to the CMY, and the
    second color space is then called CMYK (Cyan-Magenta-Yellow-blacK). This
    component is actually used because cyan, magenta, and yellow set up to the
    maximum should produce a black color. (The RGB components of the white are
    completly substracted from the CMY components.) But the resulting color
    isn't physically a 'true' black. The transforms from CMY to CMYK (and vice
    versa) are given as shown below:
    CMY -> CMYK				| CMYK -> CMY
    Black=minimum(Cyan,Magenta,Yellow)	| Cyan=minimum(1,Cyan*(1-Black)+Black)
    Cyan=(Cyan-Black)/(1-Black)		| Magenta=minimum(1,Magenta*(1-Black)+Black)
    Magenta=(Magenta-Black)/(1-Black)	| Yellow=minimum(1,Yellow*(1-Black)+Black)
    Yellow=(Yellow-Black)/(1-Black)	|

    Note, these differ to the descriptions often given, for example in Adobe
    Postscript. For more information see FIELD in section 6. This is because
    Adobe doesn't choose to use the most recent equations. (I don't know why!)

    RGB -> CMYK				| CMYK -> RGB
    Black=minimum(1-Red,1-Green,1-Blue)	| Red=1-minimum(1,Cyan*(1-Black)+Black)
    Cyan=(1-Red-Black)/(1-Black)	| Green=1-minimum(1,Magenta*(1-Black)+Black)
    Magenta=(1-Green-Black)/(1-Black)	| Blue=1-minimum(1,Yellow*(1-Black)+Black)
    Yellow=(1-Blue-Black)/(1-Black)	|

    Of course, I assume that C, M, Y, K, R, G, and B  have a range of [0;1].

5.2 - HSI and related color spaces

    The representation of the colors in the RGB and CMY(K) color spaces are
    designed for specific devices. But for a human observer, they are not
    useful definitions. For user interfaces a more intuitive color space,
    designed for the way we actually think about color is to be preferred.
    Such a color space is HSI; Hue, Saturation and Intensity, which can be
    thought of as an RGB cube tipped up onto one corner. The line from RGB=min
    to RGB=max becomes verticle and is the intensity axis. The position of a
    point on the circumference of a circle around this axis is the hue and the
    saturation is the radius from the central intensity axis to the color.

                 Green
                  /\
                /    \    ^
              /V=1 x   \   \ Hue (angle, so that Hue(Red)=0, Hue(Green)=120, and Hue(blue)=240 deg)
       Blue -------------- Red
            \      |     /
             \     |-> Saturation (distance from the central axis)
              \    |   /
               \   |  /
                \  | /
                 \ |/
               V=0 x (Intensity=0 at the top of the apex and =1 at the base of the cone)
    
    The big disadvantage of this model is the conversion which is mainly
    because the hue is expressed as an angle. The transforms are given below:

    Hue = (Alpha-arctan((Red-intensity)*(3^0.5)/(Green-Blue)))/(2*PI)
    with { Alpha=PI/2 if Green>Blue
         { Aplha=3*PI/2 if Green<Blue
         { Hue=1 if Green=Blue
    Saturation = (Red^2+Green^2+Blue^2-Red*Green-Red*Blue-Blue*Green)^0.5
    Intensity = (Red+Green+Blue)/3

    Note that you have to compute Intensity *before* Hue. If not, you must
    assume that:
    Hue = (Alpha-arctan((2*Red-Green-Blue)/((Green-Blue)*(3^0.5))))/(2*PI).

    I assume that H, S, L, R, G, and B are within range of [0;1].

    Another point of view of this cone is to project the coordinates onto the
    base. The 2D projection is:
    Red:   (1;0)
    Green: (cos(120 deg);sin(120 deg)) = (-0.5; 0.866)
    Blue:  (cos(240 deg);sin(240 deg)) = (-0.5;-0.866)
    Now you need intermediate coordinates:
    a = Red-0.5*(Green+Blue)
    b = 0.866*(Green-Blue)
    Finally, you have:
    Hue = arctan2(x,y)/(2*PI) ; Just one formula, always in the correct quadrant
    Saturation = (x^2+y^2)^0.5
    Luminosity = (Red+Green+Blue)/3
    This interesting point of view was provided by Christian Steyaert
    (steyaert@vvs.innet.be).

    Another model close to HSI is HSL. It's actually a double cone with black
    and white points placed at the two apexes of the double cone.
    I don't provide formula, but feel free to send me the formula you could
    find. ;-)

    Actually, here are many variations on HSI, e.g. HSL, HSV, HCI (chroma /
    colorfulness), HVC, TSD (hue saturation and darkness) etc. But they all
    do basically the same thing.

5.3 - CIE XYZ and gray level pictures

    The CIE (presented in the item 2) has defined a human "Standard Observer",
    based on measurements of the colour-matching abilities of the average human
    eye. Using data from measurements made in 1931, a system of three primaries,
    XYZ, was developed in which all visible colours can be represented using
    only positive values of XY and Z. The Y primary is identical to Luminance,
    X and Z give colouring information. This forms the basis of the CIE 1931
    XYZ colour space, which is fundamental to all colorimetry. It is completely
    device-independant and values are normally assumed to lie in the range
    [0;1]. Colours are rarely specified in XYZ terms, it is far more common to
    use "chromaticity coordinates" which are independant of the Luminance (Y).
    The main advantage of CIE XYZ, and any color space or color definition
    based on it, is that it is completely device independent. The main
    disadvantage with CIE based spaces is the complexity of implementing them,
    in addition some are not user intuitive. A complete description of the CIE
    system is beyond the scope of this FAQ, we simply present useful formula to
    convert between CIE values and between CIE and non-CIE color spaces. We
    cannot recommend too highly that anyone wishing to implement any of the CIE
    system in the digital domain reads the refs in section 6, specifically HUNT
    1, SPROSON, BERNS and CIE 1.

    Chromaticity coordinates are derived from tristimulus values (the amounts
    of the primaries) by normalising thus:

     x = X/(X+Y+Z)
     y = Y/(X+Y+Z)
     z = Z/(X+Y+Z)

    Chromaticity coordinates are *always* used in lower case. Because they have
    been normalised, only two values are needed to specify the colour, and so z
    is normally discarded (because x+y+z=1). Colours can be plotted on a
    "chromaticity diagram" using x and y as coordinates, with Y (Luminance)
    normal to the diagram. When a colour is specified in this form, it is
    referred to as CIE 1931 Yxy. Tristimulus values can always be derived from
    Yxy values:

    X = x*Y/y
    Z = (1-x-y)*Y/y

    For scientists and programers, it is possible to convert between RGB as
    displayed on a CRT and CIE tristimulus values.

    The first step is to ensure that you have either linear luminance
    information, or that you know the transfer function (gamma correction) of
    the display device. For further details on this see section 4. This will
    give you the luminances of the red, green and blue phosphor emissions from
    the red, green and blue pixel values that you specify.
    The second stage is to perform a matrix transform (see item 5) to convert
    the red, green, blue luminance information to CIE XYZ tristimulus values,
    essentially. We can apply Grassman's Laws to establish conversion matrices
    between the XYZ primaries and any other set of primaries, for instance (if
    we consider RGB):

    |Red  |      -1 |X|         |X|       |Red  |
    |Green| = |M|  *|Y|	  and   |Y| = |M|*|Green|
    |Blue |         |Z|         |Z|       |Blue |

    The matrix M is 3 by 3, and consists of the tristimulus values of the RGB
    primaries in terms of the XYZ primaries (phosphors on *your* CRT).

    Ideally you would measure these - if you have a colorimeter or a
    spectroradiometer / spectrophotometer handy. Alternativly you could assume
    that your system corresponds to a particular specification, e.g. NTSC, and
    use the figures given by the standard, however this is often not a valid
    assumption - and if you need to make it it's probably not worth going to
    the effort of implementing the transforms, the error's induced would outway
    any advantages of the CIE system. The third method is to derive the figures
    from other data.

    To solve this system we need some more data. The first data is the color
    reference we use. With the CIE standard the reference of your rendering is
    the white. White point is achromatic and is defined so that Y=1, and
    Red=Green=Blue. To get the white point coordinates and put it into our
    previous matrix system we use the CIE xyY diagram. This diagram is a 2D
    diagram (based on tristimuli in regard with the wave lengths) where you get
    a color as (x;y). To transform this 2D diagram into a 3D, we just consider:
    z=1-(x+y)
    X=x*Y/y
    Z=z*Y/y
    (Take care on these letters because these are case sensitive. Otherwise
    you'd get unaccurate results!)
    From there we must consider the coordinates of the vertices in your
    triangle reference. The three vertices in your triangle reference are
    of course the red, the green, and the blue in CIE xyY diagram. Those colors
    are "pure values", it means the chromacity coordinates of red, green, and
    blue are defined in the CIE xyY diagram:
    Red:   (xr; yr; zr=1-(xr+yr))
    Green: (xg; yg; zg=1-(xg+yg))
    Blue:  (xb; yb; zb=1-(xb+yb))
    And the white is defined as:
    |Xn|   |r1 g1 b1|   |Redn  |   |r1 g1 b1|   |1|
    |Yn| = |r2 g2 b2| * |Greenn| = |r2 g2 b2| * |1| (1)
    |Zn|   |r3 g3 b3|   |Bluen |   |r3 g3 b3|   |1|
    (1) becomes by invoking the white balance condition (Red=Green=Blue=1 for
    white):
    |Xn|   |ar*xr ag*xg ab*xb|   |1|   |xr xg xb|   |ar|
    |Yn| = |ar*yr ag*yg ab*yb| * |1| = |yr yg yb| * |ag| (2)
    |Zn|   |ar*zr ag*zg ab*zb|   |1|   |zr zg zb|   |ab|
    But Xn, Yn, and Zn are also defined as (xn;yn) from the CIE xyY diagram:
    zn=1-(xn+yn)
    Xn=xn*Yn/yn=xn/yn
    Yn=1 (always for white!)
    Zn=zn*Yn/yn=zn/yn
    So (2) becomes:
    |xn/yn|   |xr xg xb|   |ar|
    |  1  | = |yr yg yb| * |ag| (3)
    |zn/yn|   |zr zg zb|   |ab|
    Now, xn, yn, zn, xr, yr, zr, xg, yg, zg, xb, yb, and zb are all known
    because they are supplied. Multiplying the chromaticity coordinates by
    these values gives the the matrix in equation (1) (with a HP pocket
    computer, for example) and get ar, ag, and ab. So:
    |X|   |xr*ar xg*ag xb*ab|   |Red  |
    |Y| = |yr*ar yg*ag xb*ab| * |Green|
    |Z|   |zr*ar zg*ag xb*ab|   |Blue |
    Let's take some examples. The CCIR (Comite Consultatif International des
    Radiocommunications) defined several recommendations. The most popular
    (they shouldn't be used anymore, we will see later why) are CCIR 601-1
    and CCIR 709.
    The CCIR 601-1 is the old NTSC (National Television System Committee)
    standard. It uses a white point called "C Illuminant". The white point
    coordinates in the CIE xyY diagram are (xn;yn)=(0.310063;0.316158). The
    red, green, and blue chromacity coordinates are:
    Red:   xr=0.67 yr=0.33 zr=1-(xr+yr)=0.00
    Green: xg=0.21 yg=0.71 zg=1-(xg+yg)=0.08
    Blue:  xb=0.14 yb=0.08 zb=1-(xb+yb)=0.78
    zn=1-(xn+yn)=1-(0.310063+0.316158)=0.373779
    Xn=xn/yn=0.310063/0.316158=0.980722
    Yn=1 (always for white)
    Zn=zn/yn=0.373779/0.316158=1.182254
    We introduce all that in (3) and get:
    ar=0.981854
    ab=0.978423
    ag=1.239129
    Finally, we have RGB -> CIE XYZccir601-1 (C illuminant):
    |X|   |0.606881 0.173505 0.200336|   |Red  |
    |Y| = |0.298912 0.586611 0.114478| * |Green|
    |Z|   |0.000000 0.066097 1.116157|   |Blue |
    Because I'm a programer, I preferr to round these values up or down (in
    regard with the new precision) and I get:
    RGB -> CIE XYZccir601-1 (C illuminant)	| CIE XYZccir601-1 (C illuminant) -> RGB
    X = 0.607*Red+0.174*Green+0.200*Blue	| Red   =  1.910*X-0.532*Y-0.288*Z
    Y = 0.299*Red+0.587*Green+0.114*Blue	| Green = -0.985*X+1.999*Y-0.028*Z
    Z = 0.000*Red+0.066*Green+1.116*Blue	| Blue  =  0.058*X-0.118*Y+0.898*Z
    The other common recommendation is the 709. The white point is D65 and have
    coordinates fixed as (xn;yn)=(0.312713;0.329016). The RGB chromacity
    coordinates are:
    Red:   xr=0.64 yr=0.33
    Green: xg=0.30 yg=0.60
    Blue:  xb=0.15 yb=0.06
    Finally, we have RGB -> CIE XYZccir709 (709):
    |X|   |0.412411 0.357585 0.180454|   |Red  |
    |Y| = |0.212649 0.715169 0.072182| * |Green|
    |Z|   |0.019332 0.119195 0.950390|   |Blue |
    This provides the formula to transform RGB to CIE XYZccir709 and vice-versa:
    RGB -> CIE XYZccir709 (D65)			| CIE XYZccir709 (D65) -> RGB
    X = 0.412*Red+0.358*Green+0.180*Blue	| Red   =  3.241*X-1.537*Y-0.499*Z
    Y = 0.213*Red+0.715*Green+0.072*Blue	| Green = -0.969*X+1.876*Y+0.042*Z
    Z = 0.019*Red+0.119*Green+0.950*Blue	| Blue  =  0.056*X-0.204*Y+1.057*Z
    Recently (about one year ago), CCIR and CCITT were both absorbed into their
    parent body, the International Telecommunications Union (ITU). So you must
    *not* use CCIR 601-1 and CCIR 709 anymore. Furthermore, their names have
    changed  respectively to Rec 601-1 and Rec 709 ("Rec" stands for
    Recommendation). Here is the new ITU recommendation.
    The white point is D65 and have coordinates fixed as (xn;yn)=(0.312713;
    0.329016). The RGB chromacity coordinates are:
    Red:   xr=0.64 yr=0.33
    Green: xg=0.29 yg=0.60
    Blue:  xb=0.15 yb=0.06
    Finally, we have RGB -> CIE XYZitu (D65):
    |X|   |0.430574 0.341550 0.178325|   |Red  |
    |Y| = |0.222015 0.706655 0.071330| * |Green|
    |Z|   |0.020183 0.129553 0.939180|   |Blue |
    This provides the formula to transform RGB to CIE XYZitu and vice-versa:
    RGB -> CIE XYZitu (D65)			| CIE XYZitu (D65) -> RGB
    X = 0.431*Red+0.342*Green+0.178*Blue	| Red   =  3.063*X-1.393*Y-0.476*Z
    Y = 0.222*Red+0.707*Green+0.071*Blue	| Green = -0.969*X+1.876*Y+0.042*Z
    Z = 0.020*Red+0.130*Green+0.939*Blue	| Blue  =  0.068*X-0.229*Y+1.069*Z

    You should remember that these transforms are only valid if you have
    equipment that matches these specifications, or you have images that you
    know have been encoded to these standards. If this is not the case, the
    CIE values you calculate will not be true CIE.

    All these conversions are presented noit just as demonstrations, they can
    be used for conversion between systems ;-).
    For example, in most of your applications you have true color images in RGB
    color space. How to render them fastly on your screen or on your favorite
    printer. This is simple. You can convert your picture instantaneously in
    gray scale pictures see even in a black and white pictures as a magician.
    To do so, you just need to convert your RGB values into the Y component.
    Actually, Y is linked to the luminosity (Y is an achromatic component) and
    X and Z are linked to the colorfulness (X and Z are two chromatic
    components). Old softwares used Rec 601-1 and produced:
    Gray scale=Y=(299*Red+587*Green+114*Blue)/1000
    With Rec 709, we have:
    Gray scale=Y=(213*Red+715*Green+72*Blue)/1000
    Some others do as if:
    Gray scale=Green (They don't consider the red and blue components at all)
    Or, alternativly, you can average the three color components so:
    Gray scale=(Red+Green+Blue)/3
    But now all people *should* use the most accurate, it means ITU standard:
    Gray scale=Y=(222*Red+707*Green+71*Blue)/1000
    (That's very close to Rec 709!)
    I made some personal tests and have sorted them in regard with the global
    resulting luminosity of the picture (from my eye point of view!). The
    following summary gives what I found ordered increasingly:
    +-----------------------------+----------------+
    |Scheme                       |Luminosity level|
    +-----------------------------+----------------+
    |Gray=Green                   |        1       |
    |Gray=ITU (D65)               |        2       |
    |Gray=Rec 709 (D65)           |        3       |
    |Gray=Rec 601-1 (C illuminant)|        4       |
    |Gray=(Red+Green+Blue)/3      |        5       |
    +-----------------------------+----------------+
    So softwares with Gray=Rec 709 (D65) produce a more dark picture than with
    Gray=Green. Even if you theorically lose many details with Gray=Green
    scheme, in fact, and with the 64-gray levels of a VGA card of a PC it is
    hard to distinguish the loss.

5.4 - CIE Luv and CIE Lab

    In 1976, the CIE defined two new color spaces to enable us to get more
    uniform and accurate models. The first of these two color spaces is the
    CIE Luv which component are L*, u* and v*. L* component defines the
    luminancy, and u*, v* define chrominancy. CIE Luv is very used in
    calculation of small colors or color differences, especially with additive
    colors. The CIE Luv color space is defined from CIE XYZ.

    CIE XYZ -> CIE Lab
    { L* = 116*((Y/Yn)^(1/3)) with Y/Yn>0.008856
    { L* = 903.3*Y/Yn with Y/Yn<=0.008856
    u* = 13*(L*)*(u'-u'n)
    v* = 13*(L*)*(v'-v'n)
    where u'=4*X/(X+15*Y*+3*Z) and v'=9*Y/(X+15*Y+3*Z)
    and u'n and v'n have the same definitions for u' and v' but applied to the
    white point reference. So, you have:
    u'n=4*Xn/(Xn+15*Yn*+3*Zn) and v'n=9*Yn/(Xn+15*Yn+3*Zn)

    See also item 5.3 about Xn, Yn, and Zn.

    As CIE Luv, CIE Lab is a color space introduced by CIE in 1976. It's a new
    incorporated color space in TIFF specs. In this color space you use three
    components: L* is the luminancy, a* and b* are respectively red/blue and
    yellow/blue chrominancies.
    This color space is also defined with regard to the CIE XYZ color spaces.

    CIE XYZ -> CIE Lab
    { L=116*((Y/Yn)^(1/3))-16 if Y/Yn>0.008856
    { L=903.3*Y/Yn            if Y/Yn<=0.008856
    a=500*(f(X/Xn)-f(Y/Yn))
    b=200*(f(Y/Yn)-f(Z/Zn))
    where { f(t)=t^(1/3) with Y/Yn>0.008856
          { f(t)=7.787*t+16/116 with Y/Yn<=0.008856

    See also item 5.3 about Xn, Yn, and Zn.

5.5 - LCH and CIE LSH

    CIELab and CIELuv both have a disadvantage if used for a user interface,
    they are unintuitive to use. To solve this we can use CIE definitions for
    chroma, c, Hue angle, h and saturation, s (see section 1). Hue, chroma and
    saturation can be derived from CIELuv, and Hue and chroma but *NOT*
    saturation can be derived from CIELab (this is because CIELab has no
    associated chromaticity diagram and so no correlation of saturation is
    possible).

    To distinguish between LCH derived from CIELuv and CIELab the values of
    Hue, H, and Chroma, C, are given the subscripts uv if from CIELuv and ab
    if from CIELab.


    CIELab -> LCH

    L = L*
    C = (a*^2+b*^2)^0.5
    { H=0                                 if a=0
    { H=(arctan((b*)/(a*))+k*PI/2)/(2*PI) if a#0 (add PI/2 to H if H<0)
    { and { k=0 if a*>=0 and b*>=0
          { or k=1 if a*>0 and b*<0
          { or k=2 if a*<0 and b*<0
          { or k=3 if a*<0 and b*>0

    CIELuv -> LCH

    L = L*
    C = (u*^2 + v*^2)^0.5     or   C = L*s
    H = arctan[(v*)/(u*)]
    { H=0                                 if u=0
    { H=(arctan((v*)/(u*))+k*PI/2)/(2*PI) if u#0 (add PI/2 to H if H<0)
    { and { k=0 if u*>=0 and v*>=0
          { or k=1 if u*>0 and v*<0
          { or k=2 if u*<0 and v*<0
          { or k=3 if u*<0 and v*>0

    CIELuv -> CIE LSH

    L = L*
    s = 13[(u' - u'n)^2 + (v' - v'n)^2]^0.5
    H = arctan[(v*)/(u*)]
    { H=0                                 if u=0
    { H=(arctan((v*)/(u*))+k*PI/2)/(2*PI) if u#0 (add PI/2 to H if H<0)
    { and { k=0 if u*>=0 and v*>=0
          { or k=1 if u*>0 and v*<0
          { or k=2 if u*<0 and v*<0
          { or k=3 if u*<0 and v*>0

5.6 - The associated standards: YUV, YIQ, and YCbCr

    YUV and YIQ are standard color spaces used for analogue television
    transmission. YUV is used in European TVs (PAL) and YIQ in North American
    TVs (NTSC). These colors spaces are device-dependent, like RGB, but they
    are callibrated. This is because the primaries used in these television
    systems are specified by the relative standards authorities. Y is the
    luminance component and is usually referred to as the luma component (it
    comes from CIE standard), U,V or I,Q are the chrominance components (i.e.
    the color signals).

    YUV uses D65 white point which coordinates are (xn;yn)=(0.312713;0.329016).
    The RGB chromacity coordinates are:
    Red:   xr=0.64 yr=0.33
    Green: xg=0.29 yg=0.60
    Blue:  xb=0.15 yb=0.06
    See item 5.3 to understand why the above values.

    RGB -> YUV					| YUV -> RGB
    Y =  0.299*Red+0.587*Green+0.114*Blue	| Red   = Y+0.000*U+1.140*V
    U = -0.147*Red-0.289*Green+0.436*Blue	| Green = Y-0.396*U-0.581*V
    V =  0.615*Red-0.515*Green-0.100*Blue	| Blue  = Y+2.029*U+0.000*V

    RGB -> YIQ					| YUV -> RGB
    Y =  0.299*Red+0.587*Green+0.114*Blue	| Red   = Y+0.956*I+0.621*Q
    I =  0.596*Red-0.274*Green-0.322*Blue	| Green = Y-0.272*I-0.647*Q
    Q =  0.212*Red-0.523*Green+0.311*Blue	| Blue  = Y-1.105*I+1.702*Q

    YUV -> YIQ				| YIQ -> YUV
    Y = Y (no changes)			| Y = Y (no changes)
    I = -0.2676*U+0.7361*V		| U = -1.1270*I+1.8050*Q
    Q =  0.3869*U+0.4596*V		| V =  0.9489*I+0.6561*Q

    Note that Y has a range of [0;1] (if red, green, and blue have a range of
    [0;1]) but U, V, I, and Q can be as well negative as positive. I can't give
    the range of U, V, I, and Q because it depends on precision from Rec specs
    To avoid such problems, you'll preferr the YCbCr. This color space is
    similar to YUV and YIQ without the disadvantages. Y remains the component
    of luminancy but Cb and Cr become the respective components of blue and
    red. Futhermore, with YCbCr color space you can choose your luminancy from
    your favorite recommendation. The most popular are given below:
    +----------------+---------------+-----------------+----------------+
    | Recommendation | Coef. for red | Coef. for Green | Coef. for Blue |
    +----------------+---------------+-----------------+----------------+
    | Rec 601-1      | 0.2989        | 0.5867          | 0.1144         |
    | Rec 709        | 0.2126        | 0.7152          | 0.0722         |
    | ITU            | 0.2220        | 0.7067          | 0.0713         |
    +----------------+---------------+-----------------+----------------+
    RGB -> YCbCr
    Y  = Coef. for red*Red+Coef. for green*Green+Coef. for blue*Blue
    Cb = (Blue-Y)/(2-2*Coef. for blue)
    Cr = (Red-Y)/(2-2*Coef. for red)
    YCbCr -> RGB
    Red   = Cr*(2-2*Coef. for red)+Y
    Green = (Y-Coef. for blue*Blue-Coef. for red*Red)/Coef. for green
    Blue  = Cb*(2-2*Coef. for blue)+Y
    (Note that the Green component must be computed *after* the two other
    components because Green component use the values of the two others.)
    Usually, you'll need the following conversions based on Rec 601-1
    for TIFF and JPEG works:
    RGB -> YCbCr (with Rec 601-1 specs)		| YCbCr (with Rec 601-1 specs) -> RGB
    Y=  0.2989*Red+0.5867*Green+0.1144*Blue	| Red=  Y+0.0000*Cb+1.4022*Cr
    Cb=-0.1687*Red-0.3312*Green+0.5000*Blue	| Green=Y-0.3456*Cb-0.7145*Cr
    Cr= 0.5000*Red-0.4183*Green-0.0816*Blue	| Blue= Y+1.7710*Cb+0.0000*Cr

    Additional note: Tom Lane provided me implementation of the Baseline JPEG
    compression system but after a close look I understood that their values
    was an approximation of the previous values (you should preferr mine ;-).).

    I assume Y is within the range [0;1], and Cb and Cr are within the range
    [-0.5;0.5].

5.7 - SMPTE-C RGB

    SMPTE is an acronym which stands for Society of Motion Picture and Television
    Engineers. They give a gamma (=2.2 with NTSC, and =2.8 with PAL) corrected
    color space with RGB components (about RGB, see item 5.1).
    The white point is D65. The white point coordinates are (xn;yn)=(0.312713;
    0.329016). The RGB chromacity coordinates are:
    Red:   xr=0.630 yr=0.340
    Green: xg=0.310 yg=0.595
    Blue:  xb=0.155 yb=0.070
    See item 5.3 to understand why the above values.
    To get the conversion from SMPTE-C RGB to CIE XYZ or from CIE XYZ to
    SMPTE-C RGB, you have two steps:
    SMPTE-C RGB -> CIE XYZ (D65)		| CIE XYZ (D65) -> SMPTE-C RGB
    - Gamma correction				| - Linear transformations:
    Red=f1(Red')				| Red  = 3.5058*X-1.7397*Y-0.5440*Z
    Green=f1(Green')				| Green=-1.0690*X+1.9778*Y+0.0352*Z
    Blue=f1(Blue')				| Blue = 0.0563*X-0.1970*Y+1.0501*Z
    where { f1(t)=t^2.2 whether t>=0.0		| - Gamma correction
          { f1(t)-(abs(t)^2.2) whether t<0.0	| Red'=f2(Red)
    - Linear transformations:			| Green'=f2(Green)
    X=0.3935*Red+0.3653*Green+0.1916*Blue	| Blue'=f2(Blue)
    Y=0.2124*Red+0.7011*Green+0.0866*Blue	| where { f2(t)=t^(1/2.2) whether t>=0.0
    Z=0.0187*Red+0.1119*Green+0.9582*Blue	|       { f2(t)-(abs(t)^(1/2.2)) whether t<0.0

5.8 - SMPTE-240M YPbPr (HD televisions)

    SMPTE gives a gamma (=0.45) corrected color space with RGB components
    (about RGB, see item 5.1). With this space color, you have three components
    Y, Pb, and Pr respectively linked to luminancy (see item 2), green, and
    blue. The white point is D65. The white point coordinates are
    (xn;yn)=(0.312713;0.329016). The RGB chromacity coordinates are:
    Red:   xr=0.67 yr=0.33
    Green: xg=0.21 yg=0.71
    Blue:  xb=0.15 yb=0.06
    See item 5.3 to understand why the above values.
    Conversion from SMPTE-240M RGB to CIE XYZ (D65) or from CIE XYZ (D65) to
    SMPTE-240M RGB, you have two steps:
    YPbPr -> RGB				| RGB -> YPbPr
    - Gamma correction				| - Linear transformations:
    Red=f(Red')					| Red  =1*Y+0.0000*Pb+1.5756*Pr
    Green=f(Green')				| Green=1*Y-0.2253*Pb+0.5000*Pr
    Blue=f(Blue')				| Blue =1*Y+1.8270*Pb+0.0000*Pr
    where { f(t)=t^0.45 whether t>=0.0		| - Gamma correction
          { f(t)-(abs(t)^0.45) whether t<0.0	| Red'=f(Red)
    - Linear transformations:			| Green'=f(Red)
    Y=  0.2122*Red+0.7013*Green+0.0865*Blue	| Blue'=f(Red)
    Pb=-0.1162*Red-0.3838*Green+0.5000*Blue	| where { f(t)=t^(1/0.45) whether t>=0.0
    Pr= 0.5000*Red-0.4451*Green-0.0549*Blue	|       { f(t)-(abs(t)^(1/0.45)) whether t<0.0

5.9 - Xerox Corporation YES

    YES have three components which are Y (see Luminancy, item 2), E (chrominancy
    of red-green axis), and S (chrominancy of yellow-blue axis)
    Conversion from YES to CIE XYZ (D50) or from CIE XYZ (D50) to YES, you have two
    steps:
    YES -> CIE XYZ (D50)			| CIE XYZ (D50) -> YES
    - Gamma correction				| - Linear transformations:
    Y=f1(Y')					| Y= 0.000*X+1.000*Y+0.000*Z
    E=f1(E')					| E= 1.783*X-1.899*Y+0.218*Z
    S=f1(S')					| S=-0.374*X-0.245*Y+0.734*Z
    where { f1(t)=t^2.2 whether t>=0.0		| - Gamma correction
          { f1(t)-(abs(t)^2.2) whether t<0.0	| Y'=f2(Y)
    - Linear transformations:			| E'=f2(E)
    X=0.964*Y+0.528*E-0.157*S			| S'=f2(S)
    Y=1.000*Y+0.000*E+0.000*S			| where { f2(t)=t^(1/2.2) whether t>=0.0
    Z=0.825*Y+0.269*E+1.283*S			|       { f2(t)-(abs(t)^(1/2.2)) whether t<0.0

    Conversion from YES to CIE XYZ (D65) or from CIE XYZ (D65) to YES, you have two
    steps:
    YES -> CIE XYZ (D65)			| CIE XYZ (D65) -> YES
    - Gamma correction				| - Linear transformations:
    Y=f1(Y')					| Y= 0.000*X+1.000*Y+0.000*Z
    E=f1(E')					| E=-2.019*X+1.743*Y-0.246*Z
    S=f1(S')					| S= 0.423*X+0.227*Y-0.831*Z
    where { f1(t)=t^2.2 whether t>=0.0		| - Gamma correction
          { f1(t)-(abs(t)^2.2) whether t<0.0	| Y'=f2(Y)
    - Linear transformations:			| E'=f2(E)
    X=0.782*Y-0.466*E+0.138*S			| S'=f2(S)
    Y=1.000*Y+0.000*E+0.000*S			| where { f2(t)=t^(1/2.2) whether t>=0.0
    Z=0.671*Y-0.237*E-1.133*S			|       { f2(t)-(abs(t)^(1/2.2)) whether t<0.0

    Usually, you should use YES <-> CIE XYZ (D65) conversions because your
    screen and the usual pictures have D65 as white point. Of course, sometime
    you'll need the first conversions. Just take care on your pictures.

5.10- Kodak Photo CD YCC

    The Kodak PhotoYCC color space was designed for encoding images with the
    PhotoCD system. It is based on both ITU Recommendations 709 and 601-1,
    having a color gamut defined by the ITU 709 primaries and a luminance -
    chrominance representation of color like ITU 601-1's YCbCr. The main
    attraction of PhotoYCC is that it calibrated color space, each image being
    tracable to Kodak's standard image-capturing device and the CIE Standard
    Illuminant for daylight, D65. In addition PhotoYCC provides a color gamut
    that is greater than that which can currently be displayed, thus it is
    suitable not only for both additive and subtractive (RGB and CMY(K))
    reproduction, but also for archieving since it offers a degree of
    protection against future progress in display technology.
    Images are scanned by a standardised image-capturing device, calibrated
    accoring to the type of photographic material being scanned. The scanner is
    sensitive to any color currently producable by photographic materials (and
    more besides). The image is encoded into a color space based on the ITU
    Rec 709 reference primaries and the CIE standard illuminant D65 (standard
    daylight). The extended color gamut obtainable by the PhotoCD system is
    achieved by allowing both positive and negative values for each primary.
    This means that whereas conventional ITU 709 encoded data is limited by
    the boundary of the triangle linking the three primaries (the color gamut),
    PhotoYCC can encode data outside the boundary, thus colors that are not
    realiseable by the CCIR primary set can be recorded. This feature means
    that PhotoCD stores more information (as a larger color gamut) than
    current display devices, such as CRT monitors and dye-sublimination
    printers, can produce. In this respect it is good for archieval storage of
    images since the pictures we see now will keep up with improving display
    technology.
    When an image is scanned it is stored in terms of the three reference
    primaries, these values, Rp, Gp and Bp are defined as follows:

    Rp = kr {integral of (Planda planda rlanda) dlanda}
    Gp = kg{integral of (Planda planda glanda) dlanda}
    Bp = kb{integral of (Planda planda blanda) dlanda}

    where Rp Gp and Bp are the CCIR 709 primaries although not constrained to
    positive values.
    kr kg kb are normalising constants;
    Planda is the spectral power distribution of the light source (CIE D65);
    planda is the spectral power distribution of the scene at a specific point
    (pixel);
    rlanda glanda and blanda are the spectral power distributions of the
    scanner components primaries.

    kr kg kb are specified as kr = 1 / {integral of (Planda * rlanda) dlanda}
    as similairly for kg and kb replacing rlanda with glanda and blanda
    respectivly.

    Let's stop with the theory and let's see how to make transforms.
    To be stored on a CD rom, the Rp Gp and Bp values are transformed into
    Kodak's PhotoYCC color space. This is performed in three stages. Firstly a
    non-linear transform is applied to the image signals (this is because
    scanners tend to be linear devices while CRT displays are not), the
    transform used is as follows:
    Y (see Luminancy, item 2) and C1 and C2 (both are linked to chrominancy).
    YC1C2->RGB						| RGB->YC1C2
    - Gamma correction:					| Y' =1.3584*Y
    Red  =f(red')					| C1'=2.2179*(C1-156)
    Green=f(Green')					| C2'=1.8215*(C2-137)
    Blue =f(Blue')					| Red  =Y'+C2'
    where { f(t)=-1.099*abs(t)^0.45+0.999 if t<=-0.018	| Green=Y'-0.194*C1'-0.509*C2'
          { f(t)=4.5*t if -0.018<t<0.018		| Blue =Y'+C1'
          { f(t)=1.099*t^0.45-0.999 if t>=0.018		|
    - Linear transforms:				|
    Y' = 0.299*Red+0.587*Green+0.114*Blue		|
    C1'=-0.299*Red-0.587*Green+0.886*Blue		|
    C2'= 0.701*Red-0.587*Green-0.114*Blue		|
    - To fit it into 8-bit data:			|
    Y =(255/1.402)*Y'					|
    C1=111.40*C1'+156					|
    C2=135.64*C2'+137					|

    Finally, I assume Red, Green, Blue, Y, C1, and C2 are in the range of
    [0;255]. Take care that your RGB values are not constrainted to positive
    values. So, some colors can be outside the Rec 709 display phosphor
    limit, it means some colors can be outside the trangle I defined in
    item 5.3. This can be explained because Kodak want to preserve some
    accurate infos, such as specular highlight information.
    You can note that the relations to transform YC1C2 to RGB is not exactly
    the reverse to transform RGB to YC1C2. This can be explained (from Kodak
    point of view) because the output displays are limited in the range of
    their capabilities.

    Converting stored PhotoYCC data to RGB 24bit data for display by computers
    on CRT's is achieved as follows;

    Firstly normal Luma and Chroma data are recovered:

    Luma 	= 1.3584 * Luma(8bit)
    Chroma1 	= 2.2179 * (Chroma1(8bit) - 156)
    Chroma2	= 1.8215 * (Chroma2(8bit) - 137)

    Assuming your display uses phosphors that are, or are very close to, ITU
    Rec 709 primaries in their chromaticities, then (* see below)

	Rdisplay	= L + Chroma2
	Gdisplay	= L - 0.194Chroma1 - 0.509Chroma2
	Bdisplay 	= L + Chroma1

    Two things to watch are:

    a) this results in RGB values from 0 to 346 (instead of the more usual 0 to
    255) if this is simply ignored the resulting clipping will cause severe
    loss of highlight information in the displayed image, a look-up-table is
    usually used to convert these through a non-linear function to 8 bit data.
    For example:

    Y =(255/1.402)*Y'					
    C1=111.40*C1'+156					
    C2=135.64*C2'+137	

    b) if the display phosphors differ from CCIR 709 primaries then further
    conversion will be necessary, possibly through an intermediate device
    independedent color space such as CIE XYZ.

    * as a note -> do the phosphors need to match with regard to their
    chromaticities or is a spectral match required? Two different spectral
    distributed phosphors may have same chromaticites but may not be a
    metameric match since metarimerism only applies to spectral distribution of
    color matching functions.
    Another point to note is that PhotoCD images are chroma subsampled, that is
    to say for each 2 x 2 block of pixels, 4 luma and 1 of each chroma
    component are sampled (i.e. chroma data is averaged over 2x2 pixel blocks).
    This technique uses the fact that the HVS is less sensitive to luminance
    than chromaince diferences so color information can be stored at a lower
    precision without percievable loss in visual quality.

6 - References (most of them are provided by Adrian Ford)

    "An inexpensive scheme for calibration of a color monitor in terms of CIE
    standard coordinates" W.B. Cowan, Computer Graphics, Vol. 17 No. 3, 1983

    "Calibration of a computer controlled color monitor", Brainard, D.H, Color 
    Research & Application, 14, 1, pp 23-34 (1989).

    "Color Monitor Colorimetry", SMPTE Recommended Practice RP 145-1987

    "Color Temperature for Color Television Studio Monitors", SMPTE Recommended
    Practice RP 37

    SPROSON: "Color Science in Television and Display Systems" Sproson, W, N, 
    Adam Hilger Ltd, 1983. ISBN 0-85274-413-7
    (Color measuring from soft displays.
    Alan Roberts and Richard Salmon talk about it as a reference)

    CIE 1: "CIE Colorimetry" Official recommendations of the International
    Commission on Illumination, Publication 15.2 1986

    BERNS: "CRT Colorimetry:Part 1 Theory and Practice, Part 2 Metrology",
    Berns, R.S., Motta, R.J. and Gorzynski, M.E., Color Research and
    Appliation, 18, (1993).
    (Adrian Ford talks about it as a must about CIE implementations on CRT's)

    "Effective Color Displays. Theory and Practice", Travis, D, Academic Press,
    1991. ISBN 0-12-697690-2
    (Color applications in computer graphics)

    FIELD: Field, G.G., Color and Its Reproduction, Graphics Arts Technical
    Foundation, 1988, pp. 320-9
    (Read this about CMY/CMYK)

    POYNTON: "Gamma and its disguises: The nonlinear mappings of intensity in
    perception, CRT's, Film and Video" C. A. Poynton, SMPTE Journal, December
    1993

    HUNT 1: "Measuring Color" second edition, R. W. G. Hunt, Ellis Horwood
    1991, ISBN 0-13-567686-x
    (Calculation of CIE Luv and other CIE standard colors spaces)

    "On the Gun Independance and Phosphor Consistancy of Color Video Monitors"
    W.B. Cowan N. Rowell, Color Research and Application, V.11 Supplement 1986

    "Precision requirements for digital color reproduction", M Stokes
    MD Fairchild RS Berns, ACM Transactions on graphics, v11 n4 1992

    CIE 2: "The colorimetry of self luminous displays - a bibliography" CIE
    Publication n.87, Central Bureau of the CIE, Vienna 1990

    HUNT 2: "The Reproduction of Color in PhotoGraphy, Printing and
    Television", R. W. G. Hunt, Fountain Press, Tolworth, England, 1987

    "Fully Utilizing Photo CD Images, Article No. 4, PhotoYCC Color Encoding
    and Compression Schemes" Eastman Kodak Company, Rochester NY (USA) (1993).

7 - Comments and thanks

    Whenever you would like to comment or suggest me some information about
    this or about the color space transformations in general, please use email:
    dbourgin@ufrima.imag.fr (David Bourgin)

    Special thanks to the following persons (there are actually many other
    people to cite) for contributing to valid, see even to write some part of
    the items:
    - Adrian Ford (ajoec1@westminster.ac.uk)
    - Tom Lane (Tom_Lane@G.GP.CS.CMU.EDU)
    - Alan Roberts and Richard Salmon (Alan.Roberts@rd.bbc.co.uk,
                                       Richard.Salmon@rd.eng.bbc.co.uk)
    - Grant Sayer (grants@research.canon.oz.au)
    - Steve Westland (coa23@potter.cc.keele.ac.uk)

    Note: I'm doing my national service on 4 next months, it means up to the
    begining of july. I'll try to read and answer to my e-mails.
    Thanks to not be in a hurry ;-).

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