Transforms

As seen in the Color Space page, it's important to understand the difference between Absolute (or device-independent) Color Space (.e.g. CIEXYZ, CIExyY, CIELAB) and device-dependent colors spaces (e.g. RGB, HSL, HSB, HSV).

For example, it makes sense to convert from a device-dependent RGB space to a HSL space since, even though they are both device-dependent, they are defined in the same "dependent space".

Also, it makes sense to convert from 2 device-independent spaces like CIEXYZ and CIELAB.

Most importantly, the conversions between device-dependent color spaces and device-independent color spaces must always be accompanied by a Color Profile that appropriately describes the dependence to the device.

Device-Dependent Color Space Conversions

Here, we will list the different conversions between device-dependent color spaces.

RGB / HSL

(Source: http://www.easyrgb.com)

RGB → HSL

• Input: RGB in [0,1]
• Output: HSL in [0,1]

 var_Min = min( R, G, B )       // Min. value of RGB
 var_Max = max( R, G, B )       // Max. value of RGB
 del_Max = var_Max - var_Min    // Delta RGB value

 L = ( var_Max + var_Min ) / 2

 if ( del_Max == 0 )            // This is a gray, no chroma...
 {
    H = 0                       // HSL results from 0 to 1
    S = 0
 }
 else                           // Chromatic data...
 {
    if ( L < 0.5 ) S = del_Max / ( var_Max + var_Min )
    else           S = del_Max / ( 2 - var_Max - var_Min )

    del_R = ( ( ( var_Max - var_R ) / 6 ) + ( del_Max / 2 ) ) / del_Max
    del_G = ( ( ( var_Max - var_G ) / 6 ) + ( del_Max / 2 ) ) / del_Max
    del_B = ( ( ( var_Max - var_B ) / 6 ) + ( del_Max / 2 ) ) / del_Max

    if      ( var_R == var_Max ) H = del_B - del_G
    else if ( var_G == var_Max ) H = ( 1 / 3 ) + del_R - del_B
    else if ( var_B == var_Max ) H = ( 2 / 3 ) + del_G - del_R

    if ( H < 0 ) H += 1
    if ( H > 1 ) H -= 1
 }

HSL → RGB

• Input: HSL in [0,1]
• Output: RGB in [0,1]

 if ( S == 0 )
 {
    (R,G,B) = L;
 }
 else
 {
    if ( L < 0.5 ) var_2 = L * ( 1 + S )
    else           var_2 = ( L + S ) - ( S * L )

    var_1 = 2 * L - var_2

    R = Hue_2_RGB( var_1, var_2, H + ( 1 / 3 ) ) 
    G = Hue_2_RGB( var_1, var_2, H )
    B = Hue_2_RGB( var_1, var_2, H - ( 1 / 3 ) )
 }

 Hue_2_RGB( v1, v2, vH )
 {
    if ( vH < 0 ) vH += 1
    if ( vH > 1 ) vH -= 1
    if ( ( 6 * vH ) < 1 ) return ( v1 + ( v2 - v1 ) * 6 * vH )
    if ( ( 2 * vH ) < 1 ) return ( v2 )
    if ( ( 3 * vH ) < 2 ) return ( v1 + ( v2 - v1 ) * ( ( 2 / 3 ) - vH ) * 6 )
    return ( v1 )
 }

RGB / HSV

(Source: http://www.easyrgb.com)

RGB → HSV

• Input: RGB in [0,1]
• Output: HSV in [0,1]

 var_Min = min( R, G, B )       // Min. value of RGB
 var_Max = max( R, G, B )       // Max. value of RGB
 del_Max = var_Max - var_Min    // Delta RGB value 

 V = var_Max

 if ( del_Max == 0 )            // This is a gray, no chroma...
 {
    H = 0                       // HSV results from 0 to 1
    S = 0
 }
 else                           // Chromatic data...
 {
    S = del_Max / var_Max

    del_R = ( ( ( var_Max - var_R ) / 6 ) + ( del_Max / 2 ) ) / del_Max
    del_G = ( ( ( var_Max - var_G ) / 6 ) + ( del_Max / 2 ) ) / del_Max
    del_B = ( ( ( var_Max - var_B ) / 6 ) + ( del_Max / 2 ) ) / del_Max

    if      ( var_R == var_Max ) H = del_B - del_G
    else if ( var_G == var_Max ) H = ( 1 / 3 ) + del_R - del_B
    else if ( var_B == var_Max ) H = ( 2 / 3 ) + del_G - del_R

    if ( H < 0 ) H += 1
    if ( H > 1 ) H -= 1
 }

HSV → RGB

• Input: HSV in [0,1]
• Output: RGB in [0,1]

 if ( S == 0 )                      // HSV from 0 to 1
 {
    (R,G,B) = V
 }
 else
 {
    var_h = H * 6
    if ( var_h == 6 ) var_h = 0 // H must be < 1
    var_i = int( var_h )            // Or ... var_i = floor( var_h )
    var_1 = V * ( 1 - S )
    var_2 = V * ( 1 - S * ( var_h - var_i ) )
    var_3 = V * ( 1 - S * ( 1 - ( var_h - var_i ) ) )

    if      ( var_i == 0 ) { R = V     ; G = var_3 ; B = var_1 }
    else if ( var_i == 1 ) { R = var_2 ; G = V     ; B = var_1 }
    else if ( var_i == 2 ) { R = var_1 ; G = V     ; B = var_3 }
    else if ( var_i == 3 ) { R = var_1 ; G = var_2 ; B = V     }
    else if ( var_i == 4 ) { R = var_3 ; G = var_1 ; B = V     }
    else                   { R = V     ; G = var_1 ; B = var_2 }
 }

Device-Independent Color Space Conversions

Here, we will list the different conversions between device-independent color spaces.

XYZ / xyY

(Source: http://www.easyrgb.com)

XYZ → xyY

• Input: (Observer. = 2°, Illuminant = D65)
  • X in [0, 0.95047]
  • Y in [0, 1.00000]
  • Z in [0, 1.08883]
• Output: xyY in [0,1]

$$Y = Y \\\\ x = X / ( X + Y + Z ) \\\\ y = Y / ( X + Y + Z ) \\\\ $$

xyY → XYZ

• Input: xyY in [0,1]
• Output: (Observer. = 2°, Illuminant = D65)
  • X in [0, 0.95047]
  • Y in [0, 1.00000]
  • Z in [0, 1.08883]

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

XYZ / Lab

(Source: http://www.easyrgb.com)

Remember that CIE L*a*b* is device-independent but needs a white point reference nevertheless.

Here, the D65 illuminant is used.

XYZ → L*a*b*

• Input: (Observer. = 2°, Illuminant = D65)
  • X in [0, 0.95047]
  • Y in [0, 1.00000]
  • Z in [0, 1.08883]
• Output:
  • L* in [0,100]
  • a*, b* in [-1,1]

 var_X = X / ref_X          // ref_X = 0.95047   Observer= 2°, Illuminant= D65
 var_Y = Y / ref_Y          // ref_Y = 1.000
 var_Z = Z / ref_Z          // ref_Z = 1.08883

 if ( var_X > 0.008856 ) var_X = var_X ^ ( 1/3 )
 else                    var_X = ( 7.787 * var_X ) + ( 16 / 116 )
 if ( var_Y > 0.008856 ) var_Y = var_Y ^ ( 1/3 )
 else                    var_Y = ( 7.787 * var_Y ) + ( 16 / 116 )
 if ( var_Z > 0.008856 ) var_Z = var_Z ^ ( 1/3 )
 else                    var_Z = ( 7.787 * var_Z ) + ( 16 / 116 )

 CIE-L\* = ( 116 * var_Y ) - 16
 CIE-a\* = 500 * ( var_X - var_Y )
 CIE-b\* = 200 * ( var_Y - var_Z )

L*a*b* → XYZ

• Input:
  • L* in [0,100]
  • a*, b* in [-1,1]
• Output: (Observer. = 2°, Illuminant = D65)
  • X in [0, 0.95047]
  • Y in [0, 1.00000]
  • Z in [0, 1.08883]

 var_Y = ( CIE-L\* + 16 ) / 116
 var_X = CIE-a\* / 500 + var_Y
 var_Z = var_Y - CIE-b\* / 200

 if ( var_Y^3 > 0.008856 ) var_Y = var_Y^3
 else                      var_Y = ( var_Y - 16 / 116 ) / 7.787
 if ( var_X^3 > 0.008856 ) var_X = var_X^3
 else                      var_X = ( var_X - 16 / 116 ) / 7.787
 if ( var_Z^3 > 0.008856 ) var_Z = var_Z^3
 else                      var_Z = ( var_Z - 16 / 116 ) / 7.787

 X = ref_X * var_X     // ref_X = 0.95047     Observer= 2°, Illuminant= D65
 Y = ref_Y * var_Y     // ref_Y = 1.00000
 Z = ref_Z * var_Z     // ref_Z = 1.08883

Device-dependent / Device-independent Color Space Conversions

RGB (in sRGB) / XYZ

(Source: http://www.easyrgb.com)

Please refer to the sRGB color profile specification to understand the pseudo-gamma correction in the following routines.

RGB → XYZ

• Input: RGB in [0,1] with sRGB gamma profile
• Output: (Observer. = 2°, Illuminant = D65)
  • X in [0, 0.95047]
  • Y in [0, 1.00000]
  • Z in [0, 1.08883]

 // Apply gamma correction (*i.e.* conversion to linear-space)
 if ( R > 0.04045 ) R = ( ( R + 0.055 ) / 1.055 ) ^ 2.4
 else               R = R / 12.92
 if ( G > 0.04045 ) G = ( ( G + 0.055 ) / 1.055 ) ^ 2.4
 else               G = G / 12.92
 if ( B > 0.04045 ) B = ( ( B + 0.055 ) / 1.055 ) ^ 2.4
 else               B = B / 12.92

 // Observer. = 2°, Illuminant = D65
 X = R * 0.4124 + G * 0.3576 + B * 0.1805
 Y = R * 0.2126 + G * 0.7152 + B * 0.0722
 Z = R * 0.0193 + G * 0.1192 + B * 0.9505

XYZ → RGB

• Input: (Observer. = 2°, Illuminant = D65)
  • X in [0, 0.95047]
  • Y in [0, 1.00000]
  • Z in [0, 1.08883]
• Output: RGB in [0,1] with sRGB gamma profile

 R = X *  3.2406 + Y * -1.5372 + Z * -0.4986
 G = X * -0.9689 + Y *  1.8758 + Z *  0.0415
 B = X *  0.0557 + Y * -0.2040 + Z *  1.0570

 if ( R > 0.0031308 ) R = 1.055 * ( R ^ ( 1 / 2.4 ) ) - 0.055
 else                 R = 12.92 * R
 if ( G > 0.0031308 ) G = 1.055 * ( G ^ ( 1 / 2.4 ) ) - 0.055
 else                 G = 12.92 * G
 if ( B > 0.0031308 ) B = 1.055 * ( B ^ ( 1 / 2.4 ) ) - 0.055
 else                 B = 12.92 * B

RGB (in Adobe RGB) / XYZ

(Source: http://www.adobe.com/digitalimag/pdfs/AdobeRGB1998.pdf)

RGB → XYZ

• Input: RGB in [0,1] with Adobe RGB gamma profile
• Output: (Observer. = 2°, Illuminant = D65)
  • X in [0, 0.95047]
  • Y in [0, 1.00000]
  • Z in [0, 1.08883]

 // Gamma correction of ~2.2
 R = R ^ 2.19921875
 G = G ^ 2.19921875
 B = B ^ 2.19921875

 // Observer. = 2°, Illuminant = D65
 X =  0.57667 * R + 0.18556 * G + 0.18823 * B
 Y =  0.29734 * R + 0.62736 * G + 0.07529 * B
 Z =  0.02703 * R + 0.07069 * G + 0.99134 * B

XYZ → RGB

• Input: (Observer. = 2°, Illuminant = D65)
  • X in [0, 0.95047]
  • Y in [0, 1.00000]
  • Z in [0, 1.08883]
• Output: RGB in [0,1] with Adobe RGB gamma profile

 R =  2.04159 * X - 0.56501 * Y - 0.34473 * Z
 G = -0.96924 * X + 1.87597 * Y + 0.04156 * Z
 B =  0.01344 * X - 0.11836 * Y + 1.01517 * Z

 // Gamma correction
 R = R ^ (1.0 / 2.19921875)
 G = G ^ (1.0 / 2.19921875)
 B = B ^ (1.0 / 2.19921875)

RGB (in Adobe RGB ICC Profile v2.4) / XYZ

(Source: http://www.adobe.com/digitalimag/pdfs/AdobeRGB1998.pdf)

RGB → XYZ

• Input: RGB in [0,1] with Adobe RGB gamma profile
• Output: (Observer. = 2°, Illuminant = D50)
  • X in [0, 0.9642]
  • Y in [0, 1.0000]
  • Z in [0, 0.8249]

 // Gamma correction of ~2.2
 R = R ^ 2.19921875
 G = G ^ 2.19921875
 B = B ^ 2.19921875

 // Observer. = 2°, Illuminant = D50
 X = 0.60974 * R + 0.20528 * G + 0.14919 * B
 Y = 0.31111 * R + 0.62567 * G + 0.06322 * B
 Z = 0.01947 * R + 0.06087 * G + 0.74457 * B

XYZ → RGB

• Input: (Observer. = 2°, Illuminant = D50)
  • X in [0, 0.9642]
  • Y in [0, 1.0000]
  • Z in [0, 0.8249]
• Output: RGB in [0,1] with Adobe RGB gamma profile

 R =  1.96253 * X - 0.61068 * Y - 0.34137 * Z
 G = -0.97876 * X + 1.91615 * Y + 0.03342 * Z
 B =  0.02869 * X - 0.14067 * Y + 1.34926 * Z

 // Gamma correction
 R = R ^ (1.0 / 2.19921875)
 G = G ^ (1.0 / 2.19921875)
 B = B ^ (1.0 / 2.19921875)

RGB (in ProPhoto RGB) / XYZ

(Source: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.202.294)

RGB → XYZ

• Input: RGB in [0,1] with ProPhoto RGB gamma profile
• Output: (Observer. = 2°, Illuminant = D50)
  • X in [0, 0.96421]
  • Y in [0, 1.00000]
  • Z in [0, 0.82519]

 // Gamma correction
 if ( R > 16 * 0.001953 ) R = R ^ 1.8
 else                     R = R / 16
 if ( G > 16 * 0.001953 ) G = G ^ 1.8
 else                     G = G / 16
 if ( B > 16 * 0.001953 ) B = B ^ 1.8
 else                     B = B / 16

 // Observer. = 2°, Illuminant = D50
 X = 0.7977 * R + 0.1352 * G + 0.0313 * B
 Y = 0.2880 * R + 0.7119 * G + 0.0001 * B
 Z = 0.0000 * R + 0.0000 * G + 0.8249 * B

XYZ → RGB

• Input: (Observer. = 2°, Illuminant = D50)
  • X in [0, 0.96421]
  • Y in [0, 1.00000]
  • Z in [0, 0.82519]
• Output: RGB in [0,1] with ProPhoto RGB gamma profile

 R =  1.3460 * X - 0.2556 * Y - 0.0511 * Z
 G = -0.5446 * X + 1.5082 * Y + 0.0205 * Z
 B =  0.0000 * X + 0.0000 * Y + 1.2123 * Z

 // Gamma correction
 if ( R > 0.001953 ) R = R ^ ( 1 / 1.8 )
 else                R = 16.0 * R
 if ( G > 0.001953 ) G = G ^ ( 1 / 1.8 )
 else                G = 16.0 * G
 if ( B > 0.001953 ) B = B ^ ( 1 / 1.8 )
 else                B = 16.0 * B

RGB (in Radiance RGB) / XYZ

Radiance RGB is the default profile for Radiance files (*.HDR) and the chromaticities found in the source code are:

color	x	y
Red	0.6400	0.3300
Green	0.2900	0.6000
Blue	0.1500	0.0600
White	0.3333	0.3333

RGB → XYZ

• Input: RGB in [0,1]
• Output:
  • X in [0, 0.96421]
  • Y in [0, 1.00000]
  • Z in [0, 0.82519]

$$X = 0.5141 * R + 0.3238 * G + 0.1619 * B \\\\ Y = 0.2651 * R + 0.6701 * G + 0.0647 * B \\\\ Z = 0.0241 * R + 0.1228 * G + 0.8530 * B \\\\ $$

XYZ → RGB

• Input:
  • X in [0, 0.96421]
  • Y in [0, 1.00000]
  • Z in [0, 0.82519]
• Output: RGB in [0,1]

$$R = 2.5653 * X - 1.1668 * Y - 0.3984 * Z \\\\ G = -1.0221 * X + 1.9783 * Y + 0.0438 * Z \\\\ B = 0.0747 * X - 0.2519 * Y + 1.1772 * Z \\\\ $$

Dealing with Generic Color Profiles

XYZ Matrices

When dealing with standard profiles like sRGB, Adobe RGB or ProPhoto RGB you are given the chromaticities of Red, Green, Blue and the one for the White Point.

Also, when opening PNG file you can encounter the cHRM chunk that describes the same chromaticities. You then need to transform these 4 2D values into a 3x3 matrix to convert the RGB value to and from the XYZ master space.

First of all, remembering our basic CIEXYZ and CIExyY conversions, let's enumerate what we know:

• From the chromaticities for Red
  $$ xyz_R = \frac{XYZ_R}{X_R+Y_R+Z_R} = \frac{XYZ_R}{\Sigma_R} $$
• From the chromaticities for Green
  $$ xyz_G = \frac{XYZ_G}{X_G+Y_G+Z_G} = \frac{XYZ_G}{\Sigma_G} $$

• From the chromaticities for Blue
  $$ xyz_B = \frac{XYZ_B}{X_B+Y_B+Z_B} = \frac{XYZ_B}{\Sigma_B} $$

• From the chromaticities for White
  $$ xyz_W $$

• The standard luminance for White
  $$ Y_W = 1 $$
• The complete expression for the White Point (since we have a completely defined $xyY_W$, we can thus easily convert to $XYZ_W$, see CIE XYZ color space)
  $$ XYZ_W $$

Note that we don't know the $XYZ_R$, $XYZ_G$ and $XYZ_B$ vectors.

So we are looking for $M_{XYZ}$ so that: $$ XYZ = RGB . M_{XYZ} = RGB . \begin{bmatrix} XYZ_R \\ XYZ_G \\ XYZ_B \end{bmatrix} = RGB . \begin{bmatrix} \Sigma_R.xyz_R \\ \Sigma_G.xyz_G \\ \Sigma_B.xyz_B \end{bmatrix} $$

Using $RGB_W = [1,1,1]$ we can write: $$ XYZ_W = [1,1,1] . M_{XYZ} = \Sigma_R.xyz_R + \Sigma_G.xyz_G + \Sigma_B.xyz_B $$

Or, in matrix form again: $$ XYZ_W = \Sigma_{RGB} . \begin{bmatrix} xyz_R \\ xyz_G \\ xyz_B \end{bmatrix} = \Sigma_{RGB} . M_{xyz} $$

Solving by right composing with $M_{xyz}^{-1}$: $$ \Sigma_{RGB} = XYZ_W . M_{xyz}^{-1} $$

we thus obtain $\Sigma_{RGB}$ and since: $$ \begin{bmatrix} \Sigma_R . xyz_R \\ \Sigma_G . xyz_G \\ \Sigma_B . xyz_B \end{bmatrix} = \begin{bmatrix} XYZ_R \\ XYZ_G \\ XYZ_B \end{bmatrix} = M_{XYZ} $$

We finally obtain $M_{XYZ}$ that converts a RGB color into an XYZ color (obviously, you need to use the inverse $M_{XYZ}^{-1}$ to convert from XYZ back into RGB).

To sum up:

1. Invert the 3x3 $M_{xyz}$ matrix constructed from the 3 R,G,B chromaticities
2. Multiply with $XYZ_W$ to obtain $\Sigma_{RGB}$
3. Finish by using $\Sigma_{RGB}$ and $M_{xyz}$ to obtain $M_{XYZ}$

You can find the code to build the $M_{XYZ}$ matrix here: ColorProfile.cpp

Custom ICC Profiles

Decoding true ICC profiles is a little over the top for our purpose. I never truly had the need for it so I won't be discussing full custom profiles here although you can read the ICC Profile specs if you like.

Also, if you really need a complete CMS, you should download the excellent Little CMS by Marti Maria.