Formula to determine brightness of RGB color

Question

I m looking for some kind of formula or algorithm to determine the brightness of a color given the RGB values   I know it can t be as simple as adding the RGB values together and having higher sums be brighter  but I m kind of at a loss as to where to start

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The inverse-gamma formula by Jive Dadson needs to have the half-adjust removed when implemented in Javascript  i e  the return from function gam sRGB needs to be return int v 255   not return int v 255  5   Half-adjust rounds up  and this can cause a value one too high on a R G B i e  grey colour triad  Greyscale conversion on a R G B triad should produce a value equal to R  it s one proof that the formula is valid  See Nine Shades of Greyscale for the formula in action  without the half-adjust

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I was solving a similar task today in javascript  I ve settled on this getPerceivedLightness rgb  function for a HEX RGB color  It deals with Helmholtz-Kohlrausch effect via Fairchild and Perrotta formula for luminance correction          Converts RGB color to CIE 1931 XYZ color space     https   www image-engineering de library technotes 958-how-to-convert-between-srgb-and-ciexyz     param   string  hex     return  number        export function rgbToXyz hex        const  r  g  b    hexToRgb hex  map     gt      255  map sRGBtoLinearRGB      const X    0 4124   r   0 3576   g   0 1805   b     const Y    0 2126   r   0 7152   g   0 0722   b     const Z    0 0193   r   0 1192   g   0 9505   b        For some reason  X  Y and Z are multiplied by 100      return  X  Y  Z  map     gt      100            Undoes gamma-correction from an RGB-encoded color     https   en wikipedia org wiki SRGB Specification of the transformation    https   stackoverflow com questions 596216 formula-to-determine-brightness-of-rgb-color     param   number      return  number      function sRGBtoLinearRGB color           Send this function a decimal sRGB gamma encoded color value        between 0 0 and 1 0  and it returns a linearized value      if  color  lt   0 04045            return color   12 92       else           return Math pow  color   0 055    1 055  2 4                  Converts hex color to RGB     https   stackoverflow com questions 5623838 rgb-to-hex-and-hex-to-rgb     param   string  hex     return  number     rgb      function hexToRgb hex        const match         a-f d  2    a-f d  2    a-f d  2    i exec hex      if  match            match shift           return match map     gt  parseInt    16                   Converts CIE 1931 XYZ colors to CIE L a b      The conversion formula comes from  lt http   www easyrgb com en math php gt      https   github com cangoektas xyz-to-lab blob master src index js     param    number    color The CIE 1931 XYZ color to convert which refers to                              the D65 2   standard illuminant      returns  number          The color in the CIE L a b  color space         X  Y  Z of a  D65  light source      D65  is a standard 6500K Daylight light source     https   en wikipedia org wiki Illuminant D65 const D65    95 047  100  108 883  export function xyzToLab  x  y  z        x  y  z     x  y  z  map  v  i    gt        v   v   D65 i      return v  gt  0 008856   Math pow v  1   3    v   7 787   16   116        const l   116   y - 16   const a   500    x - y    const b   200    y - z    return  l  a  b            Converts Lab color space to Luminance-Chroma-Hue color space     http   www brucelindbloom com index html Eqn Lab to LCH html     param   number        return  number        export function labToLch  l  a  b         const c   Math sqrt a   a   b   b      const h   abToHue a  b      return  l  c  h            Converts a and b of Lab color space to Hue of LCH color space     https   stackoverflow com questions 53733379 conversion-of-cielab-to-cielchab-not-yielding-correct-result     param   number  a     param   number  b     return  number      function abToHue a  b        if  a  gt   0  amp  amp  b     0            return 0           if  a  lt  0  amp  amp  b     0            return 180           if  a     0  amp  amp  b  gt  0            return 90           if  a     0  amp  amp  b  lt  0            return 270           let xBias     if  a  gt  0  amp  amp  b  gt  0            xBias   0       else if  a  lt  0            xBias   180       else if  a  gt  0  amp  amp  b  lt  0            xBias   360           return radiansToDegrees Math atan b   a     xBias    function radiansToDegrees radians        return radians    180   Math PI     function degreesToRadians degrees        return degrees   Math PI   180           Saturated colors appear brighter to human eye     That s called Helmholtz-Kohlrausch effect     Fairchild and Pirrotta came up with a formula to    calculate a correction for that effect      Color Quality of Semiconductor and Conventional Light Sources      https   books google ru books id ptDJDQAAQBAJ amp pg PA45 amp lpg PA45 amp dq fairchild pirrotta correction amp source bl amp ots 7gXR2MGJs7 amp sig ACfU3U3uIHo0ZUdZB Cz9F9NldKzBix0oQ amp hl ru amp sa X amp ved 2ahUKEwi47LGivOvmAhUHEpoKHU ICkIQ6AEwAXoECAkQAQ v onepage amp q fairchild 20pirrotta 20correction amp f false     return  number      function getLightnessUsingFairchildPirrottaCorrection  l  c  h         const l    2 5 - 0 025   l     const g   0 116   Math abs Math sin degreesToRadians  h - 90    2      0 085     return l   l    g   c    export function getPerceivedLightness hex        return getLightnessUsingFairchildPirrottaCorrection labToLch xyzToLab rgbToXyz hex

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Below is the only CORRECT algorithm for converting sRGB images  as used in browsers etc   to grayscale   It is necessary to apply an inverse of the gamma function for the color space before calculating the inner product  Then you apply the gamma function to the reduced value  Failure to incorporate the gamma function can result in errors of up to 20     For typical computer stuff  the color space is sRGB   The right numbers for sRGB are approx  0 21  0 72  0 07  Gamma for sRGB is a composite function that approximates exponentiation by 1  2 2   Here is the whole thing in C         sRGB luminance Y  values const double rY   0 212655  const double gY   0 715158  const double bY   0 072187      Inverse of sRGB  gamma  function   approx 2 2  double inv gam sRGB int ic        double c   ic 255 0      if   c  lt   0 04045           return c 12 92      else          return pow   c 0 055   1 055   2 4         sRGB  gamma  function  approx 2 2  int gam sRGB double v        if v lt  0 0031308          v    12 92      else          v   1 055 pow v 1 0 2 4 -0 055      return int v 255 0 5      This is correct in C    Other languages may not                               require  0 5       GRAY VALUE   brightness   int gray int r  int g  int b        return gam sRGB              rY inv gam sRGB r                gY inv gam sRGB g                bY inv gam sRGB b

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To determine the brightness of a color with R  I convert the RGB system color in HSV system color   In my script  I use the HEX system code before for other reason  but you can start also with RGB system code with rgb2hsv  grDevices   The documentation is here   Here is this part of my code    sample  lt - c   010101     303030     A6A4A4     020202     010100    hsvc  lt -rgb2hsv col2rgb sample     convert HEX to HSV  value  lt - as data frame hsvc    create data frame  value  lt - value 3     extract the information of brightness  order value    ordrer the color by brightness

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Here s a bit of C code that should properly calculate perceived luminance        reverses the rgb gamma  define inverseGamma t     t   lt   0 0404482362771076      t  12 92    pow   t    0 055  1 055  2 4      CIE L a b  f function  used to convert XYZ to L a b    http   en wikipedia org wiki Lab color space  define LABF t    t  gt   8 85645167903563082e-3    powf t 0 333333333333333     841 0 108 0   t     4 0 29 0     float rgbToCIEL PIXEL p       float y     float r p r 255 0     float g p g 255 0     float b p b 255 0      r inverseGamma r      g inverseGamma g      b inverseGamma b         Observer   2    Illuminant   D65     y   0 2125862307855955516 r   0 7151703037034108499 g   0 07220049864333622685 b         At this point we ve done RGBtoXYZ now do XYZ to Lab        y    WHITEPOINT Y  The white point for y in D65 is 1 0      y   LABF y          This is the  normal conversion which produces values scaled to 100     Lab L   116 0 y - 16 0           return 1 16 y - 0 16      return values for 0 0  gt  L  lt  1 0

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RGB Luminance value       0 3 R   0 59 G   0 11 B  http   www scantips com lumin html     If you re looking for how close to white the color is you can use Euclidean Distance from  255  255  255    I think RGB color space is perceptively non-uniform with respect to the L2 euclidian distance  Uniform spaces include CIE LAB and LUV

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Rather than getting lost amongst the random selection of formulae mentioned here  I suggest you go for the formula recommended by W3C standards   Here s a straightforward but exact PHP implementation of the WCAG 2 0 SC 1 4 3 relative luminance and contrast ratio formulae  It produces values that are appropriate for evaluating the ratios required for WCAG compliance  as on this page  and as such is suitable and appropriate for any web app  This is trivial to port to other languages          Calculate relative luminance in sRGB colour space for use in WCAG 2 0 compliance     link http   www w3 org TR WCAG20  relativeluminancedef     param string  col A 3 or 6-digit hex colour string     return float     author Marcus Bointon  lt marcus synchromedia co uk gt      function relativeluminance  col          Remove any leading        col   trim  col              Convert 3-digit to 6-digit     if  strlen  col     3             col    col 0     col 0     col 1     col 1     col 2     col 2               Convert hex to 0-1 scale      components   array           r    gt  hexdec substr  col  0  2     255           g    gt  hexdec substr  col  2  2     255           b    gt  hexdec substr  col  4  2     255              Correct for sRGB     foreach  components as  c   gt   v            if   v  lt   0 04045                 components  c     v   12 92            else                components  c    pow    v   0 055    1 055   2 4                         Calculate relative luminance using ITU-R BT  709 coefficients     return   components  r     0 2126      components  g     0 7152      components  b     0 0722             Calculate contrast ratio acording to WCAG 2 0 formula    Will return a value between 1  no contrast  and 21  max contrast      link http   www w3 org TR WCAG20  contrast-ratiodef     param string  c1 A 3 or 6-digit hex colour string     param string  c2 A 3 or 6-digit hex colour string     return float     author Marcus Bointon  lt marcus synchromedia co uk gt      function contrastratio  c1   c2         y1   relativeluminance  c1        y2   relativeluminance  c2         Arrange so  y1 is lightest     if   y1  lt   y2             y3    y1           y1    y2           y2    y3            return   y1   0 05      y2   0 05

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The HSV colorspace should do the trick  see the wikipedia article depending on the language you re working in you may get a library conversion    H is hue which is a numerical value for the color  i e  red  green      S is the saturation of the color  i e  how  intense  it is  V is the  brightness  of the color

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The  quot Accepted quot  Answer is Incorrect and Incomplete The only answers that are accurate are the  jive-dadson and  EddingtonsMonkey answers  and in support  nils-pipenbrinck  The other answers  including the accepted  are linking to or citing sources that are either wrong  irrelevant  obsolete  or broken  Briefly   sRGB must be LINEARIZED before applying the coefficients  Luminance  L or Y  is linear as is light  Perceived lightness  L   is nonlinear as is human perception  HSV and HSL are not even remotely accurate in terms of perception  The IEC standard for sRGB specifies a threshold of 0 04045 it is NOT 0 03928  that was from an obsolete early draft   The be useful  i e  relative to perception   Euclidian distances require a perceptually uniform Cartesian vector space such as CIELAB  sRGB is not one    What follows is a correct and complete answer  Because this thread appears highly in search engines  I am adding this answer to clarify the various misconceptions on the subject  Luminance is a linear measure of light  spectrally weighted for normal vision but not adjusted for the non-linear perception of lightness  It can be a relative measure  Y as in CIEXYZ  or as L  an absolute measure in cd m2  not to be confused with L    Perceived lightness is used by some vision models such as CIELAB  here L   Lstar  is a value of perceptual lightness  and is non-linear to approximate the human vision non-linear response curve  Brightness is a perceptual attribute  it does not have a  quot physical quot  measure  However some color appearance models do have a value  usualy denoted as  quot Q quot  for perceived brightness  which is different than perceived lightness  Luma  Y   prime  is a gamma encoded  weighted signal used in some video encodings  Y  I  Q     It is not to be confused with linear luminance  Gamma or transfer curve  TRC  is a curve that is often similar to the perceptual curve  and is commonly applied to image data for storage or broadcast to reduce perceived noise and or improve data utilization  and related reasons   To determine perceived lightness  first convert gamma encoded R  G  B   image values to linear luminance  L or Y   and then to non-linear perceived lightness  L    TO FIND LUMINANCE     Because apparently it was lost somewhere    Step One  Convert all sRGB 8 bit integer values to decimal 0 0-1 0   vR   sR   255    vG   sG   255    vB   sB   255   Step Two  Convert a gamma encoded RGB to a linear value  sRGB  computer standard  for instance requires a power curve of approximately V 2 2  though the  quot accurate quot  transform is   Where V   is the gamma-encoded R  G  or B channel of sRGB  Pseudocode  function sRGBtoLin colorChannel               Send this function a decimal sRGB gamma encoded color value            between 0 0 and 1 0  and it returns a linearized value       if   colorChannel  lt   0 04045                 return colorChannel   12 92            else               return pow    colorChannel   0 055  1 055  2 4                     Step Three  To find Luminance  Y  apply the standard coefficients for sRGB   Pseudocode using above functions  Y    0 2126   sRGBtoLin vR    0 7152   sRGBtoLin vG    0 0722   sRGBtoLin vB     TO FIND PERCEIVED LIGHTNESS  Step Four  Take luminance Y from above  and transform to L   Pseudocode  function YtoLstar Y               Send this function a luminance value between 0 0 and 1 0             and it returns L  which is  quot perceptual lightness quot       if   Y  lt    216 24389             The CIE standard states 0 008856 but 216 24389 is the intent for 0 008856451679036             return Y    24389 27       The CIE standard states 903 3  but 24389 27 is the intent  making 903 296296296296296           else               return pow Y  1 3     116 - 16                   L  is a value from 0  black  to 100  white  where 50 is the perceptual  quot middle grey quot   L    50 is the equivalent of Y   18 4  or in other words an 18  grey card  representing the middle of a photographic exposure  Ansel Adams zone V   References  IEC 61966-2-1 1999 Standard Wikipedia sRGB Wikipedia CIELAB Wikipedia CIEXYZ Charles Poynton s Gamma FAQ

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The  V  of HSV is probably what you re looking for  MATLAB has an rgb2hsv function and the previously cited wikipedia article is full of pseudocode  If an RGB2HSV conversion is not feasible  a less accurate model would be the grayscale version of the image

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I think what you are looking for is the RGB -  Luma conversion formula   Photometric digital ITU BT 709   Y   0 2126 R   0 7152 G   0 0722 B   Digital ITU BT 601  gives more weight to the R and B components    Y   0 299 R   0 587 G   0 114 B   If you are willing to trade accuracy for perfomance  there are two approximation formulas for this one   Y   0 33 R   0 5 G   0 16 B  Y   0 375 R   0 5 G   0 125 B   These can be calculated quickly as   Y    R R B G G G  6  Y    R R R B G G G G  gt  gt 3

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The method could vary depending on your needs  Here are 3 ways to calculate Luminance   Luminance  standard for certain colour spaces    0 2126 R   0 7152 G   0 0722 B  source   Luminance  perceived option 1    0 299 R   0 587 G   0 114 B  source   Luminance  perceived option 2  slower to calculate    sqrt  0 241 R 2   0 691 G 2   0 068 B 2     sqrt  0 299 R 2   0 587 G 2   0 114 B 2    thanks to  MatthewHerbst  source     Edit  added examples using named css colors sorted with each method

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I have made comparison of the three algorithms in the accepted answer  I generated colors in cycle where only about every 400th color was used  Each color is represented by 2x2 pixels   colors are sorted from darkest to lightest  left to right  top to bottom    1st picture - Luminance  relative   0 2126   R   0 7152   G   0 0722   B   2nd picture - http   www w3 org TR AERT color-contrast  0 299   R   0 587   G   0 114   B   3rd picture -  HSP Color Model  sqrt 0 299   R 2   0 587   G 2   0 114   B 2    4th picture - WCAG 2 0 SC 1 4 3 relative luminance and contrast ratio formula  see  Synchro s answer here   Pattern can be sometimes spotted on 1st and 2nd picture depending on the number of colors in one row  I never spotted any pattern on picture from 3rd or 4th algorithm   If i had to choose i would go with algorithm number 3 since its much easier to implement and its about 33  faster than the 4th

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For clarity  the formulas that use a square root need to be  sqrt coefficient    colour value 2    not  sqrt  coefficient   colour value   2  The proof of this lies in the conversion of a R G B triad to greyscale R  That will only be true if you square the colour value  not the colour value times coefficient  See Nine Shades of Greyscale

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I found this code  written in C   that does an excellent job of calculating the  quot brightness quot  of a color   In this scenario  the code is trying to determine whether to put white or black text over the color

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This link explains everything in depth  including why those multiplier constants exist before the R  G and B values   Edit  It has an explanation to one of the answers here too  0 299 R   0 587 G   0 114 B

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Interestingly  this formulation for RGB  HSV just uses v MAX3 r g b    In other words  you can use the maximum of  r g b  as the V in HSV   I checked and on page 575 of Hearn  amp  Baker this is how they compute  Value  as well

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To add what all the others said   All these equations work kinda well in practice  but if you need to be very precise you have to first convert the color to linear color space  apply inverse image-gamma   do the weight average of the primary colors and - if you want to display the color -  take the luminance back into the monitor gamma   The luminance difference between ingnoring gamma and doing proper gamma is up to 20  in the dark grays

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I wonder how those rgb coefficients were determined  I did an experiment myself and I ended up with the following   Y   0 267 R   0 642 G   0 091 B   Close but but obviously different than the long established ITU coefficients  I wonder if those coefficients could be different for each and every observer  because we all may have a different amount of cones and rods on the retina in our eyes  and especially the ratio between the different types of cones may differ   For reference   ITU BT 709   Y   0 2126 R   0 7152 G   0 0722 B   ITU BT 601   Y   0 299 R   0 587 G   0 114 B   I did the test by quickly moving a small gray bar on a bright red  bright green and bright blue background  and adjusting the gray until it blended in just as much as possible  I also repeated that test with other shades  I repeated the test on different displays  even one with a fixed gamma factor of 3 0  but it all looks the same to me  More over  the ITU coefficients literally are wrong for my eyes    And yes  I presumably have a normal color vision

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Please define brightness  If you re looking for how close to white the color is you can use Euclidean Distance from  255  255  255

[image] Formula to determine brightness of RGB color

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