Calculate colour temperature in K

Question

I have written a lib for working with colours and stuck trying to calculate Tc(k). From what I have read working in the CIE 1931 XYZ colour space is the way to go and it can be obtained using xyY.

So far I have got everything correct to the point of figuring out the correct x and y from :

          X                 Y
x = ____________  y = ____________
    ( X + Y + Z)      ( X + Y + Z)

The numbers much up to the chart, but cant find anything that details how you go from x and y to Tc(K)

eg: For #FF0000 I get the following.

x: 0.64007449945677
y: 0.32997051063169

I have read a number of papers on the topic and litterally all the wikipedia articles. All the questions I have come across on SO simply link to a wiki article on colours, not seen one that has the actual formula for calculating Tc(k)

Answer 1

Based on the Wikipedia article I entered the approximated formula for Color Temperature calculation into Excel as

=(-449*((R1-0,332)/(S1-0,1858))^3)+(3525*((R1-0,332)/(S1-0,1858))^2)-(6823,3*((R1-0,332)/(S1-0,1858)))+(5520,33)

R1 is color space x coordinate from 0 to 1
S1 is color space y coordinate from 0 to 1

Works fine!

Answer 2

If I understand you correctly please see this pdf where a number of methods (briefly :( ) described. From this PDF:

[CCT1] Calculations are performed using CIE 1960 uniform color space coordinates u and v. Coordinates u and v are derived from x and y using the formulae: u = 4x/(12y-2x+3) and v = 6y/(12y-2x+3) The correlated color temperature is defined as the temperature of a blackbody that lies closest to the u,v coordinates of the test source. Two separate methods provide results: one is an iterative method based on the definition, and the other is the commonly used Robertson’s method involving interpolation based on a table of 30 pre-calculated u, v and inverse-slope parameters. Iteration If uS and vS are the values for a test source, and uT and vT are the blackbody values at temperature T, the correlated color temperature is the value of T where:

sqrt( (uS - uT)^2 + (vS - vT)^2 )

is minimized. Adjusting the temperature T to obtain the minimum of this function was done using spreadsheets (Quattro Pro 8 and Excel 97). Both spreadsheets gave identical values.

Don't know if it actually helps you.

Answer 3

I done some digging in some open source apps and found something in UFRaw. I have not quite figured out what is going on exactly.

Also found a paper that seems to cover the topic quite well.

Converted to php and this is what I have so far:

$temp = array(9500, 7000, 5500, 3750, 3000, 2700, 2250, 1800, 1500);
$hex = array('9DBEFF', 'E4EEFF', 'FFE4BE', 'FFA04C', 'FF7A26', 'FF6A19', 'FF500B', 'FF3403', 'FF2300');

echo '<h3>K -> RGB</h3>';
foreach ($temp as $k) {
    $rgb = ColourConverter::temperature2rgb($k);
    echo sprintf('<div style="background-color:rgb(%s); text-align: center; width: 100px; height: 25px; clear: both;">%s</div>', implode(', ', $rgb), $k);
}

echo '<h3>RGB -> K</h3>';
foreach ($hex as $v) {
    $rgb = array_values(ColourConverter::hex2rgb($v));
    $k = round(ColourConverter::rgb2temperature($rgb[0], $rgb[1], $rgb[2]));
    echo sprintf('<div style="background-color:rgb(%s); text-align: center; width: 100px; height: 25px; clear: both;">%s</div>', implode(', ', $rgb), $k);
}

My output:

Pretty close but not 100% yet. (Found a bug in my code and it is now almost perfect)

• The colours are slightly off going from k -> rgb
• It does not work doing k -> rgb -> k. You don't get back to the same value.

Code

UFRaw line 246-294

void Temperature_to_RGB(double T, double RGB[3])
{
    int c;
    double xD, yD, X, Y, Z, max;
    // Fit for CIE Daylight illuminant
    if (T <= 4000) {
        xD = 0.27475e9 / (T * T * T) - 0.98598e6 / (T * T) + 1.17444e3 / T + 0.145986;
    } else if (T <= 7000) {
        xD = -4.6070e9 / (T * T * T) + 2.9678e6 / (T * T) + 0.09911e3 / T + 0.244063;
    } else {
        xD = -2.0064e9 / (T * T * T) + 1.9018e6 / (T * T) + 0.24748e3 / T + 0.237040;
    }
    yD = -3 * xD * xD + 2.87 * xD - 0.275;

    // Fit for Blackbody using CIE standard observer function at 2 degrees
    //xD = -1.8596e9/(T*T*T) + 1.37686e6/(T*T) + 0.360496e3/T + 0.232632;
    //yD = -2.6046*xD*xD + 2.6106*xD - 0.239156;

    // Fit for Blackbody using CIE standard observer function at 10 degrees
    //xD = -1.98883e9/(T*T*T) + 1.45155e6/(T*T) + 0.364774e3/T + 0.231136;
    //yD = -2.35563*xD*xD + 2.39688*xD - 0.196035;

    X = xD / yD;
    Y = 1;
    Z = (1 - xD - yD) / yD;
    max = 0;
    for (c = 0; c < 3; c++) {
        RGB[c] = X * XYZ_to_RGB[0][c] + Y * XYZ_to_RGB[1][c] + Z * XYZ_to_RGB[2][c];
        if (RGB[c] > max) max = RGB[c];
    }
    for (c = 0; c < 3; c++) RGB[c] = RGB[c] / max;
}

void RGB_to_Temperature(double RGB[3], double *T, double *Green)
{
    double Tmax, Tmin, testRGB[3];
    Tmin = 2000;
    Tmax = 23000;
    for (*T = (Tmax + Tmin) / 2; Tmax - Tmin > 0.1; *T = (Tmax + Tmin) / 2) {
        Temperature_to_RGB(*T, testRGB);
        if (testRGB[2] / testRGB[0] > RGB[2] / RGB[0])
            Tmax = *T;
        else
            Tmin = *T;
    }
    *Green = (testRGB[1] / testRGB[0]) / (RGB[1] / RGB[0]);
    if (*Green < 0.2) *Green = 0.2;
    if (*Green > 2.5) *Green = 2.5;
}

Answer 4

Found the Temperature to RGB solution written in JavaScript and converted it into PHP.

<?php
function colorTemperatureToRGB($kelvin)
{
    $temp = $kelvin / 100;

    $red = null;
    $green = null;
    $blue = null;

    if ($temp <= 66)
    {
        $red = 255; 

        $green = $temp;
        $green = 99.4708025861 * log($green) - 161.1195681661;


        if ($temp <= 19)
        {
            $blue = 0;
        }
        else
        {
            $blue = $temp - 10;
            $blue = 138.5177312231 * log($blue) - 305.0447927307;
        }

    }
    else
    {
        $red = $temp - 60;
        $red = 329.698727446 * pow($red, -0.1332047592);

        $green = $temp - 60;
        $green = 288.1221695283 * pow($green, -0.0755148492);

        $blue = 255;
    }

    return [
        'r' => clamp($red,   0, 255),
        'g' => clamp($green, 0, 255),
        'b' => clamp($blue,  0, 255)
    ];
}

function clamp($x, $min, $max)
{
    if ($x < $min)
        return $min;

    if ($x > $max)
        return $max;

    return $x;
}

var_dump(colorTemperatureToRGB(7000));

Answer 5

The Wikipedia states that the color temperature is calculated by considering the u-v chromaticities, not x-y, so you have to perform a translation. That said, I recommend using an approximation (also explained in the Wikipedia) as suggested by @sixlettervariables.

The real question is, what are you trying to find the color temperature of?. I see a reference (#FF0000) to RGB colors, which are meaningless without stating a color space. Supposing you are in sRGB (so I can point you at Wikipedia again), you have to first obtain the linear RGB co-ordinates before proceeding to XYZ.

Answer 6

I realize this is an old question, but I was struggling to find an answer as well. I finally came across this nifty calculator that also posts the formula as follows:

n = (x-0.3320)/(0.1858-y)

CCT = 437*n^3 + 3601*n^2 + 6861*n + 5517

Hope this helps anyone else who's still looking.