visible light - What {R,G,B} values would represent a 445nm monochrome lightsource color on a computer monitor?

Is it possible to answer my question definitely (assuming the monitor is perfect)? What would be the formula for calculating RGB values for a visible monochrome light with given wavelength?

Answer

First you consult a CIE 1964 Supplementary Standard Colorimetric Observer chart, and look up the CIE color matching function values for the wavelength you want:

For your desired wavelength:

| λ   | CIE color matching functions  |  Chromacity coordinates     |
| nm  |     X    |     Y    |    Z    |    x    |    y    |    z    |
|-----|----------|----------|---------|---------|---------|---------| 
| 455 | 0.342957 | 0.106256 | 1.90070 | 0.14594 | 0.04522 | 0.80884 |

Note: The chromacity coordinates are simply calculated from the CIE color matching functions:

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

Given:

X+Y+Z = 0.342257+0.106256+1.90070 = 2.349913

So:

x = 0.342257 / 2.349913 = 0.145945
y = 0.106256 / 2.349913 = 0.045217
z = 1.900700 / 2.349913 = 0.808838

You have a color specified using two different color spaces:

• XYZ = (0.342957, 0.106256, 1.900700)


• xyz = (0.145945, 0.045217, 0.808838) *(which matches what we already had in the table)

We can also add a third color space: xyY

x = x = 0.145945
y = y = 0.045217
Y = y = 0.106256

We now have the color specified in 3 different color spaces:

• XYZ = (0.342957, 0.106256, 1.900700)


• xyz = (0.145945, 0.045217, 0.808838)


• xyY = (0.145945, 0.045217, 0.106256)

So you've converted a wavelength of pure monochromatic emitted light into a XYZ color. Now we want to convert that to RGB.

How to convert XYZ into RGB?

XYZ, xyz, and xyY are absolute color spaces that describe colors using absolute physics.

Meanwhile, every practical color spaces that people use:

• Lab


• Luv


• HSV


• HSL


• RGB

depends on some whitepoint. The colors are then described as being relative to that whitepoint.

For example,

• RGB white (255,255,255) means "white"


• Lab white (100, 0, 0) means "white"

But there is no such color as white. How do you define white? The color of sunlight?

• at what time of day?



• with how much cloud cover?


• at what latitude?


• on Earth?

Some people use the white of their (horribly orange) incandescent bulbs to mean white. Some people use the color of their florescent lights. There is no absolute physical definition of white - white is in our brains.

So we have to pick a white

We have to pick a white. Really it's you who has to pick a white. And there are plenty of whites to choose from:

I will pick a white for you. The same white that sRGB uses:

• D65 - daylight illumination of clear summer day in northern Europe

D65 (which has a color close to 6500K, but not quite because of the Earth's atmosphere), has a color of:

• XYZ_D65: (0.95047, 1.00000, 1.08883)

With that, you can convert your XYZ into Lab (or Luv) - a color-space equally capable of expressing all theoretical colors. And now we have a 4th color space representation of our 445 nm monochromatic emission of light:

• XYZ: (0.342957, 0.106256, 1.900700)


• xyz: (0.145945, 0.045217, 0.808838)



• xyY: (0.145945, 0.045217, 0.106256)


• Lab: (38.94259, 119.14058, -146.08508) (D65)

But you want RGB

Lab (and Luv) are color spaces that are relative to some white-point. Even though you were forced to pick an arbitrary white-point, you can still represent every possible color.

RGB is not like that. With RGB:

• not only is the color relative to some white-point


• but is is also relative to three color primaries: red, green, blue

If you specify an RGB color of (255, 0, 0), you are saying you want "just red". But there is no definition of red. There is no such thing as "red", "green", or "blue". The rainbow is continuous, and doesn't come with an arrow saying:

This is red

And again this means we have to pick three pick three primary colors. You have to pick your three primary colors to say what "red", "green", and "blue" are. And again you have many different definitions of Red,Green,Blue to choose from:

• CIE 1931


• ROMM RGB


• Adobe Wide Gamut RGB


• DCI-P3



• NTSC (1953)


• Apple RGB


• sRGB


• Japanese NTSC


• PAL/SECAM


• Adobe RGB 98


• scRGB

I'll pick for you. I'll pick these three colors:

• Red: xyY = (0.6400, 0.3300, 0.2126)


• Green: xyY = (0.3000, 0.6000, 0.7152)


• Blue: xyY = (0.1500, 0.0600, 0.0722)

Those were also the primaries chosen for by an international committee in 1996.

They created a standard that said everyone should use:

• Whitepoint: D65 daylight (0.95047, 1.00000, 1.08883)


• Red: (0.6400, 0.3300, 0.2126)


• Green: (0.3000, 0.6000, 0.7152)



• Blue: (0.1500, 0.0600, 0.0722)

And they called that standard sRGB - and you can see these four points plotted out on a chromacity diagram:

sRGB Chromacity Diagram (D65 & red,green,blue primaries)

Now that we have chosen our

• white-point


• three primaries

we can now convert you XYZ color into RGB, using the sRGB choices for "red", "green", "blue", and "white":

/*
    The matix values in the next step depend on location of RGB in the XYZ color space.
    These constants are for
            Observer:          2°
            Illuminant:        D65
            RGB Working Space: sRGB
*/
r = X *  3.2404542 + Y * -1.5371385 + Z * -0.4985314;

g = X * -0.9692660 + Y *  1.8760108 + Z *  0.0415560;
b = X *  0.0556434 + Y * -0.2040259 + Z *  1.0572252;

Giving you your RGB of:

• RGB = (1.47450, -65.7629, 345.59392)

Unfortunately:

• your monitor cannot display negative green (-65). It means it is a color outside what your monitor can display (i.e. outside of its color gamut)



• your monitor cannot display more blue than 255 (345). It also means that it's a color outside your monitor's gamut.

So we have to round:

• XYZ = (0.342957, 0.106256, 1.900700)


• xyz = (0.145945, 0.045217, 0.808838)


• xyY = (0.145945, 0.045217, 0.106256)


• Lab = (38.94259, 119.14058, -146.08508) (Whitepoint: D65)


• RGB - (1, 0, 255) (sRGB)

I wanted to point out that nearly everyone uses sRGB as the standard. It's a general standard for all digital cameras, for JPEGs on the Internet, and computer monitors. The goal is that every one of these devices agree on:

• the color of the red primary


• the color of the green primary


• the color of the blue primary


• the color that we will use as white

And those places outside the triangle on the sRGB chromacity diagram are still all valid colors; your monitor just can't display them.

And the very outside edge of the curve (called the locus) is the location of different pure frequencies of monochromatic light. That is where your pure 445nm monochromatic light source would be: