What we perceive as color is electromagnetic radiation coming off a surface as a spectral power distribution. A tomato appears red because its surface reflects radiation mostly around 700 nm.
The human eye has three kinds of color‑sensitive cells for perceiving color: short, medium, and long wavelength cones, each sensitive to wavelengths that correspond roughly to blue, green, and red channels.
We mathematically model this for digital display by projecting the spectrum onto a three‑dimensional color space. Projecting an infinite‑dimensional spectrum onto a three‑dimensional space results in the loss of information and leads to metamers (different spectra with the same RGB).
In 1931, the CIE established color spaces that define this relation.
The trichromatic values from spectral data are calculated as follows:
\[ X = \int R(\lambda)\,\bar{x}(\lambda)\,d\lambda \]\[ Y = \int R(\lambda)\,\bar{y}(\lambda)\,d\lambda \]\[ Z = \int R(\lambda)\,\bar{z}(\lambda)\,d\lambda \] delta_lambda = wavelengths[1] - wavelengths[0]
S = reflectance * illuminant # effective spectrum under D65
X = np.sum(S * xbar) * delta_lambda
Y = np.sum(S * ybar) * delta_lambda
Z = np.sum(S * zbar) * delta_lambda
XYZ = np.array([X, Y, Z])
Here \(R(\lambda)\) is the spectral power distribution, and \(\bar{x}, \bar{y}, \bar{z}\) are the color matching functions published in CIE 1931 based on color matching experiments. Y represents luminance, and Z is roughly equivalent to the blue of the CIE RGB curves.
RGB is computed by a linear transform of \([X\;Y\;Z]^T\).
M = np.array([
[ 3.2406, -1.5372, -0.4986],
[-0.9689, 1.8758, 0.0415],
[ 0.0557, -0.2040, 1.0570],
])
RGB = M @ XYZ
# normalize by D65 white
white_reflect = np.ones_like(reflectance)
S_white = white_reflect * illuminant
Xw = np.sum(S_white * xbar) * delta_lambda
Yw = np.sum(S_white * ybar) * delta_lambda
Zw = np.sum(S_white * zbar) * delta_lambda
XYZ_white = np.array([Xw, Yw, Zw])
RGB_white = M @ XYZ_white
RGB_normalized = RGB / RGB_white.max()